text
stringlengths 8
1.01M
|
|---|
What is Nishalspace
Nishal is my metaphoric nickname. My official name is "Prashant Kumar", but I wanted a name of meaning.
Nishal (निशाल ) is a hindi word, which means "The one without any shape and size". In mathematics this is quite similar to Topology, which generally doesn't have geometrical shape, but have a meaningful structure.
And among all 5 elements of cosmos, I am fascinated more towards space. I am fascinated in external as well as internal space. Space means where we all belong, it envelopes us. |
Einstein, Star Trek, and Gauss inspire Professor Paul Flavell
Posted on 30 Oct 2017
Earlier this year, Professor Paul Flavell, Head of Mathematics at the University of Birmingham, gave a public talk to celebrate becoming a Professor.
The talk, entitled Numbers, was part of a series of Inaugural Lectures hosted by the College of Engineering and Physical Sciences to showcase its leading academics who are pushing the boundaries in their disciplines.
His achievements, which include proving and then significantly contributing to 'very difficult' theorems, are the fulfilment of a childhood ambition – nurtured by a number of primary and secondary school teachers, who recognised his potential and enthusiasm. To Paul's delight, two of them were in the audience for his lecture.
'An Inaugural Lecture is a once-in-a-lifetime event and I was determined it was going to be something special,' says Paul, who is also Head of the School of Mathematics. 'I wanted to convey something about me and I also saw it as an opportunity to give a wider group of people an insight into what mathematicians do. Everyone knows who the great composers and writers are, but do they know who the great mathematicians are? Probably not, and I wanted to correct that. For me, Carl Friedrich Gauss is the greatest mathematician.
'One of the things people like me do is prove theorems, so I wanted to take the Fundamental Theorem of Algebra proved by Gauss in his PhD thesis and get the audience to understand it. It involves things called complex numbers, but before you get to the complex ones, you need to know about ordinary numbers – hence the title of the lecture. So I set about proving this result, using lots of equipment and visual displays.'
And did the audience, at the end, understand the theorem? 'I think so, yes.'
Paul can trace his love of mathematics back to when he was eight or nine. 'I was always interested in science and technology, particularly physics, inspired by Einstein and Star Trek. I started exploring the library and, of course, next to physics books you find maths books. So I became rather captivated. I think what captivated me was the difficulty of some things, and I developed the desire to get my head around big developments in physics, mathematics and technology, and in due course to make my own contribution.'
By the age of about 14, he was 'absolutely set' on wanting to study physics or maths at university and becoming a researcher. His ambitions were encouraged and supported by several 'wonderful' teachers, two of whom were at the lecture.
'The examples set by these teachers have remained with me always and is something I think about when I'm doing my own teaching. In fact, I'm absolutely convinced that my interest in teaching – which I discovered when I came to Birmingham – is connected to those teachers I had at school. In particular, I am keen to try to explain things to people, to demystify mathematics, to get people interested in the subject. I really love my teaching, and although I am Head of School, I still teach all year round.'
After gaining an MA(Hons) in Mathematics from Cambridge, Paul moved to Oxford to do a DPhil before getting a lectureship at Birmingham in 1990. Here, he continued to 'aggressively pursue' his research interests in group theory. In very general terms, groups are used to measure the abstract notion of symmetry. As a consequence, they appear in very many areas of science as well as being 'fascinating objects of study from the pure mathematical point of view'.
There are two themes to Paul's research: the further development of the abstract theory of finite groups, and participation in an ongoing international project to produce a new and simplified proof of the Classification Theorem for the Finite Simple Groups.
'As a mathematician, what I really like doing is proving very difficult theorems,' explains Paul. 'One of the main reasons I chose my particular field of mathematics is because I was really inspired by some of the work that had been done in the subject area.'
A particular inspiration was the Feit-Thompson Theorem, which states that every finite group of odd order is solvable. It was proved by Walter Feit and John Griggs Thompson in the early 1960s. 'I wanted to understand it, which was quite a lofty ambition: the proof is 255 pages long!'
Paul not only succeeded in understanding it, he has since significantly contributed to the subject area.
As well as making contributions to the world of pure mathematics, Paul has always been eager to contribute to the School in an administrative capacity. He has held several positions, including Head of Education, and five years ago was made Head of School. In 2016, he was appointed for a second four-year term.
Although Paul has fulfilled – and, indeed, surpassed – his childhood ambitions, there is still a lot left he wants to do.
'I certainly have ambitions for the School, including expanding it and attracting high-quality appointments,' he says. 'We are becoming even better known internationally and we are making the School a more student-centric environment.
'From a research point of view, you never reach an end, and I have plenty of ideas, building on my previous work. I am as passionate now about mathematics as I was at the beginning of my career.' |
PKU professor Xu Chenyang wins 2017 Future Science Prize
NOV
. 08 2017
Peking University, Oct. 29, 2017: On October 29, 2017, the winners of 2017 Future Science Prize were announced in Beijing. The prize in mathematics and computer science was presented to professor Xu Chenyang. Xu delivered a speech during the awards ceremony.
Xu Chenyang at the award ceremony
In his speech, Xu expressed his gratitude to the award committee for being recognized as the first winner of Future Science Prize in the field of mathematics and computer science. He pointed out that mathematics is the foundation of many cutting-edge technologies, such as the signal transmittal of mobile phone and so on. However, the importance and basis of mathematics in our daily life is not the reason why mathematicians devote themselves to mathematic study. It is the elegance and beauty inside it that attract them to dedicate themselves to the course. Thus being able to appreciate this beauty and earn his living by doing so makes him feel grateful.
The charm of mathematics also lies in the utopia it helps to create. As Grothendieck mentioned, "The sole thing that constitutes the true "inventiveness" and imagination of the researcher is the quality of his attention as he listens to the voices of things." Xu believes the quality of attention as people listen to the voices of things is equal regardless of social status. As a result, everyone is equal and owns absolute freedom while exploring the unknown. As long as you can obey the basic rules, sky is the limits. Xu believes every single progress in the history of math implicates human reaching a new summit. He mentioned winning the prize is not only a honor of himself, but also an encouragement and boosting for young mathematicians. |
Engaging Fun Pythagorean Theorem Famous MathematicianPuzzle & STEAM resource.
This fun activity is perfect for any Algebra or Geometry student. It combines practicing the Pythagorean Theorem in a fun way with learning a little bit about the |
This paper
is presented as an introduction to my art
and its geometry in the union of three areas of our experience -
mathematics, art, and physics.
1.
Introduction
Artists invent
worlds with their own laws and limitations, sometimes called style.
Within this context, artists express themselves, their culture, indeed,
a world we all share. Art as well as science and mathematics is a world
to explore and discover that which defines us and expands the parameters
of our experience. For example, in the visual arts we have various spatial
approaches such as perspective, realist, cubist, atmospheric, etc.
For the most part, they are all based on a Cartesian grid. Out of my art
has grown a different mathematical structure based on a curvilinear coordinate
system which I call Wave Space Geometry.
The two basic
facets of Wave Space Geometry to be
considered in this paper are what I call
Common
Wave Space (CWS) and Interphase
Wave Space (IWS). I will
attempt to show how these geometries are used in my painting, and share
some observations and speculations concerning my work and our own physical
world.
Phase
Line: a continuous line division of space. Phase
Track: two parallel phase lines. Phase
Field: a series of phase tracks in one plane. Cross-Phasing:
crossing of two or more phase tracks.
Cross
Phase Field: the crossing of two or more phase fields.
Wave dimensions
and wave shapes are seen in Fig. 2.2 where w
is 1/2 the basic wavelength W, and a is its amplitude - the
three basic
wave shapes assumed are the common
wave
A, the line wave
B,
and the concave wave C.
Common
Wave Space (CWS) is defined by a curvilinear coordinate system.
It is a wave gnid pattern formed using an X or Y phase field
orientation with one field a function of the other. This is what I call
a functional geometric system and should not be confused with an overlay
system where one phase field is randomly overlaid on another.
Interphase
Wave Space (IWS) is a wave pattern formed by plotting a second
generation wave field on a CWS grid.
Wave
Space Equation The working formula, or
graphic equation, used here to draw the accompanying examples is:
where each
term is drawn sequentially, Y1 and X1
indicate the primary wave field directions, the X1 field
being a function of the Y1 field to form a CWS grid configuration.
X2 is the secondary wave field taken as a function of, and interphasing
with the CWS coordinate grid. Interphasing occurs when one field track
intercepts another, going in the same direction, and returns without
crossing. The shape formed, I call a nodal
element, or node en. The other elemental shape
which is formed by two or more phase tracks crossing
each other, I call a common elementec,
as seen in Fig. 2.3. CWS is made of totally common elements;
IWS is the summation of common and nodal elements in a proportion dependent
on the interphasing geometries.
Figure 2.4 Common Wave Space
(CWS)
Figure 2.5 Interphase Wave
Space (IWS)
An example
of CWS is seen in Fig.2.4 where wY1=6,
aY1=1,
and wX1=6,
aX1=2,
or SW=G(Y16/1X16/2),
written G(Y16X16/2).
An example
of IWS is seen in Fig. 2.5 where the X2
field of wX2=4,
aX2=2
interphases with the CWS coordinate system of Fig. 2.4
in the X1 direction, the third term of Sw,
The pq superscript in Sw indicates the phase position
of the secondary wave on the primary wave and determines the rhythm of
the interphase field as shown in Fig. 2.6.There are
W
possible starting positions on wavelength W; in our example, 7 out
of 12 positions are shown in aG(X16) wave field interacting
with a secondary X26 wave field. In Fig.
2.4 there are also 12 possible positions:W=2wX1=2×6=12.
In example, Fig. 2.5, q=0 was chosen to establish
the secondary X24/2 interphase wave field.Also
note in the equation
G[Y16(X16/2X24/2)]
of Fig. 2.5 that the parenthetic bracketing indicates
what is shown, and will be so in subsequent equations.
Figure 2.6 Phase Positions
Figure 2.7 Mock Axes
The fourth
term of Sw, is what I call a mock
axes coordinate system; in other words, a IWS configuration used
as CWS grid. This is done by choosing an axes intersection point, i.e.,
an origin, with one coordinate starting in the pri'mary field direction,
the other starting in the secondary field direction and at each node alternating
to the other field direction, giving the continuous, divergent x,y
mock axes Xm and Ym as exemplified
in Fig. 2.7. (Notice Xm is drawn from
the upper left to lower right and Ym from the lower left
to upper right). The "layered" position of Xm and Ym
of SW, means that an Xminterphase
field, orYm interphase field,
or
both simultaneously can be plotted. Fig.2.7 shows the
single interphase track Xm2.
Figure 2.8 CWS
Figure 2.9 CWS
Figure 2.10 CWS
Figure 2.11 CWS
The above
examples are given to further demonstrate how Wave Space works. CWS Figs.
2.8,
2.9,
2.10,
2.11
show a progression in wave dynamics holding aX1,
and Y12 constant with wX1,
going from infinity to 2.
Isolating
the mock axis field
Xm of Fig. 2.5,
we have Fig. 2.12. The semi-colon in equations such
as Xm: G(...) and Ym: G(...)
simply means Xm or Ym of the specific
geometry G(...). Fig. 2.13 is an extension ofFig.
2.5 space in the mock
X axis direction by
Xm2.
Variation
is further increased if we consider the shape of the wave w, or
wave-train WT. Wave shape A was used in the previous
examples. The other two basic shapes B and C, Fig.
2.3, should be noted. Figs. 2.17 and 2.18
show the B
and
C waves in the Fig. 2.4 CWS
geometry; Figs. 2.19 and 2.20 show
the B and C waves in the
Fig. 2.5 IWS
geometry. The symbol above the wave dimensions indicates the wave type
and its orientation: ~ indicates a full wave; Ç
È indicates a half wave oriented
apex up or down respectively.
Figure 2.17 CWS B
Wave
Figure 2.18 CWS C
Wave
Figure 2.19 IWS B
Wave
Figure 2.20 IWS C
Wave
Another approach
is the compounding of wavelengths and/or wave shapes as seen in Figs. 2.21
and 2.22 in CWS, with a more complex CWS version in
Fig.
2.23 and its IWS extension, Fig. 2.24. The various
squiggles, as usedabove, show the specific make-up of a wave shape or wave
train.
Figure 2.21 CWS Compound
Length
Figure 2.22 CWS Compound
Shape
Figure 2.23 Compound CWS
Figure 2.24 Compound M7S
3.
Shapes in Wave Space
In Wave Space,
the geometry defines the space. The elements, which are shapes in themselves,
as squares inCartesian space, are the increments of Wave Space. To get
an idea of how shapes change from geometry togeometry. Let us take a 2×2,
4 element square, and translate it in the following two geometries:
Fig.
3.1 showsthe square in three regions of the CWS geometry G(X14/4Y14/2);
here, the square appears as a rather amorphousblob, one shape being almost
linear in its elongation. Fig. 3.2 is an example of
IWS geometry G(X12X2p22/2),which
converts the Cartesian square of our space into a 4 element circular shape
in Wave Space,
Figure 3.1 CWS
Figure 3.2 IWS
Figure 3.3 Translation
What we have
done is take a square from our reality and put it in a different geometric
reality. In a sense, it is still a square, and would be seen as such by
the inhabitants of this new geometric space; however, from our frame of
reference, it has become a circle. In this kind of physicality, it begs
the question: What is "true" shape? The answer would seem to depend on
your frame of reference, which is true; and, in this particular case, it
depends on the geometric space the object occupies. Physically and philosophically
this is interesting - considering the infinity of spatial configurations
possible; that is, of course, if such spatial realities exist, a speculation
we will consider later.
As an exercise,
if we wish to translate from our Cartesian space to a Wave Space geometry,
refer to
Fig. 3.3. Here, we have a geometry similar
to Fig. 2.5 with our mock axes arbitrarily chosen, as
shown, and numbered to parallel our Cartesian grid (inset). It is now a
matter of drawing a shape in Cartesian space and converting it to Wave
Space, or do the reverse to see what shapes created in Wave Space look
like in Cartesian space. For a demonstration, I show an imaginary character
from Cartesian space, Fig. 3.4, and show it in three
other geometric configurations, Figs.
3.4a,
3.4b,
and 3.4c.
Figure 3.4 Character
Figure 3.4a
Figure 3.4b
Figure 3.4c
However,
the idea of shapes changing outside our frame of reference is not new when
we consider Einstein's illustration of a clock approaching the speed of
light: From our frame of reference, we see the clock being flattened and
time slowing; but, from the clock's frame of reference, evenything has
remained normal.
If we assume
a shape to move in Wave Space, some curious
things occur. A shape moving in our Newtonian (Cartesian grid) space will
maintain its dimensional and geometric integrity. A shape moving in CWS
will distort to a greater or lesser degree, depending on the geometry of
the space it traverses; but the relationship of the elements will remain
the same. However, a shape moving in IWS will not only distort, but it
may fragment; that is, on an elemental level, the relationship of the elements
will not remain intact. In IWS, the elements in a shape following one of
the mock axes will keep a consistent relationship with one another regardless
of the degree of distortion of the shape, as is shown in Fig.
3.5a. Here, we have our 4-element square following the Xm
axis through a number of successive incremental changes. If the shape should
follow a specific phase track exclusively, as is the case in Fig.
3.5b, some of the elements will separate (the cross-hatched figures)
from their partners while passing through the nodal region; however, the
square will be reformed as it passes through the next nodal region, thus
following a cyclical process of fragmentation and reformation from region
to region. For larger shapes comprising a vast number of elements, the
individual incongruities would most likely be absorbed and seen as proportionally
minor disturbances, fluctuations, or pulses.
Figure 3.5a
Figure 3.5b
4.
Wave Space Painting
Wave space
art is holistic in that the shapes, themes, rhythms, and spatial textures
are defined and influenced by the geometric configuration of the chosen
space. The interweaving phase fields form the individual elements, the
basic building blocks of larger shapes which in themselves can build to
increasingly larger wholes.
The choice
of the geometry to be used depends, of course, on the artist's intent.
Some spatial configurations are more dynamic than others; in other words,
they have a greater sense of action, or vitality, than those of a more
gentle, "expressive feeling". For example, in Fig. 3.4
our Cartesian character's attitude is somewhat static; it lacks a sense
of motion that is expressed by the subsequent examples. This dynamic can
then be emphasized or diminished by the use of color and tone in the rhythmic
structure of the painting itself. For example, red in the character might
suggest a hot, aggressive, passionate individual; while a blue might express
a cool, thoughtful character; and green, a calm, comfortable personality.
Or
such
colors could be used to characterize the background environment of the
painting to enhance or diminish the drama.
What we are
doing here is creating a world of space, color,
rhythm, and texture where the shapes become living things, experiencing
the trials and tribulations of the environment they inhabit - plus their
own episodic changes and transfigurations.
In the genre
of art, my work would be considered abstract; however, the abstract
components often imply literary content - that is, an abstract
narrative or metaphor. In other words,
I try to convey an idea or story exclusively through abstract forms, as
suggested above. Indeed, our imaginary character of Fig.
3.4 could be considered venturing through the three different spatial
environments and experiencing various physical and/or emotional transfigurations
as implied by the geometric dynamic. Another example can be seen in my
IWS painting, "Come Together", Fig. 4.1.
Figure 4.1 IWS "Come Together"
Figure 4.2 "Transfiguration"
Figure 4.3 CWS "Rhapsody
for 9 Notes"
This painting
is episodic. Beginning at the left and progressing right, through a steady
rhythmic space, separate elements, or parts, come together forming a complex
whole - a shape theme, or leitmotif, at the mid-point of the painting.
It is subsequently transformed into a linear, dancing combination of like
figures, and proceeds further through a rhythmically agitated space to
the end of the painting where it is seen penetrating a vertically static,
dark rhythmic color column on the verge of entering a blank, monochromatic
space. It could be the end of the story or the end of a chapter. It could
be "read" from right to left for a different interpretation. In any event,
how the painting is approached, experienced, and interpreted is entirely
up to the individual observer.
Sometimes
my art is perceived as visual music; and indeed, in many cases, it is my
intent. In the context of the painting, my theme may be a single shape
as in Fig. 4.1 and 4.2; or as a group
of separate elements, Fig. 4.3, much in the same manner
of music notation in a score. The theme, is subsequently developed through
various color keys, color/tone harmonies, and thematic manipulations, i.e.,
turned, flipped, dissected, distorted, reversed, etc. - all depending
what I, the artist/composer, hope to convey.
5.
Similar Worlds
Exploring
the worlds of my work, patterns of our day to day experience appear, suggesting
structures or methodologies in Wave Space geometry that parallels our own
world. I have compared on many occasions, riding the local ferry, the rippling
and cross-rippling of waves with the interacting wave fields of my own
studies. Perhaps you have experienced the amusing distortion of shapes
in a carnival mirror, and how they change from one region to another. Some
other patterns observed are flame movements, as in my painting, Fig.
5.1, "Aurora", and smoke in my study, Fig. 5.2.
In a detail of my painting "Edge of Chaos" and my painting "Garden of Middle
Harmony", Figs. 5.3 and 5.4, we have
patterns reminiscent of gas and liquid currents, i.e., turbulence. Also
in Fig. 5.4, I am reminded of the spectral patterns
seen in oil slicks; I have also seen this agitated schema as a colorful,
vibrating pattern on the retinas of my eyes, which apparently is a migraine
headache phenomenon.
Figure 5.1 "Aurora"
Figure 5.2 Heat Waves
Figure 5.3 IWS Detail "Edge
of Chaos"
Figure 5.4 "Garden of Middle
Harmony"
We find interphase
fields where wave patterns are damped and reenergized as in Figure
5.5. Chaos and fractal geometries are suggested in my painting "Transfiguration",
Fig.
4.2. The wave track of Fig. 2.7 suggests the meander
of a river.
Figure 5.5 Phase Damping
Figure 5.6 Plant Pattern
Figure 5.7 Plant Pattern
Figure 5.8 Plant Pattern
By utilizing
the node as a bifurcation point, certain geometries suggest plant growth
patterns, as in Figs. 5.6, 5.7, and
5.8.
6.
The Quantum Cosmic Connection
Exploring
the many worlds of Wave Space geometry has led me to speculate on the two
extremes of our own reality, i.e., micro or quantum space and cosmic space.
The basic analogy between Wave Space and the "real" world is summed up
in Fig. 6.1. Here, the quantum world is viewed as a
high frequency, densely packed nodular space compared to our essentially
"linear", Newtonian world. This analogy seems to fit well when we consider
Einstein positing a cosmic space which takes the form of a vast curvilinear
geometric structure and the quantum scientists who view the microcosmos
as a wave/particle paradigm of incredibly high frequencies.
Figure 6.1 World Reality
Scale
Figure 6.2 Minimum IWS Geometry
Let's consider
the limits of Wave Space geometry. From the previous discussion, the macro
limit would be a Cartesian grid, Newtonian reality, which is our basic
frame of reference. Beyond our frame of reference, in the microworld, we
assume the bottom limit of Wave Space geometry.
The minimum
wave configuration is w/a=2, and its spatial configuration
in IWS is given by the two equations,
(1) G(X22) and
(2) G(X, 2X2p22), Fig.
6.2. It should be noted, here,
that the two equations give the same geometry but are physically different:
The X1 field line, the straight line portion of (1),
is considered a wave with wX1/aX1=
¥,
i.e., as a curve with an infinite radius of curvature. The straight line
portion of (2) occurs with q=2, putting the secondary wave X2,
180o out of phase. This minimum configuration gives four elements
per wave cycle, an equal distribution of common to nodal elements, or ec/en=
1, and a maximum density of nodal elements. From this, ec/en
increases, as our reference scale increases, to our macroworld level and
beyond, ec/en®
¥. Also, as ec/en
increases, the nodal density decreases and the space between the widening
nodal regions takes on the appearance of a Cartesian grid, illustrated
to some extent in Fig. 2.3. In this region, we presume
the predictable, Newtonian world we are familiar with. Let me further suggest
that in this geometry of widely separated nodal regions, that the nodal
regions are analogous to the black hole regions of our own universe. This
speculation may or may not be that far fetched if we consider Wave Space
geometry in three or more dimensions rather than the two of this paper.
Another parallel
to our world is, because of the density of nodes at the quantum level,
and their bifurcating factor, a microshape or particle/wave packet would
have more possible paths to follow in a given distance and therefore be
harder to predict its position or velocity at a given time, unless we knew
the precise geometry of the space and the laws that govern it. Also, a
microshape moving in a densely nodular space, as in Fig.
6.2, would be more difficult to identily due to its continual morphing
from nodular to non-nodular space.
The above,
of course, is pure science fiction; but even the physicists don't know
what the quantum world is like; they have a strong feeling and good reason
to believe it is quite different from our own experience. Many feel the
quantum world to be a complex paradigm of wave fields; and, even on a cosmic
scale, Einstein posits a curvilinear geometric structure. However, Einstein's
conclusion, as I understand a it is that mass, or mass as energy, interacts
with and distorts the surrounding space. In applying Wave Space to "our"
space, I have to conclude that it is the space that creates and defines
the mass, or to be more accurate, the illusion of mass. It is the interweaving
and interdependence of complex geometries that create and contribute to
the totality of reality itself. |
Mathematics : a very short introduction by Timothy Gowers(
Book
) 19
editions published
between
2002
and
2016
in
English and Dutch
and held by
961 WorldCat member
libraries
worldwide
"Mathematics is a subject we are all exposed to in our daily lives, but one which many of us fear. In this introduction, Timothy
Gowers elucidates the most fundamental differences, which are primarily philosophical, between advanced mathematics and what
we learn at school, so that one emerges with a clearer understanding of such paradoxical-sounding concepts as 'infinity',
'curved space', and 'imaginary numbers'. From basic ideas, through to philosophical queries, to common sociological questions
about the mathematical community, this book unravels some of the mysteries of space and numbers."--Jacket
<>(
Book
) 1
edition published
in
2013
in
Korean
and held by
1 WorldCat member
library
worldwide
Combinatorics in the service of mathematics by World Mathematical Year 2000 Symposium: the Legacy of John Charles Fields(
Visual
) 1
edition published
in
2000
in
English
and held by
1 WorldCat member
library
worldwide
Mathematics by Timothy Gowers(
Recording
) 1
edition published
in
2017
in
English
and held by
1 WorldCat member
library
worldwide
"The aim of this book is to explain, carefully but not technically, the differences between advanced, research-level mathematics,
and the sort of mathematics we learn at school. The most fundamental differences are philosophical, and readers of this book
will emerge with a clearer understanding of paradoxical-sounding concepts such as infinity, curved space, and imaginary numbers.
The first few chapters are about general aspects of mathematical thought. These are followed by discussions of more specific
topics, and the book closes with a chapter answering common sociological questions about the mathematical community (such
as "Is it true that mathematicians burn out at the age of 25?")"--Provided by publisher
Matematika nagyon röviden by Timothy Gowers(
Book
) 1
edition published
in
2010
in
Hungarian
and held by
1 WorldCat member
library
worldwide
Princeton Companion to Mathematics(
) 1
edition published
in
2008
in
English
and held by
0 WorldCat member
libraries
worldwide
This text features nearly 200 entries which introduce basic mathematical tools and vocabulary, trace the development of modern
mathematics, define essential terms and concepts and put them in context, explain core ideas in major areas of mathematics,
and much more |
AUSTIN (KXAN) — 3/14/15 equals National Pi Day! The first five digits of Pi line up with today's date.
Pi is the symbol used in geometry to represent the ratio of the circumference of a circle to the circle's diameter. You can use it to find the area of a circle, which is Pi times the radius of the circle squared.
To celebrate the day, there are numerous events going on throughout the country.
At 9:26 a.m. (After 3.1415, the numbers are "926" hence the time), high school seniors will find out if they got accepted into MIT. Lovebirds can tie the knot in Vegas for $314.15 and they even get an apple pie. And at the holiday's birth place in San Francisco, there will be a pizza pie tossing.
Dr. Michael Starbird, a mathematician at the University of Texas – Austin explains what Pi is.
"Pi is one of these numbers that has no pattern that repeats and so it just continues on forever. So, we'll never know all of Pi, which I think is a metaphor for our understanding of the world: that we continue to learn more things," said Starbird |
Math is the right of all free people.
A Variety of Math Jokes
Occupy Math has raised the issue of mathematical humor from time to time. In this week's blog, by request, there are nothing but jokes. There are 20 jokes here – keep track of how many you get and that is your Mathemagician Rating. Mathematicians use words a little differently from other people. That leads to the following sort of joke.
☆ Do you know what's odd? Every other number.
☆ The obtuse triangle is upset because it's never right.
☆
Mother: Son, what did you learn at university?
Son: Well, Ma, I learned that pie are squared.
Mother: That's ridiculous, son! Pie are round. Cornbread are squared.
☆ Question: What is 2n+2n? Answer: I'm not sure, but I think it's from another country.
Another source of mathematical humor is people not understanding the directions. (It is true that better directions can usually be written.)
Some math jokes require you to know special facts.
☆ There are 10 sorts of people in the world, those who understand binary and those who don't.
☆ Two is the oddest prime of all.
☆ Why are π/2 radians the right number of radians?
☆ Question: What is a polar bear? Answer: A rectangular bear after a coordinate transformation.
☆ Question: What do you get if you cross a mountain climber with a mosquito? Answer: Nothing, you cannot cross a scalar with a vector.
☆ Training in mathematics helps you to not be able to tell the difference between a coffee cup and a donut.
☆ The number 5! is equal to 120, not FIVE!
☆ Garrison Keeler (the worlds tallest radio comedian) created a wonderful mathematical joke in the sign-off for his weekly News from Lake Wobegon, to wit:
"From Lake Wobegon, where all the women are strong, all the men are good looking, and all the children are above average."
There are quite a few jokes that have to do with the way that mathematicians view those in other areas or the way people in other areas view mathematicians.
Physicist: "Three is prime, five is prime, seven is prime, nine is not prime. The proposition is false."
Mathematician: "Nine is the smallest counter example. The proposition is false."
Engineer: "Three is prime, five is prime, seven is prime. Close enough."
Chemist: "Three is prime, five is prime, seven is prime, nine is an experimental error, 11 is prime, 13 is prime, 15 suggests the proposition is false."
Computer scientist: "Three is prime, five is prime, seven is prime, SEGMENTATION FAULT…"
Student: "Dude, odd numbers are really prime!"
Biologist: "Three is prime, five is prime, seven is prime, nine is prime. The proposition is true."
Social scientist: "Three is prime, four is not prime. The proposition is false."
Administrator: "Three is prime, wait, I'll have to get back to you."
☆
A couple of physicists are at a conference with a day off in the middle. Finding that there are hot air balloons for rent, they decide to make an ascension. They get into an argument about the way that the balloon works and forget to navigate. Presently they realize they are completely lost. They let some of the hot air out of the bag and drift lower. Above an alpine meadow they see a man in shorts and a heavy coat wearing a huge backpack. He is stomping across the meadow, head down, in a straight line. The first physicist addresses him:
"Where are we?"
The man reacts with startlement and then watches the balloon drift for a while. As the two physicists are almost out of earshot, he replies:
"You are in a hot air balloon."
An exchange between the two physicists follows:
"Oh good grief! He's a mathematician!"
"How can you tell?"
"He took a long time to answer. The answer was completely correct. It was also totally useless.
☆
There are three women on a train. One of them is an psychologist, one of them is a engineer, and one of them is a mathematician. They have just crossed the border into Scotland and they see a black sheep standing in a field, though the window. The psychologist exclaims "The sheep in Scotland are black!" The engineer says, "No. There is at least one black sheep in Scotland." The mathematician replies in an exasperated tone, "No. There is at least one sheep in Scotland, with at least one black side.
This next one is a calculus joke.
☆ Question:
Answer:Log cabin. Unless you remember that an indefinite integral requires a +C in which case the answer is house boat.
☆ Occupy Math found this floating around the internet and would love a proper attribution!
The next joke is really funny to mathematicians and others really don't seem to get it. It is a null-set joke, if that helps.
Occupy Math hopes that the jokes you did not get will motivate you to diligently pursue your studies in mathematics. He also hopes the post has not made you think of calling the nice young men in their clean white coats. In any case, a tweet or comment that points out other math jokes would be appreciated!
I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics |
Post navigation
A Quadratic of Solace (or, Maybe Math Class Has a Purpose, Question Mark?)
Like, how do you factorize a quadratic? How to you differentiate a cubic? How do you solve a system of simultaneous linear equations? How do you poach an egg?
(Apparently you need a gentle whirlpool to get the egg moving. Whirlpools: the unsung hero of the breakfast table.)
Why are they so skilled at how?It's because students like procedures. They like certainty, clarity, the feeling that you know exactly what to do at every moment.
But they struggle with why. And – even more basically – they struggle with what.
For example…
I find that questions like this elicit one of two responses from students. Either this:
Or this:
These aren't questions students are accustomed to answering in math class. In history, perhaps, where they have to write IDs of historical figures and events; or even in science, where they have to understand each component's role in a theory.
But not in math. We math teachers tend to ask lots of how questions, and not so many what questions.
If you ask me, that's sort of sad. They're experts in how, and they can't even tell you what the how is for.
And in this case, it turns out, there's a pretty satisfying answer.
First, note that quadratics are much more complicated and interesting than their simple flat-brained cousins, the linears:
And then, note that quadratics are much simpler than their roller-coaster contortionist siblings, the cubics, quartics, and other high-degree polynomials:
To me, this is the appeal of quadratics. As degree-2 polynomials, they occupy a sweet spot between the dull degree-1's, and the intimidating, intractable degree-3's.
Just as Goldilocks sought the perfect bed (not too hard, not too soft) and the perfect porridge (not too hot, not too cold), so the mathematician seeks the perfect polynomial. Not too hard, not too easy. Not too complex, not too simple.
Just about right.
Of course, this line of reasoning is open to an obvious attack. Okay, a disgruntled student might say, you've convinced me that, if I'm going to study polynomials, I ought to focus on quadratics first.
But why should I study polynomials to begin with?
The answer to that is trickier, I think. You might as well ask this:
This question has as many different answers as mathematics has teachers. Some like to focus on the applications of math. Some argue it's all bout the beauty. Some just say, "Because," and then sigh, because it's been a long day.
But for me, it's about thinking.
And the role that the quadratic plays in polynomials… well, that's exactly the role that mathematics plays human thought.
In every walk of life, humans need to reason.So of course, they can learn these intellectual skills in other places. You don't need math. But gosh, does math make it easier!
You can learn to taxonomize in biology, by considering the classification of organisms. But your taxonomies will never be perfect, because life doesn't fit into neat little boxes. (I'm looking at you, protists.)
Life doesn't… but math does.
Or you can learn to dissect arguments in civics. But emotions will flare. It'll be tough to agree on premises. And even if you do, words like "justice," "freedom," and "common good" are subject to fuzzy interpretations and subtle misunderstandings. All words are like that: a little vague, tricky to pin down.
Except in math.
Logic shows up everywhere. But in math, it's the whole game. Math isolates the operations of logic and reason so that we can master them.
In short: math is the playground of reason.
This post is hastily adapted from a talk I gave yesterday at University of Birmingham, titled Death to the Quadratic Formula (or, Long Live the Quadratic Formula). Thanks to Dave Smith and the IMA for the invitation!
There's likely something to that quadratic argument – Quadratics seemingly came out on top during my Twitter poll, "Which polynomial would be the best leader of mathematics". Amusingly, I published the results of that earlier this week (in conjunction with my personified math comic) under the header "Quadratic Formula". I suspect Lyn would object to being called "simple" though.
I have a sneaking suspicion when you ask "why do you need to learn this?"
The general reaction is, "Because this is what you are teaching. If I don't you will fail me. Yes, why? Aren't you supposed to tell us why any of this is relevant?! Tell us why!"
Why quadratics?…Occams razor! The simplest model that suits your needs is best! And, too often a line just doesn't cut it.
Two linear models often interact in such a way that you need the product (force * distance, speed*time, units * price, etc.) or an area, forming a quadratic equation.
Nice idea about quadratics being the sweet spot. But in general, why study math? Besides the usual answers, I offer this one: Because math gives us one more way of thinking about the world: through mathematical lenses. The artist has one lens. The economist has a different one. So does the psychologist. And many others. Why not add one more way of looking at the world to your collection? |
Nontransitive dice (also known as "Efron Dice" after one of the inventors) have probably regained some attention since being mentioned in Simon Singh's recent book, "The Simpsons and Their Mathematical Secrets." Several different nontransitive combinations are actually possible, but the set mentioned in Singh's book, include "Die A" with sides, 3,3,5,5,7,7, "Die B" with sides 2,2,4,4,9,9, and "Die C" composed of sides 1,1,6,6,8,8. On average, a throw of Die A will beat (56% of the time) a throw of Die B, and a throw of Die B will beat (56% of the time) Die C… YET, Die C, on average, will beat out Die A (56% of the time)… How cool is THAT! or as, Singh writes, "Nontransitive relationships are absurd and defy common sense, which is probably why they fascinate mathematicians |
Good geometry is the normal identify for what we name this day the geometry of 3-dimensional Euclidean house. This e-book provides suggestions for proving a number of geometric ends up in 3 dimensions. particular realization is given to prisms, pyramids, platonic solids, cones, cylinders and spheres, in addition to many new and classical effects. A bankruptcy is dedicated to every of the next easy innovations for exploring house and proving theorems: enumeration, illustration, dissection, airplane sections, intersection, generation, movement, projection, and folding and unfolding. The booklet contains a choice of demanding situations for every bankruptcy with options, references and an entire index. The textual content is geared toward secondary university and school and college lecturers as an creation to sturdy geometry, as a complement in challenge fixing classes, as enrichment fabric in a path on proofs and mathematical reasoning, or in a arithmetic direction for liberal arts scholars.
This ebook is an advent to manifolds initially graduate point, and available to any pupil who has accomplished a great undergraduate measure in arithmetic. It includes the fundamental topological rules which are wanted for the extra research of manifolds, relatively within the context of differential geometry, algebraic topology, and similar fields.
This publication relies on a graduate path taught through the writer on the college of Maryland. The lecture notes were revised and augmented via examples. the 1st chapters enhance the basic thought of Artin Braid teams, either geometrically and through homotopy concept, and speak about the hyperlink among knot concept and the combinatorics of braid teams via Markou's Theorem.
Additional info for A Mathematical Space Odyssey: Solid Geometry in the 21st Century
Example text
2. 7. The sequence fqn g1 nD1 D f0; 1; 2; 4; 6; 9; 12; 16; 20; 25; the square and oblong numbers (starting with 0) arranged in order, is sometimes called the sequence of quarter-squares. Why? 8. 9. 3 for the first five octahedral numbers 1, 6, 19, 44, and 85. 42 CHAPTER 2. 3. Find a formula for the nth octahedral number. 10. Find the total number of cubes in an n n n cubical grid. 4 we see a 6 6 6 cubical grid with three of its 441 cubes highlighted. 4. 11. 5 we found the maximum number of parts into which space can be divided by n planes.
4 In many cases we have a sequence of configurations or patterns, and we seek to count the number of objects in each pattern in the sequence.
How large is the angle between the two diagonals? 4 |
Thursday, July 02, 2009
Mathematics and the Game Show
Last week, I saw a reference on the Coding Horror blog about an interesting problem posed and answered on the website of Marilyn vos Savant. Marilyn is the woman who, for a while, was listed in the Guinness Book of World Records as the holder of the highest recorded IQ. I've always enjoyed her column in Parade magazine. Here is the question:
You are on a game show and they show you three doors of which you can open one. Behind one of the doors is a good prize (say a new car) and the other two doors hide things you wouldn't want (I think they used goats in the example). You pick one of the doors but they don't open it yet. Then the host, knowing which door hides the car, opens one of the other doors you didn't pick that has the goats. Then the host asks you, "Do you want to keep your door or trade it for the other door that isn't open?" The question then is, "Should you change your selection or stay with the one you originally selected?"
Most people think that the answer is that it doesn't make any difference. That's what I first thought, too. It seems there is a 50-50 chance that your original door holds the car and you can't do any better by switching. But in the 1990 Parade article, Marilyn said you should change your selection. As a matter of fact, she says that you have twice as much chance of winning if you change which door you want to open!
Her original explanation of why this is true didn't convince a lot of people - including a number of math professors in college. She got a lot of nasty mail saying that she was wrong and that she was afraid to admit it. Then she did what everyone should have done from the beginning. She wrote a simple table listing every possible outcome and just counted up the good and the bad outcomes to show that it is indeed better to change your selection once the host has opened one of the doors that doesn't hold the car. There are good lessons for all of us here: 1) Math doesn't have to be hard, 2) Even when we think we know the answer, it's best to write it down, 3) We shouldn't be so sure of ourselves until we've checked the facts. I suggest you go to Marilyn's discussion of this problem and to see her table to convince yourself.
Of course, the whole premise of this is based on the idea that you would want a car more than you would want the goats :-) Perhaps you'd rather have the goats. Then I guess it's better to stay with your original selection. By the way, the picture of the goats at the top of this article is from |
SEARCH ALL LYBIO'S HERE
Song From π
Song From π
"
The Accurate Source To Find Transcript To Song From π."
[Song From π (Pi)]
[Song From Pi]Source: LYBIO.net
I created the melody for this song by taking pi and assigning each number to a note on the A harmonic minor scale. I added harmonies with the left hand.
Songs starts now!
Fun facts about pi:
Since there are 360 degrees in a circle and pi is intimately connected with the circle, some mathematicians were delighted to discover that the number 360 is at the 359th digit position of pi.
In the Greek alphabet, TT is the sixteenth letter. In the English alphabet, p is also the sixteenth letter.
Thirty-nine decimal places of pi are enough to compute the circumference of a circle the size of the known universe with an error no greater than the radius of a hydrogen atom.
[Song From Pi]Source: LYBIO.net
"Pi Day" is celebrated on March 14 (which was chosen because it resembles 3.14). The official celebration begins at 1:59 p.m., to make an appropriate 3.14159 when combined with the date.
The Bible gives pi a value of 3 in 1 kings 7:23 where it describes the altar inside Solomon's temple: "And he made a molten sea of ten cubits from brim to brim … and a line of thirty cubits did compass it round about."
In 1888, a Indiana doctor named Edwin Goodwin claimed he had been "supernaturally taught" the exact measure of the circle and even had a bill proposed in the Indiana legislature that would copyright his mathematical findings. The bill never became law thanks to a mathematical professor in the legislature who pointed out that the method resulted in an incorrect value of pi.
Many mathematicians claim that it is more correct to say that a circle has an infinite number of corners than to view a circle as being cornerless.
The first 144 digits of pi add up to 666 (which many scholars say is the mark of the Beast). And 144 = (6+6) x (6+6).
Some scholars claim that humans are programmed to find patterns in the world because it's the only way we can give meaning to the world and ourselves. Hence, the obsessive search to find patterns in TT. |
Geometry can sometimes be fun But isn't a day in the sun When I think of it I begin to fear I'm not sure I like it, it isn't that clear I'll give it a try or maybe I'll run
Limerick by Chandler McSwain & Julie Richardson
Geometry is confusing It makes us want to go snoozing Filled with many equations It certainly isn't a sensation What's up with all these "proofs"? Why don't we just make it go POOF? Maybe we'll learn to love it one day but for now our feelings are going to stay. |
The Reason Pi is a Stupid Number and Why People Are Still Celebrating It [VIDEO]
Mar 17, 2014 10:47
Mathemusician Victoria Hart—aka Vi Hart—gives an overwhelming evidence that demonstrates that Pi is a stupidly common number and the fascination of people with this number is just dumb. Check out the Anti-Pi rant |
One artist's answer to the age old question: How do you make math fun?
It's true! Discussing everything from scale factors to measurements and more, SCAPE breaks down the mathematical makeup of the mural designs in his graffiti in two excellent and colorful videos. With his refreshing take on graffiti, SCAPE shows the relevance of math in the everyday art that you love and challenges you to find the fun in it too.
Discover how SCAPE Martinez incorporates fun math in art with these two videos: |
Suggested Reading: How Mathematics Happened: The First 50000 Years
What first got me interested in this book is the "50000 years" part. I was preparing lectures notes for my course on discrete mathematics and I wanted my students to have an idea of what prehistoric maths might have been, say, 20000 years ago. Unfortunately, you wont learn much about this in this book
The book does hint about what mathematics might have been in hunter-gatherer times, and how it might have affected later developments. But that lasts for about a chapter or so, and the remainder is devoted to historical mathematics: Ancient Egyptian, Babylonian, and Classical Greek. All kinds of numerical algorithms are covered, presented in great detail, making the book more technical than historical. Some part are speculative as the historical record is incomplete at best, but it is speculative in the best way possible, with every assumption backed by an actual historical observation.
That's what I wanted to find out. Surely, there must be some. I can't believe we invented numbers at the same time as writing, and there are hints that we, indeed, had some knowledge about numbers WAY before (c.f. , that might or might not be "arithmetical"). I am also suspicious of 19th-century like assumption of man being the good savage with 1,2,many as a number system until (shortly before) the invention of writing. |
The Calculus of Crabbing
TwoPi and I, still traveling around visiting family, were just on the Oregon coast for a few days. While we were there, my brother-in-law Ken took us out crabbing. The crabbing turned out to be a wash dinner-wise since they were too small, but the day was beautiful, the beaches calm, and as a bonus there was some cool math.
It seems that the best time to go crabbing is when the currents are weakest. To find out those times, Ken used a Tide Predictor:
The weakest tides are right at High Tide and at Low Tide. Why? That's when the change in sea level — the derivative, in other words, is close to zero. Calculus in action! (Indeed, I imagine that current could be viewed as a kind of derivative of the sea height, since it is strongest when the slope of the water levels is changing the fastest.) As an aside, high tide is apparently better for catching crabs than low tide, but that has less to do with calculus and more to do with crustaceous personalities.
Edited 7/2 to add: I just realized that the second derivative also plays a role! If the second derivative is closer to zero as well, it means that the current isn't changing as quickly (in addition to not being very strong) so that gives a longer time period to check the traps and put them out again before the current gets strong enough that the crabs run back to the river sides or ocean.
If we hadn't had the tide charts, we could have used this fancy Tide Clock on the wall:
Except it wasn't working.
The tide itself leads to all sorts of other math problems. One of the neatest has to do with the cycles of the tides. The high tide peaks changed by almost 25 hours each day, not 24, so high and low tide cycle through different times of the year. It turns out that it takes 18.6 years for the pattern of high/low tide times to repeat itself. Apparently people used to be hired to take careful measurements of the tide and once the record stretched back 18.6 years, it was considered complete for that particular area. Not the most exciting job I suspect, but certainly important. |
Non-Euclidean geometry
non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclideangeometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is replaced with an alternative one. This article contains a variety of entries focusing on the history and development of the subject.
[The] empirical origin of Euclid's geometrical axioms and postulates was lost sight of, indeed was never even realised. As a result, Euclidean geometry was thought to derive its validity from certain self-evident universal truths; it appeared as the only type of consistent geometry of which the mind could conceive. Gauss had certain misgivings on the matter, but... the honor of discovering non-Euclidean geometry fell to Lobatchewski and Bolyai. ...
From the difference in geometric premises important variations followed. Thus, whereas in Euclidean geometry the sum of the angles of any triangles is always equal to two right angles, in non-Eudlidean geometry the value of this sum varies with the size of the triangles. It is always less than two right angles in Lobatchewski's, and always greater in Riemann's. Again, in Euclidean geometry, similar figures of various sizes can exist; in non-Euclidean geometry, this is impossible.
It appeared then, that the universal truth formerly credited to Euclidean geometry would have to be shared by these two other geometrical doctrines. But truth, when divested of its absoluteness, loses much of its significance, so this co-presence of conflicting universal truths brought the realisation that a geometry was true only in relation to our more or less arbitrary choice of a system of geometrical postulates. ...The character of self-evidence which had been formerly credited to the Euclidean axioms was seen to be illusory.
The decisive steps toward a clear understanding of non-Euclidean geometry were taken by Riemann, Helmholtz, and Poincaré, who recognized the essential unity of geometry and physics. However, the understanding did not come into its own until Einstein showed that such a combination of geometry and physics was really necessary for the derivation of phenomena which had actually been observed.
Philipp Frank, Philosophy of Science: The Link Between Science and Philosophy (1957)
Let us then examine the extension of this universe to ascertain whether there exists there an infinitely great. The opinion that the world was infinite was a dominant idea for a long time. Up to Kant and even afterward, few expressed any doubt in the infinitude of the universe.
Here too modern science, particularly astronomy, raised the issue anew and endeavored to decide it not by means of inadequate metaphysical speculations, but on grounds which rest on experience and on the application of the laws of nature. There arose weighty objections against the infinitude of the universe. It is Euclidean geometry which leads to infinite space as a necessity. ...Einstein showed that Euclidean geometry must be given up. He considered this cosmological question too from the standpoint of his gravitational theory and demonstrated the possibility of a finite world; and all the results discovered by the astronomers are consistent with this hypothesis of an elliptic universe.
He dwells only on broad impressions of vast angles and stone surfaces—surfaces too great to belong to any thing right or proper for this earth, and impious with horrible images and hieroglyphs. I mention his talk about angles because it suggests something Wilcox had told me of his awful dreams. He had said that the geometry of the dream-place he saw was abnormal, non-Euclidean, and loathsomely redolent of spheres and dimensions apart from ours.
Although K. F. Gauss, one if the spiritual fathers of non-Euclidean geometry... proposed a possible test of the flatness of space by measuring the interior angles of a terrestrial triangle, it remained for... K. Schwarzschild to formulate the procedure and to attempt to evaluate [curvature] K{\displaystyle K} on the basis of astronomical data... Schwarzschild's pioneer attempt is so inspiring in its conception and so beautiful in its expression...[!]
In the decades leading up to the period of relativity theory the architecture of space was revolutionized. Until then the mathematical imagination, and with it all of scientific thinking, had been dominated by a single book. ...Yet the mathematical framework the Elements espoused grants an unfounded privilege to one view, excluding the very idea of non-Euclidean geometries. The roots of a more flexible attitude to geometry reach back to the Renaissance creators of linear perspective, but the development... into the modern discipline... had to await the... great mathematicians such as Poncelet, Cayley and Klein. By the time of Einstein, non-Euclidean geometries and the even more comprehensive theory of projective geometry had broken the grip of Euclid on mathematical and spatial thinking, and a new imagination of space could be born.
Arthur Zajonc, Catching the Light: The Entwined History of Light and Mind (1993)
In geometry the axioms have been searched to the bottom, and the conclusion has been reached that the space defined by Euclid's axioms is not the only possible non-contradictory space. Euclid proved (I, 27) that "if a straight line falling on two other straight lines make the alternate angles equal to one another, the two straight lines shall be parallel to one another." Being unable to prove that in every other case the two lines are not parallel, he assumed this to be true in what is now generally called the 5th "axiom," by some the 11th or the 12th "axiom."
Simpler and more obvious axioms have been advanced as substitutes. As early as 1663, John Wallis of Oxford recommended: "To any triangle another triangle, as large as you please, can be drawn, which is similar to the given triangle." G. Saccheri assumed the existence of two similar, unequal triangles. Postulates similar to Wallis' have been proposed also by J. H. Lambert, L. Carnot, P. S. Laplace, J. Delboeuf. A. C. Clairaut assumes the existence of a rectangle; W. Bolyai postulated that a circle can be passed through any three points not in the same straight line, A. M. Legendre that there existed a finite triangle whose angle-sum is two right angles, J. F. Lorenz and Legendre that through every point within an angle a line can be drawn intersecting both sides, C. L. Dodgson that in any circle the inscribed equilateral quadrangle is greater than any one of the segments which lie outside it. But probably the simplest is the assumption made by Joseph Fenn in his edition of Euclid's Elements, Dublin, 1769, and again sixteen years later by William Ludlam... and adopted by John Playfair: "Two straight lines which cut one another can not both be parallel to the same straight line." It is noteworthy that this axiom is distinctly stated in Proclus's note to Euclid, I, 31.
The most numerous efforts to remove the supposed defect in Euclid were attempts to prove the parallel postulate. After centuries of desperate but fruitless endeavor, the bold idea dawned upon the minds of several mathematicians that a geometry might be built up without assuming the parallel-axiom. While A. M. Legendre still endeavored to establish the axiom by rigid proof, Lobachevski brought out a publication which assumed the contradictory of that axiom, and which was the first of a series of articles destined to clear up obscurities in the fundamental concepts, and greatly to extend the field of geometry.
Nicholaus Ivanovich Lobachevski['s]... views on the foundation of geometry were first set forth in a paper laid before the physico-mathematical department of the University of Kasan in February, 1826. This paper was never printed and was lost. His earliest publication was in the Kasan Messenger for 1829 and then in the Gelehrte Schriflen der Universtät Kasan, 1836-1838... "New Elements of Geometry, with a complete theory of Parallels." ...remained unknown to foreigners, but even at home it attracted no notice. In 1840 he published a brief statement of his researches in Berlin, under the title Geometrische Untersuchungen zur Theorie der Parallellinien. Lobachevski constructed an "imaginary geometry," as he called it, which has been described by W. K. Clifford as "quite simple, merely Euclid without the vicious assumption." A remarkable part of this geometry is this, that through a point an indefinite number of lines can be drawn in a plane, none of which cut a given line in the same plane. A similar system of geometry was deduced independently by the Bolyais in Hungary, who called it "absolute geometry."
Wolfgang Bolyai de Bolya... after studying at Jena... went to Göttingen, where he became intimate with K. F. Gauss, then nineteen years old. Gauss used to say that Bolyai was the only man who fully understood his views on the metaphysics of mathematics. Bolyai became professor at the Reformed College of Maros-Vásárhely, where for forty-seven years he had for his pupils most of the later professors of Transylvania. ...he was truly original in his private life as well as in his mode of thinking. ...No monument, said he, should stand over his grave, only an apple-tree, in memory of the three apples; the two of Eve and Paris, which made hell out of earth, and that of I. Newton, which elevated the earth again into the circle of heavenly bodies. His son, Johann Bolyai... once accepted the challenge of thirteen officers on condition that after each duel he might play a piece on his violin, and he vanquished them all.
The chief mathematical work of Wolfgang Bolyai appeared in two volumes, 1832-1833 entitled Tentamen juventutem studiosam in elementa matheseos puræ... introducendi. It is followed by an appendix composed by his son Johann. Its twenty-six pages make the name of Johann Bolyai immortal. He published nothing else but he left behind one thousand pages of manuscript.
While Lobachevski enjoys priority of publication, it may be that Bolyai developed his system somewhat earlier. Bolyai satisfied himself of the non-contradictory character of his new geometry on or before 1825; there is some doubt whether Lobachevski had reached this point in 1826. Johann Bolyai's father seems to have been the only person in Hungary who really appreciated the merits of his son's work. For thirty-five years this appendix, as also Lobachevski's researches, remained in almost entire oblivion. Finally Richard Baltzer of the University of Giessen, in 1867, called attention to the wonderful researches.
In 1866 J. Hoüel translated Lobachevski's Geometrische Unter suchungen into French. In 1867 appeared a French translation of Johann Bolyai's Appendix. In 1891 George Bruce Halsted, then of the University of Texas, rendered these treatises easily accessible to American readers by translations brought out under the titles of J. Bolyai's The Science Absolute of Space and N. Lobachevski's Geometrical Researches on the Theory of Parallels of 1840.
A copy of the Tentamen reached K. F. Gauss, the elder Bolyai's former roommate at Göottingen, and this Nestor of German mathematicians was surprised to discover in it worked out what he himself had begun long before, only to leave it after him in his papers. As early as 1792 he had started on researches of that character. His letters show that in 1799 he was trying to prove a priori the reality of Euclid's system; but some time within the next thirty years he arrived at the conclusion reached by Lobachevski and Bolyai. In 1829 he wrote to F. W. Bessel, stating that his "conviction that we cannot found geometry completely a priori has become, if possible, still firmer," and that "if number is merely a product of our mind, space has also a reality beyond our mind of which we cannot fully foreordain the laws a priori." The term non-Euclidean geometry is due to Gauss.
It is surprising that the first glimpses of non-Euclidean geometry were had in the eighteenth century. Geronimo Saccheri... a Jesuit father of Milan, in 1733 wrote Euclides ab omni naevo vindicatus (Euclid vindicated from every flaw). Starting with two equal lines AC and BD, drawn perpendicular to a line AB and on the same side of it, and joining C and D, he proves that the angles at C and D are equal. These angles must be either right, or obtuse, or acute. The hypothesis of an obtuse angle is demolished by showing that it leads to results in conflict with Euclid I, 17: Any two angles of a triangle are together less than two right angles. The hypothesis of the acute angle leads to a long procession of theorems, of which the one declaring that two lines which meet in a point at infinity can be perpendicular at that point to the same straight line, is considered contrary to the nature of the straight line; hence the hypothesis of the acute angle is destroyed. Though not altogether satisfied with his proof, he declared Euclid "vindicated."
J. H. Lambert... in 1766 wrote a paper "Zur Theorie der Parallellinien," published in the Leipziger Magazin für reine und angewandte Mathematik, 1786, in which: (1) The failure of the parallel-axiom in surface spherics gives a geometry with angle-sum > 2 right angles; (2) In order to make intuitive a geometry with angle-sum < 2 right angles we need the aid of an "imaginary sphere" (pseudo-sphere); (3) In a space with the angle-sum differing from 2 right angles, there is an absolute measure (Bolyai's natural unit for length). Lambert arrived at no definite conclusion on the validity of the hypotheses of the obtuse and acute angles.
Among the contemporaries and pupils of K. F. Gauss, three deserve mention as writers on the theory of parallels, Ferdinand Karl Schweikart... professor of law in Marburg, Franz Adolf Taurinus... a nephew of Schweikart, and Friedrich Ludwig Wachter... a pupil of Gauss in 1809 and professor at Dantzig. Schweikart sent Gauss in 1818 a manuscript on "Astral Geometry" which he never published, in which the angle-sum of a triangle is less than two right angles and there is an absolute unit of length. He induced Taurinus to study this subject. Taurinus published in 1825 his Theorie der Parallellinien in which he took the position of Saccheri and Lambert, and in 1826 his Geometriæ prima elementa, in an appendix of which he gives important trigonometrical formulæ for non-Euclidean geometry by using the formulæ of spherical geometry with an imaginary radius. His Elementa attracted no attention. In disgust he burned the remainder of his edition. Wachter's results are contained in a letter of 1816 to Gauss and in his Demonstratio axiomatis geometrici in Euclideis undecimi, 1817. He showed that the geometry on a sphere becomes identical with the geometry of Euclid when the radius is infinitely increased, though it is distinctly shown that the limiting surface is not a plane.
The researches of K. F. Gauss, N. I. Lobachevski and J. Bolyai have been considered by F. Klein as constituting the first period in the history of non-Euclidean geometry. It is a period in which the synthetic methods of elementary geometry were in vogue. The second period embraces the researches of G. F. B. Riemann, H. Helmholtz, S. Lie and E. Beltrami, and employs the methods of differential geometry.
It was in 1854 that Gauss heard from his pupil, Riemann, a marvellous dissertation which considered the foundations of geometry from a new point of view. Riemann was not familiar with Lobachevski and Bolyai. He developed the notion of n-ply extended magnitude, and the measure-relations of which a manifoldness of n dimensions is capable, on the assumption that every line may be measured by every other. Riemann applied his ideas to space. He taught us to distinguish between "unboundedness" and "infinite extent." According to him we have in our mind a more general notion of space, i.e. a notion of non-Euclidean space; but we learn by experience that our physical space is, if not exactly, at least to a high degree of approximation, Euclidean space. Riemann's profound dissertation was not published until 1867, when it appeared in the Göttingen Abhandlungen.
Before this, the idea of n dimensions had suggested itself under various aspects to Ptolemy, J. Wallis, D'Alembert, J. Lagrange, J. Plücker, and H. G. Grassmann. The idea of time as a fourth dimension had occurred to D'Alembert and Lagrange. About the same time with Riemann's paper, others were published from the pens of H. Helmholtz and E. Beltrami. This period marks the beginning of lively discussions upon this subject. Some writers—J. Bellavitis, for example—were able to see in non-Euclidean geometry and n-dimensional space nothing but huge caricatures, or diseased outgrowths of mathematics. H. Helmholtz's article was entitled Thatsachen, welche der Geometrie zu Grunde liegen, 1868, and contained many of the ideas of Riemann. Helmholtz popularized the subject in lectures, and in articles for various magazines. Starting with the idea of congruence, and assuming the free mobility of a rigid body and the return unchanged to its original position after rotation about an axis, he proves that the square of the line-element is a homogeneous function of the second degree in the differentials.
Helmholtz's investigations were carefully examined by S. Lie who reduced the Riemann-Helmholtz problem to the following form: To determine all the continuous groups in space which, in a bounded region, have the property of displacements. There arose three types of groups which characterize the three geometries of Euclid, of N. I. Lobachevski and J. Bolyai and of F. G. B. Riemann.
Beltrami wrote in 1868 a classical paper, Saggio di interpretazione della geometria non-euclidea (Giorn. di Matem., 6) which is analytical (and... should be mentioned elsewhere were we to adhere to a strict separation between synthesis and analysis). He reached the brilliant and surprising conclusion that in part the theorems of non-Euclidean geometry find their realization upon surfaces of constant negative curvature. He studied, also, surfaces of constant positive curvature, and ended with the interesting theorem that the space of constant positive curvature is contained in the space of constant negative curvature.
These researches of Beltrami, H. Helmholtz, and G. F. B. Riemann culminated in the conclusion that on surfaces of constant curvature we may have three geometries,—the non-Euclidean on a surface of constant negative curvature, the spherical on a surface of constant positive curvature, and the Euclidean geometry on a surface of zero curvature. The three geometries do not contradict each other, but are members of a system,—a geometrical trinity.
The ideas of hyper-space were brilliantly expounded and popularised in England by Clifford.
Beltrami's researches on non-Euclidean geometry were followed, in 1871, by important investigations of Felix Klein, resting upon Cayley's Sixth Memoir on Quantics, 1859. The development of geometry in the first half of the nineteenth century had led to the separation of this science into two parts: the geometry of position or descriptive geometry which dealt with properties that are unaffected by projection, and the geometry of measurement in which the fundamental notions of distance, angle, etc., are changed by projection. Cayley's Sixth Memoir brought these strictly segregated parts together again by his definition of distance between two points. The question whether it is not possible so to express the metrical properties of figures that they will not vary by projection (or linear transformation) had been solved for special projections by M. Chasles, J. V. Poncelet, and E. Laguerre, but it remained for A. Cayley to give a general solution by defining the distance between two points as an arbitrary constant multiplied by the logarithm of the anharmonic ratio in which the line joining the two points is divided by the fundamental quadric. These researches, applying the principles of pure projective geometry, mark the third period in the development of non-Euclidean geometry.
F. Klein showed the independence of projective geometry from the parallel-axiom, and by properly choosing the law of the measurement of distance deduced from projective geometry, the spherical, Euclidean, and pseudospherical geometries, named by him respectively, the elliptic, parabolic, and hyperbolic geometries. This suggestive investigation was followed up by numerous writers, particularly by G. Battaglini of Naples, E. d'Ovidio of Turin, R. de Paolis of Pisa, F. Aschieri, A. Cayley, F. Lindemann of Munich, E. Schering of Göttingen, W. Story of Clark University, H. Stahl of Tubingen, A. Voss of Munich, Homersham Cox, A. Buchheim.
The Non-Euclidean Geometry is a natural result of the futile attempts which had been made from the time of Proklos to the opening of the nineteenth century to prove the fifth postulate, (also called the twelfth axiom, and sometimes the eleventh or thirteenth) of Euclid. The first scientific investigation of this part of the foundation of geometry was made by Saccheri (1733), a work which was not looked upon as a precursor of Lobachevsky, however, until Beltrami (1889) called attention to the fact. Lambert was the next to question the validity of Euclid's postulate in his Theorie der Parallellinien (posthumous, 1786), the most important of many treatises on the subject between the publication of Saccheri's work and those of Lobachevsky and Bolyai. Legendre also worked in the field, but failed to bring himself to view the matter outside the Euclidean limitations.
pp. 565-566.
During the closing years of the eighteenth century Kant's doctrine of absolute space, and his assertion of the necessary postulates of geometry, were the object of much scrutiny and attack. At the same time Gauss was giving attention to the fifth postulate, though on the side of proving it. It was at one time surmised that Gauss was the real founder of the non-Euclidean geometry, his influence being exerted on Lobachevsky through his friend Bartels, and on Johann Bolyai through the father Wolfgang, who was a fellow student of Gauss's. But it is now certain that Gauss can lay no claim to priority of discovery, although the influence of himself and of Kant, in a general way, must have had its effect.
p. 566.
Bartels went to Kasan in 1807, and Lobachevsky was his pupil. The latter's lecture notes show that Bartels never mentioned the subject of the fifth postulate to him, so that his investigations, begun even before 1823, were made on his own motion and his results were wholly original. Early in 1826 he sent forth the principles of his famous doctrine of parallels, based on the assumption that through a given point more than one line can be drawn which shall never meet a given line coplanar with it. The theory was published in full in 1829-30, and he contributed to the subject... until his death.
p. 566.
Johann Bolyai received through his father, Wolfgang, some of the inspiration to original research which the latter had received from Gauss. When only twenty-one he discovered, at about the same time as Lobachevsky, the principles of non-Euclidean geometry, and refers to them in a letter of November, 1823. They were committed to writing in 1825 and published in 1832. Gauss asserts in his correspondence with Schumacher (1831-32) that he had brought out a theory along the same lines as Lobachevsky and Bolyai, but the publication of their works seems to have put an end to his investigations. Schweikart was also an independent discoverer of the non-Euclidean geometry, as his recently recovered letters show, but he never published anything on the subject, his work on the theory of parallels (1807), like that of his nephew Taurinus (1825), showing no trace of the Lobachevsky-Bolyai idea.
p. 567.
The hypothesis was slowly accepted by the mathematical world. Indeed, it was about forty years after its publication that it began to attract any considerable attention. ...
Of all these contributions the most noteworthy from the scientific standpoint is that of Riemann. In his Habilitationsschrift (1854) he applied the methods of analytic geometry to the theory, and suggested a surface of negative curvature, which Beltrami calls "pseudo-spherical," thus leaving Euclid's geometry on a surface of zero curvature midway between his own and Lobachevsky's. He thus set forth three kinds of geometry, Bolyai having noted only two. These Klein (1871) has called the elliptic (Riemann's), parabolic (Euclid's), and hyperbolic (Lobachevsky's).
pp. 567-568.
There have contributed to the subject many of the leading mathematicians of the last quarter of a century, including... Cayley, Lie, Klein, Newcomb, Pasch, C. S. Peirce, Killing, Fiedler, Mansion, and McClintock. Cayley's contribution of his "metrical geometry" was not at once seen to be identical with that of Lobachevsky and Bolyai. It remained for Klein (1871) to show, this thus simplifying Cayley's treatment and adding one of the most important results of the entire theory. Cayley's metrical formulas are, when the Absolute is real, identical with those of the hyperbolic geometry; when it is imaginary, with the elliptic; the limiting case between the two gives the parabolic (Euclidean) geometry. The question raised by Cayley's memoir as to how far projective geometry can be defined in terms of space without the introduction of distance had already been discussed by von Staudt (1857) and has since been treated by Klein (1873) and by Lindemann (1876).
The question of the truth of the assumptions usually made in our geometry had been considered by J. Saccheri as long ago as 1773; and in more recent times had been discussed by N. I. Lobatschewsky of Kasan, in 1826 and again in 1840; by Gauss, perhaps as early as 1792, certainly in 1831 and in 1846; and by J. Bolyai in 1832 in the appendix to the first volume of his father's Tentamen; but Riemann's memoir of 1854 attracted general attention to the subject... and the theory has been since extended and simplified by various writers, notably A. Cayley... E. Beltrami... by H. L. F. von Helmholtz... by T. S. Tannery... by F. C. Klein... and by A. N. Whitehead... in his Universal Algebra. The subject is so technical that I confine myself to a bare sketch of the argument from which the idea is derived.
The Euclidean system of geometry, with which alone most people are acquainted, rests on a number of independent axioms and postulates. Those which are necessary for Euclid's geometry have, within recent years, been investigated and scheduled. They include not only those explicitly given by him, but some others which he unconsciously used. If these are varied, or other axioms are assumed, we get a different series of propositions, and any consistent body of such propositions constitutes a system of geometry. Hence there is no limit to the number of possible Non-Euclidean geometries that can be constructed.
Among Euclid's axioms and postulates is one on parallel lines, which is usually stated in the form that if a straight line meets two straight lines, so a to make the sum of the two interior angles on the same side of it taken together less than two right angles, then these straight lines being continually produced will at length meet upon that side on which are the angles which are less than two right angles. Expressed in this form the axiom is far from obvious, and from early times numerous attempts have been made to prove it. All such attempts failed, and it is now known that the axiom cannot be deduced from the other axioms assumed by Euclid.
Footnote) Some of the more interesting and plausible attempts have been collected by T. P. Thompson in his Geometry without Axioms, London, 1833, and later by J. Richard in his Philo: ie de mathématique, Paris, 1903.
The earliest conception of a body of Non-Euclidean geometry was due to the discovery, made independently by Saccheri, Lobatschewsky, and John Bolyai, that a consistent system of geometry of two dimensions can be produced on the assumption that the axiom on parallels is not true, and that through a point a number of straight (that is, geodetic) lines can be drawn parallel to a given straight line. The resulting geometry is called hyperbolic.
Riemann later distinguished between boundlessness space and its infinity, and showed that another consistent system of geometry of two dimensions can be constructed in which all straight lines are of finite length, so that a particle moving along a straight line will return to its original position. This leads to a geometry of two dimensions, called elliptic geometry, analogous to the hyperbolic geometry, but characterised by the fact that through a point no straight line can be drawn which, if produced far enough, will not meet any other given straight line. This can be compared with the geometry of figures drawn on the surface of a sphere.
Thus according as no straight line, or only one straight line, or a pencil of straight lines can be drawn through a point parallel to a given straight line, we have three systems of geometry of two dimensions known respectively as elliptic, parabolic or homaloidal or Euclidean, and hyperbolic.
In the parabolic and hyperbolic systems straight lines are infinitely long. In the elliptic they are finite. In the hyperbolic system there are no similar figures of unequal size; the area of a triangle can be deduced from the sum of its angles, which is always less than two right angles; and there is a finite maximum to the area of a triangle. In the elliptic system all straight lines are of the same finite length; any two lines intersect; and the sum of the angles of a triangle is greater than two right angles.
In spite of these and other peculiarities of hyperbolic and elliptic geometries, it is impossible to prove by observation that one of them is not true for the space in which we live. For in measurements in each of these geometries we must have a unit of distance; and we live in a space whose properties are those of either of these geometries, and such that the greatest distances with which we are acquainted (ex. gr. the distances of the fixed stars) are immensely smaller than any unit, natural to the system, then it may be impossible for us by our observations to detect the discrepancies between the three geometries. It might indeed be possible by observations of the parallaxes of stars to prove that the parabolic system and either the hyperbolic or elliptic system were false, but never can it be proved by measurements that Euclidean geometry is true. Similar difficulties might arise in connection with excessively minute quantities. In short, though the results of Euclidean geometry are more exact than present experiments can verify for finite things, such as those with which we have to deal, yet for much larger things or much smaller things or for parts of space at present inaccessible to us they may not be true.
Other systems of Non-Euclidean geometry might be constructed by changing other axioms and assumptions made by Euclid. Some of these are interesting, but those mentioned above have a special importance from the somewhat sensational fact that they lead to no results inconsistent with the properties of the space in which we live.
In order that a space of two dimensions should have the geometrical properties with which we are familiar, it is necessary that it should be possible at any place to construct a figure congruent to a given figure; and this is so only if the product of the principle radii of curvature at every point of the space or surface be constant. The product is constant in the case (i) of spherical surfaces, where it is positive; (ii) of plane surfaces (which leads to Euclidean geometry), where it is zero; and (iii) of pseudo-spherical surfaces, where it is negative. A tractroid is an instance of a pseudo-spherical surface; it is saddle-shaped at every point. Hence on spheres, planes, and tractroids we can construct normal systems of geometry. These systems are respectively examples of elliptic, Euclidean, and hyperbolic geometries. Moreover, if any surface be bent without dilation or contraction, the measure of the curvature remains unaltered. Thus these three species of surfaces are types of three kinds on which congruent figures can be constructed. For instance a plane can be rolled into a cone, and the system of geometry on a conical surface is similar to that on a plane.
Note: In the preceding sketch of the foundations of Non-Euclidean geometry I have assumed tacitly that the measure of a distance remains the same everywhere.
The above refers only to hyper-space of two dimensions. Naturally there arises the question whether there are different kinds of hyper-space of three or more dimensions. Riemann showed that there are three kinds of hyper-space of three dimensions having properties analogous to the three kinds of hyper-space of two dimensions already discussed. These are differentiated by the test whether at every point no geodetical surfaces, or one geodetical surface, or a fasciculus of geodetical surfaces can be drawn parallel to a given surface; a geodetical surface being defined as such that every geodetic line joining two points on it lies wholly on the surface.
The common notions of Euclid are five in number, and deal exclusively with equalities and inequalities of magnitudes. The postulates are also five in number and are exclusively geometrical. The first three refer to the construction of straight lines and circles. The fourth asserts the equality of all right angles, and the fifth is the famous Parallel Postulate...
It seems impossible to suppose that Euclid ever imagined this to be self-evident, yet the history of the theory of parallels is full of reproaches against the lack of self-evidence of this "axiom." Sir Henry Savile referred to it as one of the great blemishes in the beautiful body of geometry; D'Alembert called it "l'écueil et le scandale des élémens de Géométrie."
Such considerations induced geometers (and others), even up to the present day, to attempt its demonstration. From the invention of printing onwards a host of parallel-postulate demonstrators existed, rivalled only by the "circle-squarers," the "flat-earthers," and the candidates for the Wolfskehl "Fermat" prize. ...Modern research has vindicated Euclid, and justified his decision in putting this great proposition among the independent assumptions which are necessary for the development of euclidean geometry as a logical system.
All this labour has not been fruitless, for it has led in modern times to a rigorous examination of the principles not only of geometry, but of the whole of mathematics, and even logic itself, the basis of mathematics. It has had a marked effect upon philosophy, and has given us a freedom of thought which in former times would have received the award meted out to the most deadly heresies.
A... fallacy is contained in all proofs [of the Parallel Postulate] based upon the idea of direction. ...
Another class of demonstrations is based upon considerations of infinite areas. [In] Bertrand's Proof... The fallacy... consists in applying the principle of superposition to infinite areas as if they were finite magnitudes.
Chapter 1. Historical, pp. 6-8.
Non-euclidean geometry has made it clear that the ideas of parallelism and equidistance are quite distinct. The term parallel (Greek... running alongside) originally connoted equidistance, but the term is used by Euclid rather in the sense "asymptotic" (Greek... non-intersecting), and this term has come to be used in the limiting case of curves which tend to coincidence, or the limiting case between intersection and non-intersection. In non-euclidean geometry parallel straight lines are asymptotic in this sense, and equidistant straight lines in a plane do not exist. This is just one instance of two distinct ideas which are confused in euclidean geometry, but are quite distinct in non-euclidean.
Chapter 1. Historical, pp. 10-11.
Among the early postulate demonstrators there stands a unique figure that of a Jesuit Gerolamo Saccheri, a contemporary and friend of Ceva. This man devised an entirely different mode of attacking the problem, in an attempt to institute a reductio ad absurdum. At that time the favourite starting-point was the conception of parallels as equidistant straight lines, but Saccheri, like some of his predecessors, saw that it would not do to assume this in the definition. ...Saccheri keeps an open mind, and proposes three hypotheses:
(1) The Hypothesis of the Right Angle.
(2) The Hypothesis of the Obtuse Angle.
(3) The Hypothesis of the Acute Angle.
The object of his work is to demolish the last two hypotheses and leave the first, the Euclidean hypothesis, supreme; but the task turns out to be more arduous than he expected. He establishes a number of theorems, of which the most important are the following:
If one of the three hypotheses is true in any one case, the same hypothesis is true in every case.
On the hypothesis of the right angle, the obtuse angle, or the acute angle, the sum of the angles of a triangle is equal to, greater than, or less than two right angles. ...
Saccheri demolishes the hypothesis of the obtuse angle in his Theorem 14 by showing that it contradicts Euclid I. 17 (that the sum of any two angles of a triangle is less than two right angles); but he requires nearly twenty more theorems before he can demolish the hypothesis of the acute angle, which he does by showing that two lines which meet in a point at infinity can be perpendicular at that point to the same straight line. In spite of all his efforts, however, he does not seem to be quite satisfied with the validity of his proof, and he offers another proof in which he loses himself, like many another, in the quicksands of the infinitesimal.
If Saccheri had had a little more imagination and been less bound down by tradition, and a firmly implanted belief that Euclid's hypothesis was the only true one, he would have anticipated by a century the discovery of the two non-euclidean geometries which follow from his hypotheses of the obtuse and the acute angle.
Chapter 1. Historical, pp. 11-13.
J. H. Lambert, fifty years after Saccheri, also fell just short... His starting point is very similar to Saccheri's, and he distinguishes the same three hypotheses; but he went further than Saccheri. He actually showed that on the hypothesis of the obtuse angle the area of a triangle is proportional to the excess of the sum of its angles over two right angles, which is the case for the geometry on the sphere, and he concluded that the hypothesis of the acute angle would be verified on a sphere of imaginary radius. ...
He dismisses the hypothesis of the obtuse angle, since it requires that two straight lines should enclose a space, but his argument against the hypothesis of the acute angle, such as the non-existence of similar figures, he characterises as arguments ab amore et invidia ducta [guided by love and jealousy]. Thus he arrived at no definite conclusion, and his researches were only published some years after his death.
Chapter 1. Historical, pp. 13-14.
About... 1799 the genius of Gauss was being attracted to the question, and, although he published nothing on the subject except a few reviews, it is clear from his correspondence and fragments of his notes that he was deeply interested in it. He was a keen critic of the attempts made by his contemporaries to establish the theory of parallels; and while at first he inclined to the orthodox belief, encouraged by Kant, that Euclidean geometry was an example of a necessary truth, he gradually came to see that it was impossible to demonstrate it. He declares that he refrained from publishing anything because he feared the clamour of the Boeotians, or, as we should say, the Wise Men of Gotham; indeed at this time the problem of parallel lines was greatly discredited, and anyone who occupied himself with it was liable to be considered as a crank.
Chapter 1. Historical, p. 14.
Gauss was probably the first to obtain a clear idea of the possibility of a geometry other than that of Euclid, and we owe the very name Non-Euclidean Geometry to him. It is clear that about the year 1820 he was in possession of many theorems of non-euclidean geometry, and though he meditated publishing his researches when he had sufficient leisure to work them out in detail with his characteristic elegance, he was finally forestalled by receiving in 1832, from his friend W. Bolyai, a copy of the now famous Appendix by his son, John Bolyai.
Chapter 1. Historical, p. 14.
Among the contemporaries and pupils of Gauss... F. K. Schweikart, Professor of Law in Marburg, sent to Gauss in 1818 a page of MS. explaining a system of geometry which he calls "Astral Geometry," in which the sum of the angles of a triangle is always less than two right angles, and in which there is an absolute unit of length.
He did not publish any account of his researches, but he induced his nephew, F.A. Taurinus, to take up the question. ...a few years later he attempted a treatment of the theory of parallels and having received some encouragement from Gauss he [Taurinus] published a small book, Theorie der Parallellinien, in 1825. After its publication he came across [J. W.] Camerer's new edition of Euclid in Greek and Latin, which in an Excursus to Euclid I. 29, contains a very valuable history of the theory of parallels, and there he found that his methods had been anticipated by Saccheri and Lambert. Next year, accordingly, he published another work, Oeometriae prima elementa and in the Appendix... works out some of the most important trigonometrical formulae for non-euclidean geometry by using the fundamental formulae of spherical geometry with an imaginary radius. Instead of the notation of hyperbolic functions, which was then scarcely in use, he expresses his results in terms of logarithms and exponentials, and calls his geometry the "Logarithmic Spherical Geometry."
Though Taurinus must be regarded as an independent discoverer of non-euclidean trigonometry, he always retained the belief, unlike Gauss and Schweikart, that Euclidean geometry was necessarily the true one. Taurinus himself was aware, however, of the importance of his contribution... and it was a bitter disappointment to him when he found that his work attracted no attention. In disgust he burned the remainder of the edition of his Elementa, which is now one of the rarest of books.
Chapter 1. Historical, pp. 14-15.
The third... having arrived at the notion of a geometry in which Euclid's postulate is denied is F. L. Wachter, a student under Gauss. It is remarkable that he affirms that even if the postulate be denied, the geometry on a sphere becomes identical with the geometry of Euclid when the radius is indefinitely increased, though it is distinctly shown that the limiting surface is not a plane. This was one of the greatest discoveries of Lobachevsky and Bolyai. If Wachter had lived he might have been the discoverer of non-euclidean geometry, for his insight into the question was far beyond that of the ordinary parallel-postulate demonstrator.
Chapter 1. Historical, p. 15.
While Gauss, Schweikart, Taurinus and others were working in Germany,... just on the threshold of... discovery, in France and Britain... there was a considerable interest in the subject inspired chiefly by A. M. Legendre. Legendre's researches were published in the various editions of his Éléments, from 1794 to 1823. and collected in an extensive article in the Memoirs of the Paris Academy in 1833.
Assuming all Euclid's definitions, axioms and postulates, except the parallel-postulate and all that follows from it, he proves some important theorems, two of which, Propositions A and B, are frequently referred to in later work as Legendre's First and Second Theorems.
Prop. A. The sum of the three angles of a rectilinear triangle cannot be greater than two right angles (π). ...
Prop. B. If there exists a single triangle in which the sum of the angles is equal to two right angles, then in every triangle the sum of the angles must likewise be equal to two right angles.
This proposition was already proved by Saccheri, along with the corresponding theorem for the case in which the sum of the angles is less than two right angles... Legendre's proof... proceeds by constructing successively larger and larger triangles in each of which the sum of the angles = π. ...
In this proof there is a latent assumption and also a fallacy. ...Legendre's other attempts make use of infinite areas. He makes reference to Bertrand's proof, and attempts to prove the necessity of Playfair's axiom...
Chapter 1. Historical, pp. 16-19.
Nikolai Ivanovich Lobachevsky, Professor of Mathematics at Kazan, was interested in the theory of parallels from at least 1815. Lecture notes of the period 1815-17 are extant, in which Lobachevsky attempts in various ways to establish the Euclidean theory. He proves Legendre's two propositions, and employs also the ideas of direction and infinite areas. In 1823 he prepared a treatise on geometry for use in the University, but it obtained so unfavourable a report that it was not printed. The MS. remained buried in the University Archives until it was discovered and printed in 1909. In this book he states that "a rigorous proof of the postulate of Euclid has not hitherto been discovered; those which have been given may be called explanations, and do not deserve to be considered as mathematical proofs in the full sense."
Just three years afterwards, he read to the physical and mathematical section of the University of Kazan a paper entitled "Exposition succinte des principes de la géométrie avec une démonstration rigoureuse du théorème des parallèles." In this paper... Lobachevsky explains the principles of his "Imaginary Geometry," which is more general than Euclid's, and in which two parallels can be drawn to a given line through a given point, and in which the sum of the angles of a triangle is always less than two right angles.
Chapter 1. Historical, p. 21.
Bolyai János (John) was the son of Bolyai Farkas (Wolfgang), a fellow-student and friend of Gauss at Göttingen. The father was early interested in the theory of parallels, and without doubt discussed the subject with Gauss while at Göttingen. The professor of mathematics at that time, A. G. Kaestner, had himself attacked the problem and with his help G. S. Klügel, one of his pupils, compiled in 1763 the earliest history of the theory of parallels.
Chapter 1. Historical, pp. 21-22.
In 1804, Wolfgang Bolyai... sent to Gauss a "Theory of Parallels," the elaboration of his Göttingen studies. In this he gives a demonstration very similar to that of [Henry] Meikle and some of Perronet Thompson's, in which he tries to prove that a series of equal segments placed end to end at equal angles, like the sides of a regular polygon, must make a complete circuit. Though Gauss clearly revealed the fallacy, Bolyai persevered and sent Gauss, in 1808, a further elaboration of his proof. To this Gauss did not reply, and Bolyai, wearied with his ineffectual endeavours to solve the riddle of parallel lines, took refuge in poetry and composed dramas. During the next twenty years, amid various interruptions, he put together his system of mathematics, and at length in 1832-3, published in two volumes an elementary treatise on mathematical discipline which contains all his ideas with regard to the first principles of geometry.
Meanwhile, John Bolyai... had been giving serious attention to the theory of parallels, in spite of his father's solemn adjuration to let the loathsome subject alone. At first, like his predecessors, he attempted to find a proof for the parallel-postulate, but gradually, as he focussed his attention more and more upon the results which would follow from a denial of the axiom, there developed in his mind the idea of a general or "Absolute Geometry" which would contain ordinary or euclidean geometry as a special or limiting case. Already, in 1823, he had worked out the main ideas of the non-euclidean geometry, and in a letter of 3rd November he announces to his father his intention of publishing a work on the theory of parallels, "for," he says, "I have made such wonderful discoveries that I am myself lost in astonishment, and it would be an irreparable loss if they remained unknown. When you read them, dear Father, you too will acknowledge it. I cannot say more now except that out of nothing I have created a new and another world. All that I have sent you hitherto is as a house of cards compared to a tower." Wolfgang advised his son, if his researches had really reached the desired goal, to get them published as soon as possible, for new ideas are apt to leak out, and further, it often happens that a new discovery springs up spontaneously in many places at once, "like the violets in springtime." Bolyai's presentment was truer than he suspected, for at this very moment Lobachevsky at Kazan, Gauss at Gottingen, Taurinus at Cologne, were all on the verge of this great discovery. It was not, however, till 1832 that... the work was published. It appeared in Vol. I of his father's Tentamen, under the title "Appendix, scientiam absolute veram exhibens."
...the son, although he continued to work at his theory of space, published nothing further. Lobachevsky's Geometrische Untersuchungen came to his knowledge in 1848, and this spurred him on to complete the great work on "Raumlehre," which he had already planned at the time of the publication of his "Appendix," but he left this in large part as a rudis indigestaque moles, and he never realised his hope of triumphing over his great Russian rival.
Chapter 1. Historical, pp. 22-23.
Lobachevsky never seems to have heard of Bolyai, though both were directly or indirectly in communication with Gauss. Much has been written on the relationship of these three discoverers, but it is now generally recognised that John Bolyai and Lobachevsky each arrived at their ideas independently of Gauss and of each other; and, since they possessed the convictions and the courage to publish them which Gauss lacked, to them alone is due the honour of the discovery.
Chapter 1. Historical, p. 24.
The ideas inaugurated by Lobachevsky and Bolyai did not for many years attain any wide recognition, and it was only after Baltzer had called attention to them in 1867, and at his request Hoüel had published French translations of the epoch making works, that the subject of non-euclidean geometry began to be seriously studied.
It is remarkable that while Saccheri and Lambert both considered the two hypotheses, it never occurred to Lobachevsky or Bolyai or their predecessors, Gauss, [F. K.] Schweikart, [F. A.] Taurinus, and [F. L.] Wachter, to admit the hypothesis that the sum of the angles of a triangle may be greater than two right angles. This involves the conception of a straight line as being unbounded but yet of finite length. Somewhere "at the back of beyond" the two ends of the line meet and close it. We owe this conception first to Bernhard Riemann in his Dissertation of 1854 (published only in 1866 after the author's death), but in his Spherical Geometry two straight lines intersect twice like two great circles on a sphere. The conception of a geometry in which the straight line is finite, and is, without exception, uniquely determined by two distinct points, is due to Felix Klein. Klein attached the now usual nomenclature to the three geometries; the geometry of Lobachevsky he called Hyperbolic, that of Riemann Elliptic, and that of Euclid Parabolic.
In general the Greeks looked upon an axiom as something which was so self-evident that no reasonable person would object... while a postulate was a request that something be allowed. Now Euclid's fifth postulate... whatever else this postulate may be, self-evident it is not, and this was early perceived. ...
The first line of attack was, naturally, the attempt to prove this postulate by the aid of others, and the axioms. Such, presumably, was Ptolemy's idea. But even if we grant that all of Euclid's axioms are self-evident, it does not... follow that he puts in his list all of the assumptions that he really uses.
The way that geometers... went about proving the fifth postulate was to smuggle in somewhere some unavowed assumption. A common practice was to assume that two straight lines could not approach one another assymptotically, that... they ultimately intersected. Or, again, it was assumed that a straight line was not a closed circuit... legitimate as long as avowed.
A franker, and so more admirable way... was to change the definition of parallel lines into something else that seemed to avoid the trouble, or else to reword the axiom in a less objectionable form. A real step in advance... is known as Playfair's axiom, though it is casually mentioned in Proclus...There are... a great many alternatives. One of the most famous is to define two coplaner lines as parallel if they are everywhere the same distance apart... but how do we know there are such pairs... A still neater method consists in defining two lines as parallel if they have the same direction, or opposite directions. But here we introduce a totally new undefined concept, direction...
A writer who clearly saw the fallacy under the constant distance assumption was Girolamo Saccheri, S. J., whose 'Euclides ab omne naevo vindicatus' [Euclid Freed of Every Flaw]... in 1733, marked perhaps the most important single step in advance ever taken in the attempt to solve the parallel difficulty. This careful logician undertook to prove the correctness of Euclid's postulate by showing that when it is replaced by another, a contradiction is sure to arise.
Having disposed, as he thinks, of the obtuse-angled hypothesis, Saccheri turns boldly to the task of destroying the acute-angle one also. He shows that under this hypothesis there passes through each point without [outside of] a given line two parallels thereto... Most unfortunately he speaks of parallels as intersecting at infinity... and then speaks of ultra-infinite points beyond them. His proof... breaks down just there. ...In Segre we find an elaborate argument to the effect that subsequent writers who approached the parallel postulate problem through the means of elementary geometry were directly, or indirectly, influenced by him. The greatest, if the least communicative, of these was Gauss.
Gauss... wrote little on the subject beyond correcting the vagaries of his friend Schumacher, but it is certain that he reflected deeply, and arrived at conclusions subsequently supported by others. His revolutionary view, that Saccheri was wrong and that a consistent geometry can be developed... was carried through with complete success by Nicholai Ivanovitch Lobachevski.
Fourteen years before Beltrami published... a greater than he had studied the whole of the non-Euclidean problem from a more lofty and difficult point of view. This was Bernhard Riemann, who offered to Gauss three topics for his projected trial lecture as Privatsozent at Göttingen. Gauss chose the most difficult, wondering what so young a man could make of such an arduous subject; he learned. ...'Ueber die Hypothesen welche der Geometrie su Grunde liegen' ...was read in 1854, but never published till 1868.
Riemann's approach is far different from anything that anyone had tried previously. ...The modern theory of relativity, on its mathematical side, is merely an elaboration of Riemann's analysis.
Riemann... made the important distinction, which had escaped previous writers, between the infinite and the unlimited. All of our experience tends to show that the universe is unlimited; a given segment may be extended indefinitely in either direction, but we know nothing as to whether it is infinite or not. If space have constant positive curvature, a geodesic surface is applicable to a Euclidean sphere where a geodesic is a circle, unlimited but not infinite. This possibility destroys the validity of Euclid's proof that an exterior angle of a triangle is greater than either opposite interior angle.
Of all methods devised for attacking the problem of the bases of geometry Riemann's has proved by far to be the most fruitful. That is probably because it is the most flexible, and applicable to the greatest number of problems. In the twentieth century reverence for Euclid has been replaced by reverence for the differential equation
Beltrami's idea was to find in space a surface with the property that if you define distance thereon in terms of geodesic length, you have the geometry of Lobachevski. An analogous idea is to find a new definition for distance such that, starting from our familiar space, if we redefine distance in this way we may have the obtuse-angled geometry, elliptic geometry, or the acute-angled, hyperbolic geometry of Lobachevski. An illuminating example of this sort was worked out by Klein following a hint dropped by Cayley. The root of the matter goes back to Laguerre... in 1858...
A scruple... has troubled conscientious writers. We take Euclidean space as we know it, we take Cartesian geometry in that space, we set up certain point functions in that space and call them distances, certain transformations and call them motions, and find at last a set of objects which obey the presuppositions of non-Euclidean geometry. But is there not here, perhaps, a vicious circle around which the kitten is chasing its tail? The basis is a Euclidean space, and a Cartesian coordinate system in that space, which is based upon Euclidean measurements, and cross ratios which depend upon distances. How do we know that without all of these it would be possible to erect a consistent non-Euclidean geometry? ...
We begin by setting up a system of axioms for a projective geometry in a space of as many dimensions as we please. The undefined elements are point, line as a system of points, and separation of pairs of collinear points. Other choices are possible... The idea of taking separation as fundamental was introduced by Vailati.
If we are to set up a system of axioms for a particular sort of geometry, two qualities are essential, and two desirable. The essential qualities are that:
1) They should be consistent.
2) They should contain all of the assumptions necessary for the purposes in hand.
3) They should be independent of one another and include nothing unnecessary.
4) The mathematical system built on them should be interesting rather than trivial.
The first work where the problem of setting up geometrical axioms in this way was Pasch in 1882. The way opened by him was subsequently followed by a goodly number of others, among whom one might mention Peano, Pieri, Vahlen, HIlbert, E. H. Moore, R. L. Moore, Veblen, Huntington, and others or lesser note.
The attempts to derive the parallel postulate as a theorem from the remaining nine "axioms" and "postulates" occupied geometers for over two thousand years and culminated in some of the most far-reaching developments in modern mathematics. Many "proofs" of the postulate were offered, but each was sooner or later shown to rest upon a tacit assumption equivalent to the postulate itself.
Not until 1733 was the first really scientific investigation... Gerolamo Saccheri received permission to print... Euclides ab omni naevo vindicatus (Euclid Freed of Every Flaw). ...Saccheri had become charmed with the powerful method of reductio ad absurdum and... easily showed... that if, in a quadrilateral... [base] angles... are right angles and [vertical] sides... are equal, then [ceiling] angles... are equal. Then there are three possibilities: [ceiling] angles are equal acute... equal right... or equal obtuse angles. The plan was to show that the assumption of either... the acute angle or... the obtuse angle would lead to a contradiction. ...Tacitly assuming the infinitude of the straight line, Saccheri readily eiliminated the hypothesis of the obtuse angle, but... After obtaining many of the now classical theorems of... non-Euclidean geometry, Saccheri lamely forced... an unconvincing contradiction.
Johann Heinrich Lambert... went considerably beyond Sacherri in deducing propositions under the hypotheses of the acute and obtuse angles. Thus, with Sacherri, he showed that in the three hypotheses the sum of the angles of a triangle is less than, equal to, or greater than two right angles, respectively, and... in addition, that the deficiency... in the hypothesis of the acute angle, or the excess, in the hypothesis of the obtuse angle, is proportional to the area of the triangle. He observed the resemblance of the geometry following the... obtuse angle to spherical geometry... and conjectured that the geometry following from... the acute angle could perhaps be verified on the sphere of imaginary radius.
It is no wonder that no contradiction was found under the hypothesis of the acute angle, for... the geometry developed from a collection of axioms comprising a basic set plus the acute angle hypothesis is as consistent as the Euclidean geometry developed from the same basic set plus the hypothesis of the right angle; that is, the parallel postulate is independent of the remaining postulates and therefore cannot be deduced from them.
In the field of non-Euclidean geometry, Riemann... began by calling attention to a distinction that seems obvious once it is pointed out: the distinction between an unbounded straight line and an infinite line. The distinction between unboundedness and infiniteness is readily illustrated. A circle is an unbounded figure in that it never comes to an end, and yet it is of finite length. On the other hand, the usual Euclidean concept of a straight line is also unbounded in that it never reaches an end but is of infinite length.
...he proposed to replace the infiniteness of the Euclidean straight line by the condition that it is merely unbounded. He also proposed to adopt a new parallel axiom... In brief, there are no parallel lines. This ... had been tried... in conjunction with the infiniteness of the straight line and had led to contradictions. However... Riemann found that he could construct another consistent non-Euclidean geometry.
Non-Euclidean geometry was the most weighty intellectual creation of the nineteenth century, or, at worst, might have to share honors with the theory of evolution.
Unlike those of science, the conclusions of mathematics had always regarded as deduced from basic truths. ...the very reason that mathematicians persisted for so many centuries in attempting to find simple equivalents for Euclid's parallel axiom, instead of entertaining contradictory possibilities, is that they could not conceive of geometry being anything else than the true geometry of physical space.
The creation of non-Euclidean geometry showed... that mathematics could no longer be regarded as a body of unquestionable truths. ...Mathematics retained its deductive method of establishing its conclusions, but it was soon appreciated that mathematics offers only certainty of proof on the basis of uncertain axioms.
What was the effect of non-Euclidean geometry on the future progress of mathematics? ...Mathematics passed from serfdom to freedom. Up to [that] time... mathematicians were fettered to the physical world. ...Had not the history of non-Euclidean geometry shown that seemingly absurd ideas may prove to be not only illuminating but of actual use to science? ...Mathematicians found their house burned to the ground only to find gold under the floor boards.
Even the mathematicians of the late nineteenth century did not take non-Euclidean geometry seriously for physical applications, though they derived a great deal of pleasure from the new concepts and relating them to other domains of mathematics. The scientific world did not awaken to the reality on non-Euclidean geometry until the creation of the special theory of relativity in 1905. |
Wednesday, January 25, 2012
Finance theoryis the study of
economic agents behavior allocating their resources across alternative
financial instruments and in time in an uncertain environment. Mathematics
provides tools to model and analyze that behavior in allocation and time,
taking into account uncertainty.
1.Louis
Bachelier's 1900 math dissertation on the theory of speculation in the Paris
markets marks the twin births of both the continuous time mathematics of
stochastic processes and the continuous time economics of option pricing.
2.The
most important development in terms of impact on practice was the Black-Scholes
model for option pricing published in 1973.
3.Since
1973 the growth in sophistication about mathematical models and their adoption
mirrored the extraordinary growth in financial innovation. Major developments
in computing power made the numerical solution of complex models possible. The
increases in computer power size made possible the formation of many new
financial markets and substantial expansions in the size of existing ones.
Louis
Bachelier
Mathematical Ideas
One sometime hears
that "compound interest is the eighth wonder of the world", or the "stock
market is just a big casino".These are colorful sayings, maybe based in happy
or bitter experience, but each focuses on only one aspect of one financial
instrument. The "time value of money" and uncertainty are the central elements
that influence the value of financial instruments. When only the time aspect of
finance is considered, the tools of calculus and differential equations are
adequate. When only the uncertainty is considered, the tools of probability
theory illuminate the possible outcomes. When time and uncertainty are
considered together we begin the study of advanced mathematical finance.Finance
is the study of economic agents' behavior in allocating financial resources and
risks across alternative financial instruments and in time in an uncertain
environment. Familiar examples of financial instruments are bank accounts,
loans,stocks, government bonds and corporate bonds. Many less familiar examples
abound. Economic agents are units who buy and sell financial resources in a
market, from individuals to banks, businesses, mutual funds and hedge funds.
Each agent has many choices of where to buy, sell, invest and consume assets,
each with advantages and disadvantages. Each agent must distribute their
resources among the many possible investments with a goal in mind.
Advanced
mathematical finance is often characterized as the study of the more
sophisticated financial instruments called derivatives. A derivative is a
financial agreement between two parties that depends on something that occurs
in the future, such as the price or performance of an underlying asset. The
underlying asset could be a stock, a bond, a currency, or a commodity.
Derivatives have become one of the financial world's most important
risk-management tools. Finance is about shifting and distributing risk and
derivatives are especially efficient for that purpose. Two such instruments are
futures and options. Futures trading, a key practice in modern finance,
probably originated in seventeenth century Japan, but the idea can be traced as
far back as ancient Greece. Options were a feature of the "tulip mania" in
seventeenth century Holland. Both futures and options are called "derivatives".
(For the mathematical reader, these are called derivatives not because they
involve a rate of change, but because their value is derived from some
underlying asset.) Modern derivatives differ from their predecessors in that
they are usually specifically designed to objectify and price financial risk.
Derivatives come in many types. There are futures, agreements to trade
something at a set price at a given dates; options, the right but not the
obligation to buy or sell at a given price; forwards, like futures but traded
directly between two parties instead of on exchanges; and swaps, exchanging
flows of income from different investments to manage different risk exposure.
For example, one party in a deal may want the potential of rising income from a
loan with a floating interest rate, while the other might prefer the
predictable payments ensured by a fixed interest rate. This elementary swap is
known as a "plain vanilla swap". More complex swaps mix the performance of
multiple income streams with varieties of risk. Another more complex swap is a
credit-default swap in which a seller receives a regular fee from the buyer in
exchange for agreeing to cover losses arising from defaults on the underlying
loans. These swaps are somewhat like insurance . These more complex swaps are
the source of controversy since many people believe that they are responsible
for the collapse or near-collapse of several large financial firms in late
2008. Derivatives can be based on pretty much anything as long as two parties
are willing to trade risks and can agree on a price. Businesses use derivatives
to shift risks to other firms, chiefly banks. About 95% of the world's 500
biggest companies use derivatives. Derivatives with standardized terms are
traded in markets called exchanges.
Derivatives
tailored for specific purposes or risks are bought and sold "over the counter"
from big banks. The "over the counter" market dwarfs the exchange trading. In
November 2009, the Bank for International Settlements put the face value of
over the counter derivatives at $604.6 trillion. Using face value is
misleading, after off-setting claims are stripped out theresidual value is $3.7
trillion, still a large figure.Mathematical models in modern finance contain
deep and beautiful applications of differential equations and probability
theory.In spite of their complexity, mathematical models of modern financial
instruments have had a direct and significant influence on finance practice.
Early History
The
history of stochastic integration and the modelling of risky asset prices both
beginwith Brownian motion, so let us begin
there too. In 17th century put options were bought on tulip bulbs in
Netherlands.The earliest attempts to model Brownian motion mathematically can
be traced to three sources, each of which knew nothing about the others: the
first was that of T. N. Thiele of Copenhagen,
who effectively created a model of Brownian motion while studying time series
in 1880 the second was that of L. Bachelier
of Paris, who created a model of Brownian motion while deriving the dynamic
behavior of the Paris stock market, in 1900,and the third was that of A. Einstein, who proposed a model of the motion of
small particles suspended in a liquid, in an attempt to convince other
physicists of the molecular nature of matter.
Though the origins of much of the mathematics in financial models traces to
Louis Bachelier's 1900 dissertation on the Theory
of speculationin the Paris
markets. This doctrate thesis was completed at the Sorbonne in 1900 under Henri Poincare, this work marks the twin births of
both the continuous time mathematics of stochastic processes and the continuous
time economics of option pricing. While analyzing option pricing,Bachelier
provided two different derivations of the partial differential equation for the
probability density for the Wiener process or
Brownian motion. In one of the derivations, he works out what is now
called the Chapman-Kolmogorov convolution
probability integral. Along the way, Bachelier derived the method of
reflection to solve for the probability function of a diffusion process with an
absorbing barrier. Not a bad performance for a thesis on which the first
reader, Henri Poincaré,gave less than a top
mark! After Bachelier, option pricing theory laid dormant in the economics
literature for over half a century until economists and mathematicians renewed
study of it in the late 1960s. Jarrow and Protter speculate that this may
have been because the Paris mathematical elite scorned economics as an
application of mathematics.
Louis
Bachelier's 1900 dissertation on the Theory of speculation
Bachelier's
work was 5 years before Albert Einstein's 1905 discovery of the same equations
for his famous mathematical theory of Brownian motion. The editor of Annalen
der Physik received Einstein's paper on Brownian
motion on May 11, 1905. The paper appeared later that year. Einstein
proposed a model for the motion of small particles with diameters on the order
of 0.001 mm suspended in a liquid. He predicted that the particles would
undergo microscopically observable and statistically predictable motion. The
English botanist Robert Brown had already
reported such motion in 1827 while observing pollen grains in water with a
microscope. The physical motion is now called Brownian motion in honor of
Brown's description.Einstein calculated a diffusion constant to govern the rate
of motion of suspended particles. The paper was Einstein's attempt to convince
physicists of the molecular and atomic nature of matter. Surprisingly, even in
1905 the scientific community did not completely accept the atomic theory of
matter. In 1908, the experimental physicist Jean-Baptiste
Perrin conducted a series of experiments that empirically verified
Einstein's theory. Perrin thereby determined the physical constant known as
Avogadro's number for which he won the Nobel prize in 1926. Nevertheless,
Einstein's theory was very difficult to rigorously justify mathematically. Let
us now turn to Einstein's model. In modern terms, Einstein assumed that
Brownian motion was a stochastic process with continuous paths, independent
increments, and stationary Gaussian increments. He did not assume other
reasonable properties (from the standpoint of physics), such as rectifiable
paths. If he had assumed this last property, we now know his model would not
have existed as a process. However, Einstein was unable to show that the
process he proposed actually did exist as a mathematical object. This is
understandable, since it was 1905, and the ideas of Borel and Lebesgue
constructing measure theory were developed only during the first decade of the
twentieth century.In a series of papers from 1918 to 1923, the mathematician
Norbert Wiener constructed a mathematical model of Brownian motion. Wiener and
others proved many surprising facts about his mathematical model of Brownian
motion, research that continues today. In recognition of his work, his
mathematical construction is often called the Wiener process.
The
next step in the groundwork for stochastic integration lay with A. N. Kolmogorov.Indeed, in 1931, two years before
his famous book establishing a rigorous mathematical basis for Probability Theory using measure theory,
Kolmogorov refers to and briefly explains Bachelier's construction of Brownian
motion ( pages 64, 102–103). It is this paper too in which he develops a large
part of his theory of Markov processes. Most
significantly, in this paper Kolmogorov showed that continuous Markov processes
(diffusions) depend essentially on only two parameters:one for the speed of the
drift and the other for the size of the purely random part (the diffusive
component). He was then able to relate the probability distributions of the
process to the solutions of partial differential equations, which he solved,
and which are now known as "Kolmogorov's equations." He also made major
contributions to the understanding of stochastic processes (involving random
variables), and he advanced the knowledge of chains of linked probabilities.
Shortly thereafter, he took an extended trip to Germany and France, and in 1933
laid out his probability theory inFoundations
of the Theory of Probability. This work secured his
reputation as the world's foremost expert in his field.
Andrey Kolmogrov, the father of
Probability Theory
Paul Samuelson
We
turn now to Kiyosi Ito, the father of
stochastic integration,no doubt an attempt to establish a true stochastic
differential to be used in the study of Markov processes was one of Ito's
primary motivations for studying stochastic integrals.His work, starting in the
1940s, built on the earlier breakthroughs of Einstein and Norbert Wiener. Mr.
Ito's mathematical framework for describing the evolution of random phenomena
came to be known as the Ito Calculus.This
random component is best modeled using a mathematics which can show the range
of possible areas.Stochastic processes and Ito
calculus make up most of the modern Financial Maths, it has important
applications in Mathematical Finance and stochastic differential
equations.Firstly, we are now dealing with random variables (more precisely,
stochastic processes). Secondly, we are integrating with respect to a
non-differentiable function (technically,stochastic processes).
Kiyosi
Ito, maker of Ito Calculus
Growth of Mathematical
Finance
Modern
mathematical finance theory begins in the 1960s. In 1965 the economist Paul Samuelson published two papers that argue
that stock prices fluctuate randomly. One explained the Samuelson and Fama
efficient markets hypothesis that in a well-functioning and informed capital
market, asset-price dynamics are described by a model in which the best estimate
of an asset's future price is the current price (possibly adjusted for a fair
expected rate of return.). Under this hypothesis, attempts to use past price
data or publicly available forecasts about economic fundamentals to predict
security prices are doomed to failure. In the other paper with mathematician Henry McKean, Samuelson shows that a good model
for stock price movements is geometric Brownian motion. The final precursor to
the Black, Scholes and Merton option pricing
formulaes can be found in the paper of Samuelson and Merton .Samuelson
noted that Bachelier's model failed to ensure that stock prices would always be
positive, whereas geometric Brownian motion avoids this error.
The
most important development in terms of practice was the 1973 Black-Scholes
terms of practice was the 1973 Black-Scholes model
for option pricing. The two economists Fischer Black and Myron Scholes (and
simultaneously, and somewhat independently, the economist Robert Merton)
deduced an equation that provided the first strictly quantitative model for
calculating the prices of options. The key variable is the volatility of the
underlying asset. Myron Scholes published a paper with Fischer Black on 'Pricing of Options and Corporate Liabilities',
incorporating suggestions from Merton Miller (of
M&M Theory fame) and Eugene Fama (father of the Efficient Market
Hypothesis). These equations standardized the pricing of derivatives in
exclusively quantitative terms. The formal press release from the Royal Swedish
Academy of Sciences announcing the 1997 Nobel Prize in Economics states that
the honor was given "for a new method to determine the value of derivatives.
Robert C. Merton and Myron S. Scholes have, in collaboration with the late
Fischer Black developed a pioneering formula for the valuation of stock
options. Their methodology has paved the way for economic valuations in many
areas. It has also generated new types of financial instruments and facilitated
more efficient risk management in society."
The
Chicago Board Options Exchange (CBOE) began publicly trading options in the
United States in April 1973, a month before the official publication of the
Black-Scholes model. By 1975, traders on the CBOE were using the model to both
price and hedge their options positions. In fact, Texas Instruments created a
hand-held calculator specially programmed to produce Black-Scholes option
prices and hedge ratios.The basic insight underlying the Black-Scholes model is
that a dynamic portfolio trading strategy in the stock can replicate the
returns from an option on that stock. This is called "hedging an option" and it
is the most important idea underlying the Black-Scholes-Merton approach. Much
of the rest of the book will explain what that insight means and how it can be
applied and calculated.
The
story of the development of the Black-Scholes-Merton option pricing model is
that Black started working on this problem by himself in the late 1960s. His
idea was to apply the capital asset pricing model to value the option in a
continuous time setting. Using this idea, the option value satisfies a partial
differential equation. Black could not find the solution to the equation. He
then teamed up with Myron Scholes who had been thinking about similar problems.
Together, they solved the partial differential equation using a combination of
economic intuition and earlier pricing formulas.
Black and Scholes had solved stochastic partial
differential equations to develop a formula for pricing European-type call
options.The result was an equation that suggested how the price of a
call option might be calculated as a function of a risk-free interest rate, the
price variance of the asset on which the option was written, and the parameters
of the option (strike price, term, and the market price of the underlying
asset.)
At this time, Myron Scholes was at MIT. So was Robert Merton,who was applying
his mathematical skills to various problems in finance.Merton showed Black and
Scholes how to derive their differential equation differently. Merton was the
first to call the solution the Black-Scholes option pricing formula. Merton's
derivation used the continuous time construction of a perfectly hedged
portfolio involving the stock and the call option together with the notion that
no arbitrage opportunities exist. This is the approach we will take. In the
late 1970s and early 1980s mathematicians Harrison, Kreps and Pliska showed
that a more abstract formulation of the solution as a mathematical model called
a martingale provides greater generality.
By
the 1980s, the adoption of finance theory models into practice was nearly
immediate. Additionally, the mathematical models used in financial practice
became as sophisticated as any found in academic financial research. There
are several explanations for the different adoption rates of mathematical
models into financial practice during the 1960s, 1970s and 1980s. Money and
capital markets in the United States exhibited historically low volatility in
the 1960s; the stock market rose steadily, interest rates were relatively
stable, and exchange rates were fixed. Such simple markets provided little
incentive for investors to adopt new financial technology. In sharp contrast,
the 1970s experienced several events that led to market change and increasing
volatility. The most important of these was the shift from fixed to floating
currency exchange rates; the world oil price crisis resulting from the creation
of the Middle East cartel; the decline of the United States stock market in
1973-1974 which was larger in real terms than any comparable period in the
Great Depression; and double-digit inflation and interest rates in the United
States. In this environment, the old rules of thumb and simple regression
models were inadequate for making investment decisions and managing risk.
During the 1970s, newly created derivative-security exchanges
traded listed options on stocks, futures on major currencies and futures on
U.S. Treasury bills and bonds. The success of these markets partly resulted
from increased demand for managing risks in a volatile economic market. This
success strongly affected the speed of adoption of quantitative financial
models. For example, experienced traders in the over the counter market
succeeded by using heuristic rules for valuing options and judging risk
exposure. However these rules of thumb were inadequate for trading in the
fast-paced exchange-listed options market with its smaller price spreads,
larger trading volume and requirements for rapid trading decisions while
monitoring prices in both the stock and options markets. In contrast,
mathematical models like the Black-Scholes model were ideally suited for
application in this new trading environment. The growth in sophisticated
mathematical models and their adoption into financial practice accelerated
during the 1980s in parallel with the extraordinary growth in financial
innovation. A wave of de-regulation in the financial sector was an important
factor driving innovation.
Quantum Mechanics in Financial Markets.
The
markets are non-linear, dynamic systems, subject to the rules of Chaos Theory. Market prices are highly random,
with a short to intermediate term trend component. They are highly dependent on
initial conditions. Markets also show qualities of fractals -- self-similar in
the sense that the individual parts are related to the whole.Due to the
non-Gaussian behavior of the markets the methods from Chaos
Theory, Fractals and Quantum Physics are being used in Finance since 1980s.
One
recent trend in the growing field of quantitative finance to apply techniques
borrowed from quantum physics to Financial models. One example is Path Integrals, which were invented by Richard Feynman in 1948 , Feynman used path
integrals along with the probability methods designed by Norbert Wiener to
reformulate methods of Quantum Physics.The path integral also relates quantum
and stochastic processes, and this provided the basis for the grand synthesis
of the 1970s which unified quantum field theory with the statistical field
theory of a fluctuating field near a second-order phase transition.Feynman
suggested that when considering the Quantum Mechanics of a moving particle,
every conceivable path can be assigned a certain complex(probability amplitude)
number called the probability amplitude for that path.
Jan Dash a particle physicist later applied
path integrals in Finance, the reason was simple ,the value of the financial
derivative depends on the "path" followed by underlying asset.
Methods of quantum mechanics for mathematical modelling of price dynamics
of the financial market. We propose to describe behavioral financial factors
(e.g., expectations of traders) by using the pilot wave (Bohmian) model of
quantum mechanics. On the one hand, our Bohmian model is a quantum-like model
for the financial market, cf. with works of W.
Segal, I.E. Segal, E. Haven, E.W. Piotrowski, J. Sladkowski. On the
other hand, (since Bohmian mechanics provides for the possibility to describe
individual price trajectories) it belongs to the domain of extended research on
deterministic dynamics for financial assets. Our model emphasizes the
complexity of the financial market: the traditional description of price
dynamics is completed by Schrodinger's dynamics for the pilot wave of
expectations of traders. This is a kind of socio-economic model for the
financial market.
Pricing an option is a complex mathematical problem which involves
diffusion processes such as Brownian motion.From this analysis, Black, Scholes
and Merton were able to derive a general Partial Differential Equation for the
value of a stock option, which turned out to look very similar to the
heat-diffusion equation.
Richard
Feynman
A path-integral description of the
Black-Scholes model was recently developed by Belal
Baaquie of the National University of Singapore and co-workers. From
this, Baaquie and co-workers went on to devise a quantum-mechanical version of
the Black-Scholes equation to describe the price of a simple,
non-dividend-paying option.
Conceptual
breakthroughs in finance theory in the 1980s were fewer and less fundamental
than in the 1960s and 1970s, but the research resources devoted to the
development of mathematical models was considerably larger. Major developments
in computing power, including the personal computer and increases in computer
speed and memory enabled new financial markets and expansions in the size of
existing ones. These same technologies made the numerical solution of complex
models possible. They also speeded up the solution of existing models to allow
virtually real-time calculations of prices and hedge ratios.
Ethical considerations
According to M. Poovey , new derivatives were developed specifically to take advantage
of de-regulation. Poovey says that derivatives remain largely unregulated, for
they are too large, too virtual, and too complex for industry oversight to
police. In 1997-8 the Financial Accounting Standards Board (an industry
standards organization whose mission is to establish and improve standards of
financial accounting) did try to rewrite the rules governing the recording of
derivatives, but in the long run they failed: in the 1999-2000 session of
Congress, lobbyists for the accounting industry persuaded Congress to pass the
Commodities Futures Modernization Act, which exempted or excluded "over the
counter" derivatives from regulation by the Commodity Futures Trading
Commission, the federal agency that monitors the futures exchanges. Currently,only
banks and other financial institutions are required by law to reveal their
derivatives positions. Enron, which never registered as a financial
institution, was never required to disclose the extent of its derivatives
trading.
In 1995, the sector composed of finance, insurance, and real
estate overtook the manufacturing sector in America's gross domestic product.
By the year 2000 this sector led manufacturing in profits. The Bank for
International Settlements estimates that in 2001 the total value of derivative
contracts traded approached one hundred trillion dollars, which is
approximately the value of the total global manufacturing production for the
last millennium. In fact, one reason that derivatives trades have to be
electronic instead of involving exchanges of capital is that the sums being
circulated exceed the total of the world's physical currencies.In the past,
mathematical models had a limited impact on finance practice.But since 1973
these models have become central in markets around the world. In the future,
mathematical models are likely to have an indispensable role in the functioning
of the global financial system including regulatory and accounting activities.
We need to seriously question the assumptions that make models of
derivatives work: the assumptions that the market follows probability models
and the assumptions underneath the mathematical equations. But what if markets
are too complex for mathematical models? What if irrational and completely
unprecedented events do occur, and when they do – as we know they do – what if
they affect markets in ways that no mathematical model can predict? What if the
regularity that all mathematical models assume ignores social and cultural
variables that are not subject to mathematical analysis? Or what if the
mathematical models traders use to price futures actually influence the future
in ways that the models cannot predict and the analysts cannot govern?
Any
virtue can become a vice if taken to extreme, and just so with the application
of mathematical models in finance practice. At times, the mathematics of the
models becomes too interesting and we lose sight of the models' ultimate
purpose. Futures and derivatives trading depends on the belief that the stock
market behaves in a statistically predictable way; in other words,that
probability distributions accurately describe the market. The mathematics is
precise, but the models are not, being only approximations to the complex, real
world. The practitioner should apply the models only tentatively,assessing
their limitations carefully in each application. The belief that the market is
statistically predictable drives the mathematical refinement,and this belief
inspires derivative trading to escalate in volume every year.
Financial
events since late 2008 show that many of the concerns of the previous
paragraphs have occurred. In 2009, Congress and the Treasury Department
considered new regulations on derivatives markets. Complex derivatives called
credit default swaps appear to have been based on faulty assumptions that did
not account for irrational and unprecedented events, as well as social and
cultural variables that encouraged unsustainable borrowing and debt. Extremely
large positions in derivatives which failed to account for unlikely events
caused bankruptcy for financial firms such as Lehman Brothers and the collapse
of insurance giants like AIG. The causes are complex, but some of the blame has
been fixed on the complex mathematical models and the people who created them.
This blame results from distrust of that which is not understood. Understanding
the models is a prerequisite for correcting the problems and creating a
future which allows proper risk
management.
A hedge fund is a fund that can take both long and
short positions, use arbitrage, buy and sell undervalued securities,
trade options or bonds, and invest in almost any opportunity in any
market where it foresees impressive gains at reduced risk.The first thing to know about hedge funds is that the
term hedge fund is not a legal term, but rather an industry term.
What a hedge fund is, therefore, is subject to some amount of interpretation.
Consider a few definitions. From Wall Street Words Houghton &
Miflin 1997:
A very specialized, volatility open-end investment company
that permits the manager to use a variety of investment techniques usually
prohibited in other types of funds. These techniques include borrowing
money, selling short and using options. Hedge funds offer investors the
possibility of extraordinary gains with above average risk.From Hedge Funds Demystified, Goldman Sachs &
Co.:
The term "hedge fund" includes a multitude
of skill-based investment strategies with a broad range of risk and return
objectives. A common element is the use of investment and risk management
skills to seek positive returns regardless of market direction.
From the Hennessee Group LLC Web page:
A hedge fund is a "pool" of capital for accredited investors
only and organized using the limited partnership legal structure... the
general partner is usually the money manager and is likely to have a very
high percentage of his/her own net worth invested in the fund.
As you can see, the definitions above focus on several
aspects of investment companies known as hedge funds:
legal structure
investment strategy
investor pool
All three components are important in understanding hedge
funds. To go further let's discuss first what hedge funds are and what
some of their salient features are:
Hedge funds are legally limited partnerships.
Hedge funds are unregistered (i.e., unregistered with
the SEC) investment companies. That is, they are not regulated by the SEC
(more on this later).
Hedge funds can be users of a variety of investment strategies
and products, including options, future, swaps and short selling.
Hedge funds often employ leverage, in that the amount
of notional market exposure often exceeds the investment capital of the
fund.
Hedge funds have limited liquidity. Typically
investors can only get into funds on certain dates and can only get their
money out of funds on certain dates.
Who invests in hedge funds?
Wealthy individuals. For an individual to invest
in an unregistered investment company, the SEC must deem an individual
to be an accredited investors (a defined by SEC rule 501 of Regulation
D. The full text of this rule may be found at
The rule includes the following points for an individual:
The individual must have at the time of investment a
net worth (or joint net worth with spouse) exceeding $1,000,000, or
The individual must have individual income exceeding
$200,000 in each of the two most recent years or joint income with spouse
exceeding $300,000 in each of the two most recent years, and must
have a "reasonable expectation" of reaching the same level in
the coming year.
And the rule contains the following points for an organization,
corporation or other such entity:
The organization must have total assets in excess of
$5,000,000, or
The organization's owners must be accredited investors.
Endowments, e.g., the Wall Street Journal (December
8, 1998) and others report that the University of Pittsburgh made a $5
million investment in Long Term Capital
The University of Michigan hired four hedge fund managers
to manage $100m of its $2.5bn pension fund, and P&I reports that University
of Michigan has a total of about $300m invested with hedge funds.
Other Hedge Funds: Some hedge funds invest in
other hedge funds, including offering Funds of Funds, that is, they
strategically allocate their capital to other funds. Pension and Investments
(November 30, 1998) reports that Grosvenor Capital Management invested
$7m in Long Term Capital Management:
Hedge Fund Industry It is important to understand the magnitude of
the hedge fund industry and the sizes of some of the key players in the
industry. "Hedge Funds Demystified" estimates that the size of
the whole industry is approximately $400bn, and that the investor pool
is dominated by wealthy individuals (accredited investors), with pension
fund interest increasing. "Hedge Funds Demystified" also notes
that it is difficult to accurately assess the size of the industry, so
this number should be read as mainly an indication of the order of magnitude.
To get a sense of where this stands, consider the pension fund industry
by contrast. Davis, in Pension Funds, reports that as of year end
1991, the US pension fund industry's assets were at least $2.9 trillion.
In other countries, the number was less for two reasons: the pension fund
industry contributes less assets as a percentage of GDP than the US (except
Germany and Switzerland) and the US GDP is much larger than other countries.
Nevertheless, the global pension fund industry (as of year end 1991) was
estimated at approximately $4.2 trillion. The numbers since then have surely
grown, but I currently do not have more up-to-date numbers.
Types of Hedge Funds
Hedge funds are generally classified according to the
type of investment strategy they run. Below we review the major
types of strategies, but refer members of the class to "Hedge Funds
Demystified" for greater detail.
Market Neutral (or Relative Value) Funds
Market neutral funds attempt to produce return series
that have no or low correlation with traditional markets such as the US
equity or fixed income markets. Market neutral strategies are characterized
less by what they invest in than by the nature of the returns. They often
are highly quantitative in their portfolio construction process, and market
themselves as an investment that can improve the overall risk/return structure
of a portfolio of investments. Market neutral funds should not be confused
with Long/Short investment strategies (see below). The key feature of market
neutral funds are the low correlation between their returns and the traditional
asset's.
Event Driven Funds
Event driven funds seek to make profitable investments
by investing in a timely manner in securities that are presently affected
by particular events. Such events include distressed debt investing,
merger arbitrage (sometimes called risk arbitrage) and corporate
spin-offs and restructuring.
Long/Short Funds
Funds employing long/short strategies generally invest
in equity and fixed income securities taking directional bets on
either an individual security, sector or country level. For example, a
fund might do pairs trading, and buy stocks that they think will
move up and sell stocks they think will move down. Or go long sectors they
think will go up and short countries they think will go down. Long/Short
strategies are not automatically market neutral. That is, a long/short
strategy can have significant correlation with traditional markets, and
surprisingly have seen large down turns in exactly the same times as major
market downturns. For example, Pension & Investments reported on November
30, 1998: Many long-short managers, which aim to profit from going
long on stellar stocks and selling short equity albatrosses, typically
use traditional stock valuation factors such as price-to-earnings and price-to-bok
value ratios in their mathematical models to cull the winners from the
losers. Unfortunately for them, when the market ran into turbulence in
late July and August, investors sought safe haven in some of the largest-but
expensive-stocks that these models had rejected as overpriced.Then, after the Federal Reserve Bank began easing interest
rates in late September, investors rushed to buy small-capitalization,
high-octane stocks that had been neglected in favor of large cap stocks
for most of the year. ... As a result, some market long-short managers
got hit with a double-whammy.
Tactical Trading
Quoting from "Hedge Funds Demystified":
Tactical trading refers to strategies that speculate on
the direction of market prices of currencies, commodities, equities and/or
bonds. Managers typically are either systematic or discretionary. Systematic
managers are primarily trend followers who rely on computer models based
on technical analysis. Discretionary managers usually take a less quantitative
approach and rely on both fundamental and technical analysis. This is the
most volatile sector in terms of performance because many managers combine
long and/or short positions with leverage to maximize returns...
The Hedge Fund Industry and Quantitative Methods
Quantitative methods have been successfully applied in
the hedge fund industry to improve returns, and control risk. That said,
there have been striking failures of seemingly quantitatively driven funds
(such as Long-Term Capital). Some of the most quantitatively driven strategies
occur in the Market Neutral/Relative Value Sector of the Hedge Fund World,
so we will exam this sector in more detail by discussing some of the specific
types of strategies they employ. The following is a list of important and
quantitatively driven market neutral/relative value strategies. I refer
you to "Hedge Funds Demystified" for a detailed description of
each:
Fixed income arbitrage
Covertible bond arbitrage
Mortgage backed security arbitrage
Derivatives Arbitrage
Market Neutral Long/Short Equity Strategies
Hedging Strategies
A wide range of hedging strategies are available to hedge funds. For example:
selling
short - selling shares without owning them, hoping to buy them back
at a future date at a lower price in the expectation that their price
will drop.
using arbitrage - seeking to exploit pricing
inefficiencies between related securities - for example, can be long
convertible bonds and short the underlying issuers equity.
trading
options or derivatives - contracts whose values are based on the
performance of any underlying financial asset, index or other
investment.
investing in deeply discounted securities -
of companies about to enter or exit financial distress or bankruptcy,
often below liquidation value.
Many of the strategies used by hedge funds benefit from being non-correlated to the direction of equity markets
Hedge Fund Returns
As hedge funds are often viewed as providing returns that
are "cheap" relative to risk, their performance is usually evaluated
on a risk-adjusted return basis. The common number that is quoted is the
Sharpe Ratio which is the ratio of annualized excess returns to
the annualized standard deviation of returns. The following repeats the
data in Table 7 of "Hedge Funds Demystified" and gives an idea
of the relative performance of hedge funds compared with some standard
indexes over the period January 1993 - December 1997. The table represents
returns on each Hedge Fund Sector, that is, the returns and standard deviations
in each column represents the returns that were realized on an equal weighted
investment portfolio of all the hedge funds in a given sector.
One important item that none of the definitions covered
was hedge fund fee structures, which is, in my opinion, a key distinguishing
feature of hedge funds versus in particular mutual funds. Hedge funds almost
always have a fee structure that includes both a fixed fee and a management
fee. The fixed fee usually ranges between 1 and 2% of assets under management
and the management fee ranges between 20 and 25% of upside performance.
As hedge funds are unregulated, these ranges are often exceeded, and can
be as high as 5% fixed fee and 25% management fee. Hedge fund fees are
often quoted in language such as "2 and 20" meaning 2% fixed
fee and 20% management fee. There are two additional important points about
hedge fund fees:
the benchmark
high water mark
The performance fee is sometimes calculated net of a benchmark.
That is, the returns that fees are paid on are sometimes only those returns
in excess of some benchmark. Sometimes the benchmark is a risk-free interest
rate such as LIBOR (often called the cash benchmark, meaning performance
fees are paid on the profit that would be made in excess of an investment
in cash) and other times it is a market index such as the MSCI World Index
or the S&P 500 index.
The biggest risk in pricing models isAssumption Risk. The
trouble with bull and bear markets is that the price behavior and the
width of bid-offer spreads can be quite different under the two regimes.
If you are in roach motel assets getting out can become expensive. Bear
Stearns was leveraged long CDOs of illiquid securities and "hedged" by
shorting liquid ABX indices. As with similar problems in the past, the
funds were long illiquid, short liquid. If a fund is leveraged and can
only sell to a limited number of counterparties that KNOW it has a
problem, getting out becomes difficult. Software for measuring risk
doesn't help when risks are unmeasurable. And aren't you supposed to
have proper risk management in place BEFORE you lose money not AFTER the fact?
You really have to know what you are doing when designing models of
prepayment and mortgage default risks; nothing in the academic
literature or public domain works. Credit is neither stochastic nor
continuous and when it jumps it really jumps. I can count the number of
good mortgage-backed securities hedge funds on one hand but I would need
many more limbs for the traders who have been blown away by not having
adequate trading AND quantitative abilities to manage ALL the exposures
in this complex field. When a product is very thinly traded, indicative
dealer prices are pretty useless. If a fund is investing in illiquid
instruments the fund valuation needs to be marked to the real bid, in
size. Mark to market is possible only when there is a market.
There is nothing inherently wrong with investing in "untraded"
assets provided the risk-adjusted returns are sufficient to compensate.
In bearish credit conditions ideally you usually want to be long the
liquid and short the illiquid but weaker credit funds and less
experienced managers do the opposite. Of course there have been skilled
hedge funds in the areas of distressed debt and collateralised loans for
a long time but their returns have justified the risks. But with some
funds, even with apparently high absolute performance, often the excess
RISK-ADJUSTED returns (the alpha!) was negative.
Just as with Long-Term Capital Management, being long the illiquid
and short the liquid works well until the market reverses and then years
of consistently positive months get given back in one massively
negative month. Leverage, liquidity and valuation risks are ONLY worth
taking if you are compensated for those risks and plainly this was not
the case. This is where investors in a hedge fund need to look at
whether the potential returns justify the potential risk. With good
hedge funds it does but NOT with the many "hedge fund" journeyman.
With public equities, liquid bonds, fx and futures valuation is
immediate, transparent, generally unarguable and there is plenty of
alpha available in these liquid arenas IF you have the tools and
expertise to find it. While liquidity is a variable even on an exchange
you have access to the widest number of potential buyers and sellers.
Venturing into illiquid areas raises the risk exponentially when there
are much fewer counterparties to trade with. Leverage just exacerbates
those problems. Funds investing in illiquid assets should be targeting
MUCH higher performance than liquid funds as compensation for that extra
risk. Yet some investors seems to compare them side by side without
modelling the non-linear risks of gearing thinly traded securities.
What's even worse than a closet index fund? A leveraged closet index
fund. And that is what most of these toxic waste CDO funds were in
effect running. Making money in BAD conditions is what hedge fund
clients pay the 2 and 20 for; long only funds are the ONLY products you
need in good times. Having criticised some of John Bogle's thinking in
my previous post let's make something clear; index equity and credit
funds are the best investment IF (and only IF!) you think the asset
class is going up. It is a waste of time and money to allocate to higher
fee actively managed funds that simply fall apart when their underlying
market falls apart.
Investors need to verify that a money management product purporting
to be a hedge fund and charging hedge fund fees actually is one. Out of
10,000 funds that claim to be hedge funds, how many actually are hedge
funds? The best estimate I have is maybe 25% tops. But of those how many
are skilled? Perhaps 500-1000 at most. In other words probably only 10%
of products that say they are hedge funds actually are GOOD hedge
funds. Skill is rare by definition. While some investors might be
discouraged by the bad news of 1/10 odds of picking a skilled fund, the
good news is that they CAN be isolated in advance.
Identifying a good hedge fund is as rare a skill as being able to
identify a good security. Some multi-manager products and weaker funds
of funds have reduced their fees because they think picking hedge funds
is easy! Most of them don't have the experience or analytical resources
to decide what is and what is NOT a hedge fund, let alone trying to find
the BEST ones. It is difficult but NOT impossible. Do "lower" fees help
if an "advisor" puts you into a fund that drops 100%?
There will always be semantically-challenged products that screw up
which is why due diligence and alignment of interests are so important.
Investors should select real hedge funds NOT leveraged beta products
that SAY they are hedge funds. The industry needs to rid itself of non
hedge funds who can't measure, manage or hedge their risks and ride beta
when proper managers aim for alpha. Fortunately we can rely on the
market to conduct these shakeouts over time. Unfortunately for some
amnesiac investors it has been a long time since difficult credit
conditions.
The colleagues of Ralph Cioffi may have liked the fund but how much personal cash did James Cayne
have in? A necessary condition for a product to be considered a hedge
fund is to verify senior management are eating their own cooking. In
bull markets many unskilled traders make money; it is bear markets that
tend to show who is good and who knows how to hedge. Many REAL hedge
funds are MAKING MONEY out of these ongoing credit events. Marketing
something is a hedge fund does not mean it is.
Andrew W. Lo,Andrew is a Professor of Finance at MIT's Sloan School of Management and he is also the Director of Laboratory of Financial Engineering at Sloan is a leading authority on hedge funds.
Andrew W. Lo
Popular Misconception The popular
misconception is that all hedge funds are volatile -- that they all
use global macro strategies and place large directional bets on
stocks, currencies, bonds, commodities, and gold, while using lots of
leverage. In reality, less than 5% of hedge funds are global macro
funds. Most hedge funds use derivatives only for hedging or don't use
derivatives at all, and many use no leverage. |
Throughout history algebra has changed in words through etymology. Etymology is an account of the history of a particular word or elements of a word. The word "algebra" is derived from Arabic writers. Algebra is a method for finding solutions of equations to the simplest possible form. Different cultures have come up with different types of names to classify algebra. Al Khwarizmi and Fibonacci contributed talented mathematic systems that shaped algebra. Al Khwarizmi was born in the town of Khwarizm in Khorason. He achieved most of his work between 813 a.d and 833 a.d. Khwarizmi contributed logical approaches to algebra and trigonometry. He came up with ways of solving linear and quadratic equations. Khwarizmi was not the only person who contributed to algebra; Fibonacci contributed to algebra has well. one by adding a number to sum up the two numbers that precedes the previous two numbers. He used this method to tie nature and mathematic together. It is formed by using a triangle whose sides' measure one number of the Fibonacci
Fibonacci contributed the decimal number system which is known as the Fibonacci sequence. The Fibonacci sequence is closely related to the golden ratio that uses the number number of the Fibonacci sequence. Fibonacci was born in Pisa, Italy around 1175. He studied mathematics in North Africa in the city of Bugia. Fibonacci's greatest achievement was the golden ratio in nature. For example, plants grow new cells in spirals such as this pattern of seeds in a sunflower. The seeds in a sunflower are packed in going left and right making a spiral affect. The number of lines in the spiral of a sunflower is almost the numbers leading to the Fibonacci sequence.
The golden ratio is an irrational mathematic constant of 1.6180339887 found in nature. When...
YOU MAY ALSO FIND THESE DOCUMENTS HELPFUL
...
The Golden Number
1.61803 39887 49894 84820 is by no means a number of memorization. However, it is a recognizable one. Never will you find a combination of numbers that is more significant than this one. This ratio is known as the Golden Number, or the GoldenRatio. This mystery number has been used throughout different aspects of life, such as art, architecture, and of course, mathematics. One may wonder where theGoldenRatio came from? Who thought to discover it? When was it discovered? And how has it been used throughout time? The Goldenratio has been used throughout different aspects of life after being discovered during the ancient times.
About two to three thousand years ago, the GoldenRatio was first recognized and made use by the ancient mathematicians in Egypt. The goldenratio was introduced by its frequent use in geometry. An ancient mathematician, sculptor, and architect named Phidias, who used the goldenratio to make sculptures, discovered it. He lived from sometime around 490 to 430 BC. None of his original works exist, however he was highly spoken of by ancient writers who gave him high praise. Hegias of Athens, Agelades of Argos, and Polygnotus of Thasos were said to have trained him.
Although not much is known about Phidias's life, he is...
...The GoldenRatio: Natures Beautiful Proportion
At first glance of the title, many may wonder: What is the GoldenRatio? There are many names the GoldenRatio has been called including the Golden Angle, the Golden Section, the Divine Proportion, the Golden Cut, the Golden Number et cetera, but what is it and how is it useful for society today? One may have heard of the number π (Pi 3.14159265…) but less common is π's cousin Φ (Phi 1.61803399…). Both Φ and π are irrational numbers, meaning they are numbers that cannot be expressed as a ratio of two whole numbers as well as the fact that they are never-ending, never-repeating numbers. The GoldenRatio is the ratio of 1:Φ (1.61803399…). The GoldenRatio is a surprising ratio that is based on the research of many composite mathematicians spanning over 2300 years, and it is found in many areas of everyday life including art, architecture, beauty and nature.
Euclid, a Greek mathematician who taught in Alexandria around 300 B.C., was one of the first to discover and record the bases for the GoldenRatio back 2300 years ago. Euclid's discovery was that if one takes a line and divide it into two unequal sections in such a way that the...
...Goldenratio ; The Definition of Beauty
"Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel." Johannes Kepler, 1571-1630
The goldenratio is present in everyday Life. The golden proportion is the ratio of the shorter length to the longer length which equals the ratio of the longer length to the sum of both lengths. It can be expressed algebraicay like :
This ratio has always been considered most pleasing to the eye. It was named the goldenratio by the Greeks. In the world of mathematics, the numeric value is called "phi", named for the Greek sculptor Phidias.
The GoldenRatio is also known as the golden section, golden mean or golden rectangle. The Golden Rectangle has the property that when a square is removed a smaller rectangle of the same shape remains, a smaller square can be removed and so on, resulting in a spiral pattern. It is a unique and important shape in mathematics which also appears in nature, music, and is often used in art and architecture.
Our human eye "sees" the golden rectangle as a beautiful geometric form....
...The GoldenRatio
By : Kaavya.K
In mathematics and the arts, two quantities are in the goldenratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The goldenratio is an irrational mathematical constant, approximately 1.6180339887. Other names frequently used for thegoldenratio are the golden section and golden mean. Other terms encountered include extreme and mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, golden number, and mean of Phidias. The goldenratio is often denoted by the Greek letter phi, usually lower case (φ).
[pic]
The golden section is a line segment divided according to the goldenratio: The total length a + b is to the longer segment a as a is to the shorter segment b.
The figure on the right illustrates the geometric relationship that defines this constant. Expressed algebraically:
[pic]
This equation has one positive solution in the algebraic irrational number
[pic]
At least since the Renaissance, many artists and architects have proportioned their works to approximate the...
...The GoldenRatio
Body, art, music, architecture, nature – all connected by a simple irrational number – the GoldenRatio. According to Posamentier & Lehmann in their work The
(Fabulous) Fibonacci Numbers, there is reason to believe that the letter φ (phi) was
used because it is the first letter of the name of the celebrated Greek sculptor Phidias (490-430 BCE). He produced the famous statue of Zeus in the Temple of Olympia and supervised the construction of the Parthenon in Athens Greece (Posamentier & Lehmann, 2007). In constructing this masterpiece building, Phidias used the GoldenRatio to create a masterpiece of work.
Figure 1: This is a model of Zeus in the Temple of Olympia. The red lines show the use of the GoldenRatio. (
Phidias brought about the beginning of the one of the most universally recognized form of proportion and style used throughout history (Posamentier & Lehmann, 2007). The irrational number Phi, also known as the GoldenRatio, has had tremendous importance. To properly understand this mathematical concept, it is important to explore the definition, history, and the relations to architecture, art, music and the Fibonacci sequence.
Figure 2: This model shows the line segments in the GoldenRatio. (Wikipedia.org)
As is...
...GoldenRatio
In mathematics, two quantities are in the goldenratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b,
Where the Greek letter phi (φ) represents the goldenratio. Its value is:
Thegoldenratio is also called the golden section (Latin: sectio aurea) or golden mean. Other names include extreme and mean ratio, medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut, and golden number.
Some twentieth-century artists and architects, including Le Corbusier and Dalí, have proportioned their works to approximate the goldenratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the goldenratio—believing this proportion to be aesthetically pleasing (see Applications and observations below). Mathematicians since Euclid have studied the properties of the goldenratio, including its appearance in the dimensions of a regular pentagon and in a golden...
...previously established.
Segment 1: The Fibonacci sequence
The Fibonacci sequence can be defined as the following recursive function:
Fn=un-1+ un-2
Where F0=0 and F1=1
Using the above we can find the first eight terms of the sequence. An example of calculations is given below:
F2=F1-F0F2=1+0=1
We are able to calculate the rest of the terms the same way:
F0 | F1 | F2 | F3 | F4 | F5 | F6 | F7 |
0 | 1 | 1 | 2 | 3 | 5 | 8 | 13 |
Segment 2: The Goldenratio
In order to define the goldenratio we need to examine the following sketch:
The line above is divided into two segments in such a way that ABAP=APPB
The ratio described above is called the goldenratio.
If we assume that AP=x units and PB=1 units we can derive the following expression:
x+1x=x1
By solving the equation x2-x-1=0 we find that: x=1+52
Segment 3: Conjecture of φn
In this segment we examine the following geometric sequence:
φ,φ2,φ3…
Since x=1+52 can simplify φ by replacing the value of x to the formula of the goldenratio we discussed before. Therefore:
φ=x+1x φ=1+52+11+52 φ=1+52
Thus φ2=1+522 φ2=3+52 and F2φ+F1=1+52+1=3+52
Therefore:
φ2=F2φ+F1
We can simplify other powers of φ the same way, thus:
φ3=2+5 and φ4=35+72
In order to from a conjecture...
...The GoldenRatio
The goldenratio is a unique number approximately equal to 1.6180339887498948482. The Greek letter Phi (Φ) is used to refer to this ratio. The exact value for the goldenratio is the following:
`
A popular example of the application of the goldenratio is the Golden Rectangle. Interestingly enough, many artists and architects have proportioned their works to apply the goldenratio in the form of the golden rectangle. A golden rectangle is a rectangle where the ratio of the longer side (length) to the shorter side (width) is the goldenratio If one side of a golden rectangle is N ft. long, the other side will be approximately equal to N(1.62) or N(Φ). One interesting attribute about the golden rectangle is that if you cut a square off it so that what remains is a rectangle, the remaining rectangle will also have the length to width properties of the goldenratio, therefore making it another golden rectangle. What happens is that if you keep cutting squares off, each time you get a smaller and smaller golden rectangle. Leonardo Da Vinci, the famous mathematician and artist from the Renaissance, featured the golden... |
FFJM competition underway
The 2010-2011 FFJM competition has started. It is one of the main Mathematical Games competitions in France (the other being the Concours Kangourou, which gets hundreds of thousands of contestants every year and is mostly for school pupils). The FFJM competition claims 100,000 contestants from ten countries.
The first round of questions (in French) is online. For this round, there is no time limit and you can send in your answers by post.
One of the most interesting characteristics of this contest is that all age groups get the same questions: for example, 2nd-graders only need to answer the first 5 questions, 8th-graders need to answer the first 14 questions, and professionals must answer all 18 questions (the last questions being the hardest). In the semi-finals and finals, this is very nice: all contestants can talk about the questions, even if they are competing in different categories.
This is the first time in almost 10 years that I will be taking part. For people who feel like taking part, answers must be sent in by 1 January. |
Comments on: Fractal Triangle
Fractals are SMART: Science, Math and Art!Fri, 19 Jan 2018 16:30:53 +0000hourly1 This Is How We Math: Fractals |
Fri, 23 Oct 2015 01:12:47 +0000 you're interested in the fractal activity, you can check it out at the Fractal Foundation's website. While you're there, I really recommend checking out all of their other activities and […]
]]>By: "Enrichment Classes" in Yucca Valley | California Desert Homeschoolers
Thu, 01 Oct 2015 17:49:48 +0000 Our new enrichment classes got off to a rocky start due some mix-ups but we still had a great time! We got on our way to India and explored fractals! I personally can't wait to see what happens in India next week and to see how many Sierpinski Triangles our group can make. Due to conflict with another event, we'll hold off on putting them all together until October 13. You can find information about making your own here. […]
]]>By: FractalMan
Thu, 21 May 2015 20:01:11 +0000 so much, Natalya!!!
]]>By: Natalya Silcott
Wed, 20 May 2015 12:02:06 +0000 I had a wonderful session doing this activity at Orwell Park Prep School in Ipswich, England. I teach Mathematics at Harrow School and would like to run a few more sessions with primary schools in our area before sending you the triangles.
]]>By: Kathy
Mon, 20 Apr 2015 17:23:54 +0000 is great! We are planning a visit to the Museum of Science and Industry's Numbers in Nature exhibit, and I was looking for activities to prep our class of 9-12 year olds for understanding several number patterns, included fractals. We did our individual triangles today and are working on assembling them all together. I will send the finished triangles to you when we've wrapped up our project and displayed them.
]]>By: FractalMan
Wed, 21 Jan 2015 07:39:43 +0000 – absolutely, we are still accepting triangles for the Trianglethon – and we always will! This is a never-ending project, we'll just keep making it bigger and bigger…! |
Fractals
Has anyone thought about the pictograph at the bottom of page 81, of
the 5D book being a prescription for a fractal. It looks like:
+-----sales------->+
| |
positive satisfied customer
word |
of |
mouth |
| |
+<-----------------+
which, if there is any random variablity, for example in sales, (which
we would anticipate,) it would be an algorithm for a fractal
process. Kind of interesting, since conventional statistical methods,
(eg., anything dealing with "bell curves," or standard deviations,)
would be an inappropriate methodology since it is a *_cumulative sum_*
of a random process-this is an issue that "program traders" exploit,
since such processes are the "engine" of speculative markets, both
capital and stock. Also interesting since it is the "engine" of the
the corporate P&L. There is a substantial mathematical infrastructure
that has been developed to analyze such phenomena. Kind of interesting
because revenue rates, organizational development, etc., could, I
would suppose at least in principle, possibly be analyzed under one
set of logical rules.
John
--
John Conover, [email protected], |
My math skills are exponentially funnier than anything
Could you please give me an explanation as to why X to the third power plus Y to the third power equals Z to the third power?
Ha, ha. That's a good one, you jokesters, you.
You don't seriously believe that I -- me, whose math skills are often called into question by my credit union in regard to the balance on my checking account -- could figure out why that formula should be so, even if it is so, which apparently it is.
I thought I had this business figured out until one of my masters called to remind me that I am an idiot and didn't have it right at all. So I will leave it up to you people to figure it out, and I'm sure you'll tell me in great detail ...
You people, you always hand me a laugh. What a bunch of kidders.
Let's talk about something less humorous, shall we?
Does washing your hands with warm water kill more bacteria than washing them with water at room temperature? If room-temperature water is just as effective, we could save a lot of water and energy.
You have obviously let your subscription to the Journal of Occupational and Environmental Medicine lapse.
In 2005, that august publication reported on a study in which people with hands contaminated with bacteria were instructed to wash their hands with soap for 25 seconds in water ranging from 40 degrees to 120 degrees.
The researchers found that there was no difference in the reduction of bacteria between those who washed their hands in the cold water and those who used the warmer water. |
>>7317928 >>7317929 don't see how that makes it more beautiful. It just makes it more complicated, and part of its beauty, aside from the fact that it uses the most fundamental numbers and operations of mathematics, is its simplicity
>>7318915 >Doesn't your "least awful contender" imply, by taking the square root of both sides, that e^i(pi) = 1? That shit is going on in another thread >>7317019
Basically, when you square a number, the number AND its negative give the same result. So the information is gone whether you started with a negative or positive number. By the same logic, the inverse of squaring –the square root– should give two results, one positive AND one negative (but we usually discard the negative one). Since you've lost the information of whether it was positive or negative to begin with, you can only reason about the magnitude of the number:
We know that , so any squaring and rooting is invalid if you end up with 1. |
Artifacts found at Blombos, about 75,000 years old, including red ochre stone with design carved in it; see
Blombos Cave Project.
Humans have long found the business of successfully articulating precise things together rewarding, both economically
and psychologically.
In addition, precise articulation is necessary in order to make things come out right, from the construction of
temple complexes to carving stone balls with curious symmetries.
The necessity and inspiration of articulation in geometric figures suggests that this ancient field of geometry really
should be called "articulatory geometry" -- and indeed, it is not difficult to find the adjective "articulatory" and
the noun "geometry" in close proximity in mechanical engineering and anatomy papers.
"Articulatory" has a more distinguished pedigree than "reticular".
Unlike the latin rete, which simply appeared in ancient Rome from who-knows-where, the coincidence of the latin
artus (joint) and articulus (joint, precise point, or division), related to armus (arm) and hence
to ars (art), seems connected to the Hittite ara (something that fits) and the Sanskrit irmas (arm):
the word seems to trace to
Indo-European roots, suggesting that the art of fitting things
together
(as one might punningly describe architecture)
has been a longstanding focus of attention in Eurasian civilization.
The wordreticularis derived fromrete, a latin word "of obscure origin"
that meansnet.
In English, an object is "reticular" if it is intricate and netlike, consisting of many articulated
components.
The wordgeometryis a Greek concoction, meaning "the measurement of the Earth"; this
etymology refers to the traditional view (advanced by
Herodotus) that geometry was invented by
Egyptian (and perhaps Mesopotamian) surveyors.
However, there is archeological evidence of interest in polyhedra and other geometric objects (e.g.,
right; in fact, all five Platonic solids are represented -- in a sense -- among these ancient
Scottish balls).
Omar Yaghi has denoted asreticular
chemistrythat part of nanoscience and materials science concerned with the assembly of large
or potentially unbounded structures composed of many articulating parts.
Similarly,
Georg Alexander Pick's
theorem on the areas of polygons formed by families of parallel lines on the plane inspired fans to
dub his work on the geometry of such net-like objectsreticular geometry, (although the
phrase is obscure enough that as of September, 2010, the AMS
MathSciNet had never heard of it.)
And so we dub asRETICULAR GEOMETRYthat area of geometry concerned with large and
complex geometric objects consisting of many articulating components.
Of course, the use of the adjective "reticular" instead of the adjective "articulatory" suggests a bias in
favor of studying the final, fully assembled geometric structures, rather than the details involved in
the design process.
This is a bias that geometers should avoid in practice, although it is quite possible that the mathematics
of the two different aspects of this geometry are different (and in fact, there is anecdotal evidence that
the mathematics of the final structures in reticular geometry is more difficult than the mathematics of
the assembly process in articulatory geometry).
On this page:
Reticular Geometry and Design.
Much of the interest in what we might call "reticular geometry" has been driven by economic demand,
by the need to design complicated objects.
Reticular Geometry as Mathematics.
Nevertheless, the problem how how to fit together many geometric objects is ultimately a mathematical one,
associated with several extant fields of mathematics.
The Sociology of Reticular Geometry.
On the other hand, there is a long history of popular interest in the sorts of shapes people try to fit
together, and how to fit them together.
My Adventures in Reticular Geometry.
I'm currently working on courses in the subject while doing some research, in particular on Euclidean
graphs as models of crystal structure.
Image of carved stone balls, approximately four millennia old, from Scotland's University of Glasgow's
Hunterian Museum; image from Wikimedia Commons.
Figurines and other decorative pottery go back several tens of thousands of years, but we do not know when
humans first started creating art, as opposed to collecting things in Bower-bird fashion.
Of course, it is likely that most of our ancestors collected artificial things made by artisans just as we
collect bric-a-brac made by local artists or manufactured by big companies.
One of the great holes in our history and archeology is what people did collect or create: just as nowadays,
people collect ugly clocks, Elvis statues, and crystal pyramids, it seems likely that ancient people collected
something.
If so, collectibles were probably collectible because of their novelty -- like fossils and crystals.
Objects associated with reticular geometry have always been popular, and have always exerted influence on our
esthetics.
Once artificial objects appeared, what were they?
And were they designed in advance or did artisans just feel their way?
Marcus Vitruvius Pollio's
Architecture,
written in the first century B.C., is the oldest architectural text known, and is an example of the interplay
between geometry and engineering.
More precisely, between reticular geometry -- in that architecture relies on the articulation of many
parts -- and the design of complex structures.
Vitruvius' Architecture is an example of applied reticular geometry: it was an account of the structural
elements, the properties of the materials used to realize those structural elements, together with some guidance
on how to assemble the desired building.
But clearly, Vitruvius' text was describing well-understood contemporary information in architecture and engineering,
as can be seen from much older and yet very sophisticated structures around the Mediterranean.
For example, the Minoan palace of Knossos in Crete, which was constructed and reconstructed repeatedly during
the second millennium B.C., clearly was the result of repeated and incremental but still intelligent design:
it had stone stairs supported by wood pillars (which meant that it required careful planning), a working
sewer system, and careful placement and sizing of windows to light interiors.
Although no accounts of the design of the palace survive, it was clearly designed by engineers who were
geometrically sophisticated.
In fact, not only the standard geometry of surveying, but the reticular geometry of fitting things together, are
apparent in ancient structures like the Anasazi complex now known as
Pueblo Bonito,
even though the societies that built these structures appear to have been at best semi-literate, and left no texts for
historians to attempt to translate.
That should not be surprising, for agriculture entails the geometry of surveying, and once one starts to build
things, one has to fit things together.
Reticular geometry may be one of the first fruits of any agricultural revolution.
Moving forward, during the
"High Middle Ages"
(roughly the 11th, 12th, and 13th centuries), Europeans making the
pilgrimage to
Santiago de Compostela
visited the embattled provinces of
Andalusia -- roughly that part of the Spanish
peninsula under Islam -- and marveled at the architecture.
The inspiration followed them home, where they built Gothic cathedrals, which were not quite as organized as
their Andalusian models (like the
Alhambra),
and which occasionally fell down during construction.
But stonemasons gradually figured out how to wing it, and during the remainder of the Middle Ages, they filled
northern Europe with
cathedrals of light.
But the Mediterranean had much better light than northern Europe, and as the Middle Ages waned into the
Renaissance, the Mediterranean also had more money.
One of the challenges symbolic of the Renaissance was the decision of the City of Florence to build a dome on
top of the
Florence Cathedral of Saint Mary,
even though no one had built a dome that large since the
Pantheon
over a millennium earlier.
(Even worse, no one knew how the Pantheon had been built, or even what, exactly, the Pantheon was made of --
it turned out to be reinforced concrete.)
Worse, the one thing that they did know was that building the scaffolding for the dome of the Florence
cathedral was impossible.
To top it all off, some silly people wanted a lantern (a structure like in the one on top of the dome at right)
on top, even though the Pantheon at least did not have that complication.
Filippo Brunelleschi's solution --
which involved careful (if secret) designs, developing a construction
process in which the partially built dome and the scaffolding supported each other, on top of the Middle
Eastern device of having a heavy but hidden dome between two light but pretty shell domes -- was a dramatic
example of the ability to build intricate structures designed in advance.
It is not coincidental that Brunelleschi was one of the pioneers of perspective drawing and painting, for
the problem of what a structure is and what it looks like was at the center of art and of architecture in the
Renaissance, for artists were working away from the abstract and symbolic art of the Middle Ages, while
architects were discovering that when they wanted to design genuinely novel buildings, it was a wise idea
to have a clear design in mind before they started building.
Albrecht Durer carried perspective into description
(of military fortifications) and so descriptive
geometry and projective geometry began, officially with
Johannes Kepler and
Gerard Desargues and
continuing up to Gaspard Monge
at the end of the 18th century, applying his work to military fortifications.
Images of an object from different positions, posted in Wikipedia Commons by Hasan Isawi.
Very loosely, projective geometry
(in at least its original incarnation) was concerned with what an object
looked like at a particular distance, while
descriptive geometry was concerned with
how it looked through
a zoom lens -- in particular, at an infinite distance through an infinitely powerful zoom lens (notice
the picture at left, considered ideally, is not from a particular distance but consists of projections
onto planes, as if through an infinitely powerful zoom lens from an infinite distance).
Descriptive geometry provided a tool for designing novel objects that some machinist would then try to make,
and it provided a critical intellectual toolbox for the Industrial Revolution of the 19th century -- and
it continues to earn a living in technical and engineering drawing.
It is no coincidence that the Nineteenth century saw the development of mass assembly, of industrial power,
and of astronomer
John Herschel's
invention of the blueprint.
The intuition had become, if you can draw it, you can make it.
A sideways glance at Frank Lloyd Wright's
fantasies suggest that that is not necessarily true: building
materials matter, and there is always the detail of the assembly process itself...
During the Nineteenth century, a new kind of structure appeared.
As scientists began to consider the possibility that matter was composed of tiny and somewhat-divisible
particles called molecules, themselves composed of even tinier and possibly indivisible particles
called atoms, people began to wonder what these molecules might look like.
Models of molecules -- including
August Wilhelm von Hoffman's
stick-and-ball models -- appeared in the later
Nineteenth century, even before there was a consensus in the scientific community about atoms and molecules.
They were, as
George Polya pointed out, combinatorial
objects, and thus
graph theory could be used to model
molecules.
But they were also geometric objects.
Isomers are molecules with the same chemical formula -- the
same number of each kind of constituent atom -- and yet are structurally different.
One notorious kind of (pair of) isomers are the
stereoisomers, i.e., two molecules that are mirror images
of each other.
This issue of chirality -- of mirror images, of
right- and left-handedness -- became one of the central
issues in stereochemistry, although it involves the spacial relationships between atoms in a molecule,
or between molecular building blocks (MBBs) in a super-molecule.
During the early Twentieth century, new techniques for determining the structure of molecules pushed geometry
into the foreground.
But during the later Twentieth century, something new arose: the ability to control molecular building blocks,
or even individual atoms, allowing chemists to design and assemble novel structures.
As
Richard Feynman predicted in an influential address on
There's Plenty of Room at the Bottom, this led to
a qualitative change in science and engineering.
During the last few decades, scientists have assembled graphite-like polyhedra, DNA complexes, artificial
proteins, and other nano-objects.
As the target products get more complicated, the mathematics of the design principles get more sophisticated,
and a demand appears for more effective mathematics for more effective design.
Image of Fullerene c540, an example of a
fullerene, i.e., a finite carbon structure in a
graphene-like
(graphite-like)
arrangment.
Fullerenes were named after architect and geometric enthusiast
Richard Buckminster Fuller.
Image posted in Wikipedia Commons.
The notion that all matter is composed of tiny particles is ancient: the Greek philosophers
Democritus and
Epicurus championed the notion, which
Aristotle refuted (at least to the satisfaction of the Europeans).
The atomic theory was resurrected by
Johannes Kepler
and Robert Boyle, and very quickly became central
to crystallography, for crystallographers from
Nicolas Steno to
Rene Hauy found that they could explain the
polyhedral shapes of crystals by assuming that crystals were composed of very regular arrays of tiny
something-or-others.
The Nineteenth century saw the protracted foodfight over the atomic theory, and crystallography was at the
center.
The structure of crystals made sense if they were composed of something like atoms or molecules.
And although the final argument for the atomic theory was based on molecular motion -- the only viable
explanation for
Brownian motion (the habit of dust particles
floating on water to jiggle about
randomly) seems to be
Albert Einstein's proposal that tiny particles are
jostled by even tinier molecules -- the success of the atomic theory in crystallography helped set the stage.
But that means that materials are themselves extended geometric structures, perhaps best envisioned as
infinite structures filling space.
If the structure was a regular array at a nanoscopic level, one could imagine that the resulting crystal
would tend to have cleavage planes and various rotational and mirror symmetries.
From cleavage planes, mathematicians like
Augustin Cauchy devised polyhedra (just as
Archimedes devised
polyhedra from cleavage planes -- although there is no surviving evidence that Archimedes was inspired by
crystals) (although, for all we know, he might have been).
From the symmetries, physicists like
Auguste Bravais and mathematicians like
Yevgraf Fyodorov devised the
crystallographic groups in a massive effort that
spanned the Nineteenth century and the careers of many scientists.
In the early Twentieth century, physicists and chemists devised methods for determining the structure of
materials.
Since then, chemists have developed methods for designing crystals and then building them, as
opposed to the alchemical approach (also known as
combinatorial chemistry)
of conducting zillions of experiments mixing things together and seeing what happened.
We are entering an age when, like Vitruvius, we design the thing we are going to make, and then we make it.
It is still a simple age, for we have not developed chemical equivalents of the architectural design
methods and construction techniques of the Renaissance.
That's what's coming next.
Meanwhile, the design of buildings has become divided between
architects,
who are theoretically concerned with the overall form and ergonomics of the building, while
structural engineers are concerned
with the integrity and soundness of the buildings.
The growing demands on the actual performance of modern buildings -- to lower energy costs,
reduce environmental footprints, enable ready navigation, etc. -- suggest that the demand for the
mathematical tools for comprehensive design will only increase.
And design considerations will include the geometry of the structure itself.
This web-page is not the first manifesto about reticular geometry, although it may be the first current
one with a mathematical slant.
For a more materials science slant, see:
Polygons, polyhedra, and other shapes permeate the artistic and engineering worlds of many cultures, but the
mathematical attention is less clear: it appears spotty, but the record is fragmentary, so we actually don't
much about this kind of geometry prior to the Renaissance, when Europeans got interested in them.
Albrecht Durer wrote much about his interests, and was a
draftsman whose work helped inspire descriptive geometry; his Painter's Manual introduced the notion
of a net of a polyhedron, which was essentially the graph of its vertices and edges.
Johannes Kepler subsequently conducted a more mathematical
investigation, particularly of atomic arrangements of matter (in A New Year's Gift of Hexagonal Snow)
and of the relationship between polyhedra and astrology -- er, astronomy -- in Harmonies of the World.
It is true that polyhedra -- or rather, their umpteen-dimensional analogues,
polytopes
-- are very big business nowadays.
Literally.
One of the primary objects in
Combinatorial Optimization
is the solution space to a system of simultaneous linear equations, which is the object of
linear programming.
If we represented each linear equation as a half-space in some dimension, the solution space to the entire
system would be a
convex polytope,
and since businesses and governments use linear programming to make optimal allocations of scarce resources,
umpteen dimensional polytopes have been all the rage ever since the British government used it to keep afloat
during World War II.
But we are interested in how (geometric) things can be taken apart and put back together.
Algebraic topology is interested in the related
problem of taking apart and gluing together topological objects -- objects that can be continuously distorted
-- and we can borrow some of the topological machinery.
Machinery that probably traces its provenance back to Durer's nets, anyway.
One of the basic and most familiar dissections of a polytope is into cells that make up its interior and
boundary.
For example, the complex (e.g., CW-complex) of a polyhedron is the
partially ordered set (poset)
consisting of its vertices, its edges, its faces, and its interior, ordered by inclusion.
Such a dissection into cells can be conducted on polytopes of arbitrary dimension.
But for our purposes, we may be even more interested in the proposal that we can have arbitrary complexes,
i.e., any poset of cells ordered by inclusion such that every boundary cell of a cell in the complex is
also a cell in the complex.
These complexes are indeed complicated arrays of geometric objects that are organized in space to fit
together, and thus would appear to be typical objects of reticular geometry.
But how to put such complicated complexes together?
In order to do this, we probably have to shock the purists by taking pliers and crowbars to the standard
definitions, but it's a free country and geometers are already taking liberties, so...
While
H. S. M. "Donald" Coxeter's primary concern seems to be more
the shapes of individual things than putting many things together to make complicated things, he did dissect
many polytopes into individual pieces and in the process helped develop much of the machinery available to the
contemporary reticular geometer.
From the point of view of most contemporary geometers and algebraists, what Coxeter created was a very useful
system for representating algebraic objects.
Coxeter's dissection techniques, applied to very nice (highly symmetric)complexes, would produce a
chamber system of cells laid out in space, complete with an adjacency relation.
Meanwhile, mathematicians developed
Group Representation Theory,
whose ultimate program is to take a horribly complicated and therefore unintelligible
group, and find an isomorphic group that was
a nice group of permutations of some nice set -- often, a nice group of linear transformations on a very
nice (perhaps even real or complex) vector space.
One collection results are the
Coxeter Groups,
the peanut butter cup of chamber systems embedded in spaces such that for such a chamber system, there is a
nice group of transformations which, when applied to the space the chamber system is embedded in, permute
orbits of chambers around.
There are all kinds of restrictions, generalizations, and other variations of the above, including
buildings (for which
Jacques Tits won the
Abel Prize, the highest prize in mathematics).
In all this, we can see that geometry is now in service to algebra, as can be seen by the very title of, say, Michael
Davis's book on
The Geometry and Topology of Coxeter Groups.
Whether Coxeter would approve of Geometry being the handmaiden of algebra is another matter.
But of course, geometers could turn the relationship around...
Imagine the plane chopped into eight pieces by four mirrors, from which we get four reflections.
Those four reflections generate a dihedral group (of order eight) of isometries of the plane.
We could regard those eight pieces as eight vertices, with an edge relation representing adjacency,
giving us a cyclic graph.
Then the four reflections generate a dihedral group (of order eight) of automorphisms of that
graph.
With only one orbit of vertex, this Euclidean graph represents a uninodal or vertex
transitive net, with which
Richard James can
model "objective structures".
Image on this website, released to the public domain.
Let me give an example.
A reflection in Euclidean space is an
affine transformation that reflects
points across some line (in two-dimensional space), plane (in three-dimensional space), or other
hyperplane in Euclidean space.
And a Reflection Group is a group generated by
reflections.
In a
Wythoff Construction,
one starts with a
fundamental region
of a reflection group (i.e., a closed polyhedron that intersects each
orbit
of the reflection group at least once while its interior intersects each orbit at most once).
Then one applies the elements of the reflection group to the fundamental region to get copies of it (which
are also fundamental regions), and these copies cover the entire space, although the intersection of any
two copies of this fundamental region intersect (if at all) only within their boundaries.
The result is a
tessellation
of the original space, and this tessellation tells the algebraist something about what the reflection
group looks like.
But one can use the same approach to build, say, a
Euclidean graph
whose vertices represent, say, atoms, and whose edges represent, say, chemical bonds.
Embedding a fragment of a graph inside a putative fundamental region, one obtains a Euclidean graph that
might be an interesting object in itself -- perhaps, as hoped by the creators of the
Symmetry-Constrained Intersite Bonding
Search (SCIBS),
a geometric representation of the atomic structure of a
zeolite crystal.
Traversing graphs -- Euclidean or otherwise -- using transformations that are (when restricted to those graphs)
automorphisms brings us to the group theory that consists of:
Taking a group and labelling a set of generators with symbols, and then
Assigning to each element of that group a list of those symbols -- a
word --
encoding a composition of generators that will produce that element.
Wythoff constructions are only one source of tessellations, and tessellations form a broader class than just
(near) partitions into fundamental regions.
Typically, a
tiling (tessellation)
is a covering of a Euclidean space by finite (bounded) closed sets (called tiles) such that:
Any two tiles are either disjoint or intersect across their boundary, and
There are finitely many congruence classes of tile.
Two of SCIBS's competitors work by enumerating tilings (or more precisely,
Delaney-Dress symbols representing
tilings), and then:
In the case of the
algorithm developed by John Huson,
Olaf Delgado-Friedrichs, et al, interpreting the tiling of 3-space as crystal, or
The most standard classification of tessellations are into the periodic and
aperiodic; a tessellation is periodic if there
is a basis of its underlying vector space such that any translation of the tessellation maps the tessellation onto
itself: it repeats in axial directions.
Periodic tessellations are often treated as models of
crystal structure,
while aperiodic tessellations have been used (depending on regularity properties) to model a range of materials from
quasicrystals to
glasses.
A related structure is the
packing,
in which structures are embedded in space like a tessellation, only they no longer have to cover the space they are
embedded in.
The standard issue is usually making all the objects fit.
Taking a larger perspective beyond assembling solid rigid objects in a vacuum to be connected at particular junctions,
there are several directions in which we can generalize:
Are these objects rigid?
If we permit them to be completely "flexible", we are in a situation more readily dealt with using topology.
However, there may be intermediate situations in which one is dealing with components that are somewhat rigid.
While the
mathematical theories of rigidity tend to
be aimed at a more binary situation -- objects are rigid or they are not -- recalling the practical example of
basket-weaving, the problem of partial rigidity seems a serious one.
Are the junctions rigid?
This brings us to the classical theory of
geometric rigidity,
which certainly goes back to Antiquity (architects have long known about bracing frameworks), and has become a
field of mathematics during the last two centuries.
Do we require contact at junctions, or do objects "fit" in accordance with some kind of action (or junction)
at a distance?
For example, an ionic crystal (like salt) consists of many positively and negatively charged components adhering
to eachother without formal conjugal relationships comparable to covalent bonding (like sugar).
If we want to model components fixed in position by potentials or stranger effects, we need some kind of geometric
representation of the potential emerging from the assembled structure and then acting on that assembled structure.
As an example of the third direction, consider the problem of
protein folding,
in which one not only has a long chain of
amino acids forming a
peptide or
protein, but then that chain wants to fold up so that "distant"
acids on the chain will interact, distorting the structure further.
The question is: what does the resulting structure look like?
This is a computationally difficult problem, and at the moment (practiced) ad hoc human intuition seems to be more
effective than contemporary algorithms: humans can practice their intuition folding proteins at the
fold.it site.
The question is: how do we compose a compact and usable mathematical description of what we see happening when we
play fold.it?
Another direction is the
Chinese box problem: frequently, an intricate object (like
a cathedral) will have intricate architectural elements (like flying buttresses) which may itself have intricate
architectural elements (like gargoyles), and so on.
The most popular mathematical example is the
fractal,
which was introduced in the early Twentieth century by
Helge von Koch as a concrete example of a continuous but nowhere
differentiable function; they were popularized by
Benoit Mandelbrot,
who sold many people, from biologists to cinematic special effects engineers, on the use of iterative processes to
generate complex structures.
In many ways, these fractals seem to be scratching the surface, for they do not seem to display the potential of, say,
Matryoshka dolls to generate objects that vary depending
on the scale.
But the field is still rather new.
And then there is the issue of the effect that such structures have on their environment, e.g., the x-ray diffraction
patterns one obtains when shining radiation of particular wavelengths through them...
Here are some books to start with.
Tilings and Patterns: An Introduction, by Branko Grunbaum and G. C. Shephard.
Still the primary source on tilings.
Groups, Graphs and Trees, by John Meier.
Introduction to geometric group theory on graphs.
Yet the appearance of highly (and often accurately!) articulated structures from the Pyramids to the Pantheon
suggest that either there was an academic tradition of reticular geometry that has been lost (quite possible!),
or that there was a folk geometry practiced by architects and engineers that attracted little attention from
scholars (also quite possible, recalling snide scholarly comments about engineers).
Or perhaps something in between.
And of course, today there is a great deal of interest among computer scientists in
computational geometry,
which may correspond to the interest ancient engineers had in the subject.
Computer scientists are also interested in the artistic and illustrative uses of
computer graphics, which is used widely by
engineers modelling their intended products.
Still, the inattention from mathematicians has been enough to irritate crystallographers into all sorts of
public grumps, like the protests (see the ScienceBase post on
Nature's Missing Crystal --
Found It!)
that met the publication of Toshikazu Sunada's article in the AMS Notices on
Crystals That Nature Might Miss Creating;
the publication of a feature article in a mathematics flagship, a feature that overlooked a stream of literature
in materials science, was treated almost as an offense instead of a goof.
This hypersensitivity by the materials scientists suggests a certain unhappiness with academic mathematicians.
Of course, materials scientists are not the only ones to whine about mathematicians, and in fact this kind of situation
is not even unique to mathematicians.
For example...
The traditional view of paleontology was that it was launched in the Renaissance when Europeans like
Leonardo da Vinci started finding strange fossils -- in Leonardo's case, sea shells on mountainsides.
But recently, folklorist
Adrienne Mayor
has found a lot of evidence of interest in and theorizing about fossils in Antiquity, suggesting that
there was a sort of folk or proto-paleontology back then -- a proto-paleontology that may have been ignored by
respectable biologists in Antiquity.
This kind of situation has recurred many times in the history of science.
Reticular geometry seems to have had a similar experience within the mathematical community itself.
One complicating factor is the peculiar position of geometry in mathematics, science, and the community at
large.
While polyhedra have provided some of the primary symbols of mathematics to the lay public, and while
geometry has long been a foundation of science, it's centrality to mathematics has declined since Newton;
while geometry and number theory were the pillars of mathematics up into the Renaissance, that was not
true afterwards, no matter what Kant said
about space (geometry) and time (number theory), and by the Twentieth century, the pillars of mathematics
were algebra and analysis.
The appearance of non-Euclidean geometries and their
applications to science (especially in the theories of relativity and quantum mechanics) seems to have led to
geometry, as a social phenomenon, being more about space than about constructions in
space.
Of course, artistic movements were inspired or partly inspired by geometry were interested in things in space,
and chemists and materials scientists doggedly concentrated on things in space, so geometric constructions
continued to appear.
Artists like
Maurits Cornelis Escher and mathematicians like
Harold Scott MacDonald Coxeter continued to be
interested in the shapes of things in space.
David Logothetti's cartoon of "Donald" Coxeter exhuming geometry, posted in a PDF by
Steven Cullinane.
Some mathematicians are interested in the shapes of things in space, although much current research
on constructions like
buildings is aimed at finding accessible
representations of groups: so again, much of reticular geometry in mathematics is in service to algebra, and
its service is to make geometric representations of algebraic objects.
This is not unlike the service that reticular geometry provides for
computer graphics and
geometric design(modelling),
but the result is that there is relatively little "pure" reticular geometry.
A lack of pure reticular geometry translates into a lack of pure reticular geometers, but on the other hand,
the fact that reticular geometry pervades several domains outside of mathematics departments was seen at two
conferences.
And it doesn't take much effort to find all kinds of polyhedral and even stranger articulating
structures in graphics scattered about the www.
Meanwhile, scientists and engineers do all kinds of reticular geometry -- without mathematical
supervision.
Just as the High Culture world distinguishes between arts (like painting and sculpture, practiced
by artists) and crafts (like jewelry and apparel, practiced by craftspeople), so there seems to
be a distinction between mathematics produced largely by mathematicians, and ("folk"?) mathematics produced
largely by non-mathematicians.
Much of reticular geometry may fall in the latter category.
I was originally trained in mathematical logic, in particular
finite model theory,
and I also dabbled in combinatorics and applied probability, but after being abducted by chemists,
I started working on the problem of designing crystals that chemists could synthesize.
I have worked on programs that would design crystals, and these led me to the theoretical issues
underlying the programs, and thus directly into reticular geometry.
My programs have found a lot of crystal nets, some already known, some new, some chemically
plausible, some not.
Image is output from my Maple program.
I got started with somewhat non-geometric but still reticular (articulation) issues in DNA computing, but
gradually turned to crystal design using reticular geometry.
During 2007, I worked with Edwin Clark on a heuristic to
generate
crystal nets (which in turn rests on the mathematical
properties of
periodic graphs), and he composed a
program, based on a
heuristic capable of producing nets
from any isomorphism class of uninodal nets.
I have since developed a heuristic capable of producing crystal nets from all isomorphism classes, using
techniques from geometric group theory and linear algebra, although the justification that it works
requires some analysis.
This heuristic is in the same family as the GRINSP and SCIBS heuristics, in that one starts with
a putative fragment of the net and applies a symmetry group to that fragment to get lots of fragments that
together make up the entire net.
In practice, I have a sort of Model A contraption that has given me lots of binodal (two orbits of
vertices) nets of one or two orbits of edges; hopefully, successor programs under development will be
more powerful.
Meanwhile, I am trying to understand reticular geometry as a field in itself, which includes getting an
idea of what it's theoretical structure looks like.
All input on this exploration is welcome (my email address is
[email protected]).
Announcements
Two things:
An extremely hostile Version 0.8 of the Crystal Turtlebug program has been posted at
Sourceforge.
It consists of several Python worksheets, and input is via a Python worksheet.
We are working on improving many aspects of the program, including I/O.
Comments are appreciated. |
Forbidden knowledge who is allowed and what can it do to you?Let him that hath understanding count the number.80,50,40,40,9,5, 7,50,90,90,5,100,100. and this is one of the easiest things i have had to figure out.Do i like it?NO |
Mathletics: How Gamblers, Managers, and Sports Enthusiasts Use Mathematics in Baseball, Basketball, and Football
Mathletics is a remarkably entertaining book that shows readers how to use simple mathematics to analyze a range of statistical and probability-related questions in professional baseball, basketball, and football, and in sports gambling. How does professional baseball evaluate hitters? Is a singles hitter like Wade Boggs more valuable than a power hitter like David Ortiz? Should NFL teams pass or run more often on first downs? Could professional basketball have used statistics to expose the crooked referee Tim Donaghy? Does money buy performance in professional sports?
In Mathletics, Wayne Winston describes the mathematical methods that top coaches and managers use to evaluate players and improve team performance, and gives math enthusiasts the practical tools they need to enhance their understanding and enjoyment of their favorite sports--and maybe even gain the outside edge to winning bets. Mathletics blends fun math problems with sports stories of actual games, teams, and players, along with personal anecdotes from Winston's work as a sports consultant. Winston uses easy-to-read tables and illustrations to illuminate the techniques and ideas he presents, and all the necessary math concepts--such as arithmetic, basic statistics and probability, and Monte Carlo simulations--are fully explained in the examples.
After reading Mathletics, you will understand why baseball teams should almost never bunt, why football overtime systems are unfair, why points, rebounds, and assists aren't enough to determine who's the NBA's best player--and much, much more.
PRINCETON UNIVERSITY PRESS
Chapter One
BASEBALL'S PYTHAGOREAN THEOREM
The more runs a baseball team scores, the more games the team should win. Conversely, the fewer runs a team gives up, the more games the team should win. Bill James, probably the most celebrated advocate of applying mathematics to analysis of Major League Baseball (often called sabermetrics), studied many years of Major League Baseball (MLB) standings and found that the percentage of games won by a baseball team can be well approximated by the formula
Consider a right triangle with a hypotenuse (the longest side) of length c and two other sides of lengths a and b. Recall from high school geometry that the Pythagorean Theorem states that a triangle is a right triangle if and only if [a.sup.2] + [b.sup.2] + [c.sup.2]. For example, a triangle with sides of lengths 3, 4, and 5 is a right triangle because [3.sup.2] + [4.sup.2] + [5.sup.2]. The fact that equation (1) adds up the squares of two numbers led Bill James to call the relationship described in (1) Baseball's Pythagorean Theorem.
Let's define R = runs scored/runs allowed as a team's scoring ratio. If we divide the numerator and denominator of (1) by [(runs allowed).sup.2], then the value of the fraction remains unchanged and we may rewrite (1) as equation (1)'.
For example, the 2006 Detroit Tigers (DET) scored 822 runs and gave up 675 runs. Their scoring ratio was R = 822/675 = 1.218. Their predicted win percentage from Baseball's Pythagorean Theorem was [1.218.sup.2]/[(1.218).sup.2] + 1 = 5.97. The 2006 Tigers actually won a fraction of their games, or 95/162 = .586. Thus (1)' was off by 1.1% in predicting the percentage of games won by the Tigers in 2006.
For each team define error in winning percentage prediction as actual winning percentage minus predicted winning percentage. For example, for the 2006 Arizona Diamondbacks (ARI), error = .469 =.490 = -.021 and for the 2006 Boston Red Sox (BOS), error = .531 = -.497 = .034. A positive error means that the team won more games than predicted while a negative error means the team won fewer games than predicted. Column J in figure 1.1 computes the absolute value of the prediction error for each team. Recall that the absolute value of a number is simply the distance of the number from 0. That is, [absolute value of 5] = [absolute value of -5] = 5. The absolute prediction errors for each team were averaged to obtain a measure of how well the predicted win percentages fit the actual team winning percentages. The average of absolute forecasting errors is called the MAD (Mean Absolute Deviation). For this data set, the predicted winning percentages of the Pythagorean Theorem were off by an average of 2% per team (cell J1).
Instead of blindly assuming winning percentage can be approximated by using the square of the scoring ratio, perhaps we should try a formula to predict winning percentage, such as
[R.sup.exp]/[R.sup.exp] + 1 (2)
If we vary exp (exponent) in (2) we can make (2) better fit the actual dependence of winning percentage on scoring ratio for different sports. For baseball, we will allow exp in (2) to vary between 1 and 3. Of course, exp = 2 reduces to the Pythagorean Theorem.
Figure 1.2 shows how MAD changes as we vary exp between 1 and 3. We see that indeed exp = 1.9 yields the smallest MAD (1.96%). An exp value of 2 is almost as good (MAD of 1.97%), so for simplicity we will stick with Bill James's view that exp = 2. Therefore, exp = 2 (or 1.9) yields the best forecasts if we use an equation of form (2). Of course, there might be another equation that predicts winning percentage better than the Pythagorean Theorem from runs scored and allowed. The Pythagorean Theorem is simple and intuitive, however, and works very well. After all, we are off in predicting team wins by an average of 162 ? .02, which is approximately three wins per team. Therefore, I see no reason to look for a more complicated (albeit slightly more accurate) model.
How Well Does the Pythagorean Theorem Forecast?
To test the utility of the Pythagorean Theorem (or any prediction model), we should check how well it forecasts the future. I compared the Pythagorean Theorem's forecast for each MLB playoff series (1980-2007) against a prediction based just on games won. For each playoff series the Pythagorean method would predict the winner to be the team with the higher scoring ratio, while the "games won" approach simply predicts the winner of a playoff series to be the team that won more games. We found that the Pythagorean approach correctly predicted 57 of 106 playoff series (53.8%) while the "games won" approach correctly predicted the winner of only 50% (50 out of 100) of playoff series. The reader is probably disappointed that even the Pythagorean method only correctly forecasts the outcome of less than 54% of baseball playoff series. I believe that the regular season is a relatively poor predictor of the playoffs in baseball because a team's regular season record depends greatly on the performance of five starting pitchers. During the playoffs teams only use three or four starting pitchers, so much of the regular season data (games involving the fourth and fifth starting pitchers) are not relevant for predicting the outcome of the playoffs.
For anecdotal evidence of how the Pythagorean Theorem forecasts the future performance of a team better than a team's win-loss record, consider the case of the 2005 Washington Nationals. On July 4, 2005, the Nationals were in first place with a record of 50-32. If we extrapolate this winning percentage we would have predicted a final record of 99-63. On July 4, 2005, the Nationals scoring ratio was .991. On July 4, 2005, (1)' would have predicted a final record of 80-82. Sure enough, the poor Nationals finished 81-81.
The Importance of the Pythagorean Theorem
Baseball's Pythagorean Theorem is also important because it allows us to determine how many extra wins (or losses) will result from a trade. Suppose a team has scored 850 runs during a season and has given up 800 runs. Suppose we trade a shortstop (Joe) who "created" 150 runs for a shortstop (Greg) who created 170 runs in the same number of plate appearances. This trade will cause the team (all other things being equal) to score 20 more runs (170 - 150 = 20). Before the trade, R = 850/800 = 1.0625, and we would predict the team to have won 162[(1.0625).sup.2]/1 + [(1.0625).sup.2] = 85.9 games. After the trade, R = 870/800 = 1.0875 and we would predict the team to win 162[(1.0875).sup.2]/1 + [(1.0875).sup.2] = 87.8 games. Therefore, we estimate the trade makes our team 1.9 games better (87.8 - 85.9 = 1.9). In chapter 9, we will see how the Pythagorean Theorem can be used to help determine fair salaries for MLB players.
Football and Basketball "Pythagorean Theorems"
Does the Pythagorean Theorem hold for football and basketball? Daryl Morey, the general manager for the Houston Rockets, has shown that for the NFL, equation (2) with exp = 2.37 gives the most accurate predictions for winning percentage while for the NBA, equation (2) with exp = 13.91 gives the most accurate predictions for winning percentage. Figure 1.3 gives the predicted and actual winning percentages for the NFL for the 2006 season, while figure 1.4 gives the predicted and actual winning percentages for the NBA for the 2006-7 season.
For the 2005-7 NFL seasons, MAD was minimized by exp = 2.7. Exp = 2.7 yielded a MAD of 5.9%, while Morey's exp = 2.37 yielded a MAD of 6.1%. For the 2004-7 NBA seasons, exp = 15.4 best fit actual winning percentages. MAD for these seasons was 3.36% for exp = 15.4 and 3.40% for exp = 13.91. Since Morey's values of exp are very close in accuracy to the values we found from recent seasons we will stick with Morey's values of exp.
These predicted winning percentages are based on regular season data. Therefore, we could look at teams that performed much better than expected during the regular season and predict that "luck would catch up with them." This train of thought would lead us to believe that these teams would perform worse during the playoffs. Note that the Miami Heat and Dallas Mavericks both won about 8% more games than expected during the regular season. Therefore, we would have predicted Miami and Dallas to perform worse during the playoffs than their actual win-loss record indicated. Sure enough, both Dallas and Miami suffered unexpected first-round defeats. Conversely, during the regular season the San Antonio Spurs and Chicago Bulls won around 8% fewer games than the Pythagorean Theorem predicts, indicating that these teams would perform better than expected in the playoffs. Sure enough, the Bulls upset the Heat and gave the Detroit Pistons a tough time. Of course, the Spurs won the 2007 NBA title. In addition, the Pythagorean Theorem had the Spurs as by far the league's best team (78% predicted winning percentage). Note the team that underachieved the most was the Boston Celtics, who won nearly 9% fewer (or 7) games than predicted. Many people suggested the Celtics "tanked" games during the regular season to improve their chances of obtaining potential future superstars such as Greg Oden and Kevin Durant in the 2007 draft lottery. The fact that the Celtics won seven fewer games than expected does not prove this conjecture, but it is certainly consistent with the view that Celtics did not go all out to win every close game.
APPENDIX
Data Tables
The Excel Data Table feature enables us to see how a formula changes as the values of one or two cells in a spreadsheet are modified. This appendix shows how to use a One Way Data Table to determine how the accuracy of (2) for predicting team winning percentage depends on the value of exp. To illustrate, let's show how to use a One Way Data Table to determine how varying exp from 1 to 3 changes the average error in predicting a MLB team's winning percentage (see figure 1.2).
Step 1. We begin by entering the possible values of exp (1, 1.1, ... 3) in the cell range N7:N27. To enter these values, simply enter 1 in N7, 1.1 in N8, and select the cell range N8. Now drag the cross in the lower right-hand corner of N8 down to N27.
Step 2. In cell O6 we enter the formula we want to loop through and calculate for different values of exp by entering the formula = J1.
Step 3. In Excel 2003 or earlier, select Table from the Data Menu. In Excel 2007 select Data Table from the What If portion of the ribbon's Data tab (figure 1-a).
Step 4. Do not select a row input cell but select cell L2 (which contains the value of exp) as the column input cell. After selecting OK we see the results shown in figure 1.2. In effect Excel has placed the values 1, 1.1, ... 3 into cell M2 and computed our MAD for each listed value of exp.
Chapter Two
WHO HAD A BETTER YEAR, NOMAR GARCIAPARRA OR ICHIRO SUZUKI?
The Runs-Created Approach
In 2004 Seattle Mariner outfielder Ichiro Suzuki set the major league record for most hits in a season. In 1997 Boston Red Sox shortstop Nomar Garciaparra had what was considered a good (but not great) year. Their key statistics are presented in table 2.1. (For the sake of simplicity, henceforth Suzuki will be referred to as "Ichiro" or "Ichiro 2004" and Garciaparra will be referred to as "Nomar" or "Nomar 1997.")
Recall that a batter's slugging percentage is Total Bases (TB)/At Bats (AB) where
We see that Ichiro had a higher batting average than Nomar, but because he hit many more doubles, triples, and home runs, Nomar had a much higher slugging percentage. Ichiro walked a few more times than Nomar did. So which player had a better hitting year?
When a batter is hitting, he can cause good things (like hits or walks) to happen or cause bad things (outs) to happen. To compare hitters we must develop a metric that measures how the relative frequency of a batter's good events and bad events influence the number of runs the team scores.
In 1979 Bill James developed the first version of his famous Runs Created Formula in an attempt to compute the number of runs "created" by a hitter during the course of a season. The most easily obtained data we have available to determine how batting events influence Runs Scored are season-long team batting statistics. A sample of this data is shown in figure 2.1.
James realized there should be a way to predict the runs for each team from hits, singles, 2B, 3B, HR, outs, and BB + HBP. Using his great intuition, James came up with the following relatively simple formula.
runs created = (hits + BB + HBP) ? (TB)/(AB + BB + HBP). (1)
As we will soon see, (1) does an amazingly good job of predicting how many runs a team scores in a season from hits, BB, HBP, AB, 2B, 3B, and HR. What is the rationale for (1)? To score runs you need to have runners on base, and then you need to advance them toward home plate: (Hits + Walks + HBP) is basically the number of base runners the team will have in a season. The other part of the equation, TB/(AB + BB + HBP), measures the rate at which runners are advanced per plate appearance. Therefore (1) is multiplying the number of base runners by the rate at which they are advanced. Using the information in figure 2.1 we can compute Runs Created for the 2000 Anaheim Angels.
Actually, the 2000 Anaheim Angels scored 864 runs, so Runs Created overestimated the actual number of runs by around 9%. The file teams.xls calculates Runs Created for each team during the 2000-2006 seasons and compares Runs Created to actual Runs Scored. We find that Runs Created was off by an average of 28 runs per team. Since the average team scored 775 runs, we find an average error of less than 4% when we try to use (1) to predict team Runs Scored. It is amazing that this simple, intuitively appealing formula does such a good job of predicting runs scored by a team. Even though more complex versions of Runs Created more accurately predict actual Runs Scored, the simplicity of (1) has caused this formula to continue to be widely used by the baseball community.
Beware Blind Extrapolation!
The problem with any version of Runs Created is that the formula is based on team statistics. A typical team has a batting average of .265, hits home runs on 3% of all plate appearances, and has a walk or HBP in around 10% of all plate appearances. Contrast these numbers to those of Barry Bonds's great 2004 season in which he had a batting average of .362, hit a HR on 7% of all plate appearances, and received a walk or HBP during approximately 39% of his plate appearances. One of the first ideas taught in business statistics class is the following: do not use a relationship that is fit to a data set to make predictions for data that are very different from the data used to fit the relationship. Following this logic, we should not expect a Runs Created Formula based on team data to accurately predict the runs created by a superstar such as Barry Bonds or by a very poor player. In chapter 4 we will remedy this problem.
Ichiro vs. Nomar
Despite this caveat, let's plunge ahead and use (1) to compare Ichiro Suzuki's 2004 season to Nomar Garciaparra's 1997 season. Let's also compare Runs Created for Barry Bonds's 2004 season to compare his statistics with those of the other two players. (See figure 2.2.)
What People are Saying About This
Nemhauser
I really enjoyed this unique book, as will anyone who is a serious sports fan with some interest in mathematics. Winston is very knowledgeable about baseball, basketball, and football, and about the mathematical techniques needed to analyze a multitude of questions that arise in them. He does a very good job of explaining complex mathematical ideas in a simple way. — George L. Nemhauser, Georgia Institute of Technology
Pete Palmer
People who want the details on the analysis of baseball need to read Mathletics. This book provides the statistics behind Moneyball. — Pete Palmer, coeditor of "The ESPN Baseball Encyclopedia" and "The ESPN Pro Football Encyclopedia"
KC Joyner
Wayne Winston's Mathletics combines rigorous analytical methodologies with a very inquisitive approach. This should be a required starting point for anyone desiring to use mathematics in the world of sports. — KC Joyner, author of "Blindsided: Why the Left Tackle Is Overrated and Other Contrarian Football Thoughts"
Mark Cuban
Winston has an uncanny knack for bringing the game alive through the fascinating mathematical questions he explores. He gets inside professional sports like no other writer I know. Mathletics is like a seat at courtside. — Mark Cuban, owner of the Dallas Mavericks
Michael Huber
Mathletics offers insights into the mathematical analysis of three major sports and sports gambling. The basketball and sports bookies sections are particularly interesting and loaded with in-depth examples and analysis. The author's passion seems to jump right off the page. — Michael Huber, Muhlenberg College
Daryl Morey
Winston has brought together the latest thinking on sports mathematics in one comprehensive place. This volume is perfect for someone seeking a general overview or who wants to dive into advanced thinking on the latest sports-analytics topics. — Daryl Morey, general manager of the Houston Rockets
Editorial Reviews
"."--Booklist
"."--Ken Berger, CBSSports.com
"[A] terrific read for anyone trying to model markets statistically and make trading decisions based on statistical data. . . . Reading Winston's book is a mind-opening experience."--Brenda Jubin, Reading the Markets blog
From the Publisher
.
Booklist
. — Ken Berger
Word search puzzles are far more than an inexpensive source of entertainment. They are also
a fun and challenging way to:• Stay mentally fit• Improve your spelling and vocabulary • Enhance your pattern recognition skillsIn this book, the 101 puzzles ...
Athlon Sports Fantasy Football is your guide to prepare you for a fantasy football draft.
400-plus in-depth player reports, a 280-player big board, 20-round mock draft, cheat sheet for draft day and team-by-team analysis from NFL beat writers around the ...
In a rousing trip through the worlds of Basketball and Football, George Thomas Clark explores
the professional basketball league in Mexico, the Herculean talents of Wilt Chamberlain, the difficulties and humor of attempting to play basketball in middle age, and ...
This volume, the fifth in the Biographical Dictionary of American Sports, provides biographical and bibliographical
information on 620 distinguished American sports personalities. The majority of the entries represent team sports, including baseball, football, and basketball. Other entries treat individual sports, ...
Get ready to know more about football, basketball, and baseball with the first three titles
of the Game-Day Goddess Sports Series. Use your knowledge of sport to advance your business effectiveness, break the ice with new contacts, be accepted as ...
Johnny Rutledge, former NFL athlete, scholar, and coach tells a unique story where fatherhood, football,
and math all meet. Grant learns a valuable lesson about education and how math coincides with football. Math and Football is a fun, page-turning tale ...
The last time a Philadelphia professional sports team won a championship, Ronald Reagan was in
the White House and Return of the Jedi was number one at the box office. No city with all four major sports has gone longer ... |
Friday, July 21, 2006
Is This Really Math? Or Is This Just Crap?
I stumbled upon a website today: . It was a funny name... after all, math + drinking age = João Magueijo?
No, 21 does not refer to the age, but the 21ST CENTURY! Upon this realization, I reminded myself that I should approach this page with an open mind, and crap alarms armed.
The page was impossible to navigate, and unreadable in Firefox because the XML was screwed up. I clicked links and ended up going in circles. Everything is in outline format, which is usually great and easy. This, however, was outline format from hell, because there were ten outlines with no distinct starting position. I had to pick up this thing in pieces... in Internet Explorer...
This is what I managed to get from this mathematical revolution (because of the outlines):
This will change mathematics, immediately.
Everyone should be accquainted with this... because it's better.
It involves substituting common algorithms for mathematical objects.
These mathematical objects... are better.
This page so poorly designed--I'm getting crap vibes from it, and worse. He claims this new method is great; all it seems to be is confusing. Essentially, it's avoiding math, and replacing math with variables that represent math, but really have no practical application. What happens when you integrate one of these objects? Does it do anything? No. You've still gotta figure out the integral.
I'm keeping an open mind...
One of the big claims of this... d00d is that this new math will transcend "AXIOMATIC METHOD." AXIOMATIC is a big word. AXIOMATIC sounds archaic and evil. I had heard the term before but couldn't put my finger on its definition. I checked Wikipedia and found out that "An axiom is a sentence or proposition that is taken for granted as true, and serves as a starting point for deducing other truths. In many usages axiom and postulate are used as synonyms."
For example, this is an axiom, stated by Euclid, a dead Greek guy, "Things which are equal to the same thing are also equal to one another." So, that means that if x=2 and b=2, x=b. Ok, duh. According to this guy, he's throwing AXIOMATICISM AWAY. Let's wait a minute and see if 2 is still equal to 2...
I'm really trying to keep an open mind.
Here are 2 ampersands (&& ). Here are 2 more (&& ). && = && Yea... as best I can tell, 2 still equals 2. The axiom remains in place. The world is saved.
Things descended, soon into even deeper crap--into Crap Lake. I've spent a jolly good bit of time programming; this guys says he can make it easier. (I attempted, at this point, to find the page where I read this on. I can't. It's too big of a goddamn mess to find anything. How did this guy write this much? Why?) He says his form of math uses algorithms as data structures. This makes no sense. If you want to get anything practical out of such a data structure, you would have to recalculate the result over-and-over again, thereby limiting your processing speed, especially for very large or slow algorithms.
Has he written a program before? More than one? Has he ever written one "just because?" I have.
The author of this webpage (whose name I omit, for his own sake) really slid below the "Free Energy Line," where people begin to sound more ignorant than, if you're unfamiliar with them, those total pseudoscientists who claim to pull electricity out of their ass (where they have hidden, a small generator burning bullshit). Pardon the run-on, but to the point, he said this when talking about using his method to create massive libraries of information with his "method:"
"These repositories can also count how often a given person or entity generates new ideas. This can be used as a mean for finding and chosing clever persons for high positions and to make the amounts of job earnings more fair. (I deem that it will decrease World unfairness because the criteria in greater degree than now will be based on inherent objective mind abilities rather than on subjective criteria of education and subjective personal characteristics.)"
Hmm... somebody fucked around in high school and wasn't admitted to college. Oops... but his SAT was so high!
He started an online journal for POST-AXIOMATIC Mathematics. He is the only author published.
He claims that nobody has ever studied formulas, presuming you equate formulas to functions. This guy has. Ooh! and this guy. They invented this branch of mathematics, which deals entirely with functions.
Note: If you don't believe, for some reason, that a formula is a function, take any formula, solve it for the dependent variable, and plug in all values possible for the independent variable. Graph the results. Look! a function!
Open mind... closed.
This guy is painfully, ridiculously wrong. He's using a "theory" that is not actually a "theory" to try to revolutionize mathematics; it doesn't. All it does is navigates around the truly challenging and exciting part of math--to reach the heights of human understanding, wisdom, and ability--at times through the worst misery one can experience (I've been there). Through fancy lettering and jibberish rhetoric, he has attempted to appear intelligent. However, those who can see through his garbage can tell that it is nothing more than weak sauce. Period. |
1
Math Professor: What you're looking at here is widely thought to be the hardest math equation in the world, and has yet to be solved. What we are doing today is far less difficult, and... Tony Stark: Sir? 42.
2 |
7.28.2010
While trying to convince Alz to sacrifice sleep in order to watch Inception with me and Luce this past Saturday morning, I ended up sort of blowing my own mind. This is what happens when I voluntarily bring MATH into a conversation (which is close to never).
We chatted about various works in progress, which brought up my last year's NaNoWriMo story. The story, if you're curious, involves an esteemed old family, whose baggage includes an equally old curse that hasn't become any less deadly with age. It's supposed to sort of be my summer project, but I got distracted (okay, so really this is just my procrastination in action, but I digress). Alz and I then had the following conversation:
KRISPY: Part of what's keeping me from writing it is I still haven't figured out how the problem is solved. Like I know enough of everything leading up to the climax, but nothing afterward.
ALZ: What's after the climax? Falling action!
KRISPY: Well, maybe it's not really the climax. It's like I have from point A to point B, but not-
ALZ: But not B to Z?
KRISPY: Um, no. I was going to say B to C. There's only 3 points on my scale! A, B, and C! The whole alphabet is too complicated!
ALZ: Okay, so you can't get from B to C.
KRISPY: Right. I can get to the point where the big plot twist/problem is revealed and things are going wrong, but I can't resolve it. I even plotted with Luce! I told her everything, and you know how I hate to divulge secrets.
ALZ: And she couldn't help you figure it out?
KRISPY: No! She helped me un-complicate some things, but we got up to that point and both of us were stumped. I don't know how to explain more without spoiling you. It's like we're at the tip of the iceberg and we can't get down.
ALZ: ...
KRISPY: Or like, you know in math, like in Calculus, when you're trying to - wait, I take that back. You didn't take Calculus and I don't remember enough of it to make this analogy work. Um, oh! Okay, this one's a BETTER analogy. It's like in GEOMETRY, you know, when you have to do those proofs. You have the two statements and you have to use theorems and stuff to show how one gets to the other.
ALZ: Yes...
KRISPY: So it's like when you're doing those and you know which theorems to use to start off with. Then you get halfway through the proof and you realize you have no idea what theorem to use next to get to the end! IT'S LIKE THAT.
ALZ: YES! I know exactly what you mean.
KRISPY: I can't believe I just used a math analogy for writing.
ALZ: Yeah, that was pretty good. You should blog it.
KRISPY: I do need a post idea...
ALZ: DO IT! And write your story.
KRISPY: So, you know what's more mind-blowing than math analogies for writing? Inception. You should totally come with.
I still haven't figured out the rest of that geometry WIP proof. Sad times.
5 comments:
I wish I had some time to work on my WiP, but I'm back to revisions. I usually know how things are going to end before I get started, but I still don't know exactly where that story is going yet. It'll be fun when I finally get back to it, though.
Tere- True, of all the math I had to endure, proofs were one of the only things I kind of liked.
I usually don't know the ends of my stories. Meh.
Xixi- Go see it! I didn't use to have a crush on Joseph Gordon Levitt, but I do now. XP It's the suits; whoever dressed him in the movie did an A+++ job. But also the movie is very good with amazing visuals and also a really cool fight scene.
Do not feel like an infant! I know I act like a 5 year old. ;) But srsly, this whole time I thought I was like 2-ish years older, but NO, it is FOUR. |
Books > Pi, the world's most mysterious number September 24, 2006
Pi: A Biography of the World's Most Mysterious Number By Alfred S. Posamentier and Ingmar Lehmann
Prometheus Books
ISBN 1-59102-200-2
AUD$52.95 324 pages
Good old pi. The ratio of the circumference of a circle to its diameter. According to this book, pi's been known, in an approximate form, since about 2000bc. And, of course, because the thing is 'irrational' (a number which can't be expressed as a finite decimal number) we will never have anything but approximations. It's just that nowadays that approximation goes to 1.24 trillion decimal places!
This book has chapters on the history of pi (an exhaustive chronicle of the increasing accuracy of the approximation of pi across many cultures, including Ancient Egyptians, the Babylonians, Ancient Greeks, Chinese andRomans, right up to the present) as well as chapters about the paradoxes, curiosities and the applications of pi. It has proofs and ways of calculating the value of pi. It also touches on some of the greatest mathematical minds because they were fascinated by pi. And to top it all off, Posamentier and Lehmann provide us with a lovely 27 pages filled with nothing more than pi to one hundred thousand decimal places.
This is a book for enthusiasts, trivial pursuit addicts or wannabe nerds. It has too many equations and too much mathematical notation to attract a general lay readership and it's too basic and flippant to interest professional mathematicians. As well, it doesn't come anywhere close to approaching the beauty or scope of books such as Mario Livio's The Golden Ratio, John McLeish's Number, or John Barrow's The Book of Nothing.
I suppose you can learn something from almost anything. From this book, I learned that International Pi Day is March 14, not because this is Einstein's birthday, which it is, but because this date can be written as 3.14, which is pi to two decimal places.
If you like that kind of stuff, you'll find something to enjoy in this book.
Pi chart Pi first became known as such in 1706 when mathematician William Jones used it in his own method of relating the circumference of a circle to its diameter. Why did he opt for Π rather than some other symbol? Phonetics: pi sounds like 'p' (for 'perimeter'). The rough numerical value of pi was known as long ago at 1650bc, as demonstrated on a surviving papyrus scroll |
Quantity
Two basic divisions of quantity, magnitude and multitude, imply the
principal distinction between continuity (continuum ) and
discontinuity . Under
[...More...]
Set (mathematics)
In mathematics , a SET is a collection of distinct objects,
considered as an object in its own right. For example, the numbers 2,
4, and 6 are distinct objects when considered separately, but when
they are considered collectively they form a single set of size three,
written {2,4,6}. The concept of a set is one of the most fundamental
in mathematics. Developed at the end of the 19th century, set theory
is now a ubiquitous part of mathematics, and can be used as a
foundation from which nearly all of mathematics can be derived. In
mathematics education , elementary topics such as Venn diagrams are
taught at a young age, while more advanced concepts are taught as part
of a university degree. The German word Menge, rendered as "set" in English, was coined by
Bernard Bolzano Bernard Bolzano in his work
The Paradoxes of the Infinite [...More...]
Scalar (mathematics)
A SCALAR is an element of a field which is used to define a vector
space . A quantity described by multiple scalars, such as having both
direction and magnitude, is called a vector . In linear algebra , real numbers or other elements of a field are
called SCALARS and relate to vectors in a vector space through the
operation of scalar multiplication , in which a vector can be
multiplied by a number to produce another vector. More generally, a
vector space may be defined by using any field instead of real
numbers, such as complex numbers . Then the scalars of that vector
space will be the elements of the associated field. A scalar product operation – not to be confused with scalar
multiplication – may be defined on a vector space, allowing two
vectors to be multiplied to produce a scalar. A vector space equipped
with a scalar product is called an inner product space . The real component of a quaternion is also called its SCALAR PART
[...More...]
Euclidean Vector
In mathematics , physics , and engineering , a EUCLIDEAN VECTOR
(sometimes called a GEOMETRIC or SPATIAL VECTOR, or—as
here—simply a VECTOR) is a geometric object that has magnitude (or
length ) and direction . Vectors can be added to other vectors
according to vector algebra . A
Euclidean vector Euclidean vector is frequently
represented by a line segment with a definite direction, or
graphically as an arrow, connecting an initial point A with a terminal
point B, and denoted by A B . {displaystyle
{overrightarrow {AB}}.} A vector is what is needed to "carry" the point A to the point B; the
Latin word vector means "carrier". It was first used by 18th century
astronomers investigating planet rotation around the Sun. The
magnitude of the vector is the distance between the two points and the
direction refers to the direction of displacement from A to B
[...More...]
Tensor
In mathematics , TENSORS are geometric objects that describe linear
relations between geometric vectors , scalars , and other tensors.
Elementary examples of such relations include the dot product , the
cross product , and linear maps . Geometric vectors , often used in
physics and engineering applications, and scalars themselves are also
tensors. A more sophisticated example is the
Cauchy stress tensor T,
which takes a direction V as input and produces the stress T(V) on the
surface normal to this vector for output, thus expressing a
relationship between these two vectors, shown in the figure (right). Given a reference basis of vectors, a tensor can be represented as an
organized multidimensional array of numerical values. The order (also
degree or rank) of a tensor is the dimensionality of the array needed
to represent it, or equivalently, the number of indices needed to
label a component of that array
[...More...]
Variable (mathematics)
In elementary mathematics , a VARIABLE is an alphabetic character
representing a number, called the value of the variable, which is
either arbitrary, not fully specified, or unknown. Making algebraic
computations with variables as if they were explicit numbers allows
one to solve a range of problems in a single computation. A typical
example is the quadratic formula , which allows one to solve every
quadratic equation by simply substituting the numeric values of the
coefficients of the given equation to the variables that represent
them. The concept of a variable is also fundamental in calculus .
Typically, a function y = f(x) involves two variables, y and x,
representing respectively the value and the argument of the function.
The term "variable" comes from the fact that, when the argument (also
called the "variable of the function") varies, then the value varies
accordingly
[...More...]
John Tukey
JOHN WILDER TUKEY ForMemRS (/ˈtuːki/ ; June 16, 1915 – July
26, 2000) was an American mathematician best known for development of
the FFT algorithm and box plot . The Tukey range test , the Tukey
lambda distribution , the Tukey test of additivity , and the
Teichmüller–Tukey lemma all bear his name. CONTENTS * 1 Biography * 2 Scientific contributions * 2.1 Statistical practice * 3 Statistical terms
* 4 See also
* 5 Publications
* 6 Notes
* 7 External links BIOGRAPHYTukey was born in
New Bedford, Massachusetts New Bedford, Massachusetts in 1915, and obtained a
B.A. in 1936 and
M.Sc. in 1937, in chemistry, from
Brown University ,
before moving to
Princeton University where he received a
Ph.D. in
mathematics . During
World War II World War II , Tukey worked at the Fire Control Research
Office and collaborated with Samuel Wilks and William Cochran
[...More...]
Counting
COUNTING is the action of finding the number of elements of a finite
set of objects. The traditional way of counting consists of
continually increasing a (mental or spoken) counter by a unit for
every element of the set, in some order, while marking (or displacing)
those elements to avoid visiting the same element more than once,
until no unmarked elements are left; if the counter was set to one
after the first object, the value after visiting the final object
gives the desired number of elements. The related term enumeration
refers to uniquely identifying the elements of a finite
(combinatorial) set or infinite set by assigning a number to each
element.
Counting Counting using tally marks at
Hanakapiai BeachCounting Counting sometimes involves numbers other than one; for example, when
counting money, counting out change, "counting by twos" (2, 4, 6, 8,
10, 12, ...), or "counting by fives" (5, 10, 15, 20, 25, ...)
[...More...]
Observable
In physics , an OBSERVABLE is a dynamic variable that can be
measured. Examples include position and momentum . In systems governed
by classical mechanics , it is a real -valued function on the set of
all possible system states. In quantum physics , it is an operator, or
gauge, where the property of the system state can be determined by
some sequence of physical operations . For example, these operations
might involve submitting the system to various electromagnetic fields
and eventually reading a value. Physically meaningful observables must also satisfy transformation
laws which relate observations performed by different observers in
different frames of reference . These transformation laws are
automorphisms of the state space, that is bijective transformations
which preserve some mathematical property
[...More...]
Gérard Debreu
GéRARD DEBREU (French: ; 4 July 1921 – 31 December 2004) was a
French-born American economist and mathematician . Best known as a
professor of economics at the
University of California, Berkeley University of California, Berkeley ,
where he began work in 1962, he won the 1983 Nobel Memorial Prize in
Economic Sciences . CONTENTS * 1 Biography
* 2 Academic career * 3 Major publications * 3.1 Books
* 3.2 Book chapters
* 3.3 Journal articles * 4 References
* 5 External links BIOGRAPHYHis father was the business partner of his maternal grandfather in
lace manufacturing, a traditional industry in
Calais Calais . Debreu was
orphaned at an early age, as his father committed suicide and his
mother died of natural causes. Prior to the start of
World War II World War II ,
he received his baccalauréat and went to
Ambert to begin preparing
for the entrance examination of a grande école
[...More...]
Infinitesimal
In mathematics , INFINITESIMALS are things so small that there is no
way to measure them. The insight with exploiting infinitesimals was
that entities could still retain certain specific properties, such as
angle or slope , even though these entities were quantitatively small.
The word infinitesimal comes from a 17th-century
Modern Latin Modern Latin coinage
infinitesimus, which originally referred to the "infinite -th " item
in a sequence. Infinitesimals are a basic ingredient in the procedures
of infinitesimal calculus as developed by Leibniz, including the law
of continuity and the transcendental law of homogeneity . In common
speech, an infinitesimal object is an object that is smaller than any
feasible measurement, but not zero in size—or, so small that it
cannot be distinguished from zero by any available means. Hence, when
used as an adjective, "infinitesimal" means "extremely small"
[...More...]
Argument Of A Function
In mathematics , an ARGUMENT of a function is a specific input in the
function, also known as an independent variable . When it is clear
from the context which argument is meant, the argument is often
denoted by the abbreviation arg. A mathematical function has one or more arguments in the form of
independent variables designated in the function's definition, which
can also contain parameters . The independent variables are mentioned
in the list of arguments that the function takes, whereas the
parameters are not. For example, in the logarithmic function f ( x
) = log b ( x ) {displaystyle f(x)=log _{b}(x)} , the base
b {displaystyle b} is considered a parameter. A function that takes a single argument as input (such as f ( x )
= x 2 {displaystyle f(x)=x^{2}} ) is called a unary function .
A function of two or more variables is considered to have a domain
consisting of ordered pairs or tuples of argument values
[...More...]
Calculus
CALCULUS (from
Latin Latin calculus, literally "small pebble used for
counting on an abacus ") is the mathematical study of continuous
change, in the same way that geometry is the study of shape and
algebra is the study of generalizations of arithmetic operations . It
has two major branches, differential calculus (concerning rates of
change and slopes of curves), and integral calculus (concerning
accumulation of quantities and the areas under and between curves);
these two branches are related to each other by the fundamental
theorem of calculus . Both branches make use of the fundamental
notions of convergence of infinite sequences and infinite series to a
well-defined limit . Generally, modern calculus is considered to have
been developed in the 17th century by
Isaac Newton Isaac Newton and Gottfried
Leibniz . Today, calculus has widespread uses in science , engineering
and economics
[...More...]
Density
The DENSITY, or more precisely, the VOLUMETRIC MASS DENSITY, of a
substance is its mass per unit volume . The symbol most often used for
density is ρ (the lower case Greek letter rho ), although the Latin
letter D can also be used. Mathematically, density is defined as mass
divided by volume: = m V , {displaystyle rho ={frac
{m}{V}},} where ρ is the density, m is the mass, and V is the volume. In some
cases (for instance, in the United States oil and gas i |
Finally got a chance to share this site with my younger tonight. This site is fantastic to share with kids – my son enjoyed playing around with the tiling patterns, and it was also really interesting to hear him try to describe what he was seeing.
Here's his initial look at the site:
Here's his reaction and play with the part of the site that allows you to create and manipulate new quadrilaterals:
This is a wonderfully easy site and a really fun idea to play with. I think with older kids it would be nice to see them try to think through why the cyclic quadrilaterals have this hinged tiling property, but I thought that might be a little much for my younger son. We'll do a follow up exploring those ideas soon, though.
This problem (#22 from the 2014 AMC 10a) gave my son some trouble this morning:
We ended up having a nice talk about the problem this morning. To see if the ideas really sunk in, I asked him to talk through the solution tonight, and he did a nice job:
After we finished, I wanted to go back to the 2014 AMC 10 and just happened to notice that google was also showing that Art of Problem Solving had a video about the problem. So, I thought it would be fun to watch Richard Rusczyk's solution. Turned out to be a lucky decision since his solution was totally different than the one we found:
It was neat to see this second solution – I learned a lot about 15-75-90 triangles today!
Moon Duchin gave a talk about math and gerrymandering in San Diego yesterday that generated an enormous amount of excitement. One lucky bit of that excitement for me was that Francesca Bernardi shared the teaching resources from a math and gerrymandering conference in Madison, Wisconsin organized by Moon Duchin and Jordan Ellenberg:
This morning I decided to try out some of those ideas with my kids. The boys are in 6th and 8th grade and really enjoyed working through the project this morning. Overall, my impressions are that:
(i) The math all by itself is both interesting and accessible for middle school and high school kids.
(ii) Working with a larger group would produce some fascinating discussions about the tetris-like shapes involved in this project. For example, what sorts of shapes do kids consider natural and what sorts of shapes seem unnatural when dividing up a square?
(iii) The project is great for showing why gerrymandering is a difficult math problem. I think that students will see quickly that creating 6 "winning" regions out of 10 for a group that has only 40% of the population seems unfair. However, they'll also see quickly that it isn't as easy as they might think for the math to flush out that unfairness.
So, here's how things went with my kids today. I started by trying to give a simple explanation of gerrymandering – a concept that they'd not heard of before:
Now I had them each work on one of the exercises from the materials that Bernardi shared yesterday. In this exercise you start with 10×10 grid that has 40 orange squares and 60 purple squares. The first challenge is to divide the large square into 10 connected regions of 10 small squares each in which exactly 4 regions have majority orange squares. The next challenge is to try for exactly 6 majority orange regions.
Here's how the boys explained their approaches to the two exercises. You'll see that this problem is a great way to get kids to talk and think about some basic ideas in geometry.
Now we moved on to the part of the exercise that tries to use geometric ideas to identify gerrymandering. Again, working through these different math ideas in this part of the exercise is a fantastic exercise for kids.
Before diving into this part of the project I explained three of the geometric ideas just to make sure they boys understood them prior to diving into the calculations:
The boys did their calculating work off camera. I had them pick 3 regions from each of the two shapes and work through 3 of the different metrics.
Here's what my older son had to say after he finished:
And here's what my younger son had to say (if you look really carefully you'll see that he was confused on some of the calculations, but that shows, I think, why this exercise can be a great activity for kids – this could easily be a week long activity in a 6th grade math class):
Wow is this a great project for kids – and we barely scratched the surface!
One surprise for me was that the ideas of "packing and cracking" didn't come up in the conversation with the boys. Maybe looking at the different shapes while simultaneously noting the different colors inside of those shapes is a harder exercise for kids than I guessed. Introducing the "packing and cracking" ideas would make a good follow up project.
Anyway, I think the educational project from Wisconsin's math and gerrymandering conference is absolutely fantastic. Huge thanks to Francesca Bernardi for sharing these resources. The exercises and ideas will make a great addition to just about any middle school or high school math class – I hope they are shared widely!
If you are looking for additional resources, here a few that I've found to be helpful from the last year:
I thought that the boys would love reading the book and asked them to each read it twice prior to today's project. Here are some of the things that they thought were interesting (ugh, sorry for the focus problems . . . .) :
The first thing the boys wanted to talk about was the "smallest" infinity -> . Here we talked about that infinity and other sets of integers that were the same size.
Next we moved on to talk about the rational numbers – we had a good time talking through the argument that the "size" of the rational numbers was the same as the positive integers.
This argument is represented in the book by a painting of a shark!
Now my older son wanted to talk about Cantor's diagonal argument. He was a little confused about the arguments presented in the book, but we (hopefully) got things straightened out. I think this shows kids can find ideas about infinity to be really interesting.
Finally, we wrapped up by talking about the implications of the infinity of binary strings being larger than the infinity of counting numbers.
Definitely a fun project. I love the content of the book and so do the kids. The only problem is that the quality of the binding is awful and although we've only had the book for a few days, it is falling to pieces. Boo 😦
My son was working on a few old AMC 10 problems yesterday and problem 17 from the 2016 AMC 10a gave him some trouble:
I thought this would be a nice problem to go through with him. We started by talking through the problem to make sure that he understood it:
In the last video he had the idea to check the cases with 10 and 15 balls in the bucket, so we went through those cases:
Now we tried to figure out what was happening. He was having some difficulty seeing the pattern, so I spent this video trying to help him see the pattern. The trouble for me was that the pattern was 0, 1, 2, . . ., so it was hard to find a good hint.
Finally he worked through the algebraic expression he found in the last video:
This isn't one of the "wow, this is a great problem" AMC problems, but I still like it. To solve it you have to bring in a few different ideas, and combining those different ideas is what seemed to give my son some trouble. Hopefully going through this problem was valuable for him.
probabilities
I'd forgotten about that project, but when I mentioned to my younger son that we'd be looking at Markov chains today he told me he already knew about them!
So, I started today by having the boys watch the PBS Infinite Series video again. Here's what they thought:
Next I introduced the "COVFEFE" problem. I was really happy how quickly the boys were able to pick up on how Markov chains could be used to solve this problem.
Next we looked at Nassim Taleb's Mathematica code – that code is so clear that the problem becomes instantly accessible to kids, which is pretty amazing.
Finally, since things were going so well this morning, I introduced the word that we'll study tomorrow -> ABRACADABRA. The kids were able to see why the transitions in this word were different. I'm excited to see how they think through the "ABRACADABRA" problem tomorrow!
The math behind this problem really was the most interesting math that I learned in 2017. It is really important math, too, and I'm excited that the Mathematica code makes some of the ideas accessible to kids. This was a fun one!
This is probably my favorite 3d printing project that we've done on our own. I didn't do a specific project with the boys using the shape because it is really fragile (in fact, I have 3 other broken ones . . . ).
The problem is -> can you cut a hole in a cube large enough so that you can pass another cube of the same size through the first cube?
An old project where we talk about the problem (without 3d printing) is here:
Here's a really fun shape to play with – the rattleback. It wants to rotate one way, but not the other way. There's very little indication when you look at it that it would have such an odd property:
(10) James Tanton's tetrahedron problem
This one has a special place in my heart because it was one of the first times we used 3d printing to solve a "new to us" problem. I loved how these shapes came together. The problem involved understanding the locus of points that were 1 unit away from a tetrahedron:
Another one of my all time favorites projects came from Laura Taalman. Right after the discovery of a 15th type of pentagon that tiles the plane, Taalman created 3d print models of all 15 of the pentagons so that anyone could explore this new discovery: |
It's the latest straw in a mounting pile of evidence that subtle environmental factors can have a profound influence on how people think.
And, of course, the researchers often appear to forget that people are individuals, with individual differences. I know from my own work with clients that different people represent numbers differently, though with similarities across cultures.
So there's an experiment here, a bit like the pronouns exercise I showed on YouTube last week.
When you think of the number 1, where do you think of it? (By the way, does it have a size or a shape?)
When you think of the number 6, where is that?
When you think of the number 100, where is that?
And is there a relationship between those locations?
Please post your answers in the "comments" below!
(By the way, if you're thinking of posting a comment for the first time, don't be concerned about the fact that comments need to be "approved". It's just to block spammers, not to stifle discussion.)
6 replies to "How People Think About Numbers"
I have a hard time connecting w/numbers the way I do w/words that to me piece together in pictures, feelings, and moods etc. With words, I can create stories or acronyms– essentially stories– that have that stickiness factor. For example, if I parked my car in level p2A of the garage, I'll remember that by making up, say, "Peggy 2 arms" (that's my little sis, and this is actually a reminder I used last wk!). It's silly, given I suspect most humans born w/their expected sets of all appendages have 2 arms, but things and people, and even the silliness factor– life, relevance– etch memory. I don't really know what numbers "feel" like. They're abstract figures to me, which makes me wonder how they feel and look like for people who have mastery w/advanced math etc. Whereas I have such difficulty even trying to juggle a few double-digit ones for simple addition without pen and paper handy. However, attempting your above survey of numbers, the single digits, I think, "feel" closer. e.g., taking a stab, I'd say 1 hovers in my head, 6 is in my chest, and 100, teetering on the high end, floats maybe above my head to the side, can't decide whether left or right… out of reach? Although, 100 being a round, even number ending w/0's does make it somewhat accessible… just thinking out loud.
~ Anna (twit: @shummyShummy)
Meg
11/11/2011
Thinking about this, I realize I don't really have sizes, shapes or positions for them but more relationships among them. For instance, 1 and 2 are friends, but 1 doesn't like 3. 5 and 8 are often together but don't really get along. 6 and 7 are best friends. 4 is female, 5 is male. 9 is a loner. Zero is empty, bland. I recall thinking this way since I was a child. Letters don't have anything like those characteristics, though.
1 is small and close by- left and down a bit and faded..6 is larger, black and up to the right a bit. 100 is far away, grey and like down a tunnel. It makes me wonder about how easy or difficult that makes it to eg generate £100, contact 100 people on my list, interest 100 people in my services. I also wonder if 1/6/100 will be in different places/spaces tomorrow. I think I will play with 100 – I have just made it gold, bigger that 1 and 6 and brought it in front of me..6 is not very pleased withthis change ..1 is diffident…
Stewart Mason
12/11/2011
I must be a little odd, because my numbers are uniform in boldness and size, and are arranged in a line across like a vintage car speedometer! (vauxhall viva?) oh dear…. |
Category Archives: math fact tricks |
Crocheted chaos
December 16, 2004
The famous Lorenz equations that describe the nature of chaotic systems – such as the weather – have been turned into a beautiful real-life object, by crocheting computer-generated instructions.
It is not often that a serious mathematics journal contains a crochet pattern, but the current issue of the Mathematical Intelligencer has instructions on how to crochet your very own model of chaos. It looks like a large Christmas decoration.
Dr Hinke Osinga and Professor Bernd Krauskopf, both in Bristol University's Department of Engineering Mathematics, have turned the famous Lorenz equations that describe the nature of chaotic systems – such as the weather or a turbulent river – into a beautiful real-life object, by crocheting computer-generated instructions.
Dr Osinga explained: "Imagine a leaf floating in a turbulent river and consider how it passes either to the left or to the right around a rock somewhere downstream. Those special leaves that end up clinging to the rock must have followed a very unique path in the water. Each stitch in the crochet pattern represents a single point (a leaf) that ends up at the rock."
Together all the points (stitches) define a complicated surface, according to the Lorenz equations. Osinga and Krauskopf have developed a method to describe such surfaces using a computer. After months of staring at animations on a screen, they suddenly realised that in fact their computations had naturally generated crochet instructions.
Osinga, who learnt to crochet at the age of seven, was ready for the challenge. "The computer-generated crochet instructions were remarkable. Simply by looking at the real-life surface I would never have designed it the way the computer did. After all those months of trying to create it on screen, it was fascinating to see the surface grow under my own hands," she said. "And it was truly amazing to see a floppy object fall into its desired shape when it was mounted with steel wire," Krauskopf added.
The final result consisted of 25,511 crochet stitches and took Osinga about 85 hours to complete. It now hangs in their house as a Christmas decoration.
But this wasn't just done for fun. Osinga and Krauskopf's work gives much-needed insight into how chaos arises and is organised in systems as diverse as chemical reactions, biological networks and even your kitchen mixer. Their crocheted model, called the 'Lorenz manifold', is a very helpful tool for understanding and explaining the dynamics of the Lorenz system.
If you would like to crochet your own Lorenz manifold in time for Christmas, the pattern and mounting instructions are available online here |
Desert Diary
Mathematics/Easy Math
One of the roles of science and mathematics is to determine
relationships set by natural law. Confusion between such relationships and matters
unbound by such restrictions affects politics and people's vision of the natural
world.
A hopefully apocryphal story emphasizes the point. A state legislator
wanted to increase the efficiency of engineering works and noted that many projects
required calculation of the relationship between a circle and its circumference. Being
told that the value of pi was necessary, he immediately realized that its cumbersome
value of 3.1416 slowed things down. The result? A law stating that henceforth, the
value of pi would be set at the easily usable figure of 3.0.
This, of course, couldn't really change the relationship. On the
other hand, changing the length of a human-mandated mile to exactly 5000 feet, although
raising economic havoc, would violate no natural relationships. Do you suppose the
Texas legislature would help out school children by decreeing that the angles of a
triangle will add up to the nice, round number of 200 rather than the clumsy 180
degrees?
Listen to the Audio (mp3
format) as recorded by KTEP, Public Radio for the Southwest. |
Wednesday, March 22, 2006
Is our conventional decimal number system perfect? Learn Octomatics!: "what do you think: why do we have the decimal system in our western world? because of our 10 fingers? why do we have 7 days a week? why are 60 seconds 1 minute and 60 minutes 1 hour? why do we have 24 hours a day? and 31 or 30 days a month? do you think thats a really good solution? well, here is another one: ...welcome to octomatics |
Topic:math
Learn how to calculate the day of the week for any date you can think of with this impressive mental trick and some practice. In this It's Okay to Be Smart video, Universal Calendar Puzzle, Joe Hanson demonstrates theWhen sharing cake between two people, the envy-free recommendation is for one person to cut the cake, making the two pieces as equal as possible, while the other person picks their preferred piece. With this agreementArtist, designer, inventor, and Stanford professor John Edmark creates sculptures that are driven by precise mathematics, but his interest in spiral geometries is driven by something more enigmatic... "a search for unIn 2016, bricklayers in Teralba, NSW Australia filmed their domino-style technique for capping a concrete brick wall. The video went viral thanks to the surprising second part of the chain reaction: After the bricks f |
Of Dimensions' Dissimilarly Finite Distances -- Of One, Then Three, Then Two
Between The Zeros And The Zeronesses
In All Aspectlessness Of Anything Are Aspects Maybe As Mathematical
[Whoops, I already had an image
with a name somewhat similar to
"Between The Zeros And The
Zeronesses" in my previous post.]
--------------------------------
--------------------------------
Anagrams:
(14 -- fourteen.)
This hourglass exists.
=
A helix's sight rusts so.
---
This pearl became this:
=
Hemispherical, at best.
---
Fewer loops are circled
less so than else is pi.
=
As ellipses and eclipses
were of their colors.
---
This yet created this
repetition yet inside this.
=
That periodicity is the
entirety then is its seed.
---
Space is oval;
or a sphere's tangent
is parallel.
=
As all overlapping spectra
are as those lines.
---
Math's abacus is of
these metaphors.
=
For, this sum so became
as that shape.
---
All vectors are shapes.
=
These scalars overlap.
---
As yellowish-reds
with orange:
=
A shadow's silly
green or white.
---
All is as the dire cone:
=
Its nodes are helical.
---
Utopian entirety:
=
Upon it, yet near it.
---
This rotation was less so
cosmically eternal here.
=
It is all as most.
As those wheels are
only incorrect.
---
Therefore, as in
a clearest lens,..
=
All these refractions
are seen.
[^I must have posted
something very close
to this anagram before,
maybe even multiple
times already.]
---
Periodicity:
=
Tripe-idiocy.
---
Index:
=
Nixed.
--------------------------------
--------------------------------
Perfect repetition without
any variation or deviation:
'Pure-iodicity'.
--------------------------------
What do friends call
nameless people by?
Their 'nix-names'!..
--------------------------------
The principle of cookie-entropy:
It becomes what it..
be-crumbs..
(And it cannot ever
be de-crumbled.)
--------------------------------
--------------------------------
I was wondering the other day..
"Hmm,..
Given the post-death decay of
dead people's teeth and gums,
and with the increased growth
of bacteria in their mouths due
to their body's disintegration,
and with their (obvious) lack
of tooth-brushing since dying,
...
I wonder then, do dead people
thus have bad-breath?"..
Then I soon realized,..
"Oh,.. No, no they do not."..
--------------------------------
Love the phrase which is sometimes
used in news-reports when a dead
body has been found and foul-play
is suspected by the investigators:
It is.. "a suspicious death."
I get this image:..
Yes,
hooded Death, with his sickle,
is so very very paranoid and
suspicious of everybody..
Everybody is out to get HIM, see,
and end all death with their
medical advancements and whatnot.
So he trusts nobody.
Yes,.. a very suspicious Death..
--------------------------------
Ironically,
one's 'moral-failings'
can too often lead to
one's 'life-successes'.
(So life is immoral, then?
And thus, the MORAL thing to
do is to try to end all life
and lives?..
See, irony is fun!..)
--------------------------------
--------------------------------
Peter The Choker
is a cheater at poker.
He's a choker of the pips
and of the poker chips.
As he does the jacks of poker,
he also packs a joker.
And when he plays strip-poker,
he's a pip-stroker.
But when he plays his poker,
what's that he sips?
It's what he uses as a soaker
of both himself and those pips.
Since the words "calculus" and
"calculate" etymologically both
come from the old word for small
stone (which is also related to
the word "calcium") [because of
the stones moved within abaci
to calculate sums long ago],...
then maybe, to be more logical,
the phrase "number-crunching"
should have instead been
"number-crushing"?
--------------------------------
--------------------------------
The term "dial-up internet"
I have got to love, because
it literally it implying the
phone-connection is via a
phone with an olde-timey
rotary-dial.
Better yet, why not.. this?..
The internet-user tries to
log-on (analog-on) by, first,
quickly turning the crank on
his telephone-device, and then
yells into it, "Hello, operator!
Please get me the [so-and-so]
ISP at the [such-and-such]
telephone exchange!"
Operator:
"Please [crackle, crackle], sir,
would you repeat that!?"
Then, finally with any luck,
the operator plugs the caller's
phono-jack into the ISP's input
on the big board before her;
and then and only then
does all become well.
("But you trendy kids just had
to go and get your newfangled
push-button phones, with their
beeps and their tones and
their spiffy buttons and
whatnots.")
--------------------------------
I don't do much social-network
crap online because.. (rather
than vice-versa) I prefer the
life I live to be more so real..
A pearl lodged within
a venus-flytrap (as if,
but instead of, within
an oyster).
(A Venus persontrap?)
----
So,..
if Venus is on the half-shell,
as in that one famous painting,
but Mars is standing on another
half-shell, then which of these
half-shells is it that Venus
is standing upon?
The better-half, of course!
--------------------------------
--------------------------------
A chamber-pot is a..
nocturnal urinal.
--------------------------------
I may have seen this somewhere
already, but a good nickname,
maybe:
"Kraken-On-Crack".
"I be attackin'..
like a kraken
-(on-crack)."
--------------------------------
--------------------------------
There are only two types
of people in the world.
(And here I am..
stereo-typing.. them.)
The winners..
and the whiners.
Guess which side the good people
are almost always on, though!..
(But the bad people are
often of both types!)
--------------------------------
[Warning: Politics!
Warning:
Next three items may offend.]
America today: Where the
prisons are often more so privately run..
than how privately the lives
of any free Americans are lived.
[More irony:
I am betting that there is
not much 'privacy' for any
prisoners in those so-called
"private-prisons".]
--------------------------------
Hey, I had a fun idea for an
email/social-media provider:
"Blabsurdness.com".
(^Unless the name already exists,
which it likely already does..
or surely will soon.)
Here are some screen-shots*
from their non-real promotion
(which exists only as a warning,
not as a handbook, as they say):
*[Bonus anagram:
Screen-shot:
=
Short scene.]
(Some polychromatism has been very
necessary for this blog, ever since
I had to stop making computer art
a half-year or so ago.
It is ironic that all this color
here is for a bunch of text,
however.)
--------------------------------
Speaking of
schizonoid-paraphrenia...
Science vs Religion:
BOTH Science and
Religion are lies.
But Science is as the liar of the
famous "Liar's Paradox", claiming
all it has ever claimed (including
anything regarding its nature as
such a liar) is a lie.
However, Religion is as a liar
who claims to always be telling
the truth. Not always is religion
credible, therefore, but at least
it has been consistently logical,
if mostly only so on a..
higher level.
Science created the Paradox
which equals our Universe
(by both denying the universe
and also knowing it is so).
And that Paradox then (thus?)
created Religion.
--------------------------------
One last thing today --
about circles.. again:
Some people who do math with
angles and circles use degrees
(360 per complete turn), while
some prefer to use radians
(2*pi per complete turn).
But I would like to see a world
where more often the unit of an
angle's measure was the fraction
the angle is of one complete
rotation around the circle.
So, 1 'circumferian' =
360 degrees,
which = 2*pi radians.
(A right angle would be of .25 or
1/4 circumferian, for example.
And the angles of an equilateral
triangle would each be of
1/6 a circumferian.)
There is a direct linear relation
between these 3 types of units,
so the conversions would be super
easy.
[Whoops, I see now that this unit
is also called a "full angle" or
"perigon". But I really like
my term, "circumferian", for it.]
-------------------------------- |
Logarithms and their use in the real world
Hello, I have been studying Logarithms in University. I understand it's how many of ONE number to get another number, and I see how it is rearranged to find these "missing" links. But maybe I am overlooking something, but I don't quite see the bigger picture here with how to use Logarithms. How can they be applied to something in real life? If someone could present a problem and then a solution to how logarithms could be used in real life that would be fantastic. |
goodreading 19 up close And yet at the heart of modern physics is a language most of us still find boring, baffling or terrifying: mathematics. I spoke to three writers and mathematicians – Margaret Wertheim, Robyn Arianrhod and Clio Cresswell – about their books and their passion for maths. Margaret Wertheim lives in Los Angeles, where she recently established the Institute for Figuring ( to showcase the 'aesthetic and enchanted' constructs of science, mathematics and technology. Her bestselling book Pythagoras'Trousers: God, physics and the gender wars (1996) is a radical rereading of the his- tory of physics, maths and religion. Her second book, The Pearly Gates of Cyberspace:A history of space from Dante to the Internet (1999), tracks Western conceptions of space, both physical and spiritual, from medieval times. In the opening paragraph of Pythagoras'Trousers, Wertheim refers to a mystical experience she had in a maths class aged ten. This is so tantalising that my first question is about her early expe- rience of maths. One day when she was in Grade 6 her teacher, Mr Marshall, gave a lesson on circles. He told his students about 'a number hidden in the circle which is the secret of its properties', then conducted some exercises so they could discover pi for them- selves. Some did, and Wertheim was one of them. 'It was an extraordinary experience,' she says in her rich, earthy voice over the phone. 'I remember being struck forcefully by what we call Platonism (the belief that there is a transcendent math- ematical reality beyond the physical world) – that hidden in the physical circles which we see manifested in physical objects around us, there was this transcendent mathematical entity called pi.' When she left school Wertheim completed degrees in phys- ics (University of Queensland), and maths and computing (University of Sydney). But she began to feel she was living one life at university and another life beyond it, and these two lives became increasingly difficult to reconcile. So after university Wertheim worked as an assistant film editor for a year. Then she read Chaos by James Gleick, which inspired her to become a science writer. Her articles have since appeared in a range of publications, from The NewYork Times to Vogue.When Wertheim moved to LA – 'to be with the man who eventually became my husband' – she decided to write the accessible book on physics she'd been planning to write for her friends. Four years later she completed Pythagoras'Trousers, a very different book from the one she had set out to write. Like many people,Wertheim had believed that science and religion were ancient adversar- ies. But quite unexpectedly her research uncovered a Western mystical tradition that dated back to Pythagoras, a fusion of religion and mathematically-based science that proved her belief to be wrong – and Wertheim found it had been incorporated into Christianity during the Middle Ages and sub- sequently 'woven into Christian thinking'. According to Wertheim, this is why physicists like Hawking can talk so freely about the mind of God. As she points out, 'when scientists talk about God, a lot of people who wouldn't dream of hearing a priest talk about God will pay attention.Why? Because I think our culture has been very receptive to the conception that the mathematical relations in the world around us are transcendent, divine, eternal truths.' Wertheim's discovery of this connection between maths, physics and God led her to another unexpected insight: that women's absence from physics has been profoundly shaped by their exclusion from the Church. 'I stumbled across something that I think was a real insight into why it has been so difficult for women to break into this field,' she says. The role of the imagination in theoretical physics is the sub- ject of Wertheim's next book. She believes Hawking's brilliant imagination is what has made him 'the most famous living scien- tist on the planet'. As she says, the imaginative universe Hawking presents 'gives us the power to dream' – and physics has become 'in some sense a new form of fiction.Through the language called mathematics we bring fabulous worlds into being.' Which is just how Robyn Arianrhod sees mathematics – as an 'amazing and elegant' language. Arianrhod's book Einstein's Heroes: imagining the world through the language of mathematics was published in 2003. Einstein's Heroes tells the story of maths through the life and revolutionary work of the 19th-century mathematical physicist Robert Clerk Maxwell, a legendary mathematician. Also present in this comprehensive book are the two men whose work made Maxwell's discovery possible: Michael Faraday and Isaac Newton. And woven throughout is a lucid history of maths from ancient to modern times. Arianrhod and I met at the Sydney Writers' Festival to talk about her work. Petite and fair, Arianrhod is as dreamy-faced as any poet or novelist.Which is appropriate, because one of the most striking things about Einstein's Heroes is the consider- able novelistic talent of its author. 'I've always loved writing and language,' Arianrhod says. At school, she dreamt of writing like Grahame Greene or Tolstoy. 'Then in about Year 11, when I first encountered mathematical proof in the algebraic, linguistic sense, I was entranced.' Robyn Arianrhod |
If your math teacher told you that mathematics is everywhere, believe him. Almost all the things that we see around (even things that we do not see) are related to mathematics — even potato chips. Yes, even potato chips.
Some potato chips, particularly Pringles (I hope they give me 500 bucks for this), are in a shape of a saddle. In mathematics a saddle-shaped graph is called a hyperbolic paraboloid (see left figure).
A hyperbolic paraboloid quadratic and doubly ruled surface given by the Cartesian equation . Now, whatever that means will be discussed when you take your analytic geometry course.
For now, let's be happy that we know that even potato chips can be modeled by graphs. 🙂
3 thoughts on "Potato chips and mathematics"
Fantastic! As a math professor and textbook author, I love finding math in everyday life. Question, what program or plugin did you use to add the equations to your post? I've been looking for one to use on my blog but haven't had any luck with the ones I've tried so far.
I am using Latex, but it's not a plugin since it is integrated to WordPress.com (I am still using WordPress.com). If you are using WordPress.org, there is a Latex plugin, but I haven't tried it yet. If you are using blogger, then this post might help you. |
zero
Zero is the integer, denoted 0, which, when used
as a counting number, indicates that no objects are present. It is the only
integer that is neither negative nor positive: it is smaller than any positive
number but larger than any finite negative number. It obeys
x ± 0 = x x × 0 = 0
0/x = 0 x0 = 1.
Division by zero is an undefined operation. Zero may be regarded as the
identity element for addition in the field of real numbers.
Zero is both a number and a numeral.
The number zero is the size of the empty set but it is not the empty set itself, nor is it the same thing as nothing.
The numeral or digit zero is used in positional number
systems, where the position of a digit signifies its value, with successive
positions having higher values, and the digit zero is used to skip a position.
Zero in history
The earliest roots of the numeral zero stretch back 5,000 years to the Sumerians
in Mesopotamia, who inserted a slanted double wedge between cuneiform characters
for numbers, written positionally, to indicate a number's absence. The symbol
changed over time as positional notation made its way to India, via the
Greeks (in whose own culture zero made a late and only occasional appearance).
Our word "zero" derives from the Hindi sunya for "void" or "emptiness,"
through the Arabic sifr, (which also gives us "cipher") and the Italian zevero. As a number in its own right, aside from it use as a position
marker, zero took a much longer time to become established, and even now
is not equal in status to other numbers: division by zero is not allowed. |
Written for: Imaginary Gardens With Real Toads: "Let's Count on Our Fingers and Toes" Do you have a number that is special to you? Perhaps you would like to learn the significance of your Birthday Number. Approach this challenge from any angle that inspires.
Seven is an odd number with an odd number of letters. When I write the number, I slant the stem, like an uneven hem. Rome boasts of seven hills, continents number seven, and in the seventh month of the year, my father was born. I am happy gazing at a rainbow's seven colors.
I must add, I do not gamble, or play number seven in any game of chance. Also, I have never liked 7-Up. Still, seven is an odd number, and I am an odd person |
Integer
Integer
The integers (from the Latin integer, literally "untouched," hence "whole": the word entire comes from the same origin, but via French) are formed by the natural numbers (including 0) (0, 1, 2, 3, ...) together with the negatives of the non-zero naturalnumbers (−1, −2, −3, ...). Viewed as a subset of the real numbers, they are numbers that can be written without a fractional or decimal component, and fall within the set {..., −2, −1, 0, 1, 2, ...}. For example, 21, 4, and −2048 are integers; 9.75, 5½, and √2 are not integers.
The set of all integers is often denoted by a boldface Z (or blackboard bold, Unicode U+2124 ℤ), which stands for Zahlen (German for numbers, pronounced ).
The integers (with addition as operation) form the smallest group containing the additive monoid of the natural numbers. Like the natural numbers, the integers form a countably infinite set.
In algebraic number theory, these commonly understood integers, embedded in the field of rational numbers, are referred to as rational integers to distinguish them from the more broadly defined algebraic integers.
Integer - Cardinality ... The cardinality of the set of integers is equal to (aleph-null) ... This is readily demonstrated by the construction of a bijection, that is, a function that is injective and surjective from Z to N ...
FRACTRAN ... list of positive fractions together with an initial positive integer input n ... The program is run by updating the integer n as follows for the first fraction f in the list for which nf is an integer, replace n by nf repeat this rule until ... Starting with n=2, this FRACTRAN program generates the following sequence of integers 2, 15, 825, 725, 1925, 2275, 425, 390, 330, 290, 770.. ...
Second Moment Method - First Moment Method ... method is a simple application of Markov's inequality for integer-valued variables ... For a non-negative, integer-valued random variable X, we may want to prove that X = 0 with high probability ... P(X = 0), we first note that since X takes only integer values, P(X > 0) = P(X ≥ 1) ... |
Sunday, March 1, 2015
Found a book
Today I found a book on Amazon that was on sale called the" Joy of X, the guided tour of math from one to infinity"by Steven Storgatz for $2.99. The description indicates the author goes through and shows real world applications of mathematics connecting math to people's lives. It says he explains why you need to turn your mattress on a regular basis or how Google searches the net. I am going to read this and see if I can use some of the material in my math class. I have students who ask me when are we going to use this and i don't have any answers for most of the math that I teach. I would like to be able to say something like, you will use linear equations to help you decide the best price for your book that you are selling on Amazon, or we can determine the best speed on your snowmachine for fuel consumption. This is actually one of my goals in math is to provide these examples my students can relate to because too many of the examples in the math book almost feel contrived.
Back to flipping your mattress to get maximum wear, I am willing to wager that most of my students do not know they are supposed to do that. I know about it but the mattress I have is one with extra padding on one side and I can flip it over and over. I can only rotate it. My mother used to have us do it but I never understood why and I certainly didn't realize there was math associated with it. I thought it was something my mother did because her mother did it |
Copyright, 1936, by Ed L
Comments
Tau, ed uses it lots. If we think ed was a mason, than the triple tau is a knights templar symbol... Royal Arch Interesting because on page 10 of MC, ed uses triple T's or tau's, to start the three paragraphs. Ed does this, as we both agree, GME is an example, and I believe I mentioned YHWH, err YIWI, on the back cover of MVAL, but for good measure. Edm does it all through his chapter 8, starting with SW S SILHOUETTES HI and Chapter 9, LIT I SHY o SW, or mirrored of the mason's NE, ed's tower's cornerstone S or 19, Tau is 19th greek letter, which I showed could be 3, and the masons triple tau is in an equilateral triangle. There's more to this, hebrew actually, but I still dont follow it well, so I'll say that prime numbers go with hebrew here too. Silhouettes seems to go back to the pics in ABIEH, as they are all silhouetted, but there is that triangle with the 4 in it on the cover that seems to be HI, and silhouetted... doesn't all tie together exactly, but worth giving a think. Speaking of edm, ever notice his self publishing title Passels Information Network Easy to miss it... Pass Els Information Net Work When edm says net... he says it a few times, in his intro, and on his last page of his hidden pages. Edl mentions 'net' too usually as mag- net but it's there, he also mentions spider webs, and spiders are the eternal weavers, and net makers that inspired us... since we are still on greek, maybe more can be found in the story of Athena and Arachne? But I like to think edl and edm are trying to get us to see more along the infinite, such as in Indra's Net. Not sure, food for thought.
The alpha and omega The Greek alpha corresponds to Hebrew aleph The Greek omega(last letter of the alphabet) corresponds to the Tav (last letter of Hebrew alphabet) in English A - T in his " AT WORK " picture.. several layers of meaning. Truth in being/ truth of knowing
Passels - Els Pass? Is EDM giving a pass for access? Passels being a large group or net.. grouping.. of numbers, letters? perhaps hinting at a large numerical els technique
------------
AT is aleph and Tav in Hebrew, I'm not experienced enough yet but these two together are important in older.hebrew writings
Time is precious. Edm says it several times. He also says edl's messages are tailored to the background and knowledge of the reader. How is that? It's on page 15, can't miss it. Edm says, How would you pre-select into who's hand it would be placed? According to my friend Stephen Hawking, time and space are measured in light and gravity. If edl could manipulate gravity, could he manipulate time? The bible starts with Light, and edl's MC ends with Light. Edm makes the point to show us light is electromagnetic. Radio waves are made from EM fields, and radio waves, especially short wave, travel FAAAARRRR... like I could get russian short wave from New York, with the right antenna, right? Net could be any number of things, ed was a Mason, so he was schooled in double/triple meanings. sengA talked about 153 lots, the number of fish in the net, Vulcan's net, internet. World wide web, spiders web, web of deceit. How could George Orwell have such an intimate knowledge of things happening today, on an exaggerated scale of course, I mean exaggerated now... compared to the actual 1984, where iphones, and 2 way monitoring devices in your breast pocket was still a fantasy dream, Imagine how close his novel will seem in another 20 years, at the pace of our privacy slowly disappearing. But back to the net, it isn't going away. Internet I mean. Every time we post something, it's leaving a semi-permanent trail right here in the electric aether. In 50 more years, will this post still be findable? Will poughkeepsieblue be dead and gone, but my time spent posting and speculating on edl live on forever? Is the internet the only legacy we will leave behind, after books are banned as a waste of trees and paper, all that will be left are these ever changable electronic 1's and 0's stored on chips and boards and transistors. Is the only thing the future will know of me a stack of silicon a few microns thick? If time is just gravity and light, what makes right now different from 50 years ago? Is it the way the waves of light and gravity change wavelength and frequency over time? Wavelength is measured in time, but actually isn't time measured by wavelength? Ed is standing next to a giant wavelength lambda in his at work picture. Why is it upside down? Again, I won't even try to understand anymore than this, yet, I'm no expert on EM and the like, just trying to understand. But like charlie said, kids learn fast, cause it's easy to fill an empty head. My head is pretty empty the more I look around it. I guess the point I'm making, is that; is it easier to ponder how ed knew about, us, and how he knew about a WW2 enigma machine, at a time when only nazis knew about it, and how he seemed so advanced in his simple ways, or is the simplest explanation that he was getting his knowledge from the future? Did he figure out how to manipulate his radio to pick up our future left over em signatures? Or when edm says that edl's sign says edl is 'older', did ed live the earlier part of his life before 1887? Or after 2016? Or perhaps, if he was spending time in other times, those days, weeks, even months would add up to more physical age in ed's body, but not passing time in his 'reality'. If i figured out how to go back and visit ed in 1936, and spend a year with him before returning to right now because I have work tomorrow, I would be a year older, but no time would have passed here, here I would be the same age, but my body would have been one year older. How do you keep track of your age that way? Of course, I'm violating occam's razor, because the simplest explanation is time travel, but the most complicated explanation is explaining time travel. But, if the so called greatest minds of science; Hawking, Einstein, Sagan, all put serious time and thought into the possibility of time travel, why shouldn't we consider it? Especially because, theoretically, it's been postulated as possible. More food for thought, edm seems to make it a point to keep you guessing, for example on page 18 of edm's book he says 'At least there isn't a CC here. One could get quite suspicious if you would se e=mc^2, wouldn't you?' I think edm is calling attention to the fact that edl may have known his place would someday be called Coral Castle, as opposed to his Rock Gate. That would explain the discrepancies between his statistics page, and the legend that the next owner renamed the place Coral Castle, was the statistics page added after the new owners? No, because it's rife with code, so did ed name CC, Coral Castle? Or did the second owners? That's an open question, because there seems to be some debate on it. I propose ed wrote the statistics page, and therefore he was quoting the future, or he made up the name. What do you think? Since I'm on page 18, anyone here have any musical background, especially piano, can figure out the right song at the top of the page? Been working on it all week. On page 17 he mentions, 'one more thing of note', and he also mentions the 'comma'. Does it mean anything? Google 'comma, music'. The link to the right wiki isn't working, so you're on your own here. But, it sure does, it's a minute (my-nute) interval between notes, sorta like drop below 10 cents. Ever see my first 1984 video? I go over minute (as 60 seconds) and minute (my-nute) as 'less than' in that video because Orwell uses the wordplay in his code too, when he says 'Two minutes hate' which turns s or 19 into 17, by making it minute by 2. Thus, with the title 'nineteen eighty-four' becomes 'seventeen eighty-four', and 1780-4=1776, the birth of a nation... But minute is not important here, just making mention. And the list of numbers at the top of the page 5115113423111 translates to notes in the musical scale eaaeaacdbcaaa I had a friend look at it, and I have a guitar tablature for the basic tune, but I believe ed is talking violins and pianos as 'my vio' can be found in the title of ABIEH, and the 10 C in the ADM sign indicates middle C on a piano, as the cents mark is also a center line in building. On a piano, it sounds like the Bridal March from Richard Wagner's 'Lohengrin'. Which would be very relevant to ed and agnes, but the beginning eaaeaa also matches the very famous opening to the Flying Dutchman Overture, also by Richard Wagner. Shit, I just realized both were Richard Wagner written... Must be about Wagner. Disregard that, I may have just figured it out... Wagner. Edm was right, they may just come to you, the answers that is... I'll get back to the copyright soon, keep that dime handy...
Pretty sure I'm not sure I get it, like I said, lots of space to fill... I do like Hank though, he comes up on my tube a lot. As for Richard Wagner, I am pretty sure I'm getting closer there. Richard Wagner, wrote 'The Ring Cycle.' A 4 night opera event. What's interesting about The Ring Cycle you say? Well, it happens to be about the Norse legends, Odin, Thor, etc... The Ring of the Nibelung. hart72294 looking in your direction more now, as the connection with the eye is perfect to the yin yang the more I find out about it. Mimir owes odin an eye, and Mimir is said to drink from the well from the Gjallarhorn. But the Gjallarhorn is better associated with Heimdallr, who is an interesting cat as I read more. The loud horn, and the water of the well of knowledge, deep stuff. Heimdallr is also called Rig, I remember reading, and craig, if you're out there, I'm looking in your direction now, rig 'thor's pref' and 'odin i o u' on the author's preface? interesting as the 'Ring' reference. Edl says to 'fasten a ring on top' of your electric magnet. Edm mentions a ring a few times, he draws extra attention to the 'camera stand' as a 'ring within a ring'. Or an o within an O. He also mentions on level 3 of the hidden pages, there's a first letter of every paragraph cipher in the last section about 'The Tree'. It says SEE IT A RING EDM or SEE IT A RINGED M He also ends it with SOUND BASE that's the kicker that ties the ring message here into the notes on page 18 and then the same excerpt from edl he ends with on page 18 of edm's book. He discusses tree rings heavily in these paragraphs of the hidden pages as well. The Tree, the rings, the reeds, the seed, all goes along with the copyright subject, and the ADM sign, both at the 'beginning' of ed's works, his writings and park, if you're watching along. The tree also fits good with the world tree Yggdrasil, edl mentions trees doesn't he? It will go along even more after we bring in the dime, as the dime is a ring, and it's edges have 'reeds' I promise there's more to the dime, so much, i have no time yet, but i will, there's so much. So anyway, that's the thinking of it as I see it, this week.
"Lightning strikes the tallest tree..." The tallest tree would definitely be the world's tree.
Lightning, we all know who is famous for that. Thor. Another Thor reference. I got in to Norse M after I saw the first Thor movie. They leave a lot of things out, but thats the Marvel Comics adaptation.
Dimes: One of the older dimes is called a "Mercury Dime." Not to be confused with the messenger god, according to Wiki.
Yess, good point, I will have to revisit that, it's not fresh on my mind. That's funny, my wife got into Norse M after seeing the film Thor as well, and after watching the 'Vikings' series. She seemed to enjoy the movie more than me... and she's always giddy when Vikings is on, hmm... Anyhow, now I'm into it, seeing the tie ins to the bible, and ed, and the symbolism. Interesting Nibelung, Nephilim, hmm... There is a lot I could go about, like Thor's death in Ragnarok, against a serpent, or coil, taking 9 steps, and falling to the earth. Yeah, that's interesting. I have some silver, a few dimes, but they're the same size as today's dime, which matters most you know. I can't keep up exactly with the chronology of the silver coinage of the US at the time, but I will find out soon enough, I know some serious collectors. They know better than me, and who better to learn from. Liberty was the half dollar I believe, and it was designed by the same fella who designed the dime, If I'm correct. Dollar was the peace dollar, but that wasn't made after 1935 or something like that, because there were so many, and the US was changing the currency all the time due to war and economy and trying to get gold and silver out of the public's hands. They succeeded in 65, and got rid of silver at last, but it's still out here, I get it sometimes. Size matters. No matter what you hear.
That's interesting, both got into it by the same reason. Also, Thor: Ragnarok comes out next year, can't wait to see it.
Interesting you mention Heimdallr; in the mythology and movies, he is the one who opens the rainbow bridge to Midgard (Earth) and the other realms. In the movies, any one from Asgard is referred to as an Asgardian.
I wouldn't be surprised if Ed was able to communicate with the Norse gods and goddesses, or even visited Asgard. The thought of that gives me shivers, but its just a thought. Or is it?
Happy 7/29 Everyone! It's ed's prime day! Almost over for me, but not yet... 7/29, or 7129 is ed's first 'immigration' number. It's a magic number with lots of different meanings. Let's look at some. 7129 is GABI or GLI add 7+1+2+9=19, 19 is the 8th prime number 29 is the 10th prime number one of ed's birthdays is 8/10 71 is a prime number, the 20th prime 71 +29=100, and ed wants 'one hundred per cent good' so, 7/29 July 29 is the 210th day of the year, or 211 on a leap year 211 is the 47th prime number 210 is a triangular number, the 20th triangular number 3 more than ed's 17th triangular number, or the magic 153 17 the 7th prime back to 210 though, it's an even more fun number because it's the product of the first 4 prime numbers 2x3x5x7=210 add the first 4 prime numbers you get 2+3+5+7=17, there's 17 again whether or not it's a leap year, there's 155 days of the year left. 1+5+5=11, the 6th prime 155 is 31x 5, or the 11th prime x the 3rd prime, which would give magic 33 if you wanna see it that way. 366 days on a leap year divided by 211 = 1.734, which is interestingly close to the sqrt3 there's interesting play in these numbers, rearrange 211, and you get 1 12, ed's other birthday, or 121, which is 11^2 but let's start thinking in primes and gematria prime sequencing is the mathematical key to the torah, and ed's number references are always somehow linked to gematria get yourself a 'Strong's', or use 'bible wheel', it works good 210 is 'ayin, or eye in hebrew, makes it's first appearance in Genesis 3:6.' 'ayin in hebrew normally, not as it's printed in gen 3:6, is valued at 130, or 13 normally. The first greek appearance is in Matthew 5:41, as 'milion', or 'mile' " And whosoever shall compel thee to go a mile, go with him twain." What's interesting about milion, strong's defines it as; " a mile, among the Romans the distance of a thousand paces or eight stadia" when I read 1000 and 8, I thought of ed's '8000 word booklet' but what about 155?? this gets better 155 in hebrew, makes it's first appearance as 'yacaph' or 'more', in Genesis 8:12 "And he stayed yet other seven days; and sent forth the dove; which returned not again unto him any more." in this case the offending phrase is "not again", but synonyms for yacaph include 'again', 'add', 'henceforth', 'continued', so let's just see what the next 155 is It's in Genesis 13:14, and it's 'qedem' or 'east' U read it, 21 look east qedem is usually valued at 144, or 12^2, or ADD, or ADM, or ADV the verse reads "And the LORD said unto Abram, after that Lot was separated from him, Lift up now thine eyes, and look from the place where thou art northward, and southward, and eastward, and westward" in greek, it's 'pous' (soup?) or 'foot', first found in Matthew 4:6 "And saith unto him, If thou be the Son of God, cast thyself down: for it is written, He shall give his angels charge concerning thee: and in their hands they shall bear thee up, lest at any time thou dash thy foot against a stone." East is very important, it makes it's first appearance in Gen 2:8, when describing Eden, right before it goes into describing the 4 rivers of eden as a heads up, some of the best math is found in the parts about the rivers of eden just as a tidbit, because I have to mention it 4^3, 4 cubed is 64, the age ed died, and 64 makes it's first appearance in Gen 2:11 as Havilah "The name of the first is Pison: that is it which compasseth the whole land of Havilah, where there is gold" 'Gold' has a value of 19, again 19 is the 8th prime, and 8^2 is 64 The gold is the numbers I believe. Enjoy 7/29 everyone.
Doing a little reading, and got thinking about sign language. I know a little about the sign language alphabet, American sign language that is. Ed does appear to be making masonic gestures, none of them ever seemed to be sign language to me, but I never investigated it either. I thought about modern sign language, but ed never did much 'modern' anything, he likes references to 'classic' works, I noticed. So I used my quick google education to learn some more about the history of sign language. Thank god it's not 1936, and I actually had to go to the library or something. Unfortunately, the internet isn't a 'valid' place to 'learn' anything, so I didn't actually get any smarter after reading anything... But before I forget where I found the info, thought I would point this out. The 'modern' history of sign language begins about 1620, in a book by Juan Pablo Bonet; "Reducción de las letras y arte para enseñar a hablar a los mudos," ("Summary of the letters and the art of teaching speech to the mute".) Allegedly the first recorded modern use of sign language to teach speech to the mute, as stated. In his book, it contained a sign alphabet, the basics of which were adapted to modern use in Spanish, French and American Sign language. Bonet's name interested me, because of the 'net' in his name. In MC ed ends and starts many sentences with 'mag-' and 'net', respectively. I felt like I saw it before, but not in MC, it was in ABIEH, page 2. If 'you read ititit' and you 'look east', you will see 'Bonet', kind of, and a reference to 'hand' and 'body.' Ed's pictures are in ABIEH, so I looked closer at ed's cover. You can see ed standing on 'ED' under his feet, which I call the 'seed'. Look at his hands, they look a little, uh, a little, contorted(?) perhaps. Contrived you might say. But let's look at Bonet's sign alphabet. That is not in order I'm sure you can tell. It's similar to the ASL alphabet I know, but not the same. For starters, it has 22 letters, like Hebrew has. Ed was very into hebrew. Looking at Bonet's 'E' and his 'D', I would dare to say ed is making the sign for 'ED'. His left hand as 'E', his right as 'D'. You can sign with either hand, depending on which hand dominant you are, so either hand is correct. Poking around, ed doesn't seem to have much of his hands showing, but in his generator pic he's making the beginnings of the 'Devil horns' hand gesture, as I know it (KISS ARMY for life). Which is a mason's gesture, allegedly. Well, the devil horns are Bonet's 'Q'. AG make EZ. |
STORIES, HELPFUL INFO, MISTAKES & MORE
GOLDEN RATIO
Mona Lisa conforms to it. So does Michelangelo's David. And so does "The Creation of Adam" on the famous Sistene Chapel ceiling. These are, the scholars enlighten us, the very best works of art in the world because of the Golden Ratio. That is precisely what makes them so beautiful, so owe-inspiring and so memorable. Golden Ratio? Really?
Much has been said about the use of the Golden Ratio in design, architecture, nature, works of art, etc. And much praise has been given and odes sung. It has even been called the Divine Proportion. The learned professors have found it in the Great Pyramid of Giza, the Acropolis, Taj Mahal, the Eiffel Tower, the Aston Martin cars, Salvador Dali's paintings, and even in TOYOTA's logo. They tell us, "Any thing that has the golden ratio is perceived as aesthetically pleasing.Such is the nature of the human brain".
The golden ratio can be defined this way: A line divided in two sections in such a way that the whole line is to the longer section as the longer section is to the shorter section. What? Say that again, please. The whole is to the longer as the longer is to the shorter. Weird, isn't it? Let's try one more time. Split a line in two and you'll get a long section and a short section. Now, if you divide the length of the whole line by the length of the long section, you'll get a number. Let's call it X. Then, divide the length of the long section by the length of the short section and you'll get another number. Let's call it Y. In all cases, except one and one only, X will be different from Y. But when X = Y, then, boy, oh boy, do we have something really special. Little wonder some called it the Divine Ratio. Wikipedia (Sectio Aurea) gives a very good illustration and much additional explanation.
For our purposes, all we need here is the number itself. The magic "golden" number when X = Y. Care to take a guess? No, it's not 2. And it is not 7 either. It is 1.61803398875…., which is an "irrational" number, like π.
Obviously, every line has a golden ratio point. Let's call it the reference point. Easy to find and easy to deal with, if you are looking at a line. But what about a more complex form or object? Especially one irregular in shape? Well, it's not easy to do that objectively. In evaluating a three-dimensional multi-faceted object such as a motorcycle, for example, choosing such a reference point is purely arbitrary.
A popular consensus in evaluating motorcycles for the presence of the golden ratio is to use the "juncture" between the seat and tank as the reference point. If the distance from this juncture to the front end and the distance from the juncture to the back end are in the golden ratio,…bingo! The bike is perfect, they claim. According to the above definition, BLACKSQUARE is exactly at the golden ratio as we can see from the image below.
83.5/51.6 = 1.618
51.6/31.9 = 1.618
Another popular consensus on looking for the golden ratio in motorcycles is to compare the tank height to the seat height. BLACKSQUARE meets this criterion as well as demonstrated below.
8.5/5.25 = 1.619
But, why stop there? Let's take a different look and see what else we might find. When viewing BLACKSQUARE form behind, we discover the golden ratio present in height vs. width as shown below:
32/19.8 = 1.616
And we also find the golden ratio in the taillights width vs. tank width.
17.1/10.5 = 1.628
Taking a look at the front, we see the golden ratio pop up a lot. It is in the total height vs. wheel height.
40.5/25.2 = 1.607
And the golden ratio is in the handle bar's width vs. engine width.
28/17.4 = 1.609
In terms of basic shapes, the triangle shown below has a near perfect golden ratio between sides and base.
Is BLACKSQUARE perfect then?
I prefer not to rely on numbers (however magical or golden) for the answer. My approach to the answer is simple and, some may say, old fashioned. If it looks good, it is good. And if it doesn't, it isn't.
Simple enough principle, isn't it? But it begs the question, "If it does look good, is this then due, at least partly, to the golden ratio?"
Not an easy question to answer. So, let's use the old trick: when presented with a challenge, change the subject.
A motorcycle is intended to be ridden more than it is intended to be looked at. And if there's pleasure in riding it, so much the better. Because when you are sitting in the saddle and your eyes are on the road coming at you and moving underneath you at 80 MPH, the last thing you see is any of the motorcycle itself.
Ah, sorry. Forgot about the front wheel. It is not the original CB550 wheel, but a front wheel from a HONDA CL450. Amazingly, it conforms to the golden ratio as seen below. Kudos to the HONDA designers!
Before closing, let's remind ourselves that linear proportions depend on perspective. When viewed from different angles, some of the linear golden ratios illustrated above will simply vanish. Meaning they will no longer be "golden" at 1.618. The image below demonstrate this well. The perfect golden ratio, present in a direct side view, has now become almost 1 : 1. Interestingly, other features, that were not in the golden ratio in previous views, can, now, from this particular perspective, pop into the magic of the perfect 1.618 : 1. Fun, isn't it?
NOTE: All dimension shown in inches were actual measurements taken using a tape measure. Accuracy was perhaps 1/8″ or so. In the photos showing just 1.618 and 1.0, no actual measurements were taken. The ratio was determined using Adobe Acrobat Pro's Measuring Tool. |
We can write ANY essay for you and make you proud with the result!
Example essay writing, topic: Imaginary Numbers Number Root Complex
1, Imaginary Numbers The origin of imaginary numbers dates back to the ancient Greeks. Although, at one time they believed that all numbers were rational numbers. Through the years mathematicians would not accept the fact that equations could have solutions that were less than zero. Those type of numbers are what we refer to today as negative numbers. Unfortunately, because of the lack of knowledge of negative numbers, many equations over the centuries seemed to be unsolvable. So, from the new found knowledge of negative numbers mathematicians discovered imaginary numbers.
Around 1545 Girolamo Car dano, an Italian mathematician, solved what seemed to be an impossible cubic equation. By solving this equation he attributed to the acceptance of imaginary numbers. Imaginary numbers were known by the early mathematicians in such forms as the simple equation used today x = +/- ^-1. However, they were seen as useless. By 1572 Rafael Bombe li showed in his dissertation Algebra, that roots of negative numbers can be utilized. To solve for certain types of equations such as, the square root of a negative number (^-5), a new number needed to be invented.
They called this number i. The square of i is -1. These early mathematicians learned that multiplying positive and negative numbers by i a new set of numbers can be formed. These numbers were then called imaginary numbers. They were called this, because mathematicians still were unsure of the legitimacy. So, for lack of a better word they temporarily called them imaginary.
Over the centuries the letter i was still used in equations therefore, the name stuck. The original positive and negative numbers were then aptly named real numbers. What are Imaginary Numbers An imaginary number is a number that can be show as a real number times i. Real numbers are all positive numbers, negative numbers and zero. The square of any imaginary number is a negative number, except for zero. The most accepted use of imaginary numbers is to represent the roots of a polynomial equation (the adding and subtracting of many variables) in one variable.
Imaginary numbers belong to the complex number system. All numbers of the equation a + bi, where a and b are real numbers are a part of the complex number system. Imaginary Numbers at Work Imaginary numbers are used in a variety of fields and holds many uses. Without imaginary numbers you wouldnt be able to listen to the radio or talk on your cellular phone. These type of devices work by receiving and transmitting radio waves. Capacitors and inductors are used to make circuits that are used to make radio waves.
In order to determine the right values of capacitors and inductors to use in the circuits, designers need to use imaginary numbers. Another use of imaginary and complex numbers is in physics, quantum mechanics to be exact. In quantum mechanics a big problem is to find the position of a particle. Unfortunately, only the probability distribution of its position is possible to find. The only way to calculate this is to use imaginary and complex variables.
Lastly, electrical engineers use imaginary numbers. However, instead of using i in their equations they use j. This is because in the equations they commonly use, i means current, so to represent imaginary numbers they use j. Four Most Familiar Number Concepts There are four of the most common numbers that we, the common person, know about and can understand why they exist. At one point or another you might have used one of these four concepts in your math classes.
The first concept are Natural Numbers, which are abstract numbers that answer questions, like how many. They are able to describe sizes and sets. The second concept are Integers, they describe the relative sizes between two sets. They answer questions, like how many more does A have than B Rational numbers are what describes ratios and fractions.
For example you might tell Karen that you ate 3/4 of an apple pie. This will let Karen know you ate three quarters out of a four quarter pie. A real number is a number that will describe a measurement like weight, length and fluid. However, in none of the four concept can you see the square root of -1 fall into place. There exist a fifth concept which is referred to as a complex number.
As mentioned earlier a complex number equation = a+bi. It is a real number with an imaginary number. Quadratic Formula and Imaginary numbers Throughout our lifetime, teachers have informed students that negative numbers cannot be squared. With imaginary numbers we are able to do so. With a very simple example it can be shown how this is true. With an equation like y = x^2+ 4 x+29, we can get the x intercept by using the quadratic equation.
By following all appropriate steps you will find out where the x intercepts are at. Roots are all places that a graph will touch the x intercept. The quadratic equation = -b+- square root of b^2-4 (a) (c) / 2 (a). Therefore, x^2 = a, 4 x = b and 29 = c. -4+- square root of 4^2 - 4 (1) (29) 2 (1) -4+- square root of 16-116 2 -4+- square root of -100 2 -4 +-10 i 2 = -2+-5 i The answer -2 +-5 i, lets you know that it is a complex root, meaning that it does not touch the x intercept. By graphing the equation y = x^2+4 x+29, you will see the parabolas location.
This parabola will not touch the x intercept. This table will show you how: i^1 = i i^4 = 1 i^2 = -1 i^5 = i i^3 = -i i^6 = i Complex Root and Complex Conjugate Root This is an ongoing cycle that will help you solve problems that deal with i^n power. Another amazing technique you can use is when you are given a complex root and the complex conjugate root and you need to derive the equation by the root given and complex root. A complex conjugate root is that exact opposite of a complex root. For example if you are given one complex root of 2-5 i, and you are asked to find the equation you simply multiply 2-5 i by the conjugate root of 2+5 i. By using foil method you will find out the equation.
For example: (2-5 i) (2+5 i) = 4+10 i-10 i-25 i elimination will give you = -25 i^2+4 you know i^2 = -1 therefore = -25 (-1) +4 a negative times a negative equals a positive = 25+4 = 29 You then add the complex root (2-5 i) with its complex conjugate root (2+5 i). 2+5 i = 2-5 i = 4 This will let you know that your equation is y = x^2+4 x+29 and you are able to graph and see how and where the complex roots are located on the graph. Making An imaginary Number A Real Number You can multiply, add, divide, subtract and even take the square root of a negative number. Like mentioned in the History of Imaginary Numbers, negative numbers were not believed to be a valid answer. However, we know that a negative number does have meaning and is a valid answer. A negative number will let us determine many different things.
We see them in our check books, when graphing, and even when finding the expected number of a roulette game. Complex numbers can be added to show you how they can become real numbers. For example: 5 i^2 + 4 i^2 = 9 i^4 9 (1) = 9 The answer is a real number that we obtained after adding it to imaginary numbers. You can refer to the imaginary number cycle. It is known that i^4 = 1, nine is then multiplied by 1 to get a positive nine. Weather you get a negative or positive number they are real numbers.
Conclusion Imaginary numbers are in fact very real. They have common uses and very intricate uses. Little does the average person know the imaginary number is one of the oldest and greatest discoveries ever found. Dr. Anthony, Ask Dr. Math The Math Forum, 1994-2001.
Imaginary numbers have had a great effect whether
direct or indirect on the microwave oven. Although
the concept has come out of left field, there is
still a plausible explanation for their influence
on the microwave oven. The fact remains that
complex numbers have much less direct relevance to
real...
Just having a Social Security number is no longer
a symbol of adulthood, the numbers use is no
longer confined to working and paying taxes.
Government agencies, business and schools rely on
Social Security numbers to identify people in
their computer systems. Many people receive their
Social Securit...
Numerology is the study of numbers used to
predicting future happenings, judge of one's
character, or attempting to acquire connections
with the occult. Based on a simple number, and
alphabet, system, Numerology is probably one of
the easiest arts of the occult to use and
understand. Predating back ...
Still cannot find the paper you need? Buy essay or research paper tailored exactly to your instructions and demands -- original, written from scratch for you!
Free essay examples, how to write essay on Imaginary Numbers Number Root Complex |
Mathematics The Arabs developed the concept of irrational numbers, made algebra an exact science, founded analytical geometry, plane and spherical trigonometry, and incorporated into mathematics the...
Posts
Ibn Battuta's Hajj: Experience in Maps and Timelines
posted on: Jul 13, 2016
Ibn Battuta's Haj Through Map
Ibn Battuta was one of history's greatest travelers, journeying through much of Asia, Africa and the Middle East. What most people don't know is that his journeys began with Hajj, the Islamic pilgrimage to Mecca. Explore below as we follow Ibn to Mecca through interactive map and timeline,
Ibn Battuta's Haj Through Narrative
All Content is sourced from Wikipedia and Ibn Battuta's written accounts of his journeys as found in Fordham University's Medieval Sourcebook: Ibn Battuta: Travels in Asia and Africa 1325-1354 ( |
Which one doesn't belong? How would you defend your answer? Cam you pursuade others to share your perspective/belief? And this lead nicely into our Lego Robotic Helicopters. @MathBeforeBedpic.twitter.com/vJodUNXr2O |
More From People Who Like Mathematics
Everything inside you, outside of you, around you, away from you... Everything you see and touch is all mathematics. It's everywhere, the algebra, the measurements, the statistics & probability, the perimeter, area, surface area, volume capacity, the analytical geometry and much...
Believe it or not, maths can be fun. Its also very functional, but some mathematics can be absolutely stunning!
"What's the link between the Fibonacci sequence and the Golden Ratio?"
I am posting this answer in response to the above question from an EP fan who wishes to remain... |
Mathematics for the Liberal Arts
Creating Escher-Like Tilings with Software
What we're going to learn to day is how to take your computer and create some art, and what you will make will astound you. First things first. What we are going to create will look a lot like wallpaper, with repeated shapes. However, the more someone looks at it, the more they'll see. In fact, what we shall make are called tilings or tessellations. A tiling is a way to divide up the plane into geometric figures, each of which is of an identical shape (you may also have a few different shapes, but they must repeat in a pattern and fill the plane).
In the text you've seen many tilings. One example is with squares. Another one can be achieved with hexagons (six sided figures) and triangles. I've created one of my own below.
Note that there are two different variations of one tile, in different colours. I'm going ot describe how to use software to create you own tiling. You'll need a painting program. There are many free ones; we'll use thePaint program available in Windows under the "Accessories" menu.
STEP ONE
Start up your paint program. If you're using Paint, it will automatically open with a blank page to work with. You'll probably want to increase the size of the page by dragging the bottom right corner of the page out as far as you can.
STEP TWO
Now you want to draw a black rectangle, one that is of medium size (in the end, they should be small enough that you should be able to place at least six of the squares across and down without any overlapping). In Paint this is fairly straightforward: After making sure that both "Color 1" and "Color 2" are black (if one iusn't, select it and then select the black box to the right), all you do is click on the rectangle shape and drag out a rectangle on the page (if you want the rectangle to be a square, hold down the Shift key whie dragging).
STEP THREE
This is the key step. What you want to do here is alter the shape of your tile from a square to a more interesting one. If you think of what you need to be able to put your tiles together, what you'll see is that whatever pieces are indented into the left side must project from the right side in the same way, and vice versa. The same is true for the top, and as long as this happens, the tiles should fit together like a hand in a glove!
Here is how to proceed. You need to click on the Free-from selection tool under the "Select" menu (by clicking on the arrow beneath it). Now click with the mouse to the left of the rectangle you've drawn, and drag out some closed shape inside the left side of the square, making sure to get a bit of the top or bottom edge (this will make the subsequent alignment easier later). By closed I mean that when you let the mouse up, the end point of the mouse is close to where you started dragging.
Now choose Control-X to cut the piece out, and then Control-V to paste it. You'll need to drag the pasted piece to the right side of the tile and carefully release it so that it is exactly fits. If you click on the page away from the rectangle, you should see how the left side of your tile will fit exactly into the right side. Now go about and do the same with the top and bottom (select, cut, paste and drag a piece from the bottom up to the top, or vice versa). An example is shown below. You've just drawn your basic tile.
STEP FOUR
Now select your tile with the rectangular selection tool (it is the dotted rectangle on the left), drag a rectangle around your tile, and then click down in the middle of your tile and drag the whole tile down to the bottom of the page. You can take your tile and paint on it however you like with the painting tools. If you don't know how to use the paint tools, ask someone, or just try them out – it's pretty easy! Make sure all your painting is within the tile. Here is how I painted mine:
STEP FIVE
Use the rectangular selection tool to select your tile, cut and paste a copy of the tile. Drag it far enough away that you've left plenty of space between it and the original tile. Now you can recolour the new tile. One of the best ways is simply to change some of the colours. The easiest way to do this is first to click on the bucket tool. Then select a color for "Color 1". Select the piant bucket tool and click on any colour on your copy of the tile; the colour you selected should replace the old one. Do this several times until you're happy with how the second tile looks. (You can use Control-Z to undo any choices you regret!)
STEP SIX
Select aone of the two tiles with the rectangular selection tool and move it to fit into the other tile; this is your two tile pattern. Move the copy down to the bottom of your window.Then copy and paste it. Move this to the left side to be you first tile. You can then repated paste and drag this combined tile several times to fill in across the page, fitting it in snuggly as you go, and then continue beneath the first row, and so on. Continue to alternate, dragging copies of the two tiles at the bottom and placing them successively across the top so that they fit neatly into one another. The tiles will always be of opposite colour to the ones directly beside it (on all sides).
STEP SEVEN
Save your picture and print or post it. It should look wonderful! Feel free to start all over again. Here are some ideas to explore once you've got the idea:
You don't need to start off with a black tile; any solid colour will do.
Try colouring the second tile quite differently from the original (see my Elephants and Ghosts tiling).
In most painting programs, you can "crop" your picture (that means, you can cut it down in size to a smaller rectangle). You can then add a frame so it really looks like a picture!
Finally, you can start with a parallelogram rather than a rectangle provided your program allows you to create one. The hard part will be aligning the selected parts as you move them across the screen (here's a hint how to do this – find a way to draw lines joining the midpoints of opposite sides with a brand new colour. When you drag the pieces, as long as you've selected some of these lines, you'll be able to line things up properly).
Potato Printing!
George Escher, M.C. Escher's youngest son, has described a game that his father invented in 1942. You'll need the same items as described previously, along with a rag or firm sponge and some tissue paper. Here is what you do:
You'll need a grid, that is a piece of paper that's covered with squares of the same size. Xerox several copies of the page, or, if you've created the grid on your computer, print off several copies. You may want to hide the lines of the grid, and only place dots on the corners of the squares.
What you want to do now is draw some lines (not necessarily straight) across one of the squares, so that when two copies of the square are placed side-by-side, the lines will connect to one another (one easy way to do this is to make sure, for example, that the lines run across from the middle of one side to the middle of the opposite side).
Cut a large single potato right through. Use the knife and one of the squares to pare away more of the potato halves so that each half's flat portion is a identical in shape to one of the squares.
Take the piece of paper with the square you've designed and place it on the flat portion of one of the potato halves. Carefully use the knife to cut through from the paper onto the potato half in order to transfer the edges of the design onto the potato. Then use the knife to pare away the parts of the potato that are to be white when you print (these are the 'non-design' parts of the potato half).
Make an 'ink pad' by spreading some of the fingerpaint onto a piece of paper, and then use a rag or sponge to absorb the paint (you'll find using just one colour works best).
Press the potato half with the design onto the 'ink pad'; the raised parts of your design will be covered in ink. Use some tissue to absorb some of the excess paint.
Press the other potato half firmly against the inked half (aligning them up precisely, of course) so that you get a mirrored image of the design. Use your knife now to cut the second stamp out (you'll pare away all the non-inked areas); this second stamp is a mirror image of the first.
Now take a new grid page, and use your stamps (after inking them) on each square. You're free to rotate each square as you like before pressing it onto a square of the page.
You'll be surprised at the beauty you can create. To create true 'escher'-like patterns, you'll need to make a little 'notch' on each potato half so that you can keep track of the design's 'orientation'. Then you can try any arrangement for a 2x2 set of squares, and be careful to repeat the pattern you use over and over. It's a little complicated, so if you want to know more, I suggest you go to the library and see George Escher's article in M.C. Escher: Art and Science. |
14, 2015
The art of Pi - celebrating Pi day 3-14-15
This one will take some time, so get a cup of coffee and a piece of pie and enjoy.
Today, March 14,2015 is a math geeks holiday! It's Pi day! Pi is a value used to determine the area of a circle, which is 3.14159265359 in short form so the date 3/14 is Pi day annually and once every hundred years the 15 comes into play along with the time 9:26:53, so there is a lot of attention to today. It is known as an irrational number since there is no end point. Since Pi looks and sounds like the English word pie, part of the celebration is baking and eating pie. Its a day where math and food coincide. In fact if you look at 3.14 in a mirror, you get the word pie.
Which leads to the art of Pi. The cartoonist come up with all sorts of pun offerings and Pi day works |
…be constant
If you're looking for the perfect Pi Day celebration, you need look no further than the same page in the dictionary—pie. It's the perfect way to celebrate the day dedicated to pi.
And while you're enjoying your Pi Day pie perhaps take a moment to appreciate the usefulness of this infinite, irrational, transcendental, and constant figure…perhaps by solving for the circumference of your Pi Day pie (C=2πr)…or you could just eat it. |
The class blog for Math 3010, fall 2014, at the University of Utah
Base 1 and ancient Egyptian math
Earlier in class, we started to discuss different bases for mathematical systems. I brought up base 1 and we all got into a debate as to whether or not base 1 actually makes sense.
The answer is, it depends on who you ask. To understand why, we need to really know what it means for a number to be represented in base "n". A way of rigorously defining this is to define a sequence of whole numbers a0, a1, a2,…, with each a being less than n, such that the number you want to represent is equal to a0+a1*n+a2*n2+… For example, representing 5 in base 2 would give us a0=1, a1=0 and a2=1, because 1 + 0*2 + 1*4 = 5. Then, it is a simple matter of writing the number as a0a1a2…, or in this specific case, 101.
We want to have 4 properties when doing this:
1. Each number has at least 1 representation.
2. Each number has no more than 1 representation.
3. Each representation corresponds to at least one n.
4. Each representation corresponds to no more than 1 n.
Some of these properties are more important than others. As it turns out, properties 1 and 4 are key. Property 1 guarantees that anything we want to write in our base, we can. It'd be silly if we had a numerical system that couldn't list numbers over 1,000,000, for example. Property 4 is even more important: it guarantees that no two different numbers can look the same. It would be really silly if we looked at a number and had to guess whether or not it was 3 or 4.
As it turns out, to have ALL of these properties, we must specify that n is greater than or equal to 2. Why? If we set n equal to 1, then the restriction that each a must be less than n requires ALL a's to be equal to 0. Meaning that 0 becomes the ONLY number we can represent. Not very useful.
Well, let's see if we can salvage this. Let's say we drop the restriction that each of the a's has to be less than our n. Now our situation becomes more hopeless. In base 1, we could, for example, write 6 as 51, or 2211, or 312, or 501, and so on. While it may be clear what each of these numbers is, there'd be no point in being able to write it so many ways. And if we picked the simplest one to represent our system, why not just pick the number itself? And if we're representing a number by simply itself, why complicate things and work in this odd base-1 system anyways?
This is why when mathematicians and computer scientists talk about base-1, they are mostly referring to the unary mathematical system.
If you want to represent a number in the unary system, say N, all you do is repeat the same symbol N times. So 10 becomes ||||||||||, and so on. This type of numerical representation is a lot like counting on your fingers or tallying a certain number of objects. It's also useful for birthday cakes.
Now, let's see if this satisfies the above 4 conditions:
1. Each number has at least 1 representation. Clearly true
2. Each number has no more than 1 representation. True
3. Each representation corresponds to a number. If a representation has N slashes, the number it represents is simply N. True
4. Each representation corresponds to no more than one number. If a number represents N, it can't represent anything else in this system. True.
Alright, it looks like we have a good system. Now, let's take a look at how one would add, subtract, multiply or divide. Addition and subtraction are quire easy in the unary system: For addition, just combine the two sets of slashes, and for subtraction take away a certain number of slashes from one of them. In fact, division and multiplication are also quite easy. Calculating N/M is as easy as crossing out M slashes from the number N, marking that as a 1, and repeating until you no longer have M slashes left. N*M is as simple as writing down N slashes M times.
The advantages of the unary system is that you do not need to memorize any multiplication tables to do multiplication. As long as you know the system, it's incredibly simple. The obvious downside is that even for relatively small numbers multiplied by one another the result can become extremely large extremely quickly. You can think of it sort of as a trade-off: In base 10, for example, multiplication and division are much more difficult, but can be much, much quicker to calculate and much easier to understand.
How this actually relates to math history is through ancient Egyptian mathematics. What's interesting about their style of mathematics is that it incorporates elements of both the unary system and the base 10 system, and isn't truly one or the other, but a hybrid of both. In the ancient Egyptian system, a 1 is represented as |, 2 as ||, and so on. But 10 gets its own symbol, represented as an upside down U, and 100, 1,000, 10,000, 100,000 and 1,000,000 all have their own representation. This is somewhat "in-between" the two bases: on the one hand, the order in which you place the digits doesn't matter, but on the other, numbers become quite a bit longer than they need to be.
This goes to show one of the cooler parts of mathematics: when you take an idea such as "dividing by 0" or "base-1" to its logical extreme, you start to understand deeper relationships between numbers, and can come up with some very interesting results. |
There is currently a push on to get more children interested in mathematics. Personally, I reckon it would be easier to get them to drink liver flavoured milkshakes, but I wish them luck anyway.
You see, my generation was caught between two math systems, Metric and Imperial, and as a result, I'm hopeless at both. Eventually I morphed into someone who measures height in feet, distance in metres, and body parts in inches, eg: I'm six feet tall, my finger joints are an inch long, and each foot is nearly one foot. I don't know the lengths of any other bits…
Every now and then I'll have another crack at my arithmetical nemesis, but as soon as I open one of my daughters' math books and see the words, "Train A is approaching the station at an unknown speed, while Train B…" I feel my skin crawl as the old terror comes creeping back.
Suddenly, I'm back in school gazing at test questions that made as much sense to me as a cars' wiring diagram. Are the trains' diesel, electric or steam? What colour is train B? How fast was the station moving again? And should we be worrying about a hypothetical train smash when President Reagan was about to nuke the Russians and plunge us into WW3?
I recall late nights hunched over exercise books filled with crossed out equations, rubber shavings, and dried tear stains. Eventually I'll close the math book and pick up a novel, preferring the fun of fantasy to a book full of frustrating formulas. My mathematical illiteracy remains a dark cloud in an otherwise sunny existence.
Words and wordplay thrill me; I can 'see' words, or what they represent, eg: mountain, green, numbskull, etc. But for the life of me, I just can't put a picture to the number 647, or any of its' numerous associates. Fractions continue to remain a complete mystery to me, and Roman numerals have only come in handy for working out what year movies were made.
So while I heartily cheer the efforts of those passionate mathematicians who are hoping to improve our children's calculating confidence, I think I'll stick to wrangling with words. The only time I'm ever going to get slightly enthusiastic about numbers is when I see my Lotto numbers come up; and then I'll pay one of those whiz kids to do my sums for me. |
Create your own at Storyboard ThatGupta Empire Pi Mathmatics Vaccines Round Earth Tajma Hall India's golden age. 1. Pi was created during the golden age of India. 2.This is pi 3.14159265359 and so on. 1.Aryabhata was the founder of algebra, decimal system, and the concept of zero. 2. The decimal system is still used today. 1. A vaccine to cure smallpox was found. 2.Herbs and oils were used to cure ailments. 1. Aryabhata was the one who thought of this idea. 2.His theory is, that the night and the day are caused by the Earth's rotation. 1.The Tajma Hall was a tomb for Mumtaz Mahal. 2.Shah Jahan built the Tajma Hall. The Guptains created Pi. We invented algebra. Many vaccines were created in Gupta. The Guptains started the idea that the Earth is round. The Tajma Hall was built in India. Earth |
Category: factors
Most probability resources contain a familiar type of question: the two-dice probability distribution problem. Often times, it is accompanied with questions concerning the sums of the faces that appear on each dice. For example: Roll two fair, 6-sided dice. What possible sums can be made by adding the faces together? What is the probability that: a) the sum is 6 b) the sum is a multiple of 4 c) the sum is greater than 15? I think the obsession with this specific subdomain of probability questions stems from the elegant way in which a table of outcomes (pictured below) leads to a …
A while ago I wrote a post on embedding atomic skills into tasks so that the basic skills are developed and used as tools of mathematics rather than the ultimate goal of mathematics. I try to develop tasks that follow this framework. I want the student to choose a pathway of thought that enables them to use basic skills, but doesn't focus entirely on them.Recently, I was reading Young Children Reinvent Arithmetic: Implications of Piaget's Theory by Constance Kamii and came across one of her games that she plays with first graders in her game-driven curriculum.The game was called double …
Every so often, an idea comes out of left field. I woke up with this on my mind–must have been a dream. Back in the day, my family had a dilapidated copy of the game "Guess Who?" My siblings and I would take turns playing this game of deduction. You essentially narrowed a search for an opponent's person by picking out characteristics of their appearance. I vividly remember playing with my younger sister one time at a family cottage. She–foolishly–chose a female person for me to identify. Anyone who has played the game before knows that the males far …
There is two hour parking all around University of Saskatchewan. I once went to move my car (to avoid a ticket) and found that the parking attendant had marked–in chalk–the top of my tire. I wanted to erase the mark so began driving through as many puddles as possible. I then convinced myself to find a puddle longer than the circumference of my tire–to guarantee a clean slate and a fresh two hours. As I walked back to campus, I got thinking about the pattern left behind by my tires. For simplicity, let's take the case of a smaller vehicle–a …
I came across the following situation while shopping for paint at a local home improvement store: Admittedly, the three varieties were not positioned like this, but this positioning does raise an interesting question. "We can see the packages are the same height, what is that height?" I see this question going one of two ways: The students realize that really any conceivable measurement is possible. (Barring, of course, zero and the negatives) One could make the argument that it also cannot be irrational, but this would be nit-picking. Can a roll of tape have a width of pi/6? Exactly? The …
I have only been teaching for 2 years, but am already beginning to encounter the recursive nature of the profession. I have had several repeat classes in my 4 semesters of teaching, and they require the achievement of the same outcomes. This does not bother me, in general, because I am excited to see the improvement in my teaching. There is one unit, however, that has already frustrated me. Its ability to sabotage creative exploits is unrivaled throughout the mathematics curriculum; I am speaking of the unit on polynomial factoring. The topic was taught in isolation of numerical factors until this …
I completed school before manipulatives were in vogue. I am still not sure that they are today (where I teach). I know that my department's manipulatives are locked up in a cupboard. In this Potter-like clandestine state, I didn't even learn of their existence until the end of the year. I was moving classrooms, and found a pile of algebra tiles that the previous teacher had left behind. I didn't discover that I had manipulatives available to me until, ironically, I inquired where I could dispose of this rather large supply of algebra tiles. When I opened the doors of …
I am teaching 5 new classes next year. I am trying not to think of it that way; rather, I am taking it one step at a time. Unfortunately, most of these steps need to be taken during my summer vacation. This isn't the end of the world; I am fairly stationary, and enjoy a mental workout as much as some enjoy time on the beach or in a foreign shopping mall. I began my massive preparation marathon with a unit for Grade 10 Precalculus on factoring. As I dove into the curriculum and textbooks, I found myself actually enjoying …
This week marked my baptism by fire into the twitter world. It was not long until I was neck deep in tweets, favorites, re-tweets, and followers. The eternal nerd awoke inside me when I was confronted with my first NCTM "Problem of the Day". A simple, yet dangerously deep, question was posed. Wanting to cement my reputation as a responsible twit, I sat down and began to tinker with the theory. The question was as follows: How many different fractions can you write using only the digits 1,2,3 & 4? Be sure to include fractions greater than 1. Immediately, I … |
(Original post by meatball893)
There is a book on zero which is worth reading. Zero is infinity's brother, after all. |
Wednesday, June 10, 2015
CALL FOR ENTRY: GEOMETRY
Concept: Geometry
is the branch of mathematics concerned with the properties and relations of
points, lines, surfaces, solids, and higher dimensional analogs. In the arts it
has referred to the shape and relative arrangement of the parts of something
and includes concepts such as the golden ratio and the use of geometric
symbols. It can be found historically in Islamic & Renaissance art, folk
art quilts, and was embraced as an aesthetic by
the great masters of modern art in the 20th Century. It continues to thrive
today as exemplified by artists utilizing algorithms and expressing them
through visual means. When art and math are used in conjunction, moments of
true interdisciplinary coordination occur.
This national juried exhibition at
Hera Gallery seeks to exhibit a varied
perspective on contemporary views of geometry including, but NOT limited
to, pattern (whether in data or aesthetic), fractals, string theory, along with
the formal vocabulary of geometry from mathematicians like Euclid to artists
such as Mondrian. |
Category Archives: MathematicsSurface***/*****
These 6 matches enclose an area with size 2 (2 standard triangles).
Rearrange them so that:
a) they enclose an area of exactly 4 triangles
b) they enclose an area of exactly 6 trianglesCalcdoku problems, also called K-doku or Calcudoku were invented in 2004 by Japanese math teacher Tetsuya Miyamoto, who intended the puzzles to be an instruction-free method of training the brain. He used the Japanese name KenKen, which could be translated as 'Cleverness'. In his classes, he sets aside about 90 minutes each week for solving puzzles. He believe that when students are motivated, they learn better, and he lets them do so at their own pace.
Other names used for this type of puzzle are Kendoku and Kashikoku naru Puzzle. The names KenKen and Kenduko are trademarked. Books are in Japan published by Gakken Co. In the USA the New York Times started publishing them in 2008. In my native Netherlands they appear regularly in both puzzle magazines and general magazines.
A calcudoku puzzle consists of a latin square – a latin square can have any size. If its size 4, the numbers 1 to 4 should appear exactly once in every rown and column exactky once. Similarly, if its size 5, the numbers 1-5 should appears exactly once in every row and column.
The square is subdivided into smaller areas, and the sum, product, difference or division result is given in the top left cornerTakuzu (often called Binairy/Binairo or occasionally Tic-Tac-Toe) puzzles have been around for several years. In my native Netherlands they have been published by both Sanders puzzles and by Denksport, and also in daily newspaper Algemeen Dagblad.
In France, they have been published under the name Takuzu in le Figaro, at least in 2011.
The puzzle consists of a rectangle or square with an even number of rows and columns. This is partly filled with 0's and 1's. It should be completed by the solver with 0's and 1's in such a way that:
a} There are never more than two consecutive 0's or 1's in any row or column;
b) There are an equal number of 0's and 1's in every row and column;
c) All rows and columns are different;
This last condition seems to be rarely used in the commercial publications. Sizes are usually 10×10, 12×12 or 14×14, though 6×6 and 8×8 are often offered as introduction puzzles |
The Golden Rules Sketchbook by Olivia Lee
Posted on June 10, 2010
As a mathematician and designer I could not resist the urge to talk about this product. The Golden Ratio first emerged as a mathematical term. It has fascinated mathematicians for thousands of years since Euclid first defined the concept in 300 BC. The Golden Ratio has many implications in geometry and mathematical properties. The Golden Ratio is an irrational number: 1.6180339887498948482…
Architects, artists and designers often used the golden ratio in their creations because it is believed to create aesthetics pleasing proportions. If you wish to explore the virtues of the Golden Ratio, you can buy a unique sketchbook designed by Olivia Lee. Her Golden Rules sketchbook contains grids that can help you draw objects with the golden proportions.
Refer to the review of Dezeen if you wish to better understand how to utilize the sketchbook. |
Forked
Wow. It's kind of unsettling how quickly two weeks can pass in between posts. I don't know how Dr. Allain finds the time to be so consistent posting to his blog.
One of the television programs that gets watched on in my house is Minute to Win It. You know, the 6 minute show they somehow stretch out to an hour with that annoying host with the terrible hair. Fortunately, DVR allows the show to be watched in under 10 minutes, as God intended.
Anyway, a lot of the contests they have challenge one's skill, balance, concentration, etc. But when the stakes are high, the contestants are usually subjected to a game of luck. A recent case in point was a game called "Forked". In this, the contestant is required to roll a quarter in between a fork tine from 16 feet away.
Now, this seemed kind of unfair, but I figured I'd calculate just how difficult it was.
First, we'll see how hard it is just to hit anywhere on the fork. Then we'll check out how precise one needs to be to lodge in a single tine. This analysis isn't so different from the field-goal analysis from an earlier post. For our purposes, we'll assume that once released, the quarter travels in a straight line.
According to the show, each tine is separated by . I'll assume that each of the tines is also wide. The total span is therefore .
What does look like from 16 feet away? We'll there are a couple of ways to do this, but I'll rock a trigonometric approach, using the tangent operator to calculate the angular span of the fork from 16 feet.
Where the units mean 24 arc-minutes or 24/60th of a degree. I'm not sure what you make of this, but it seems small to me. Compare that to the span of the moon and the sun in the sky; they each span approximately 30 arc-minutes in the sky. So follow me here, but that means if we were to build a plank all the way to the moon/sun, it would be easier to roll a quarter to hit either of them (again, assume the quarter stays straight and can make it the whole way), than it is to hit the fork. Holy cow.
What about actually rolling it in between the space between two tines? Now, since the quarter has finite width (0.069″), the quarter actually only has of play and still be entirely within each tine.
Ok, 10 times smaller! Granted, there are three such openings in which you can lodge the quarter, but you'd have to aim less carefully to roll a quarter between Jupiter and Europa! |
Jake's Guest Lecture
Our well-regarded local junior college is the top destination for my high school's graduates, a number of whom are more than bright enough to go to a four-year university but lack the money or the immediate desire to do so. Case in point: Jake, my best case for the hope that subsequent generations of Asian immigrants will adopt properly American values towards education, now at the local community college with a 4.0 GPA. He earned it entirely in math classes, having taken every course in the catalog–and nothing else. This from a kid who failed honors Algebra/Trig for not doing homework, and didn't bother with any honors courses after that.
Jake visits four or five times a year, usually coming during class to see what's up, working with other students as needed, then staying afterwards to chat. This last week he showed up to my first block trig class, with the surly kids who mouth off. We were in the process of proving the cosine addition formula.
The day before, I started with the question: "cos(a+b) = cos(a) + cos(b)?" and let them chew on this for a bit before I introduce remind them of proof by counterexample. A few test cases leads to the conclusion that no, they are not equal for all cases.
Then we went through this sketch that sets up the premise. I like the unit circle proof, because the right triangle proofs just hurt my head. So here we can see the original angle A, the original angle B, and the angle of the sum. Moreover, the unit circle proof includes a reminder of even and odd functions, a quick refresher as to why we know that cos(-B) = cos(B), but sin(-B) = -sin(b).
Math teachers often forget to point out and explain the seemingly random nature of some common proof steps. For example, proving that a triangle's degrees sum up to 180 involves adding a parallel line to the top of the triangle and using transversal relationships and the straight angle.
Didn't I make that sound obvious? You have this triangle, see, and you wonder geewhiz, how many degrees does it have? Hmm. Hey, I know! I'll draw a parallel line through one vertex point! Who thinks like that? The illustration of a triangle's 180 degrees is much more compelling than any proof.
So when introducing a proof, I try to make the transition from question to equation….observable. Answering the question requires that we define the question in known terms. What is the objective? How does the diagram and the lines drawn get us further to an answer?
Point 1 in the diagram defines the objective. Points 2 and 4 allow us to represent the same value in known terms–that is, cos(A) and cos(b). And thanks to some geometry that is intuitively obvious even if they've forgotten the theorem, we know that the distance between Point 1 and Point 3 [(1,0)] is equal to the distance between Point 2 and Point 4.
So I'd done this all the day before in first block, setting up the equation and doing the proof algebra myself, and the kids were lost. In my second block class, I turned the problem over to the kids at this point.
I grouped the second block kids by 5 or 6 instead of the usual 3 or 4 (always roughly by ability), giving each team one distance to simplify (P1P3 or P2P4). Once they were done, they joined up with kids who'd found the other distance, set the two expressions equal and solve for cos(A+B). The group with the strongest kids were tasked with solving the entire equation, no double teaming.
Block Two kids worked enthusiastically and quickly. I decided to retrace steps and do the same activity with block 1 the next day. Which is when Jake—remember Jake? This is a story about Jake—showed up.
"Hey, Jake! You here for the duration? Good. I'm giving you a group."
Jake got those who had either been absent or were too weak at the math to be comfortable doing the work. I kept a watchful eye on the rest, who tussled with the algebra. I tried not to yell at them for thinking (cos(A) + cos(B))2 = cos(A)2 + cos(B)2, even though they all passed algebra 2 (often in my class), even though I've stressed binomial multiplication constantly throughout the year but no, I'm not bitter. Meanwhile, Jake carefully broke down the concept and made sure the other six understood, while they paid much more attention to him than they ever did to me but no, I'm not bitter.
Result: much better understanding of how and why cos(A+B) = cos(A)cos(B) – sin(A)sin(B). One of my most hostile students even thanked me for "making us do the math ourselves" because now, to her great surprise, she grasped how we had proved and thus derived the formula.
And then she went on to ask "But we have calculators now. Do we need to know this?" She looked at me warily, as I'm prone to snarl at this. But I decided to use my helper elf.
"Jake?"
Jake, mind you, gave exactly the same answer I would have, but he's just twenty years old, so they listened as he ran through the process for cosine 75 (degrees. 75 degrees. Jake's a stickler for niceties.)
"But why is this better?" persisted my skeptic.
"It's exact," Jake explained. "Precise. When we use a calculator, it rounds numbers. Besides, who programs computers to make the calculations? You have to know the most accurate method to better understand the math."
"Class, one thing I'd add to Jake's answer is that depending on circumstances, you might want to factor the numerator, particularly if you are in the middle of a process." and I added that in:
"Yeah, that's right," Jake confirmed. "like if you were multiplying this, I can think of all sorts of reasons a square root of two might be in the denominator. But other times you need to expand."
I suddenly had another idea. "Hey. How about if we use right triangles?"
"Like how?"
I sketched out two triangles.
"Oh, good idea. Except you forgot the right triangle mark."
I sighed. "Class, you see how Jake is insanely nitpicky? Like he's always making me write in degrees? He's right. I'm wrong. I've told you that before; I'm not a real mathematician and they have conniptions at my sloppiness. But…" I'm struck by an idea. "I don't need to mark it here! These have to be right triangles. Neener." (I nonetheless added them in, although I left them off here out of defiance.)
"This is good. So suppose you want to add the two angles here. These right triangles have integer sides, but their angle measures are approximations. Let's find those values using the inverse."
Me: "Just checking–does everyone understand what Ahhmed did?" I wrote out cos-1(3⁄5). "He used the inverse function on the calculator; it's just a reverse lookup."
" Let's keep them rounded to integers. So 53 + 67 is 120 degrees, which has a cosine of ….what?" Jake paused, waiting for a response. Born teacher, he is.
By golly, my efforts on memorization have paid off. Several kids chimed in with "negative one half."
"Meanwhile, if we multiply all these values using the cosine addition formula…" he worked through the math with the students, "we get -33⁄65".
Dewayne punched some numbers and snorted. "-0.507692307692. That's practically the same thing!" .
I had another idea. "You know how I said you should look at things graphically? Let's graph this out on the unit circle."
Jake was pleased. "This is excellent. So where would cosine(A+B) show up? We need to find the sine of each to plot it on the circle." We worked through that and I entered the points.
Isaac: "Yeah, Dewayne is right. The two points are the same on the graph!"
"But this is a unit circle," Jake said. "Just a single unit. As the values get bigger….I wish we could show it on this graph. Could we make a bigger circle? Or that probably wouldn't scale."
"How about if we just show all the values for every x? We could plot the line through that point? From the origin?"
"What would the slope be?" Gianna asked.
"Yeah, what would the slope be? Rise over run. And in the unit circle, the rise is sine, the run is cosine, so…"
"Tangent!" everyone chorused.
Jake was impressed. "See, this is why I should have taken trigonometry. I never thought about that."
"OK, so I'm going to graph two lines. One's slope is the tangent of 120, the other's is the tan(cos-1(-33⁄65))), which is just using the inverse to find the degree measure and taking the tangent in one step. Shazam."
We then looked more closely at different points on the graph and agreed that yes, this piddling difference became visible over time.
"So the lines show how far apart the points would be for 120 and the addition formula number if you made the circle to that radius?" Katie asked.
"Yep. And that's just what we can see," Jake added. "The difference matters long before that point."
When second block started, after brunch, Abdul rushed in, "Ahmed said we had a genius guest lecturer? Where is he?"
I faced a cranky crowd when I told them the genius had to go to class, so Jake will have to come back sometime soon.
Two months ago, Jake stopped by for a chat and I asked him about his transfer plans.
"Oh, I don't know. Four year universities, I'll have to take other classes, instead of what interests me."
"You can't be serious."
"Well, maybe in a few years. But I have to wait a while for the computer programming classes I need to take, and the math classes are more fun."
"Computer programming?"
"Yeah. That's what I want to….what. Why are you laughing."
"Do you know anything about computers?"
"No, but it's a good field, right?"
"I think you're one of the most gifted math students I've bumped into, and you've never shown the slightest interest in technology or programming."
Jake sat up. "My professor told me that, too. He said I should think about applied math. Is that what you mean?"
"Eventually, probably, but let's go back to why the hell you don't have a transfer plan."
"Well, should I go to [name of a local decent state university]?"
I brought up his school website, keyed in "transfer to [name of elite state university system]".
Jake looked on. "Wait. There's a procedure to apply to [schools much better than local decent state university]?"
"You will go to your counselor, tell her or him you want to put together a transfer plan. Report back to me with the results in no less than 2 weeks. Is that clear?"
"OK," meekly.
Just five days later, Jake's cousin, Joey, my best algebra 2 student, reported that Jake had a transfer plan started and was getting the paperwork ready.
So after this class, I asked him about transfer plans.
"Oh, yeah. I'm scheduled to transfer to [extremely elite public university] in fall of 2017. I've been taking all math classes, so I have a bunch of GE to take. But it's all in place." He grinned wryly. "I didn't think I'd be eligible for a school that good."
"And that's just the guarantee, right?"
"Yes, I want to look at [another very highly regarded public]. Do you think that's a good idea?"
"I do. You should also apply to a few private universities, just for the experience. It's worth learning if they give transfer students money." I named a few possibilities. "And ask your professors, too."
"Okay. And you don't think I should major in computer programming?"
"Do you know anything about programming right now? If not, why commit?"
"I don't know. I never knew about applied math possibilities. It sounds interesting."
"Or pure math, even. So you've got some research to do, right? And keep your GPA excellent with all that GE."
"Right."
"And at some point, you're going to think wow, I never would have done any of this without my teacher's fabulous support and advice."
"I already think that. Really. Thanks."
Just in case you think his visits pay dividends in only one direction.
14 responses to "Jake's Guest Lecture"
The sine and cosine formulas are the analytic expression of the fact that a rotation of angle theta1 followed by a rotation of angle theta2 is equivalent to a rotation of angle theta1 + theta2. That is to say the use of sine and cosine reduce composition of rotations to addition of real numbers.
The actual formulas are the result of composing the formula for a rotation of angle theta1 with the formula for a rotation of angle theta2. Just plug the formulas for a rotation of angle theta1 into the formulas for a rotation of angle theta2 and you get the formulas for a rotation of angle theta1 + theta2.
Yes it's kind of interesting. The sine-cosine addition formulas don't seem very intuitive but they are making in analytical language exactly the same statement as that the angles of rotations add up when the rotations follow one another which is a very intuitive geometric statement.
I am so glad to see teacher like you who really cares students' "understanding". Following is my idea about the sin-cosine addition law:
Let us see things in a triangle (the starting point of trigonometry).
sinA, sinB are actually the lengths of the opposite sides of the angle A and B in the triangle ABC (just take 2R=1 in the law of sine).
Therefore, sinA cosB = length of side b times cos(adjacent angle) = projection of side b on side c.
sinAcosB+sinBcosA = sum of projections = length of side c = sinC
C= pi -A-B, we get sin(A+B)=sinAcosB+sinBcosA. QED.
I think this is the "real geometrical" proof of the addition formula
Remark: sinA originally is the opposite side of A, in a right triangle; in view of the law of sin, sinA IS the opposite side of A in any triangle, as long as we see things geometrically—-taking 2R=1 means an expansion of the figure and does NOT change the GEOMETRY of the whole picture.
Hope that you like the geometric proof (which I invented some years ago)
That is a nice proof. But aside from the details of any proof of the sine-cosine addition formulas these formulas have a meaning and significance in themselves. And this meaning and significance is exactly that the assignment of real numbers to rotations as angles makes the addition of real numbers correspond to the composition of rotations thus serving to reduce the study of the geometry of the plane to the study of the addition of real numbers.
The fundamental law of trigonometry is
Rotation(theta1 + theta2) = Rotation(theta2) o Rotation(theta1)
in words – "A rotation of angle theta1 followed by a rotation of angle theta2 is equal to a rotation of angle theta1 + theta2".
The sine-cosine addition formulas are a somewhat disguised version of the above fundamental law. They are just the analytic equivalent of the fundamental law using the linearity of rotations.
[…] some tremendous discussions throughout the math community. I couldn't have been more pleased. Jake's Guest Lecture and The Test that made them go Hmmmm is an accurate representation of my classroom discussions. The […]
[…] to many students of all races and ethnicities who couldn't otherwise afford it–even after graduation. There are kids in top colleges today who once never had a thought of attending, because I had the […] |
Math club earns Guinness World Record certification
Students at a North Carolina middle school learned about scale, proportion, volume and other mathematical concepts during a yearlong project. The students built a paper pyramid based on the Sierpinski triangle fractal -- a mathematical design in which matching smaller pieces make up each large piece. The 9-foot-tall pyramid -- made of 1,024 smaller paper pyramids -- has earned the math club a place in the Guinness World Records. |
Science Articles You Can Use. Authors welcome
Pi Derivation
In mathematics number of math symbols and Constants are used. The word pi is one of the major constant in mathematics. The symbol pi is the Greek alphabet and pi symbol is denoted as (Pi, pi). The symbol pi is the 17th letter in the Greek system. The word pi is used in so many fields like mathematics and science operation. Pi is the irrational number that specifies value cannot be expressed exactly as fraction that is 'a/b' in these case a and b is the two integers. More about Pi Derivation:
Pi equation can be calculated by using various algorithms. Some of the pi equations are given below,
Geometric Derivation for pi:
Pi is empirically estimated by sketching a large circle, and then determine its diameter and circumference and then dividing the circumference by the diameter.
Consider Euclidean plane geometry pi is specified as the ratio of a circle circumference 'C' to its diameter 'd'
'Pi=C/d'
In above formula c/d is constant. A specifies the area of the square and derivation of pi is
'Pi=A/r^2' Brent-Salamin algorithm:
In 1975 when Richard Brent and Eugene Salamin independently find the Brent-Salamin algorithm which uses only arithmetic to double the number of correct digits at The algorithm consists of setting
'a_0=1' , 'b_0=1/sqrt(2)' ,' t_0=1/4′ , 'p_0=1'
Use iterating process
'a_(n+1)=(a_n+b_n)/2'
'b_(n+1)=sqrt (a_n,b_n)'
't_(n+1)=t_n-p_n(a_n-a_(n+1))^2'
'P_(n+1)=2p_n'
Using the 25 iterating process to find the pi equation
'Pi=(a_n+b_n)^2/4t_n'
Using BBB formula to find the Derivation for pi is
'Pi=sum_(k=0)^(oo)p(k)/(b^(ck)q(k))'
In the above equation b and c are positive integer and p, q are polynomial
Substitute 'b=2 ' and 'c=4 ' find the constant pi
'Pi^k=sum_(n=1)^(oo)(a/(q^n-1))+(b/(q^(2n)-1))+c/(q^(4n)-1)'
Using the above equation to find the pi value. Value for Derivation of Pi:
Pi does not have any physical unit. Pi value is already derived. The derived pi value is given below
Rational value of Pi (pi) =3,'22/7′ ,'355/133′ ,'103993/33102′ ,…
Binary value of Pi (pi) = 11.00100100001111110110…
Decimal value of pi (pi) =3.14159265358979323846264338327950288…
Hexadecimal value of pi (pi) = 3.243F6A8885A308D313….
Continued fraction of pi (Pi) =3,7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,…
The number π (/paɪ/) is a mathematical constant that is the ratio of a circle's circumference to its diameter. The constant, sometimes written pi, is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century. π is an irrational number, which means that it cannot be expressed exactly as a ratio of two integers (such as 22/7 or other fractions that are commonly used to approximate π); consequently, its decimal representation never ends and never repeats. Moreover, π is a transcendental number – a number that is not the root of any nonzero polynomial having rational coefficients. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straight-edge. The digits in the decimal representation of π appear to be random, although no proof of this supposed randomness has yet been discovered.
For thousands of years, mathematicians have attempted to extend their understanding of π, sometimes by computing its value to a high degree of accuracy. Before the 15th century, mathematicians such as Archimedes and Liu Hui used geometrical techniques, based on polygons, to estimate the value of π. Starting around the 15th century, new algorithms based on infinite series revolutionized the computation of π, and were used by mathematicians including Madhava of Sangamagrama, Isaac Newton, Leonhard Euler, Carl Friedrich Gauss, and Srinivasa Ramanujan.
In the 20th century, mathematicians and computer scientists discovered new approaches that – when combined with increasing computational power – extended the decimal representation of π to over 10 trillion (1013) digits. Scientific applications generally require no more than 40 digits of π, so the primary motivation for these computations is the human desire to break records, but the extensive calculations involved have been used to test supercomputers and high-precision multiplication algorithms.
Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, or spheres. It is also found in formulae from other branches of science, such as cosmology, number theory, statistics, fractals, thermodynamics, mechanics, and electromagnetism. The ubiquitous nature of π makes it one of the most widely known mathematical constants, both inside and outside the scientific community: Several books devoted to it have been published; the number is celebrated on Pi Day; and news headlines often contain reports about record-setting calculations of the digits of π. Several people have endeavored to memorize the value of π with increasing precision, leading to records of over 67,000 digits rogergordon If they pursue subjects like physics, economics, engineering, accounting and business in the college years, mathematics forms a part of the coursework. Guidance of experts is required so that students can understand the concepts with clarity. Qualified teachers in schools offer assistance to students and help them with the numerals. You can buy […]
by C_Dave A particle accelerator is a device that uses electromagnetic fields to propel charged particles to high speeds and to contain them in well-defined beams. An ordinary CRT television set is a simple form of accelerator. There are two basic types: electrostatic and oscillating field. In the early 20th century, cyclotrons were commonly referred Moms Clean Air Force As often seems the case, less developed countries seem to receive the bulk of the impact when it comes to negative world developments. The third world and climate modification is no different. Third World and Climate modification By definition, climate modification impacts the globe as a whole. That being said, […] |
The common Mandelbrot set is really a 2-dimensional slice of a 4-dimensional object identified by both the combination of the complex numbers Z0 and C in the canonical Zn+1 = Zn^2 + C. The mandelbrot set lives in the plane where Z0 = 0 + 0i, while the Julia sets live on infinitely-many-squared orthogonal planes in the remaining two dimensions, each one intersecting Mandelbrot's plane in a single point of complex coordinates C. |
According to the World Population Clock there are currently about 7.191 billion people alive. This year there have been 118 million births (or 264 per minute) and 49 million deaths (or 110 per minute), resulting in a net growth of 69 million people. Where will this end? Nobody can say for sure. But what we can be certain about is that the explosive growth has been slowing down for the past 40 y...
Read more of this blog post »
"what practical uses are there for functions? We can use them to establish a connection between the value of one physical quantity and another. For example, through experiments or theoretical considerations we can determine a function f(p) that links the air pressure p to the air density D. It would allow us to insert any value for the air pressure p and calculate the corresponding value for the air density D, which can be quite useful."
―
Metin Bektas,
Math Shorts - Exponential and Trigonometric Functions
"There are C(n, k) ways to distribute a given event k-times in a string of n events. Hence, and now comes the grand moment, the probability that a certain event, occurring with the probability p, occurs k times within n trials is thus: P = C(n, k) · pk · (1-p)n-k And that! Ladies and gentleman, is the formula for the binomial distribution in all its glory. If"
―
Metin Bektas,
Math Concepts Everyone Should KnowScience and Inquiry
— 3203 members
— last activity Jan 19, 2018 06:57PM
This Group explores scientific topics. We have an active monthly book club, as well as discussions on a variety of topics including Science in the NewThis Group explores scientific topics. We have an active monthly book club, as well as discussions on a variety of topics including Science in the News, new science book reviews, and any general science topic imaginable including history of science, physics, astronomy, biology, medical and environmental studies, to name but a few.
...more
Goodreads Authors/Readers
— 28030 members
— last activity 43The Magical Universe of Books
— 1201 members
— last activity 2 hours, 49 min ago
Ever lost track of time because you got lost in a book? This group has three purposes: 1 - Help authors to promote their books 2 - Help readers to conEver lost track of time because you got lost in a book? This group has three purposes:
1 - Help authors to promote their books
2 - Help readers to connect with authors.
3 - Act like a Library bulletin board where where people come, look, take what they need and post what they want to share.
Authors' Corner
General chat
Promotions on Ebooks/Freebies
And more...Come in, be welcome.
Remind me to add your book onto our virtual bookshelf
...more
Goodreads Librarians Group
— 79494Readers Delight
— 496 members
— last activity Dec 26, 2017 09:58AM
A group dedicated to find best deals and events for readers. From free to heavily discounted books to giveaways and competitions, Readers Delight aimsA group dedicated to find best deals and events for readers. From free to heavily discounted books to giveaways and competitions, Readers Delight aims at covering all of these and help readers find amazing delights.
Quality discussions are no less than a delight for readers. Readers Delight aims at hosting and promoting quality discussions on books and related topics as well....more
Amazon Kindle Prime Readers & KDP Select Authors
— 802 members
— last activity Dec 21, 2017 11:32AM
This is a group for readers who own a Kindle and are enrolled in Kindle Prime. Authors who have their books currently in the KDP Select program are enThis is a group for readers who own a Kindle and are enrolled in Kindle Prime. Authors who have their books currently in the KDP Select program are encouraged to post their book by genre and let readers know the dates that their book will be available to borrow, as well as the five promotional days when it is free to purchase. Hopefully this will mutually help readers have access to free books without having to sift through thousands and will help KDP Select authors get their books noticed. ...more
Science Fiction Aficionados
— 2259 members
— last activity Jan 22, 2018 02:44AM Study of the Mind: A Psychological Book Club
— 1973 members
— last activity Dec 18, 2017 10:27AM
This is a book club for those who love to read books about Psychology! Each month we will pick a book dealing with psychological topics, read it, andThis is a book club for those who love to read books about Psychology! Each month we will pick a book dealing with psychological topics, read it, and then discuss. ...more
Making Connections
— 11490 members
— last activity 5 hours, 10 your blog |
Introduction: Mouse Toe Numbering Scheme
Step 1: Look at the Cute Little Mouse Toes! (background)
Toe clipping is one of many methods of mouse-numbering used over the years in large lab colonies. The IACUC now recommends against toe clipping, instead preferring techniques that are less invasive and/or require lower levels of training. Properly clipped toes are easy to read, and make for unambiguous numbering. Even though you'll probably never need (or want) to clip a mouse's toes, the counting method is still of interest.
When you hold a mouse, they quite obligingly splay their toes for you to observe and count. (Because I don't keep rodents anymore, I found this nice picture on the internet.) They have four fingers and a thumb. This numbering method utilizes 1-2 fingers per paw, leaving the thumb intact for proper mouse motility.
Step 2: Scheme
The idea: toes are clipped to represent 0-9, with each foot representing a different power of ten.
Thus, the front right paw is the "ones paw", 1-9, (100); the front left paw is the "tens paw", 10-90, (101); the rear right paw is the "hundreds paw", 100-900, (102), and the rear left paw is the "thousands paw", 1000-9000 (103). Mix and match.
The mouse is held as shown, with splayed toes facing the reader. Toes are read left to right in all cases. The following demonstration only uses the 1's and 10's paws (since I can't do much with my hindpaws) but hopefully you can extrapolate.
Step 3: Count to 9
Pictures of 1-9, using my fingers. Not clipped, otherwise I couldn't really show you all the numbers.
1-4 are obvious, right? Then you move onto combinations of two, walking along left to right. Follow along on your own fingers if the progression doesn't immediately make sense.
Step 4: Count by 10's
Same thing, left forepaw.
Step 5: Test
Roll over to check your answers.
Now use this technique to signal your friends across a room.
Comments
This seems a LOT like the American Sign Language numbering system. I'd post a link, but that would require more effort. Just google ASL numbering and you should do just fine. ...
People clipped the toes of mice to number them? Definitely not the first thing I'd think of . . . the world is truly a strange, strange place.
I wonder why didn't they use binary. Using the binary system you get more numbers. If each toe is a binary digit you get 2 to the power of 16 = 65,536 combinations. ( more than six times 9,999). With a little practice each group of 4 toes could be converted into hexadecimal. (1-2-3-4-5-6-7-8-9-A-B-C-D-E-F) then you can represent the numbers in a compact , readable form ranging from 0000 to FFFF.
Thanks for the comment. I find that many IACUC policies on marking (and sacrificing) are often designed with the ultimate comfort of the human in mind. Having performed all of these procedures, and seen them all done both well and poorly, I have some complicated opinions that I won't get into here.
For convenience, let's consider toeing an old technique that gave rise to an interesting method of numbering. To avoid distracting from the neat counting method, I'll do some editing on the main instructable. |
People who bought this also bought...
Mysticism and MathematicsInfinitesimal
How a Dangerous Mathematical Theory Shaped the Modern World
By:
Amir Alexander
Narrated by:
Ira Rosenberg
Length: 12 hrs and 15 mins
Unabridged
Overall
75
Performance
68
Story
68
Pulsing with drama and excitement, Infinitesimal celebrates the spirit of discovery, innovation, and intellectual achievement - and it will forever change the way you look at a simple line....
An intriguing and underappreciated bit of history
By
Marino
on
09-22Finding Zero
A Mathemetician's Odyssey to Uncover the Origins of Numbers
By:
Amir D. Aczel
Narrated by:
Stefan Rudnicki
Length: 5 hrs and 59 mins
Unabridged
Overall
32
Performance
31
Story
31
The story of how we got our numbers - told through one mathematician's journey to find zero....
Not what I expected but I loved it just the same.
By
Darren
on
08-24A Mind for Numbers
How to ExcelHow to Bake Pi
An Edible Exploration of the Mathematics of Mathematics
By:
Eugenia Cheng
Narrated by:
Tavia Gilbert
Length: 8 hrs and 18 mins
Unabridged
Overall
246
Performance
218
Story
212
In How to Bake Pi, math professor Eugenia Cheng provides an accessible introduction to the logic of mathematics....
Mathematics is easy, life is hard.
By
Bonny
on
08-06-15Publisher's Summary
No number has captured the attention and imagination of people throughout the ages as much as the ratio of a circle's circumference to its diameter. Pi is infinite and, in The Joy of Pi it proves to be infinitely intriguing. With incisive historical insight and a refreshing sense of humor, David Blatner explores the many facets of pi and humankind's fascination with it - from the ancient Egyptians and Archimedes to Leonardo da Vinci and the modern-day Chudnovsky brothers, who have calculated pi to eight billion digits with a homemade supercomputer.
The Joy of Pi is a book of many parts. Breezy narratives recount the history of pi and the quirky stories of those obsessed with it. Sidebars document fascinating pi trivia. Dozens of snippets and factoids reveal pi's remarkable impact over the centuries. Mnemonic devices teach how to memorize pi to many hundreds of digits (or more, if you're so inclined). Pi-inspired poems, limericks, and jokes offer delightfully "square" pi humor.
A tribute to all things pi, The Joy of Pi is sure to foster a newfound affection and respect for the big number with the funny little symbol.
Story
A great start for those who love math.
I think that the book "A beautiful mind" did a great deal to rekindle the love for mathematics in many of us. In school I certainly remember the drudgery of mathematics, and with the exception of my 10th grade math teacher most instructors were simply un-inspiring.
I have recently started to read about the history and the theory of mathematics in my leisure and have found that it is a relaxing, albeit unorthodox diversion.
This book is excellent in terms of giving the history and providing interesting pieces of the fascinating people who have worked with this number. Unlike other reviewers, I found it captivating. As with many audio books, I would recommend obtaining a copy of the print version also, because some of the equations need to be "seen" rather than just heard to truly appreciate them.
This book is clearly too basic for people who are acquainted with mathematical history or theoretical aspects of recent math theory, but for someone like myself, who finds this kind of information interesting and challenging I give it my highest recommendation.
Interesting, if superficial and myopic
An enjoyable enough romp through the mystery, history, and personalities surrounding the elusive ratio, but after illustrating and celebrating the many paradigm-shifts involving the search for and understanding of pi, e.g., Archimedean or electronic, the author spends an entire chapter making fun of cyclometricians (circle-squarers), never entertaining (or admitting) that the next leap in pi studies (if there is such a thing) MIGHT be among them, and that those who in retrospect are now called visionaries in mathematics, were at one time considered cranks by the establishmentarians they displaced. Also, it could be difficult for someone not well versed in mathematics to follow the formulas recited in the audio format, but this is kept to a minimum, and you can always "rewind."
Did Not Reach Its Promise
Got off to a good start, but then sputtered. Not enough explanation on how the number is actually calculated or how it's used in the real world. Too much discussion on trivial things, not enough on how this concept came about and how it helps us.
Lost interest halfway through the book
Interesting concept of a book but sort of redundant. Even for a math major, this book is somewhat dry. There is only so many ways you can try to describe a number or theory without repeating yourself. I quit listnening halfway through the book.
Not what I expected
I had hoped for a thorough review of the place of pi in our understanding of the world, it's use throughout mathematics and science. But, alas, all I got was a basic human history of pi, evidently gleaned from a few other histories. the "pi on the side" asides became annoying, repetitive, and difficult to link to the main text. For audio "readers" the recitation of numbers and formulas may be too diffficult to hold in your head to follow the point being made. This book belongs in the history section so if you are reading it for the math or science, skip it.
Get A Life, People!
Wasting time listening to this audiobook is as non-productive as the people who spend time calculating Pi, squaring the circle or memorizing the digits of Pi. This book is absurd. The first third is an endless repetition of all the people who have calculated Pi all the way back to Genesis. The next third is about two Russian brothers that have wasted their lives calculating Pi to 8 bazillion digits. The last third is about people who waste their time memorizing the digits of Pi and the circle squarers who figure they have an answer for Pi. The mindless recitation of useless facts about Pi is made even more unbearable by the fact that the narrators feel compelled to imitate the accent of the characters they quote such as the Russian brothers, Norwegians, Indians, etc.
The only worthwhile thing about the book is the good 3 1/2 hours of walking I got in while enduring this agony. If you enjoy watching paint dry, give this one a try.
The Joy of Pi
Needless arrogance
Although the historicity of PI is fascinating and well represented by this book, the occasional slams against religion (the Bible mainly) were totally out of line and uncalled for. Just write books people and keep your agenda to yourselves. |
Unfortunately, there aren't many resources focused on analogies, especially for math, so you have to make your own. (This site exists to share mine.)
Modifying the Learning Order
It seems logical to assume we can present facts in order, like transmitting data to a computer. But who actually learns like that?
I prefer the blurry-to-sharp approach to teaching:
Start with a rough analogy and sharpen it until you're covering the technical details.
Sometimes, you need to untangle a technical description on your own, so must work backwards to the analogy.
Starting with the technical details:
Can you explain them in your own words?
Can you solve an example problem, describing the steps in your own words?
Can you create a diagram that represents how the concept fits together for you?
Can you relate the concept to what you already know?
With this initial analogy, layer in new details and examples, and see if it holds up. (It doesn't need to be perfect, but iterate.)
If we're honest, we'll admit that we forget 95% of what we learn in a class. What sticks? A scattered analogy or diagram. So, make them for yourself, to bootstrap the rest of the understanding as needed.
In a year, you probably won't remember much about imaginary numbers. But the quick analogy of "rotation" or "spinning" might trigger a flurry of recognition.
The Goal: Explanations That Actually Work
I'm wary of making a contrived acronym, but ADEPT does capture what I need to internalize a new concept. Let's stop being shy about thinking out loud: does a fact-only presentation really work for you? What other components do you need? I have a soft, squishy brain that needs the connecting glue, not just data.
Scott Young uses the Feynman Technique to explain concepts in everyday words and work backwards to an analogy and diagram. (Richard Feynman was a world-class expositor and physicist, and one of my teaching heroes.)
Beyond any technique, raise your standards to find (or create) explanations that truly work for you. It's the only way to have concepts stick.
Distributed Version Control is like sharing changes to a group shopping list with your friends.
Diagram / Example
Plain-English
We check out, check in, branch, and share differences ("diffs").
Technical
git checkout -b branchname git diff branchname
Combine ingredients with your own style. Steps might merge, but shouldn't be skipped without a good reason ("Zombies coming, no time for biochem, use this serum for the cure."). The site cheatsheet has a large collection of analogies.
Leave a Reply
64 Comments on "Learn Difficult Concepts with the ADEPT Method"
Sort by:
newest |
oldest
| most voted
mj deyoung "the man!" Please, keep on writing! You put many so called college "math professors" to shame! It's clear you have a Passion for what you do, where most of the phony Profs are merely there for the paycheck, no wonder this country (US) lacks serious engineering talent; it's our own fault for tolerating much too powerful teacher unions where tenure is a ticket to do whatever the hell one pleases, good for the profs, BAD for the students who PAY their salaries Really like ADEPT Method […]
Vote Up1Vote Down Reply
3 years 3 months ago
Mike Ryan appreciate your skill and style Khaled. Thank you for another great lesson in process.
Vote Up1Vote Down Reply
3 years 3 months ago
Sai Kalid,
You are doing a great job by making difficult to understand subjects intuitive and colorful. I wish my professors had done this.
Spending adept like methods to learn the interesting subjects in college days, took the time away from understanding the complex ones. Reading your articles give the joy of understanding some of those missed out subjects (curl, dot product). You have presented them in the most simple and lucid way.
Thanks a lot !!!
Cheers,
Sai
Vote Up1Vote Down Reply
3 years 1 month ago
Bhaskar Jha coming up with this wonderful concept of ADEPT. I also like to thank you for recommending the 'Learning about learning' course. It is very useful course and I am learning and enjoying it a lot.
Vote Up1Vote Down Reply
3 years 7 days ago
Evan Nicholson taken the Learning How to Learn course, Kalid, and was happy but not surprised to see you show up as the windup interview. I'm glad to see you offer it on this site, as well. ADEPT is a great tool Evan! I was really honored and humbled to be able to share my thoughts with the class. Glad the method's clicking for you =).
Vote Up0Vote Down Reply
3 years 3 months ago
Indra Lukman I recently discovered your absolutely marvelous site. I am struggling on note taking technique right now. What tools did you use on drawing those diagrams? They looked great Indra, thanks so much. I use PowerPoint to draw all the diagrams: there's many nice default formatting styles that look very clean out of the box. Here's an example Khalid,
Thank you! Much appreciated! Richard Feynman is my Guru too! His 'lectures on Electromagnetism' were recommended as our text book for one of our courses. Reading the lectures was itself so enlightening! I would go back and read it all over again like a favorite novel. I wonder what it would feel like to have been his student Khaled- I liked the diagram which explains negative numbers with a circle along which a number can travel, from plus 1 through zero, via i, than on below the line, where I lost you a bit. But I was wondering, instead of a circle, what if we represented the journey of a number with a sphere? Obviously cannot be shown easily in two dimensions. In a circle, we divide up the possible points into degrees,minutes and seconds. In a sphere, how do we divide up the much greater number of possible points important thing that you have to learn is to focus on the current task that you do. Try to concentrate on it so that you will be able toilliant stuff thanks! It confirms what I try to do on my best days, and then some! It'll be a really useful format to consider when I'm figuring out how to introduce new topics, especially for A-levelmj: Thanks for the encouragement =). Yes, unfortunately there's a pressure on professors to publish or perish (not teach or perish). I think positions should deb split into research-only and teaching-only. It's like forcing a chef to be a waiter, or a writer to be an actor too.
@Jonathan: Great question. As you suspected, we might want to move a number around in a 3d sphere, vs. a 2d circle. There are special numbers which have been invented to go to higher dimensions (called quaternions – which are actually 4d) and I hope to do some follow ups on them.
You can also skip labeling the individual dimensions, and start giving the angles you want (go forward 1.0, rotate 30-degrees in this direction, then rotate 45 degrees up, etc.). Part of math is figuring out the right system that helps describe what you want, or inventing a better one :).
@Anthony: Awesome, glad it'll be helpful. It's now my mental checklist that I have to fill out before I'm satisfied I really understood a topic.
@Mike: Thanks!
Vote Up0Vote Down Reply
3 years 3 months ago
NANDEESH H N Kalid,
How to explain that the sum of all positive integers is -1/12.
Will BE ADEPT help?
Vote Up0Vote Down Reply
3 years 3 months ago
Harish Dobhal way you put all your work in one single post! I really like this ADEPT approach which I think should be adopted by serious learners who want to grasp the thing and not just memorize a bunch of methods and formulae.
I am a big admirer of great physicist Richard Feynman and definitely your approach matches his style. Keep up the goodandeesh: Yep, in that case you might want to look at the existing proofs and work backwards. When explaining something to others, you start with the analogy and go forward, but with an existing result, you have to build up the plain-English, diagrams, etc. yourself.
@Harish: Thank you!
Vote Up0Vote Down Reply
3 years 3 months ago
Tony found this site when I was searching for a better explanation of imaginary numbers for myself and my students. WOW! this is how I like to teach. Sometimes I hit the mark, sometimes I don't, but with your site, I think the bulls-eye just got a whole lot bigger. I am definitely going to hang out here a lot! Glad you found the site, that's great to hear. I love sharing ideas & material with other teachers, feel free to poke around. Welcome aboard = questions. My booklet, All About Energy, makes for a nice introduction. Taking inspiration from BetterExplained's ADEPT method of introducing a concept, I've filled the booklet with analogies, pictures and some real factsThough I am usually not a phan of phorced acronyms ADEPT has actually been working for me as I try to explain just what and how it is that better explained explains. Keep up the awesome Mark =). I'd been struggling to describe the difference myself (I think I got lucky that I didn't have to search for more letters).
Vote Up1Vote Down Reply
3 years 2 months ago
Eric Vance post! Is there some place on this website (now or in the future) where readers can contribute their own ADEPT explanations? I'm sure many of your readers have great explanations and are willing to! I have a few projects underway to help make the site more collaborative — stay tuned : Cosines" gets you thinking about the mechanics of the formula, not what it means. Part of my learning strategy is rewording ideas into sense that make sense to […] |
"Looking at the style of writing, it was thought to be maybe eighth to 12th century. Everyone was deeply shocked by how old this is. There was so much mathematics going on in this region, bubbling away already in the third and fourth century."
In the manuscript, thought to have been a training manual for merchants, the symbol for zero appears as a dot.
Du Sautoy said: "Zero is one of the most important numbers, but it had to be invented. It's a very abstract idea: why do you need a number to count nothing? This was quite a shock for Europeans.
"European mathematics was very practical. It was about counting things. But the use of zero democratised mathematics — it allowed people to write down mathematics and to keep track of things. The authorities were using the abacus and could trick people because no record was being kept.
"It means that the common person can keep track of calculations."
The manuscript will go on show at the Science Museum in London on October 4 for its exhibition Illuminating India: 5000 Years of Science and Innovation, before returning to the Bodleian.
Bodleain librarian Richard Ovenden said: "Determining the date of the Bakhshali manuscript is of vital importance to the history of mathematics and the study of early South Asian culture, and these surprising research results testify to the subcontinent's rich and long- standing scientific tradition." |
No Right to Believe
Mathematics could save your life December 6, 2010
Three prisoners are brought to jail, but the jail is full. So the warden suggests the following procedure: Each of the three prisoners will have a colored hat placed on his head, either white or black or red. (Each color may appear any number of times — they may all get red hats, or two could be white and one black, etc.) Each prisoner will be able to see the others' hats, but not his own. Without communicating with each other in any way, they must each write down on a piece of paper what color they think their own hat is. If at least one of them guesses correctly, they will all go free; otherwise, they will all be executed. The warden gives the prisoners a few minutes to talk it over before the procedure begins. What should they do to ensure their survival?
I like this puzzle because at first it seems impossible: there is no way for a prisoner to deduce anything about the color of his own hat based on the hats of the others, so what strategy could possibly ensure that at least one of them will guess correctly? The key is to think like a mathematician, and look at the underlying structure of the scenario. Try to solve it on your own before reading on.
Let us assign a number to each of the colors: zero for white, one for black, two for red. Now, consider the sum of the three numbers corresponding to the colors of the prisoners' hats. We don't know what this sum is, but each of the prisoners knows two out of its three components. Next, think of the remainder we will get if we divide that sum by three: it must be either zero or one or two. For example, if there are two red hats and one white hat, then the sum is four, and the remainder when dividing by three is one. The idea is for each of the prisoners to be assigned one of the possible remainders, and each will guess the color which, when added to the other two colors he sees, yields a sum whose remainder matches the one he was assigned.
According to the strategy described, in this instance prisoner #0 would guess black, prisoner #1 would guess red, and prisoner #2 would guess black.
For instance, if prisoners zero and one are wearing red hats, then prisoner two will guess that his hat is black (bringing the sum to five and the remainder to two). He could be wrong, of course — but exactly one of the prisoners will always be right: the one who was assigned the remainder that matches the actual sum of the three colors. In the above case, if prisoner two's hat is white then prisoner one will guess correctly, and if prisoner two's hat is red then prisoner zero will guess correctly.
In case you were worried, this strategy can naturally be generalized for n prisoners and n hat colors. |
I wrote graphed this set of numbers years ago but neglected to note the crtieria upon which it is based. Perhaps someone more mathematically inclined could help me figure it out. The interesting thing about it is certain points of symmetry that exist within it (ie the left side mirrors the right... endlessly. If you graph it out you'll see what i mean, the values don't match exactly but the points at which peaks, valleys and plateaus do eerily). I've marked those points in italics. here goes: 0,1,1,1,2,1,3,1,4,2,3,1,8,1,3,3,8,1,8,1,8,3,3,1,20,,2,2,3,8,1,13,1,16,3,3,3, 26,1,3,3,20,1,13,1,8,8,3,1,48,2,8,3,8,1,20,3,20,3,3,1,44,1,3,7,32,3,13,1,8, 2,13,1,76,1,3,8,8,3,13,1,48,8,3,1,44,3,3,3,20,1,44..... there is more but i don't have the 2nd page on hand... gratsi ai
I am a nerd. Period. I love the entire universe of school, and everything about it starting with learing going into infinity. To prove my point, I have nine classes as opposed to the normal seven (but I'm studing for the equivalent of twelve classes). I seem to always be three or four steps ahead of my math teachers, and run to my science classes out of curriosity of what is going to be taught that day. I write my own philosophy and function on patterns which I point out left and right. Speak two languages fluently (English and Russian), one language I understand (Belarussian), and one I am currently learing (French-I started this year in French I, but got bored and skipped into French II-sad fact: I am doing better than the majority of the class, the way it is). I am so interested in joining this community because there are not enough people in school who I can just collaborate about these things (one friend, in particular, is excluded because he's the next Einstien).
If you take the number 6, and square it and then triangle that, you'll get 666.
By "triangle it" I mean 4+3+2+1 (Which is the same as [X²+X/2] if X is 4) So I mean 36+35+34+33+32+31+30+29+28.... I'm pretty sure there's a name for doing that that's better than "triangling", but I don't know it. In fact I'm kinda hoping some one can tell me. Anyway, [(6^4 + 6^2) / 2] = 666
can someone explain how phi fits in with the golden means spiral? are they two completely different things and is there a different number symbolising the golden MS or is phi representative of it? i googled but didnt come up with much
two weeks late and no one said happy Pi day!! Well today is Easter so it reminded me of the forgotten holiday (march 14th). What did you do on PI day? I am a high school math teacher so I had my students bring pie to eat and hand in pi posters they made.
I just joined because I like math. I particularly like math in relation to art, and art that has geometrical themes, like Escher and labyrinth patterns and spirals and stars and what not. I don't know very much math. I'm in a class called 'intermediate algebra' right now. I like to do tedious arithmetic.
I also think things about math, and I don't have an outlet for any of it, so I joined this community. ( For example...Collapse )
Does 4714280 have any significance? I might've done this wrong, but in hexadecimal notation I think it's 47EF28 (point zero), which is a funny coincidence if you ignore the 'F'. |
It's funny how we tend to find patterns in things. Left alone, we tend to gather, order, and number everything we can. Take the stars in the sky. Or the stock market. Even gambling. I loved it when I talked to my parents after an academic trip overseas. Here's how it would start: "How long […] |
AVAILABLE Last checked: 51 Minutes ago!
Dynamical systems theory is an area of mathematics used to describe the behavior of the complex dynamical systems usually by employing differential equations or . The simple question of how do you run is largely unanswered in the running community you have a bunch of pseudo guru styles like pose or chi but the key to running |
finding unexpected ways mathematics pops up around us
'I'm not a math person' no longer a valid excuse
This article was sent to me via the Jerry P. Becker listserv, and was originally published in the Business Insider on Monday, November 18.
Enjoy!
I'm Not A Math Person' Is No Longer A Valid Excuse
By Kelly Dickerson
Contrary to popular opinion, a natural ability in math will only get you so far in studies of the subject.
Research published in Child Development found that hard work and good study habits were the most important factor in improving math ability over time.
But bad attitudes about math are holding us back.
Most of us would never think that "I'm bad at reading," is a good excuse to stop taking English classes, so why is it ok, even normal, to say "I'm bad at math"?
A survey in 2010 conducted by Change the Equation found that three out of 10 Americans said they consider themselves bad at math. Over half of the 18 to 34-year-old bracket find themselves regularly saying they can't do math. Almost one-third of Americans reported they would rather clean a bathroom than solve a math problem. [See ]
Generally, people believe their learning ability works in one of two ways, according to research conducted by Patricia Linehan from Purdue University. [See ]We classify our learning abilities in a given subject as "incremental orientation" – the belief that we can continually improve our ability by studying and practicing, or we think about our learning as an "entity orientation" – the belief that we can't get any better no matter how hard we try. One person can have different orientations for different subjects.
Entity orientation toward math – basically saying, "I'm not good at math and so I never will be" – is a dangerous thing. When someone with entity orientation about learning math gets a math problem wrong, they think it's just an indication of the poor math ability they were "born with," according to a study published in Personality and Individual Differences in 2010. [See ]
This can have a very negative impact on motivation. If we don't believe we can improve, we won't bother trying.
Research shows that hard work, not natural ability, is the most important factor
The study mapped the progress of math ability in 3,520 students for five years – from grade five until grade 10. Students' math ability was measured by their performance on the PALMA Mathematics Achievement Test. Questions included basic arithmetic, algebra, and geometry. The researchers also asked the students to answer questions about their study habits and interest in math.
In the early grades, a high IQ generally meant a high math score. But it turns out natural talent will only get you so far. How students study made a big impact on how much their math ability improved. Students who simply relied on memorization when studying, and didn't attempt to make deeper connections with other areas of math, didn't show much improvement over time.
The researchers also found that where a student's motivation came from made a difference in their improvement. Students who said they wanted to get better at math simply because they were interested in the subject ended up improving more than those who wanted to get better in the interest of good grades.
"While intelligence as assessed by IQ tests is important in the early stages of developing mathematical competence, motivation and study skills play a more important role in students' subsequent growth," Kou Murayama, the lead researcher on the study, said in a press release [See ]
You can see the difference it made in the chart to the left. Students listed as high-growth believed they could get better at math the more they practiced and used in-depth study techniques. Students listed as low-growth were more likely to believe that math ability is something you're born with and it can't be improved, and they relied more on memorization when studying.
How can we change our attitude about math?
Not only do we hear "I'm bad at math" from our peers, but we're bombarded with messages that it's OK to be bad at math. For instance, there are shirts made for young girls that check off shopping, music, and dancing as their best subjects, but deliberately leave the box next to math unchecked. There are also shirts that say "Allergic to Algebra" and "4 out of 3 people are bad at math." [See ] |
Search This Blog
Experience the union between Math & Art in a RAFT workshop!
While Geometry often describes and measures shapes like a cone, sphere or triangle, does it define the shape of a cloud, a mountain, a coastline, or a tree?
Starfish and Broccoli
are a few examples of fractal patterns in nature!
It does! Most patterns in nature, called "fractals" describe curves, surfaces and objects that have some very peculiar properties so irregular and fragmented, that it takes more than spheres, cones, circles, triangles, smooth or straight lines to describe them!
These fractals might resemble arteries, coral, a heart, a brain, tree branches and other such 'designs' that have symmetry, 'self-similarity' and are scale-free!
Fern fractals - with symmetry
and self -similarity!
Look around you – Fractals are everywhere! Have fun with us and
discover everyday Fractals at the upcoming RAFT workshop 'Fractals and Beyond' on Feb 9th, at RAFT San Jose. This workshop will demonstrate how
fractals can be related to Sierpinski's gasket, to patterns, to
equations, to graphing, and to even a broccoli!
Break off a branch of the whole broccoli and what do you see? The smaller branch looks just like a miniature copy of the whole broccoli! Now think of self-similarity in ferns, the formation of shells, mountains, lightening, river estuaries, fault patterns, galaxies, musical compositions, and other nature's designs.
A fractal's dimension indicates its degree of detail, or crinkliness. Simple curves, such as lines, have one dimension. Squares, rectangles, circles, polygons and others have two dimensions, while solid objects such as cubes, spheres, and other polyhedra have three dimensions.
All those dimensions are integers: 1, 2, 3… But a fractal could have a non-integer dimension of 1.4332. By understanding fractal dimension, mathematicians can now measure shapes such as coastlines that once were thought to be immeasurable.
If you want to explore the world of Fractals, a science that marries Art with Math, join us for the upcoming workshop this week.
Discover your own fractal designs with RAFT's Hands on Activity Kit 'Freaky Fractals'!
Share your experiences with everyday Fractal patterns in nature here! Add your comments below |
Saturday, May 22, 2010
A few posts back, I described a session with some arts faculty to construct assessments for their new programs. I showed this graphic to represent the trial and error process of going from a question to an answer in a creative realm. During this week I worked on a problem that makes a good example of how this works in math.
The problem I wanted to solve is related to this one. Imagine you're building a space ship, which is composed of many components that all have to work together for the ship to function. Also assume that you know the probability that any one of these will fail. Those probabilities are fixed, but you can create redundancies to ameliorate them. If you only have a total number of components you can distribute among these redundancies, what's the optimal distribution? An example will make it clear.
Imagine a simple case where you only have two components A and B, with probabilities of failure 1% and 4%, respectively. You can use a total of 10 components in your design, so you could use five As and 5 Bs, or one A and nine Bs, or any other distribution that totals ten. Each redundancy reduces the chance of catastrophic failure. What's the best solution, to minimize this risk?
Intuitively we need more redundancy on the Bs, since they are four times as likely to croak. For this simple problem, we can see the answer by plugging everything into a spreadsheet and listing all the possibilities.
Our intuition is right. The best distribution is four As and six Bs. How do we find this in general? I worked on this problem over the course of a couple of days in my spare time. I scanned the notes and created a Prezi presentation out of it, to illustrate the interplay between
knowledge of different techniques (analytical)
trial and error
making connections and exploiting them
using examples to simplify the problem
using examples to check work for plausibility
seeking symmetries that make the problem easier to understand and solve |
The Slide Rule Explorer User's Guide
The slide rule explorer (SRE)
let's you explore more than 100,000 mathematical expressions that can
be evaluated on a slide rule. You enter an expression and then click
on Find to identify
slide rule procedures that will evaluate that expression.
One way to view the SRE is as
a device to look up the expressions described on
my slide rule
page and in my article
Many Formulas, Journal of the
Oughtred Society, vol. 18, No. 2, 2009, pp. 18-21. However, it
can also be used to explore more restricted scale combinations and to
analyze the capabilities of existing or newly designed sliderules.
To get started
SRE runs on any computer that supports
java. Download the file
SRE.zip, unzip it, and put the files you'll get into
a suitable directory. To start the software connect to that directory and type
java rule
in a suitable Command Window. (To get a command window in
Windows 10, for example, right click on the start button and then
click on "Command Prompt" in the resulting menu.)
A panel should appear that looks like the image on the top of this web
page. We will refer to it as the Control Panel. Using any of
the variables x, y, and z, enter a mathematical
expression in the Expression
Text Field in the second row of the control window and
click on the green button labeled Find (or just click on
Find if you wish to
know how to compute x*y*z). In the Procedure Text Field in the third row of the
Control Panel you will find a short description of how to
compute your expression with a slide rule. Click on the button
labeled Verbose to
see a more detailed description appear in the Command Window.
With the SRE, you can enter
expressions and see if and how you can evaluate these expressions with
your slide rule. You can also see a list of expressions that can be
so evaluated, and you can select the scales that are present on the
slide and body of your slide rule. The remainder of this page
describes how to use the controls in the
Control Panel, what kind of sliderule procedures are available,
how to select what scales you assume are present on your sliderule,
and how to log the results of your investigation.
Available Procedures
Depending on whether an expression has 1, 2, or 3 variables,
the SRE investigates one of four
procedures. These are described here briefly, and more fully on
my
slide rule page.
1 Variable: Table Lookup
You align all scales, find the variable x on Scale 1, and look
up the result on Scale 2.
Examples: If the two scales are the same, you won't be surprised to find x as
the result. If Scale 1 is the C scale and Scale 2 the CF scale you
find pi*x. Reversely, if Scale 1 is the CF scale and Scale 2 the C scale you will find x/pi.
2 Variables: Multiplication or Division
These are modeled after the ordinary multiplication and division
procedures which are probably the most frequently employed
slide rule procedures. However, instead of using the C and D scales
exclusively, the SRE considers
the use of other scales as well. For example, suppose you have a
sliderule with an A (x^2) and D (x) scale on the body,
and a K (x^3) scale on the slide, and you want to compute
x^(1/2)*y^(1/3) or x^(1/2)/y^(1/3). Proceed as follows:
Example, Multiplication: Find x on the A scale, align
the index of the K scale with x, move the hairline to y
on the K scale, and read the result
x^(1/2)*y^(1/3) under the hairline on the D scale.
Example, Division: Find x on the A scale, align
with y on the K scale, move the hairline to the index of the K
scale, and read the result
x^(1/2)/y^(1/3) under the hairline on the D scale.
3 Variables
This procedure is obtained from either of the two 2 variable procedures
by replacing the index of scale 2 with a number on another scale.
Example: To compute the triple product x*y*z you can
save one alignment over the ordinary repeated multiplication procedure
by proceeding as follows: Find x on the D scale, align x
with y on the CI scale, move the hairline over z on the
C scale, and read the expression x*y*z under the hairline on
the D scale.
Available Scales
All scales are referenced to the C (on the slide) and D (on the body)
scales. We refer to these two scales collectively as the CD scale, and to a value on
either of those two scales as x. Those scales are logarithmic, the logarithm
of x is proportional to the distance of x from the index of the scale.
The SRE incorporates a total of
13 scales. They are briefly described here, for more information see
Table 1 on the
slide rule page.
CD, x, the C and D scales, C on the slide, D on
the body.
CDI, 1/x, inverted scales, the CI scale on the
slide, DI on the body.
CDF, pi*x, the folded scales, CF on the slide, DF
on the body.
CDIF, 1/(x*pi), folded and inverted scales, CIF on
the slide, DIF on on the body.
AB, x^2, square scales, A on the body, B on the
slide.
W, sqrt(x), square root scale.
ABI, 1/x^2, inverted square scales, AI on the body, BI on the
slide.
K, x^3, cube scale.
KI, 1/x^3, inverted cube scale.
LL, E, exp(x), exponential scale.
L , log(x), logarithmic scale, the only scale to have a constant
increment.
S arcsin(x), measured in degrees.
T arctan(x), measured in degrees.
P, sqrt(1-x^2), the Pythagorean Scale.
H, sqrt(1+x^2).
SH, sinh(x) hyperbolic sine
CH, cosh(x) hyperbolic cosine
TH, tanh(x) hyperbolic tangent
Selecting Scale Combinations
The radio buttons
in the last 2 rows of the control panel let you select the scales that
are present on the body and the slide of a hypothetical slide rule.
The SRE will search only for
expressions that can be computed with those scales by the above
procedures. By default all scales are assumed to be present on both
slide and body. The total number of expressions, including
duplications, that can be evaluated with those scales are 324 for 1
variable expressions, 11,664 for 2 variable expressions, and 104,976 for
3 variable expressions. (The SRE
tries even more approaches by permuting the sequence of input
variables for the 2 and 3 variable expressions.)
The scale menu
in the first row of the control panel, next to the quit button, let's
you preselect the following combinations:
All, the default.
None, to start setting up a very basic sliderule with few scales.
Arithmetic, having the CD, CDI, CDF, CDIF, AB, W, and K
scales on both body and slide.
Aristo Scholar, the sliderule I used in High School, having
CD, AB, K, L, S, and T on the body, and CD, CDI, and AB on the
slide.
FC 2/83N, Faber Castell, the perhaps fanciest slide rule
ever made, see the pictures
on the main
page.
FC 5789, a much more basic Faber Castell, having CD, AB, K,
S, and T on the body and CD, CDI, AB, and L on the body. (It also
comes with a straightedge and a centimeter scale!)
K+E 68-1617, a simple rule with few scales,
having CD, AB, and K on the body, and CD, CDI, AB, L, S, and T on
the slide
Staedtler Mars 544A, another simple rule with
CD, CDF, AB, and K on the body, and CD, CDI, CDF, and CDIF on
the slide
Any of these selections can of course be modified by the radio buttons.
Listing Available Expressions
By clicking on one of the three List Buttons in the top row of the control panel
all 1, 2, or 3 variable expressions that can be computed with the current scale
combination are listed in the command window. You can also save them
in a log file, see the next section.
Logging
By clicking on the Logs Button in the first row of the control panel
you can cause the SRE to log its
output in a file called SRE.n.log . The log is an exact copy of what
is also printed in the command window. The number n is an
integer that gets incremented every time you log a session. (The
bigger it is the more you have worked, and the better you can feel.)
When logging is active the Logs Button is green, otherwise it is red. By
default logging is inactive. To close the log file properly make sure
you exit the session with the Quit button rather than by some other means (such
as turning off your computer.)
Finding Working Scale Combinations
Pressing on Find in
the second row of the control panel will find and print the next
working scale combination in the procedure text field and in the
command window. Pressing Find again will find the next expression, and so
on. Clicking on All will find all (remaining) scale combinations
and list them in the command window. You can go back to the beginning
of the list by clicking on
start over. For
each working scale combination there is a brief description in
the Procedure Text
Field. A more detailed descriptions of the last listed
scale combination can be printed in the command window by clicking on
the
Verbose button.
The SRE proceeds through all
available scale combinations and identifies those (if any) that work for the
current expression. For the 2 variable expressions it then tries
interchanging x and y. For example, x*y = y*x,
and so you can first enter x and then y, or vice versa,
using the same procedure. Thus, having all scales available on body
and slide, the product x*y can be computed in 42 different
ways. For the 3 variable expressions all 6 permutations of the input
variables are investigated. Thus the product x*y*z can be
evaluated in any of 96 distinct ways.
Entering Expressions
Expressions can be entered in the text fields in the second and third
rows of the control panel, or by selecting scale combinations in the
fourth, fifth, or sixth row. For example, if you wonder what you get when using the C
and D scales exclusively select CD on each of the scale menus in the fourth,
fifth, and sixth row of the control panel. (Select Multiply in
the last menu of the 2 variable row.) The expressions
x , x*y, and x/y*z will appear in the associated
text fields. (You can also use these text fields to store expressions
for later examination. In that case make sure not to touch the
associated scale menus after entering your expressions.) To find more ways to evaluate any of those expressions
click on the pertinent
Transfer button, and then
use the
Find,
All,
and
Verbose
buttons, as desired.
When entering expressions in a text field, note that for your input to
be effective it needs to be finalized by pressing the Enter key on the
keyboard.
Expressions are of the ordinary algebraic kind. By clicking on the
button labeled Syntax you get the following information in the
command window.
Note that while you can use the usual notations to input the hyperbolic functions and their inverses, they are represented internally in terms of exponentials and logarithm and will be so displayed on output.
If an expression cannot be interpreted, for example because there
is a syntax error, there will be an appropriate message in the command
window. If the expression cannot be evaluated with the currently
available scales the message
no working scale combos found
is printed in the text field in the third row of the control panel,
and in the command window.
Documentation
The main documentation of the SRE
is this web page. Clicking on Help prints this very brief help information in the
command window:
You may find it interesting to look at the files table1.data,
table2.data, and table3.data which come with this software. They
contain tables with available expressions in algebraic form,
equivalent versions in reverse polish notation, their scale
combinations, and information on how to test whether an expression entered by
you is equivalent. Do note that modifying these files in any way
may interfere with the proper operation of the
SRE software!
How It Works
The tables of computable expressions were generated with the symbol
processing language
Maple .
The SRE software converts those
to reverse polish notation for faster evaluation. It does the same to
any expression entered by you. To compare two expressions it
evaluates each at two sets of random numbers and compares the results.
If they agree within a certain tolerance the software deems the two
expressions equivalent. Thus it does not matter (within the narrowly
defined SRE syntax) how you enter
your expression. For example, it will recognize that any procedure to
compute x*y also works to compute ((x+y)^2-(x-y)^2)/4 if
you choose to enter that latter expression.
On the other hand, you want to enter expressions without unduly
limiting their domain. For example, the software will find 2 scale
combos for the expression
sqrt(1-x^2)/sqrt(1-y^2) but 4 for the seemingly equivalent
expression
sqrt((1-x^2)/(1-y^2)). The reason for this is that the first
expression cannot be evaluated in real arithmetic if the absolute
values of x and y are greater than 1, whereas the second expression
can.
Caveats
The SRE is restricted to the four
procedures described on this page. Of course many, in fact,
infinitely many, others are possible. For example, a product of any
number of factors can be computed in a straightforward fashion, but
the SRE will tell you that it
cannot find a scale combination to compute 2*x*y, and it cannot even
recognize expressions with more than 3 variables (and those 3
would have to be x, y, and z).
It is possible that some valid scale combinations are not recognized
by the software because some parts of an equivalent expression cannot be evaluated at the random numbers provided in the data files. It is also possible that
the software will deem two expressions equivalent when mathematically they aren't. These cases should be rare.
It is well known
to any slide rule enthusiast that there are many restrictions on the
ranges of the numbers that can be entered and manipulated on the
various scales of a slide rule. Scales may be split into two or more
subscales and it is up to you to figure out which is relevant for a
particular calculation. Scales may not contain information about the
location of the decimal point. In any calculation, slide rule based
or otherwise, you have to understand your problem before you compute
an answer, you need to check your answers, and you need to be vigilant
about the possibility of errors. So don't just identify a procedure,
apply it, and assume you have the right answer! |
Finding Patterns in the World
NUMBER: THE CONNECTING PATTERN OF EUROPEAN-DERIVED AUSTRALIAN LIFE
English-speaking mothers and fathers teach their babies the number names, counting pegs or people or pieces of toast. They riddle, they chant and they sing: 'One, two, three, four, five. Once I caught a fish alive .. .'. In school, learning the number names and how to use them becomes a central activity of the child's day. Counting money, telling time, following a recipe-the number system mediates all but the most intimate of human relations in the Western world. Numbers can be put on just about anything, and so give whatever it is that is being talked about a specific value. Using its value, people understand something of where it fits into the scheme of things.
In its most basic sense, a number series is a linguistic pattern that enables one to count or to add up separate things, encoding the practice of tallying on fingers. It might be considered a kind of 'linguistic trick' that provides a means of counting beyond ten. Numerals constitute an infinite series by having a base about which repetition occurs and rules by which any element may be derived from the element which precedes it. Decimal number systems associated with Indo-European languages have ten as their base unit. Thus, in a decimal system, ten is the point in the series which marks the end of the basic set of numerals. As each ten is reached, the basic series is started again. Other cultures have used other base units like duodecimal (base twelve) or vigesimal (base twenty).
One instinctively understands the use of fingers and toes for tally-keeping in a nonlinguistic way. One separated finger codes for one separated item, indicating, perhaps, the passing of a sheep through a gate or the filling of a vessel with grain. But if we then extend the coding operation and say a word which codes for the finger or toe, we have done something much more complex and much more useful. In saying a word as a finger is held up, we understand that the word does not name either the item, or the finger. It names a position in a progression. Thus, numerals, or number names, are a linguistic code with which we may record how far through the series of fingers and toes we have progressed, and how many times we have done it.
Notions of equivalence and hierarchy (and many other notions as well) are enabled through the system of number names. Each succeeding number has a higher value than all antecedent numbers. The number system encompasses Western life. Though it carries the appearance of utter neutrality, it also seems to fit 'naturally' with notions of economic hierarchy and competition over value. The working of the number system carries a powerful ideology: a particular image of orderliness across space and time which lends itself to particular political and economic ends. Imagine, for example, what it might mean to build capitalism in a culture in which kinship took precedence over number.
Many, perhaps most, everyday transactions in Western industrial society require the use of the number system:
'I didn't get up till twenty past eight this morning, so 1 was half an hour late for work.'
'Turn left at the second light, about a hundred metres down the road, the tax office is in the third building on your right, fifth floor.'
'One box of cereal – ninety-eight, chicken soup – two for one dollar, three hundred grams of hamburger – two fifty; that comes to four forty-eight; change of ten; four fifty, five, and five is ten.'
'Uncle Joe sold off the the back paddock. It's been subdivided into half-acre blocks, going for fifty thousand each.'
The apparent triviality of the examples above is an indication of how much the number system is taken for granted. The first case shows that everyday life is minutely divided into mathematical cycles which are themselves numerically related to lunar (monthly) cycles and solar (annual) cycles. The second case shows the spatial grid which (as we shall see in the next exhibit) is conceived of as covering the planet and, at least in cities, is also extended vertically in a precise numeric way. The third case shows, in everyday life, both the complexity and power of the number system as applied to trade. In a bartering system, such a transaction might have taken hours to negotiate and involved far richer and more intricate social interactions. The cereal would have been grain from the fields and the beef, an animal, thus ensuring that both parties to the transaction had some particular knowledge of the natural world, utterly unnecessary to a supermarket employee and customer. The final example (which will also be amplified in the next exhibit) shows how the land itself, to a significant degree, is understood in terms of numeric cycles: length and width establishing area within selected boundaries, and the market establishing monetary value.
Learning to count comes first, then the Western child learns to measure: to give numeric value to parts of things, and to subdivide 'the flowing face of nature' into arbitrary units which can be counted. Thus, the notion of size is made exact by developing concepts, or qualities, such as length, width, area and volume. Again, these concepts are so taken for granted that adults tend to assume them to be qualities of nature itself, rather than conceptual tools which society has developed for talking about and manipulating the natural world. It is primary school teachers in contemporary Western society who must regularly explain what these qualities actually mean. They are wise enough to teach children to understand things like length by asking them to carry out the practices employed when numbers are put on length (that is, giving them a ruler and setting them to measuring). The qualities themselves are elusive and mysterious things, and so they are objectified and naturalised. Though not 'facts of nature', they are what we might call 'social facts'. They are used in good faith by Westerners as 'real things' to construe the natural world as so divided.
Obviously it is easier to integrate numbers, counting and measuring into the social scheme of things if the ordinary language that people use has them focusing on spatially individuated things, as we saw English doing in Exhibit 2. In this situation, counting seems to follow naturally from talking. This leads us on to the question of 'natural kind', which we also considered earlier. Part of using the number system is deciding to what units numbers should be applied; the decision must be made as to precisely what to count, and what constitutes a unit in measuring.
In Exhibit 2 we considered some of the conceptual problems associated with the questions of where to draw lines and how to establish natural categories within nature. Some types of things in nature are much more amenable to quantification than others. For example, it is much easier to assign numeric value to trees and to the board feet of timber they contain than it is to put numbers onto the complex ecological relationships which those trees sustain. Furthermore, a monetary value can immediately be placed on cut timber so measured, whereas we have not yet learned to measure the monetary loss in terms of land degradation, long-term job loss, tourist amenity, cultural equilibrium and ecological health. This is only one well-known example of how our choice of what to quantify in nature has a direct and often devastating impact on our lives and those of our children, on our culture and on nature itself. It shows how the use of number conjures up the notion of value in a particular way.
We have seen how the number system constitutes one kind of grid which can be 'placed over' the lumpy, bumpy landscape and make that landscape easier for people to deal with and control. Numbers mediate our involvement with the world, and their use is intimately tied up with the power and remarkable achievement of modern industrial society. But using numbers is helpful in specific sorts of ways. For example when numbers are used in statistical ways, as they have been used to assemble the maps in ITEM 4.2, vast ideas can be precisely conveyed. Yet at the same time something is lost.
Let's assume you are somewhere in Australia on a particular day in the year, holding in your hand the grid maps of ITEM 4.2. You wish to predict the weather for the next few days; you wish to know what vegetation grows just over that rocky ridge; you wish to find fresh food. Maps of this sort-based on averages, on highly generalised calendrical cycles, focusing on particular variables but entirely ignoring others-such maps tell you virtually nothing about what you as an individual will actually find in the real world on that day. (Refer to Maps are territories, pp.19-27, for the distinction between indexical and non-indexical texts.) It is no exaggeration to say that a person skilled in bushcraft has no need whatever for these sorts of data, except perhaps to start a campfire. But at the same time let us not forget that for particular purposes they enable a land never visited to be known and controlled from a great distance. And that has profound implications for us all.
Knowledge which is superior for some purposes may be useless, possibly even counterproductive, for other purposes. The uses to which knowledge can be put are directly related to the conditions and assumptions of its production and the form in which it is communicated. The trick is to remember that knowledge, no matter how well dressed in number, is never value-free. This insight opens up the understanding that both of the knowledge systems coming together in ganma have their limitations and their uses.
We could not extricate number from Western culture, even should we wish to do so. Solving the problems which number has created in Western society is a twofold project: firstly, we must find ways of letting number speak forcefully for values other than monetary aggrandisement and short-term economic growth. Secondly, we must learn how to revitalise the values which are more readily understood and preserved within other ways of patterning knowledge, for example, the kinship system. |
ABSTRACT:An
intrepid gambler plays a simple game for a dollar. With probability
"p" he wins any trial and with probability "1-p" he loses
the trial. He begins the game with an initial fortune F and plays repeatedly
until either he has won
his goal of G
dollars (he would then have F+G dollars) or until he has no money left.We will try to compute the probabilities of
these events given various values for "p", F, and G. The problem is
easy to understand and the interest is in the various ways to solve the
problem, including guessing, matrix manipulations, and summing series. As an
analyst, I will choose the series approach.
ABSTRACT:Mathematical symbolism generally-and symbolic algebra in particular-is
among mathematics' most powerful intellectual and practical tools. Knowing
mathematics well enough to use it effectively requires a degree of comfort and
ease with basic symbolics. Helping students acquire symbolic fluency and
intuition has traditionally been an important, and sometimes daunting, goal of
mathematics education. Cheap, convenient, and widely available technologies can
now handle a good
share of the
standard symbolic operations of undergraduate mathematics.Does it follow that teaching these topics,
and even some of the techniques, is now a waste of time?The short answer is "no."The key question is how to help students develop
"symbol sense" and, above all, a feeling for mathematical structure.An answer concerns choosing mathematical
content and pedagogical strategies wisely, in light of technology, to highlight
what matters most.
9:40 - 10:00Art
Inspired by Mathematics in Minnesota
Lisl
Gaal; University of Minnesota, Minneapolis
ABSTRACT:In the
December 2000 issue of the MAA Focus Magazine there is an article on "Art
Inspired by Mathematics in New York" by Ivars Peterson.The only illustrations in this article are of
sculptured Moebius strips, although the show also included paintings.I shall show six simple examples of
illustrations from combinatorics (counting), geometry, group theory, probability
,and transfinite arithmetic.
10:05 – 10:25Betting
on the Outcome of the NBA Final: Can One Make Money?
M.
B. Rao; North Dakota State University, Fargo
ABSTRACT:In this
talk, I will explain how to set up a system of linear equations for betting on
NBA teams in winning the final. The technique is applicable to betting in horse
racing and some casino games. The treatment is accessible to any one who has
some rudimentary knowledge of linear algebra.
10:30 – 10:50Take
Putnam Problem B-1, 2001 for Example
Loren
Larson: Northfield, MN
ABSTRACT:I'll
present a number of solutions, including a natural approach that several
students attempted, but none successfully. I'll conclude with some
related problems of a recreational nature. The point of the title is that
a good problem produces good mathematics.
Miller
Center B-31: Morning Session 2
9:15
– 9:35Gnomonic Pythagorean
Triples
Dale
Buske; St. Cloud State University
ABSTRACT:A gnomon is a connected figure G which when suitably
attached to figure F produces a third figure similar to F. A characterization of all Pythagorean triples having
Pythagorean triangles as their gnomons is given. From this
characterization it will follow thatfundamental
Pythagorean triples do not have Pythagorean gnomons.
9:40 – 10:00A Partial Differential-Difference
Equation
Namyong
Lee; Minnesota State University, Mankato
ABSTRACT:In this
paper, we study a certain partial differential-difference equation that arose
from a mathematical modeling project.We
show the idea of how to construct the solution and its asymptotic behavior.
10:05
– 10:25Bijections Needed:Some Open Problems in Partition Theory
Tina
Garrett; Carleton College
ABSTRACT:We will
review the basic definitions in partition theory.We then describe several of the traditional
notations that are commonly used in bijective proofs of partition theorems,
including the Ferres diagram and Frobenius
notation.Several known theorems and conjectures are
stated for which bijective proofs may be expected but do not exist.
10:30
– 10:50New Ways of Teaching
Mathematics of Interest and Life Contingencies
Ken
Kaminsky; Augsburg College
ABSTRACT:As part of our quantitative literacy program at Augsburg College,
I teach a course on the mathematics of interest. The course is a popular
one for non-majors as the topic of money is fascinating to students but the
traditional actuarial notation used by existing texts often obscures the key
results. For example, although simple annuities-certain differ from one another
only by a factor related to the valuation date, actuaries have given a distinct
notation for virtually every particular case. Recently, I decided to buck
this trend and unify the study of annuities using a comprehensive
formula. In this talk I will present new approaches to teaching using
this result. I will also discuss extensions to varying annuities and
insurances and I will discuss the implications for teaching both non-majors and
majors.
BREAK
Ann Watkins,
President of the MAA; California State University, Northridge
ABSTRACT:We will
have some fun demolishing several enticing examples that commonly are used in
elementary statistics textbooks to illustrate the mean, median, and mode. Some
mathematics backed up by a little data show that these concepts are not as
intuitive as they appear.This talk is
actually more sophisticated than it sounds and includes some nice applications
to calculus.
1:40 – 2:00Distinguishing
Gamblers from Investors at the Blackjack Table
David
Wolfe; Gustavus Adolphus College
ABSTRACT:A
skillful blackjack player, one who counts cards, maintains some information
about the distribution of cards remaining in the deck at all times.The player adjusts both betting style and
play based on this "count" information.Depending on the rules used by a particular
casino, the skillful player may have a slight edge over the casino.Without knowing exactly what the player is
counting, we would like to write a program which is able to assess the player's
playing skill
2:05 – 2:25WeBWork – A Web-Based Homework
System
Charles
Pastor; Gustavus Adolphus College
ABSTRACT:WeBWorK
is an online system for creating and grading homework problems.Problems can be individualized and students
receive immediate feedback.In this
presentation I will introduce WeBWorK, demonstrate how our
Department is
using this program and discuss student responses. WeBWorK is distributed for
free by the University of Rochester.
Miller
Center B-31 Afternoon Session 2
1:40
– 2:00End Base Discriminatin
Now!
Tom
Sibley; St. John's University
ABSTRACT:Do
you find it unfair that relatively awkward fractions like 17/32 can have finite
decimal representations, whereas seemingly simpler fractions like 1/7 confront
us with infinitely repeating decimals?Would you like familiar numbers such as e and pi to have easily recalled
decimal representations?We will explore
alternative bases to achieve nice representations and encounter some pretty
mathematics and open conjectures along the way.
2:05
– 2:25Disjunctive Rado
Numbers
Dan
Schaal (presenting) and Brenda Johnson (student); South Dakota State University
ABSTRACT:Rado
numbers are an area of discrete mathematics with applications in computer
science.In this talk we will present a
new variation of the classical Rado numbers.Brenda Johnson is an undergraduate student at SDSU.This talk is based on joint research
conducted by the authors as part of Johnson's Honors Program. |
Day 2
A 12-fold quasicrystal: a two-dimensional arrangement of atoms that can never fully repeat itself. The arrangement can be duplicated using three types of tiles, shaped like squares, flattened rhomboids (figures with four equal sides), and equilateral triangles. [Credit: Wolf Widdra]
Many solids are crystalline: the atoms are arranged in repeating patterns, such as cubes (with several variants), hexagons, and other regular structures. Others are irregular, and we call that type "glass". Between those are the quasicrystals: solids that have regular structures, but ones that never repeat.
Think of it this way: if you have a set of square tiles of a single color, you can lay them out on a flat surface with no gaps, and the same pattern repeats forever, like a checkerboard. You can do the same with equilateral triangle tiles or hexagonal tiles (which are commonly used in board games as well). Square tiles exhibit four-fold symmetry; hexagonal tiles exhibit six-fold symmetry.
However, five-fold symmetry doesn't work if you want an endlessly repeating pattern: if you try to tile a surface with pentagons of the same size, you'll leave gaps. The solution is to create five-fold symmetry without repetition; you can do it most simply with two sizes of rhombus, though other options also exist. You'll fill an entire surface with these shapes, but the pattern will never repeat, even if you had an infinite number of tiles. |
In the ancient Chinese book Chiu-chang Suan-shu (Nine Chapters on Arithmetic, 九章算术), estimated to have been written some time around 200 B.C., there appears a problem:
A bunch of roosters and rabbits are prisoned in the same cage. One counts from the top, and sees thirty-five heads. One counts from the bottom, and sees ninety-four feet. How many roosters and rabbits are there?
A solution by Sun Zi is as follows:
Suppose one chops off half of the feet for each rooster and rabbits, and is left with forty-seven feet, which accounts for one foot each rooster, and two feet each rabbit. If all the thirty-five heads belong to roosters, then there are supposed to be thirty-five feet. However, for each rabbit, the count for feet increase by one. Since there are twelve more feet in the actual count than thirty-five, the number of rabbits is twelve. There are twenty-three roosters. |
Described even today as "unsurpassed," this history of mathematical notation stretching back to the Babylonians and Egyptians is one of the most comprehensive written. In two impressive volumes -- first published in 1928-9 -- distinguished mathematician Florian Cajori shows the origin, evolution, and dissemination of each symbol and the competition it faced in its rise to popularity or fall into obscurity.
Illustrated with more than a hundred diagrams and figures, this "mirror of past and present conditions in mathematics" will give students and historians a whole new appreciation for "1 + 1 = 2."
Swiss-American author, educator, and mathematician FLORIAN CAJORI (1859-1930) was one of the world's most distinguished mathematical historians. Appointed to a specially created chair in the history of mathematics at the University of California, Berkeley, he also wrote An Introduction to the Theory of Equations, A History of Elementary Mathematics, and The Chequered Career of Ferdinand Rudolph Hassler. |
Pages
What is mathemafiction?
Sunday, January 9, 2011
January dribbles - 7
They'll change the world with their words, with their recital of others' words, with their endless urge like tides to make more words. They'll change if they can ever agree, or nothing changes, like waves washed under the sea. |
National Pi Day: History's Great Mathematicians
Today is National Pi Day, and we want to celebrate by highlighting some of history's most amazing mathematicians (in addition to eating a big slice of pie!).
Some cool facts about Pi:
It has been represented using the Greek letter "π" for the past 250 years.
It is a mathematical constant that's special, unique, and significant in its own way.
It is the ratio of a circle's circumference to its diameter.
It never ends or settles into a repeating pattern.
It is the most recognized mathematical constant.
Computing the value of Pi is a stress test for computers.
Five of history's most interesting mathematicians:
Grace Hopper (aka "Amazing Grace") was an American computer scientist and United States Navy rear admiral. As a child Hopper would dismantle household gadgets, specifically alarm clocks, to figure out how they worked. During WWII Hopper decided to take a leave of absence from Vassar where she was working as an associate professor of math and was sown into the U.S. Navy Reserve as a volunteer. A pioneer in her field, she worked at Harvard University for the navy and was one of the first programmers to work on a computer called Harvard Mark I that was used in the war effort. On top of it all, she invented the first compiler for a computer programming language.
William Playfair was the founder of graphical methods of statistics, in other words charts and diagrams. He was a Sottish engineer and political economist who invented four types of diagrams: the line graph, the car chart, the pie chart, and the circle graph. Born during the Enlightenment – a Golden Age when the arts, sciences, industry, and commerce were all thriving – Playfair was involved in many different careers. He was an engineer, accountant, inventor, silversmith, merchant, investment broker, economist, publicist, land speculator, editor, journalist, the list goes on.
Ada Lovelace is considered to be the world's first computer programmer. She earned this title after working on one of the earliest mechanical general-purpose computers called the Analytical Engine. The notes she took on this project are recognized as the first algorithm intended to be carried out by a machine. This has earned her the title of "first computer programmer." As a young child Lovelace showed signs of being highly influence by math and science, and her parents pushed her to pursue this talent.
Isaac Newton is best known for having developed the theory of gravity and physics, but he also invented calculus (as did Gottfried Leibniz, who he had many disputes with over this topic during his life). This Englishman formulated laws of motion and universal gravitation using mathematical processes. Born on Christmas Day, Newton was known to be an independent person who never married. His work in science and math are some of the core foundations on which many other developments were made.
Sofia Kovalevskaya was the first major Russian female mathematician. She contributed major original advances to analysis, differential equations, and mechanics. She was the first woman to ever be appointed to full professorship in Northern Europe and was one of the first women to work for a scientific journal as an editor. Born in Moscow, Kovalevskaya studied in Germany by auditing courses at a German university. For a long time she tried to build up her career but because she was a woman she was unable to. Finally she was accepted as a professor in Stockholm, Sweden.
Catherine, the co-founder of Carpe Juvenis, recently graduated from the George Washington University. When she's not writing or editing for Carpe Juvenis or promoting "Youth's Highest Honor," Catherine can be found searching for the best bubble tea in town. Suggestions are welcome on Twitter @CathMeowJessen! |
Nearly every American who has become a parent in the last decade has heard the slogan, "breast milk is best," and has likely been encouraged to offer breast milk to newborns. Among other things, breast milk contains naturalInfants who were breastfed for less than six months before starting infant formula milk and infants who had mothers who were obese at the start of pregnancy, were much more likely to develop nonalcoholic fatty liver disease ...
Formula
In mathematics, a formula (plural: formulae or formulas) is an entity constructed using the symbols and formation rules of a given logical language.
In science, a formula is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. Colloquial use of the term in mathematics often refers to a similar construct.
Such formulae are the key to solving an equation with variables. For example, determining the volume of a sphere requires a significant amount of integral calculus; but, having done this once, mathematicians can produce a formula to describe the volume in terms of some other parameter (the radius for example). This particular formula is:
Having obtained this result, and knowing the radius of the sphere in question, we can quickly and easily determine its volume. Note that the quantities V, the volume, and r the radius are expressed as single letters. This convention, while less important in a relatively simple formula, means that mathematicians can more quickly manipulate larger and more complex formulae.
Expressions are distinct from formulae in that they cannot contain an equals sign; whereas formulae are comparable to sentences, expressions are more like phrases.
In a general context, formulae are applied to provide a mathematical solution for real world problems. Some may be general: F = ma, which is one expression of Newton's second law, is applicable to a wide range of physical situations. Other formulae may be specially created to solve a particular problem; for example, using the equation of a sine curve to model the movement of the tides in a bay. In all cases however, formulae form the basis for all calculations. |
The Joy of x: A Guided Tour of Math, from One to Infinity pdf
The Joy of x: A Guided Tour of Math, from One to Infinity by Steven Strogatz
Language: English
ISBN-10: 0544105850
Publisher: Mariner Books; Reprint edition (October 1, 2013)
Paperback: 336 pages
"Delightful . . . easily digestible chapters include plenty of helpful examples and illustrations. You'll never forget the Pythagorean theorem again!"— Scientific American Many people take math in high school and promptly forget much of it. But math plays a part in all of our lives all of the time, whether we know it or not. In The Joy of x , Steven Strogatz expands on his hit New York Times series to explain the big ideas of math gently and clearly, with wit, insight, and brilliant illustrations. Whether he is illuminating how often you should flip your mattress to get the maximum lifespan from it, explaining just how Google searches the internet, or determining how many people you should date before settling down, Strogatz shows how math connects to every aspect of life. Discussing pop culture, medicine, law, philosophy, art, and business, Strogatz is the math teacher you wish you'd had. Whether you aced integral calculus or aren't sure what an integer is, you'll find profound wisdom and persistent delight in The Joy of x .
Click on the below link below for The Joy of x: A Guided Tour of Math, from One to Infinity pdf free download, whole book
Recommended books
One the most helpful guidebooks ever written about New Jersey, this guide covers hundreds of day trips--including New York City, Philadelphia, Bucks County, and more. Visit zoos, museums, historic sit...
This is a heart warming tale of a medical mission to Malawi and how the experience of helping others changed a life. Operation Smile is an international organization that performs corrective surgery o...
See the best of Prague with this streamlined walking guide, complete with 12 step-by-step itineraries and maps, to help you explore the city like a pro and navigate like a local. Created in a handy, t... |
Struik: A Concise History of Mathematics: The Orient after the Decline of the Greek Society
by Carson Reynolds
Despite Hellenistic influence, Near Eastern thought remained intact, as is evidenced by work in Alexandria, India, and Constantinople. The Byzantine Empire served as a guardian for Greek culture while the Indus region and Mesopotamia became independent. The sudden growth of Islam ended Greek domination. Arabic administration and language competed with and conquered Greek culture in much of the Mediterranean.
As the roman empire declined the center of math research shifted from Alexandria to India and Mesopotamia. The Surya Siddhanta shows an influence of Greek and Babylonian astronomy. Aryabhata (c. 500) and Brahmagupta (c 625) were the best known. Mahavira considered rational triangles and quadrilaterals. General solutions for indeterminate equations of the first degree (ax+by =c) is found in Brahmagupta. Bhaskara admitted negative roots of equations and his Lilavati became a standard text for arithmetic and mensuration. Nilakantha (c. 1500) had already found the Gregory Leibniz series for pi/4.
Our present decimal-position system first appeared in China and was used increasingly in India (c. 595). The word sunya although the use of a dot predates this in Babylonian texts. "0" probably comes from the Greek ouden (nothing). However, in Hindu math, zero was equivalent to 1…9, not just a holder as with the Babylonian dot. Translation of the Siddhantas into Arabic introduced the Hindu system to the Islamic world, where permutations of it (the gobar system made their way to Spain and to the West.
Persia and Baghdad were taken by Arabs, causing Arabic to be instated as the official language, although other cultures remained. Islamic math was influenced by the same factors as Alexandria and India. The caliphs promoted astronomy and math, creating libraries and observatories. Muhammand ibn Musa al-Khwarizmi (c. 825) wrote a book whose Latin translation (Algorithmi de numero Indorum) spread the decimal position. The word "algorithms" is a latinization of his name. Similarly his Hisab al-jabr wal-muqabala (science of reduction and confrontation or science of equations) introduced al-habr or algebra into the lexicon. Although lacking formalism and mostly geometric, his examples (i.e. x^2+10x=39) were a thread appearing in algebras for several centuries. He also included trigonometric tables. His geometry, while simple can be traced to a Jewish text of 150 CE. His work lacked the axiomatic foundation, but was important for the introduction of decimal position to the West.
Arabic scholars also faithfully translated Greek classics into Arabic: Apollonius, Archimedes, Euclid, Ptolemy (Almagest being the familiar name for his Great Collection). Arabic math was particularly interested in trigonometry (sinus is a latinization of the sanksirt jya). Sines were half a chord, and were thought of as lines. Al-Battani provided extensive cotangent tables (for every degree) Abu-l-Wafa inroduced secant and cosecant, and derived the sign theorem of spherical trig. Al-Karkhi (d. c. 1029) was monomaniacal interested in Greek and wrote an algebra inspired by Diophantus and was interested in surds (sq roots).
Omar Khayyam (c. 1038-1123), who lived in northern Persia near Merv, was notable for a reformed Persian calendar with an error of one day in 5000 years (compared to 330 years w/ the Gregorian). His Algebra examined cubic equations and determined root as the intersection of two conic sections. He also introduced a non-Euclidean geometry. Nasir al-din separated trig from astronomy and attempted to prove Euclid's parallel axiom, which was made use of in Renaissance Europe by John Wallis. Nasir followed Khayyam's approach to theory of ratio and the irrational. Jemshid Al-Kashi (d. c. 1436) was influenced by Chinese mathematics and knew of (what is now called) Horner's Method, iterative methods. He also provided the binomial formula for a general positive integer exponent. Al-Kashi had pi to 16 decimals. Ibn Al-Haitham, whose Opticswas influential, solved the problem of Alhazen. He also employed the exhaustion method. Abu Kamil (a follower of Al-Khwarizmi) had influenced on Al-Karkhi and Leonardo of Pisa. Al-Zarqali, was notable for the Toledan tables, which influenced the Alfosine tables, which were authoritative trig tables for centuries.
Chinese mathematics was not isolated. It developed at least by the Han Dynasty, the decimal position system was probably invented there. Pi was found to many decimal places (Liu Hui had two digits, Tsu Ch'ung-Chih, had seven [22/7]). During the Tang dynasty, imperial examinations made use of math books, spurring the printing of Nine Chapters as early as 1084. The Sung dynasty saw greater progress. Wang Hsio Tung exceeded the Nine Chapters by looking at cubic equations of a higher complexity. Ch'in Chui -Shao used successive approximation to solve higher degree polynomials (similar to Horner's work of a much later date). Yang Hui (c. 1261) used a decimal notation similar to our modern style. He also provided the earliest extant pascal's triangle. Chi-Shih-Cieh, the most important Sung mathematician, extended "matrix" methods to solve linear equations with several unknowns and of a high degree. The post-Sung period saw a decline, but diffusion of these developments westward. |
New Scientist presents...
Instant Expert: Mathematics in the Real World
Mathematics underpins almost everything. From the algorithms that rule the digital age to the basic principles of nature, mathematics is front and centre. So come to our one-day masterclass to really get to grips with the language of the universe, as six experts lead you through the wonderful world of mathematics.
Speakers:
Helen Wilson, professor of applied mathematics at University College London
Holly Krieger, lecturer, director of studies and fellow in maths at Murray Edwards College, University of Cambridge
Chris Budd, professor of applied mathematics at the University of Bath
Adam Kucharski, assistant professor and Sir Henry Dale Fellow at the London School of Hygiene and Tropical Medicine
James Grime, mathematician and panel tutor in cryptography, formerly of University of Cambridge
Eugenia Cheng, mathematician and scientist-in-residence at the School of the Art Institute of Chicago and a Honorary Fellow in Pure Mathematics at University of Sheffield
Hosted by Timothy Revell, New Scientist technology editor and reporter
Overview:
The impact of mathematics in the real world is hard to understate. Public health, economics, gambling, the universe, computers, quantum mechanics, social media, and more are all understood through the lens of mathematics. Despite this mathematics is often ignored, loathed, or misunderstood.
To remedy this injustice, our six experts with guide you through a smorgasbord of tasty mathematical treats to blow your mind. How intelligent bots beat human poker players, sending secret messages using the Enigma machine, music, Euclid, and art, will all be covered.
But does all the mathematics we need already exist? Far from it. Mathematicians are squirreling away day and night to crack the toughest mind-melting mathematical challenges. Our speakers will talk you through the latest news from the frontiers of mathematics, and equip you with the knowledge needed to truly penetrate some of the most exciting developments in recent years.
Topics covered will include:
Why the world is chaotic
How gambling changed the world
The secret codes that govern the internet
The Enigma machine
The statistics that shape our lives
And much more
Who should attend?
You are curious about mathematics and how it impacts on our lives. Popular articles have whetted your appetite, yet you have struggled to understand textbooks. You want to learn about the frontiers of research from scientists at the coalface.
What's included in your ticket:
In-depth and engaging talks from six leading experts in mathematics
Ask-an-expert Question Time session
Your chance to meet our six speakers and host Timothy Revell
Delicious lunch, plus morning and afternoon refreshments
Exclusive Instant Expertcertificate
Exclusive on-the-day New Scientist subscription deal, book and merchandise offers
Booking information:
The event will be held in the Brunei Auditorium at RCGP / 30 Euston Square.
Doors will open at 9:15am, with talks commencing at 10am sharp. The event will finish at 5pm.
We require the name of each person attending - please ensure this is provided at the time of booking. If you need to change the name of an attendee, please notify us as soon as possible: [email protected]
Eventbrite will email you your ticket(s) immediately after purchase. Please remember to bring your ticket(s) with you as you'll need it to gain entry. We can scan tickets from a print out, or off the screen of a phone / tablet / smartwatch.
The ticket price includes a buffet lunch, as well as morning and afternoon refreshments.
The schedule / exact running order for the day will be confirmed closer to the event, and will be emailed to all ticket holders.
All tickets are non-refundable, however if the event has sold out then we may be able to offer a refund if we are able to sell your ticket(s) to someone on the event waitlist (please note that there is no guarantee that we will be able to sell on your ticket(s) or that the event will sell out). Please email us at [email protected] to notify us. In the event of a sell out, all requests of this nature will be dealt with in the order they are received.
New Scientist reserves the right to alter the event and its line-up, or cancel the event. In the unlikely event of cancellation, all tickets will be fully refunded. Neither New Scientist nor its parent company will be liable for any additional expenses incurred by ticket holders in relation to the event.
Tickets are subject to availability and are only available in advance through Eventbrite.
A minimum of 100 early bird discount tickets are available priced at £129 (saving £20 on the full ticket price of £149).
There are a very limited number of discounted student scholorship tickets available - please email [email protected] to enquire about availability. No other concessionary tickets are available for this event. |
Mathematical Vistas: From a Room with Many Windows
Hardcover | January 8, 2002
Pricing and Purchase Info
$81.74 online
$90.95list pricesave 10%
Earn 409 plum® points
Quantity:
In stock online
Ships free on orders over $25
Not available in stores
about
The goal of Mathematical Vistas is to stimulate the interest of bright people in mathematics. The book consists of nine related mathematical essays which will intrigue and inform the curious reader. In order to offer a broad spectrum of exciting developments in mathematics, topics are treated at different levels of depth and thoroughness. Some chapters can be understood completely with little background, others can be thought of as appetizers for further study. A number of breaks are included in each chapter. These are problems designed to test the reader¿s understanding of the material thus far in the chapter. This book is a sequel to the authors¿ popular book Mathematical Reflections (ISBN 0-387-94770-1) and can be read independently.
Details & Specs
Title:Mathematical Vistas: From a Room with Many WindowsFormat:HardcoverDimensions:351 pagesPublished:January 8, 2002Publisher:Springer New YorkLanguage:English
The following ISBNs are associated with this title:
ISBN - 10:0387950648
ISBN - 13:9780387950648
Customer Reviews of Mathematical Vistas: From a Room with Many Windows Paradoxes in Mathematics * Not the Last of Fermat * Fibonacci and Lucas Numbers: Their Connections and Divisibility Properties * Paper-Folding, Polyhedra-Building and Number Theory * Are Four Colors Enough? * From Binomial to Trinomial Coefficients and Beyond * Catalan Numbers * Symmetry * Parties
Editorial Reviews
From the reviews:MAA ONLINE".much of the material is about the authors' own research. This is a positive thing; its good for students at all levels to get information straight from the horse's mouth, not only for accuracy but also for enthusiasm and authenticity. The authors put their writing where their talents are, and students get to see just how alive mathematics is.there. In elementary school, before first-year algebra had made me conscious of being a mathematician, I called my favorite pastime 'number tricks.' The section 'A Number Trick and its Explanations' brought back fond memories." P.Hilton, D.Holton, and J. PedersenMathematical VistasFrom a Room with Many Windows"The authors put their writing where their talents are, and students get to see just how alive mathematics is . . . there."-THE MATHEMATICAL ASSOCIATION OF AMERICA"Readers will find in . Vistas more than another couple of collections of popular tidbits. . a lively selection of mathematical topics delivered in a casual writing style, in which technical arguments are richly interspersed with comments on the guiding principles of mathematical investigation . . Finally, a word of praise is due for the many illustrations appearing throughout . these are carefully rendered and appropriate, never gratuitous. . Enjoy!" (John Grant McLoughlin, Crux Mathematicorum Mathematical Mayhem, 2003)"All chapters have in common that they speak about their subject in a light manner. But in each chapter there are so-called breaks consisting of problems designed to enable the readers to test their understanding of the material. Answers to some of the problems appear at the end of the book. The reading of this book can be recommended for high school and college students who are interested in mathematics, and it may stimulate them to read more about it." (Helmut Koch, Zentralblatt MATH, Vol. 1011, 2003)"The purpose of this book . is to provide a relaxed and informal treatment of several mathematical topics that will convey to the reader . a sense of the pleasurable excitement that accompanies genuine mathematical discovery. . topics range from Andrew Wile's famous proof of Fermat's Last Theorem . to the discussion of paradoxes in which all relevant arguments are given in full. Each chapter is provided with 'Breaks', in which readers can test their understanding of the material just presented by answering questions." (Zentralblatt für Didaktik der Mathematik, August, 2003)"Peter Hilton, Derek Holton and Jean Pedersen have produced a gem . . The authors have written a set of independent mathematical essays . for which there is rarely room in the undergraduate curriculum. . To be sure, there is a lot . that forces reader participation and that is, frankly, fun. . a joy to read, and I'm certain . a delight to teach from! . The writing is seamless, thought-provoking and entertaining. You and your students deserve to read . !" (Marvin Schaefer, MAA Online, July, 2002)"The reader should not only approach this delightful book with pen and paper but also will find it worthwhile to consider attempting the geometric constructions." (F. J. Papp, Mathematical Reviews, Issue 2003 e)"It's good for students . to get information straight from the horse's mouth, not only for accuracy but also for enthusiasm and authenticity. The authors put their writing where their talents are, and students get to see just how alive mathematics is. . there is much to commend the book. It contains plenty of interesting mathematics, often going in unusual directions. . And readers with some mathematical experience are sure to find things here and there that will delight them." (Marion Cohen, MAA, May, 2003)"The first purpose of this book is to stimulate the interest . in mathematics. . One of the features of the book is the presence in each chapter of a number of breaks consisting of problems that permit to the reader to test his understanding of the given material. This book is a good motivation for mathematics and has to be recommended to all students beginning studies in mathematics." (Yves Félix, Belgian Mathematical Society - Simon Stevin Bulletin, 2003)"This trio of authors will be familiar to many . . Their new book is a sequel to the original . . Vistas contains three chapters which build on subject matter in the earlier book. . These two volumes are an impressive achievement. There are many books on the market . but few of them are prepared to go into as much detail as you will encounter here. . I thoroughly recommend both books . ." (Gerry Leversha, The Mathematical Gazette, Vol. 87 (509), 2003)"The present book . consists of nine chapters devoted to different mathematical problems. . One of the special features of the text is that a number of 'breaks' are included in each chapter; these consist of problems so that the readers can test their understanding of the presented material. Selected answers . are given at the end of the book. . The book can be recommended for students and for all people who consider mathematics as a part of human culture." (European Mathematical Society Newsletter, December, 2002)"The goal of 'Mathematical Vistas' is to stimulate the interest . in mathematics. The book consists of nine related mathematical essays which will intrigue and inform the curious reader. . Some chapters can be understood completely with little background, others can be thought of as appetizers for further study. A number of breaks are included in each chapter. These are problems designed to test the reader's understanding. This book is a sequel to the author's popular book 'Mathematical Reflections' and can be read independently." (L'Enseignement Mathematique, Vol. 48 (1-2), 2002)"The book aims to inspire readers with the pleasure that comes from genuine mathematical discovery. Much of the discussion is of the mathematics itself, rather than its applications. . I found the book very enjoyable to read, and I imagine that many other physicists will as well. . One of the strengths of the book is its frequent text breaks, which consist of questions that have been designed to test the readers' understanding of the preceding information." (Sally Jordan, Physics World, September 2002)"The book is written in an economical yet 'chatty' and relaxed style. The subject matter is developed from the concrete to the abstract with many clear examples and frequent diagrams and illustrations. The book will hold the interest of the bright interested person. . There are undergraduate mathematics students with sufficient self-motivation to read and profit from the book . . There is also the opportunity for the sympathetic instructor to include the book or chapters into required reading for formal courses . ." (W.P. Galvin, European Mathematical Society Newsletter, December, 2002)"This book presents a broad spectrum of exciting developments in mathematics that will intrigue the curious reader with an interest in mathematics. The nine related essays cover topics, such as paradoxes in mathematics, Fermats last theorem, Fibonacci numbers, paper-folding, polyhedra building, and much more. This book is written in the same spirit as the authors highly popular Mathematical Reflections." (Amazon.de, May, 2002)"Mathematical Vistas covers a single mathematical problem in each of its nine sections, including gems such as the Four-Colour Problem and Fermat's Last Theorem. The authors . describe the ideas involved while encouraging the reader to investigate it through regular 'breaks'. References provide pointers to more detailed information. Such an approach seems well suited to undergraduate teaching, giving students an overview of topics, then enabling them to investigate further the ones that interest them most. . mathematics students will enjoy this book." (Andrew Bowler, New Scientist, March, 2002) |
The tetrabrot fractals are 4 dimensional objects, which are generated from the recursion relation:
Tn+1=Tn*Tn+T0
The elements T which do not diverge under this iteration are the members of the Tetrabrot set. For the case where T is a complex number (2D), this is the famous Mandelbrot set, shown to the right. This same image is the 1 vs. i or j axes in the hypercomplex tetrabrot, or 1 vs. i,j,or k in the quaternion tetrabrot.
There are a few ways to extend complex numbers to higher dimensions, creating "hypercomplex" numbers. These numbers can be represented as sums of the "basis unit vectors", which have multiplication tables such as shown below:
Note that the videos below are NOT images of Julia sets, or exponent variations, or other fractal manipulations. Only T^2+T is used here! To generate the video, the 2D "slice" of the 4D set is shifted or rotated, in addition to any zoom or color cycling. |
This is a quiz on math and logic. A few guidelines for you: x*y is x times y, x/y is x divided by y, sqrt(x) is the square root of x, x^y is x to the power y, lim(n>&) is the limit with n approaching infinity. A calculator isn't necessary.
Mathematics is considered as the language of the universe. Truly, it has wide applications in daily life. Get a quick peak into the world of Mathematics and enjoy! This quiz is a mixture on formulae, problems and history of Maths. |
Mathematical Principles of Natural Fig Snacks
A recent One Big Happy has Joe cutting corners on a book report:
(#1)
Classic kid behavior.
Joe's title echoes the famous long palindrome "A man, a plan, a canal — Panama" (attributed to Leigh Mercer in 1948). And of course it confounds Isaac Newton — author of the monumental Philosophiæ Naturalis Principia Mathematica ("Mathematical Principles of Natural Philosophy"), first published in 1687 — and the excellent snack the Fig Newton (discussed at some length in my 5/20/15 posting "Fig time"), which is (remarkably) named after the town of Newton MA.
So it all goes back to the place name Newton 'new town' (given to a huge number of places) and the toponymic surname Newton derived from it.
Isaac Newton gave his name to the unit of force, the newton, and to a number of other things, most memorably the Apple Newton. From Wikipedia:
The Newton is a series of personal digital assistants developed and marketed by Apple Inc. An early device in the PDA category – the Newton originated the term "personal digital assistant" – it was the first to feature handwriting recognition.
On the name:
The first thing [Steve Sakoman] did was select a name for the project. Because Apple's original logo … had a rendering of Isaac Newton sitting beneath an Apple tree, Sakoman decided to name the project Newton. (link)
Newton himself often told the story that he was inspired to formulate his theory of gravitation by watching the fall of an apple from a tree. Although it has been said that the apple story is a myth and that he did not arrive at his theory of gravity in any single moment, acquaintances of Newton (such as William Stukeley, whose manuscript account of 1752 has been made available by the Royal Society) do in fact confirm the incident, though not the cartoon version that the apple actually hit Newton's head. |
The Symbolic Universe: Geometry and Physics 1890-1930
Hardcover | June 1, 1999
Pricing and Purchase Info
$253.09 online
$390.00list pricesave 35%
Earn 1265 plum® points
Quantity:
In stock online
Ships free on orders over $25
Not available in stores
about
With the development of the theory of relativity by Albert Einstein, physics underwent a revolution at the end of the 19th century. The boundaries of research were extended still further when in 1907-8 Minkowski applied geometrical ideas to this area of physics. This in turn opened the door toother researchers seeking to use non-Euclidean geometrical methods in relativity, and many notable mathematicians did so, Weyl in particular linking these ideas with broader philosophical issues in mathematics. The Symbolic Universe gives an overview of this exciting era, giving a full account forthe first time of Minkowski's geometric reformulation of the theory of special relativity IIntroductionGeometrizing configurations. Heinrich Hertz and his mathematical precursorsEinstein, Poincare, and the testability of geometryGeometry-formalisms and intuitionsPART IIIntroductionThe non-Euclidean style of Minkowskian relativityGeometries in collision: Einstein, Klein and RiemannHilbert and physics (1900-1915)The Gottingen response to general relativity and Emmy Noether's theoremsPART IIIIntroductionRicci and Levi-Civita: from differential invariants to general relativityWeyl and the theory of connections
Editorial Reviews
'This volume provides a wide-ranging and detailed survey of this exciting era... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.