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THE MATHWRIGHT LIBRARY NEWSLETTER, Oct 2003, VOL 5,
#6
A publication of Bluejay Lispware
James E. White, Editor
The official publication of the New Mathwright Library and Café:
In this issue:
Free
Interactive Web Book
at the MATH Café:
Color Portraits of Complex Mappings
Free Interactive Web Book at the MATH Café: Color Portraits of Complex Mappings
Complex
Analysis is one of the venerable pillars on which modern mathematics rests,
and to which such diverse fields as Algebraic Topology and Algebraic Geometry
owe their origins. And while it enjoys a long and distinguished tradition
that parallels the analysis of real functions, analytic maps also have some
deep connections with geometry that are somewhat distinct from those of Real
Analysis. The condition of differentiability for maps from
has an interesting geometric interpretation (the Cauchy-Riemann equations) that leads to such beautiful novelties as their conformality away from singularities, Cauchy's Integral Theorem, the Riemann mapping theorem, and Lucas' Theorem on the disposition of the zeros of the derivative of a polynomial in relation to the disposition of the zeros of the original function. Marden's theorem, which we illustrate and prove in this Microworld, is a surprising special case of Lucas' Theorem. But maps from the plane to the plane are in general difficult to visualize, simply because they require four dimensions (degrees of freedom) to represent them.
This
3D Mathwright Microworld aims to give you the opportunity to investigate
some of these things for yourself. Along the way, you can make your
own gallery of color portraits. It is a 3D Microworld that uses orthographic
rather than perspective projection. Everything happens in a plane, so there
is no need for the third dimension (Four would be nice, alas).
One of the great strengths
of 3D Graphics is its versatile use of color and light. Imagine that each
point in the complex plane is represented in polar coordinates: 
where 
is the distance from the origin or the modulus, and 
is the angle that the positive real axis makes with a directed ray from the
origin to the point, the argument. If we let 
represent intensity and 
represent hue as an additive combination of the three basic colors, Red, Green,
and Blue, then each point in the plane is uniquely represented by a color
with black at the origin. While such a choice is rather arbitrary, once we
establish a correspondence between complex numbers and colors, we will have
a bridge to the visualization of these maps. The following is one such association:
Given
this correspondence, any function mapping the complex plane to itself may
be represented by a picture in which each domain point is given the
color of its image. It is probably easy to understand the rotation
given by multiplication by 
.
For
example, the polynomial function 
has the following picture:
It
has 6 easily discernible zeros, the sixth roots of unity, and those zeros
interact with one another in interesting and colorful ways. For example, the
center of the picture is dominated with function values that are close to
negative reals, as one would expect. Since 
is a primitive sixth root of unity, we see the "snowflake" or hexagonal
symmetry of the map under rotations through 
and its multiples.
Another
fact, easily portrayed by the picture, is the simple nature of the zeros.
Around each zero, the function makes a single circuit of the origin. Each
zero has order 1, and this implies that the function is locally invertible
in the vicinity of each zero. It is reflected in the fact that there is around
each zero, a single circuit of the colors from red to yellow to green to blue
and violet and back to red.
If a complex mapping has only zeros and poles of finite order,
then for any simple closed differentiable curve in the plane that does not
pass through any zeros or poles of the complex mapping, we can ask what number
of zeros and poles, counted algebraically with their multiplicities, are contained
within the interior of the curve. Visually, it is the integer number
of times the curve is "wrapped" around the origin by the complex
mapping. We see that number easily with a color portrait, simply by reading
how many times the color spectrum
is traversed in one counterclockwise traversal of the curve. A pole gives color traversal in the opposite sense. This integer is the algebraic sum of the number of zeros and poles, counted with their multiplicities, that are contained within the curve. When there are multiplicities, small perturbation of the function yields bifurcation where only simple zeros and poles close to the original ones are found.
The
Microworld features a new capability of Mathwright (available since version
2.13, Sept 14, 2003) that makes use of 3D graphics and the built-in
complex number type to draw fast color portraits of complex maps, thereby
giving the reader the chance to experiment and to explore the properties of
complex functions at her own pace, and with her own questions.
Analytic
functions can be the source of beautiful geometric patterns and forms, precisely
by virtue of their rich geometric structure. Even polynomial and rational
functions disclose graceful symmetries, tensions and attractions among their
zeros and poles that have the allure of fine art. It is in the spirit of developing
an eye for the beauty and grace of analytic functions that we develop this
Microworld. If the heuristic of this book (which is to represent analytic
functions with characteristic "pictures" or "color portraits")
leads to a deeper appreciation of the mathematics, that will be a good thing.
But what we are after is a style of visualizing them that is impressionistic
and visually intuitive, and that leads to a variety of experiments for the
reader.
These
ideas apparently originated in the literature with the work on "Domain
Coloring" of Frank
Farris. He recommends the highly acclaimed book by Tristan Needham entitled
Visual Complex Analysis. Larry Crone
at the American University has developed some powerful algorithms for color
maps on his page: Color
Graphs of Complex Functions, and we illustrate one at the end of
the book. A later reference to these ideas is the web page of Hans Lundmark
entitled Visualizing
Complex Analytic Functions Using Domain Coloring. The latter author develops
a powerful idiom for analyzing the mapping properties of analytic functions
in a serious way. This Microworld offers the reader the opportunity to experiment
and to design her own posters of her favorite complex maps.
We
thus explore the heuristic value of representing such functions as "pictures."
As always, the pictures can convey at a glance many of the qualitative geometric
properties of such mappings, especially the polynomial and rational functions,
but also a few transcendental functions. Among those qualitative features,
we shall be especially interested in the interactions among zeros and poles,
the orders of zeros and poles, and the relation in simple cases, between a
complex mapping and its derivative. For the latter, we will illustrate and
prove Marden's Theorem, which discloses a surprising geometric fact about
zeros of the derivatives of cubic polynomials and their relation with the
original polynomial.
The
following is the Table of Contents for this 11 page Interactive Web Book.
- Color Portraits of Complex Functions
- Derivatives as Rotations/Expansions
- Color Portraits of Cubic Polynomials
- Lucas' Theorem for Cubics
- Illustrating Marden's Theorem
- Proving Marden's Theorem using the Adjoint Quadratic
- Transforming Curves by Complex Maps
- Interactions of Zeros and Poles
- Crone's Representation of the Moduli