Iterative
Fractals Abstract
graphs have both vertices and edges. Each graph can be given a matrix
which can be turned inot a polynomial. The symmetries of the graph
lead to factorization of the polynomial. Very interesting and very
fun.
An iterative process is on that is repeated over and over, each new
step beginning with the output of the previous step. Here
is a page on iterative systems in general, includingthese. Here
is a page showing the Mathematica code that gives the results.
Here is a page showing these fractals in triangle geometry.
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Iterations
with Jacobian 1 Back when everyone was discoverting fractals
and strange attractors, I was discovering the same iterations
but with Jacobian 1, whose orbits often close to form beautiful
patterns, at least I think so.
This page leads you through the process of iteration dynamics for the equations
x' = y– x2, y = a - x. Neat movies at the end.
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Artmatic,
the uitmate in implicit graphing Go
here to download a trial copy. My tutorial below explains how
it works. You should see for yourself the incredible complexity
and variety of mathematical equations. This is a Mac only program.
Here
is my tutorial, explaining some of the ways Artmatic graphs equations.
Here is some
work by Mollie Maloy .
By
Jenny Middleton By
Mia DeSimone
and Amy
Hailes and Ari
DeSimone |
Symmetry
of Abstract Graphs Abstract graphs have both vertices and edges.
Each graph can be given a matrix which can be turned inot a polynomial.
The symmetries of the graph lead to factorization of the polynomial.
Very interesting and very fun.
Introduction
to Graphs and Polynomials.
Mathematica techniques for graphs and their polynomials.
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MetaSynth, digital
sound creation Increasingly
music is being produced from sounds as well as notes. Methsynth lets
you focus on sound creation while letting you put those sounds in
a traditional musical context if you wish. Here are movies created
in Artmatic with scores created in Metasynth for the Being Digital
course here at Paideia. Check out the lower bandwidth versions (formated
for iPod) and then look at the full quality versions of the ones
you like.
Movie #1: Ferrari Movie by current students.
The assignment here was to make a movie from a single picture and a single sound,
digitally manipulated as much as one wishes. Now what will 14 year old boys want
to make a move of? A car, of course. This movie was made from a single picture
of a Ferrari and, except for the startup sounds, a single beep from a Ferrari
horn. A conceptual tour de force.
Move#2:
Movie#3: |
Historical
Pitch of AAs as physicist I know
that the frequency of an oscillation in air corresponds to the
acoustic variable pitch. I also know that frequency is a continuous
variable, so that nature provides an infinte number of pitches
to an octave. In our wisdom, we have reduced this number to 12,
the western equally tempered scale. Still the physicist, I wonder
about this. Why 12? What makes this work? Is it a fancy compromise
or deeper? I think it is a compromise that makes instruments like
the piano possible. Perhaps I will put up some webpages on the
acoustic variable pitch.
This page is about our attempts to standardize pitch. In 1955
some commission or other attempted to make 440 cycles per
second the official standard pitch, corresponding to the A above
middle C on a piano. Most Americal orchestras conform to this (but
I have heard whispers that the Boston symphony plays at 448). Most
European orchestras play slightly higher at 442 or 445. (continued
on page)
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Proving
the existence of fish B. Kliban. Heh...heh...heh
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John and
Adam and Steve's excellent adventure Put 6 points on a conic.
The opposite sides of the inscribed hexagram meet at 3 colinear
points, which determine a Pascal line. Three Pascal lines meet
at a Steiner point. Four Steiner point determine a Plucker line.
But there ismore: 3 Pascal's also meet at a Kirkmen point. Three
Kirkman's determine a Cayley line; four Cayley's meet in a Salmon
point. The full figure is knon as the Hexagrammum
Mysticum. John
Conway has worked out a new way of looking the Hexagrammum at
by extending the abstract algebra of the Desargues
configuration.
Adam Marcus and I looked at the combinatorics. Fun, Fun, Fun. Its
here now, and will be improved many more times. |
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Implicit graphing Function graphing
is very restrictive, because few shapes, especially interesting ones,
pass the vertical line test. A much more interesting type of graphing
(and much more demanding on the computer that creates it) is implicit
function plotting. When an equation is written as f(x,y) = 0, we
say that it is written implicitely. All explicit function y = f(x)
can be written implicitely, but most implicit equations cannot be
written explicitly. Here is my implicit function
slideshow. Many of the graphing techniques shown above are
implicit ones.
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Steiner
quadrilateral theorems In 1827 Jacob Steiner submitted
a chain of amazing theorems about 4 arbitrary lines. Here is
work by the seminar class illustrating those theorems.
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Parametric graphing Just as implicit
graphing is more general then explicit function graphs, parametric
graphs are more general still. Here we express x and y interms of
one or more parameters. If x = f(t) and y = g(t), x and y are both
expressed interms of a single parameter t, which is topologically
a line. Hence this graph is a line in 2D space. If we add z = h(t),
then the graph is a 1D curve in 3D space.
If x, y, and z are functions of two variable, this will be a two
dimensional surface in a 3D space. These graphs are extremely interesting.
Here
is the 1D parametric graphing page. Here is the slideshow,
and here is the
page that tells how to create them.
Have fun.
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The original "monster" and
my discussion of continuity and epsilon deltas (pdf) The functions
one learns in mathematics are very untypical in that they are very
nice in that they possess and infinte number of derivatives at each
point. We now know that nice functions are rare. Functions which
can possess no derivatives anywhere (such as fractals) are more common.
To deal with all types of functions, a method, known as the epsilon-delta
method has been developed. Here
is my discussion of it in pdf format. |
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Graphs
as algbraic structures The
implicit function that describes a straight line is Ax + By + C.
Two functions of this type can be added making another straight line.
A Circles added to a straight line is another circle. These
pages show that graphs can be treated as algebraic
structures and how to create conic functions
from 4 lines. These pages explain oriented
lines. |
Design your own coordinate system You
know about rectangular coordinate and polar ones. Are their othere.
Yes an infinite number of them. Each function of a complex variable
defines its own unique two dimenstional coordiante system. Let f(x+iy)
= u(x, y) + i v(x,y). For example (x+iy)2 = x2 - y2 + i 2xy. The
equations u = x2 - y2 and v = 2xy give curves that meet orthogonally
and so define a coordinate system. To create it graphs the above
equations for u and v = 0, 1, 2,3, 5 ... and also for netagives.
The u and v values are the coordinates. |
Life John Conway's
Life game is very interesting. See the Seminar
class for a way to
get started and to begin to get advanced. Here you will find links
understand how to create objects in life and Margit Zwemer's (who
graduated from Paideia a few year ago and it now at Stanford) construction
of an adding machine.
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Trilinear
lines (pdf) This is a new
way of thinking about lines. I wrote this about
10 years ago when I was beginning to work on Triangle geometry. I
purchased a second hand book on Conics by Issac Todhunter (one of
my favorite old book writers). I looked at the early chapters just
to get used to the notation. The first two chapters
were about straight lines. To my surprise there was a lot that I
did not knw about straight lines ( I thought I know everything about
them -- after all they are straight, they have slope and intercepts,
there are vector forms -- what else is there?).
Well, straight lines can be writen using homogeneous coordinates;
this was new to me so I wrote it up both to understand it myself
and to pass it on. This is pdf file rather than a web page. |
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abstract algebra that mimics the derivative Ever wonder what
happens if we relax some of the rules of high school algebra? Here
we let all rules be the same but x and y do not commute. This leads
to an algebra that is formally equivalent to the derivative, but
without the objects being numbers or any question of continuity.
This is the same algebra used in quantum mechanics. |
Quantum
mechanics uses complex numbers Just the use of complex numbers
leads to interference; i.e., wave like properties. It does not
require the actual existence of waves but gives wavelike effects.
In addition complex numbers require the existence of other dimensions
and, say, rotations (called "gauges") in them. |
Rational
Functions Rational functions are polynomials divided
by polynomials. There is a special agebra technique that pertains
only to them called "partial fractions," which splits a complex
rational function into a sum of ones with simple denominators.
As usual there are structural reasons why algebra techniques work.
Here is a simple way to do this, called the residue theorem in
higher mathematics.
Rational functions have real asymptotes if their denominator is
zero. Usually these asymptotes are vertical, but if the degree
if the numerator is one higher than that of the denominator, the
asymptote is oblique. I always thougth that the algebra involved
was more difficult than the concept, so here is an easy way to
find oblique asymptotes. |
Mineur's
Anallagmatic cubics An anallagmatic structure includes it
opposite. In triangle geometry the opposite used is called the
conjugate. These pages transcribe a 1922 monograph by Adolph
Mineur. As geometry goes this is pretty advanced, using barycentric
coordinates and general knowledge of triangle geometry, especially
projective triangle geometry. |
Barycentric
coordinates This is a method of letting a two dimensional
point in the plane of the triangle be represented by three coordinates.
Rather than have an origin for the coordinate system, we use a
reference triangle instead. This type of coordinate system is useful
when the numerical value of distances is not important, and objects
are not dependent on scale.
There are two parts. One show how to
find Barycentric coordinates. Finallly there is a long email
message from John Conway on the superiority of the barycentric
system. |
Geometry
Pages When I was learning triangle geometry, I
wrote artices for myself and for friends. These articles are all pdf
format.
Conics -- a description of conics using homogeneous
coordinates. I learned this from English books from the 1880's.
Modern
Geometry of the Triangle -- a description
of basic things a person should know about the more recent (1800's)
developments in geometry.
Pictures -- I like making pictures; here are some
I did about 8 years ago.
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The
most important picture in triangle geometry This
shows two triangle in perspective and the dual nature of points
and lines in the triangle.
This page on Desargues
configuration, of which the above is a specific
instance, is also interesting as is the Desargues
slideshow.
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Fractals
from complex bases In school the numbers we lean
are in base 10, presumably because we have 10 fingers. Counting
in base 10 increases numbers consecutively. But if we are good
mathematicians we push every idea as far as our imaginationns
are capable. This section is about using a complex numbers as
the base of your number system.
Complex numbers can be plotted as points. These If you count numbers
consecutively, and then plot them on graph paper, very interesting
patterns are formed.
Each set of digits produces a complex number
and a point that is plotted. As one goes to higher and higher
orders of digits, the self-similar nature of the patterns is
evident. This is a property often used to describe fractal images.
By clicking on the title of this section, you get a slideshow
of the patterns described above for base 1+i. Below are pictures
created by Paideia students in the Seminar class. Some explain
the process. Some are beautiful. Some both. The pages that show
"work" or "technique" show how the pictures were created using
Mathematica. The have pictures at the bottome of the files. The
others are pictures only.
Kurt's technique
Jason Ku's technique
Kurt Ude's pictures
Jason's pictures
Matt
's work
Hobie's work
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Number
Lines and the size of the world Each time a variable is used
in an equation, one should spend time understanding that variable.
The understanding of equations often depends on understanding the
variables tha reside in them.
It helps to show the ranges of these variables on a number line. We are very
used to the human scale of meterse, feet, seconds, minutes, and hours. Nature
is a different story. Its sizes and times and energies are on a different scale
altogether, a logarithimic scale.
This page gives
Energy
scales.
and here is the
How
far? number line web page.
Here is a page explaining logarithmic number lines.
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Indra's
Perls -- Mobius transformationsIn '03-'04 the Seminar class
used David Mumford et al's book Indra's Perls as
a text. We know that each complex number is both a number and a
geometric transformation, and this book implements this in a beautiful
way. We used Mathematica to implement some of the transformations
in the book. Here is work done by the named students.
Bobby McKeon and Jeffrey Holtzberg (both '04)
My Mathematica introduction with pictures by Jeffrey
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Odom's
new construction of the golden section and Conway's construction
of the dodecahedron From an equilateral triangle and its circumcircle,
extend a midline in one direction until it meets the circumcircle.
The two lengths on this line are in the golden section ratio.
This ratio can be used to construct the dodecahedron from the cube.
Here is work of my students doing just that.
Odom
Construction
of the platonic solids from the cube. |
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