Elsevier

Computers & Graphics

Volume 28, Issue 3, June 2004, Pages 417-430
Computers & Graphics

Polynomiography and applications in art, education, and science

https://doi.org/10.1016/j.cag.2004.03.009Get rights and content

Abstract

Polynomiography is the art and science of visualizing approximation of the zeros of complex polynomials. Informally speaking, polynomiography allows one to create colorful images of polynomials. These images can subsequently be re-colored in many ways, using one's own creativity and artistry. Polynomiography has tremendous applications in the visual arts, education, and science. The paper describes some of those applications. Artistically, polynomiography can be used to create quite a diverse set of images reminiscent of the intricate patterning of carpets and elegant fabrics, abstract expressionist and minimalist art, and even images that resemble cartoon characters. Educationally, polynomiography can be used to teach mathematical concepts, theorems, and algorithms, e.g., the algebra and geometry of complex numbers, the notions of convergence and continuity, geometric constructs such as Voronoi regions, and modern notions such as fractals. Scientifically, polynomiography provides not only a tool for viewing polynomials, present in virtually every branch of science, but also a tool to discover new theorems.

Introduction

Polynomiography is defined to be “the art and science of visualization in approximation of the zeros of complex polynomials, via fractal and non-fractal images created using the mathematical convergence properties of iteration functions”. An individual image is called a “polynomiograph”. The word polynomiography is a combination of the word “polynomial” and the suffix “-graphy”. It is meant to convey the idea that it represents a certain graph of polynomials, but not in the usual sense of graphing, say a parabola for a quadratic polynomial. Polynomiographs are created using algorithms requiring the manipulation of thousands of pixels on a computer monitor. Depending upon the degree of the underlying polynomial, it is possible to produce beautiful images on a laptop computer in less time than a TV commercial.

Polynomials form a fundamental class of mathematical objects with diverse applications; they arise in devising algorithms for such mundane task as multiplying two numbers, much faster than the ordinary way we have all learned to do this task (Fast Fourier Transform). According to the Fundamental Theorem of Algebra, a polynomial of degree n, with real or complex coefficients, has n zeros (roots) which may or may not be distinct. The task of approximation of the zeros of polynomials is a problem that was known to Sumerians (third millennium BC). This problem has been one of the most influential in the development of several important areas in mathematics. Polynomiography offers a new approach to solve and view this ancient problem, while making use of new algorithms and today's computer technology. Polynomiography is based on the use of one or an infinite number of iteration functions designed for the purpose of approximation of the roots of polynomials. An iteration function is a mapping of the plane into itself, i.e. given any point in the plane, it is a rule that provides another point in the plane. Newton's iteration function is the best known:N(z)=z−p(z)p′(z).An iteration function can be viewed as a machine that approximates a zero of a polynomial by an iterative process that takes an input and from it creates an output which in turn becomes a new input to the same machine.

The word “fractal”, which appears in the definition of polynomiography, was coined by the world-renowned research scientist Benoit Mandelbrot. It refers to sets or geometric objects that are self-similar and independent of scale. This means there is detail on all levels of magnification. No matter how many times one zooms in, one can still discover new details. It turns out that some fractal images can be obtained via simple iterative schemes leading to sets known as the Julia set and the famous Mandelbrot set. The simplicity of these iterative schemes, which may or may not have any significant purpose in mind, has resulted in the creation of numerous web sites in which amateurs and experts exhibit their fractal images. Many fractal images pertain to the famous Mandelbrot set.

Polynomiography, on the other hand, has a well-defined and focused purpose in mind. It makes use of one or an infinite number of iteration functions for polynomial root-finding. A polynomiograph may or may not result in a fractal image. This is one of the reasons, it was necessary to give a new name to this process. A second reason is because its purpose is the visualization of a polynomial, via approximation of its roots and the way in which the approximation is carried out. Even when a polynomiograph is a fractal image it does not diminish its uniqueness. The assertion that an image is a fractal image is no more profound than asserting that an image is a picture, or a painting. In particular, that assertion does not capture any artistic values of the image. Indeed, even in terms of fractal images polynomiography reveals a vast number of possibilities and degrees of freedom and results in a wider variety of images than typical fractal images.

Polynomiography could become a new art form. Working with polynomiography software is comparable to working with a camera or a musical instrument. Through practice, one can learn to produce the most exquisite and complex patterns. These designs, at their best, are analogous to the most sophisticated human designs. The intricate patterning of Islamic art, the composition of Oriental carpets, or the elegant design of French fabrics come to mind as very similar to the symmetrical, repetitive, and orderly graphic images produced through polynomiography. But polynomiographic designs can also be irregular, asymmetric, and non-recurring, suggesting parallels with the work of artists associated with abstract expressionism and minimalism. Polynomiography could be used in classrooms for the teaching of art or mathematics, from children to college-level students, as well as in both professional and non-professional situations. Its creative possibilities could enhance the professional art curriculum.

The “polynomiographer” can create an infinite variety of designs. This is made possible by employing an infinite variety of iteration functions (which are analogous to the lenses of a camera) to the infinite class of complex polynomials (which are analogous to photographic models). The polynomiographer then may go through the same kind of decision-making as the photographer: changing scale, isolating parts of the image, enlarging or reducing, adjusting values and colors until the polynomiograph is resolved into a visually satisfying entity. Like a photographer, a polynomiographer can learn to create images that are esthetically beautiful and individual, with or without the knowledge of mathematics or art. Like an artist and a painter, a polynomiographer can be creative in coloration and composition of images. Like a camera, or a painting brush, a polynomiography software program can be made simple enough that even a child could learn to operate it.

Despite the significant role of the root-finding problem in the development of fundamental areas, today it is not considered to be a central problem in pure or computational mathematics. According to a 1997 article in SIAM Review by Pan [1], a leading authority on the computational complexity aspects of the root-finding problem, in practice often one needs to compute roots of polynomials of very moderate degree (10 or 20), except possibly in computer algebra which is applied to algebraic optimization and algebraic geometry. In view of the above and since there are already efficient subroutines for computing roots of moderate size polynomials, it would not be surprising that many may view the polynomial root-finding problem as one that has basically reached a dead-end. However, I believe that polynomiography will change all of this, not only from the mathematical or scientific point of view, but from the educational and artistic point of view.

Quoting the great American mathematician Smale [2], “There is a sense in which an important result in mathematics is never finished”. The Fundamental Theorem of Algebra is one of those results. To me polynomiography is a good evidence in support of Smale's statement. With the availability of a good polynomiography program, the user (who may be a high school student, an artist, or a scientist) is quite likely to wish to experiment with much larger degree polynomials than degree 10 or 20. In particular this is true because in polynomiography there is a “reverse root-finding problem”: given a polynomial whose roots form a known set of points, find an iteration method whose corresponding polynomiograph would take a desired pattern. In principle, using this reverse problem, I can conceive of the blue print to some of the most elegant patterns, for instance carpets designs, yet to be woven. Their complexity and beauty could very well increase by increasing the degree size. Thus, it is conceivable that the implementation of such designs would demand working with much more computer power than that offered by a laptop computer, perhaps a supercomputer, or a network of computers.

Section snippets

Mathematical foundation of polynomiography

Consider the polynomialp(z)=anzn+an−1zn−1+⋯+a1z+a0,where n⩾2, and the coefficients a0,…,an are complex numbers. The problem of approximating the roots of p(z) is a fundamental and classic problem. A large bibliography can be found in [3]. For an article on some history, applications, and new algorithms, see [1]. Some of the root-finding methods first obtain a high-precision approximation to a root, then approximate other roots after deflation, see e.g., [4]. Many such root-finding methods make

Basins of attraction and Voronoi regions of polynomial roots

Consider a polynomial p(z) and a fixed natural number m⩾2. The basins of attraction of a root of p(z) with respect to the iteration function Bm(z) are regions in the complex plane such that given an initial point a0 within them, the corresponding sequence ak+1=Bm(ak),k=0,1,…, will converge to that root. It turns out that the boundary of the basins of attraction of any of the polynomial roots is the same set. This boundary is known as the Julia set and its complement is known as the Fatou set.

Polynomiography and visual arts

I will first describe some general techniques for the creation of polynomiographic images and exhibit sample images produced via a prototype software for polynomiography, as well as provide an explanation of how some of these images were created. Subsequently, I will consider other applications of polynomiography.

An exhibition

This section presents an exhibition of some polynomiographic images that I have created through the above four techniques. The reader may notice the diversity of these images. In particular, they contrast with literally hundreds of fractal images exhibited at web sites. It should be noted that the images displayed are not necessarily symmetric. These are given as Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11, Fig. 12, Fig. 13, Fig. 14. It should be mentioned that none

Polynomiographs of numbers

One interesting application of polynomiography is in the encryption of numbers, e.g., ID numbers or credit card numbers into a two-dimensional image that resembles a fingerprint. Different numbers will exhibit different fingerprints. One way to visualize numbers as polynomiographs is to represent them as polynomials. For instance a hypothetical social security number a8a7a0 can be identified with the polynomial P(z)=a8z8+⋯+a1z+a0. Now we can apply any of the techniques discussed earlier. A

Public use of polynomiography in arts and design

Through various software programs, polynomiography could grow into a new art form for both professional and non-professional art-making, and in the teaching of both art and mathematics. The recent proliferation of craft supply stores in the United States reveals the deep seated urge in the general population to create. Having polynomiography software available on home computers could release that creative power even more; people could invent their own knitting and needlepoint patterns, design

Polynomiography and education

Polynomiography has enormous potential applications in education. A polynomiography program could be used in the mathematics classroom as a device for understanding polynomials as well as the visualization of theorems pertaining to polynomials. As an example of one application of polynomiography, high school students studying algebra and geometry could be introduced to mathematics through creating designs from polynomials. They would learn to use algorithms on a sophisticated level and to

Extensions of polynomiography and iteration techniques

In this paper, polynomiography has been introduced in terms of the properties of a fundamental family of iteration functions, namely the Basic Family. However, more generally polynomiography can be defined with respect to any iteration function for polynomial root-finding. While experimentation with polynomiographs of other iteration functions is likely to be interesting, many unique mathematical and computational properties of the Basic Family members makes them much more interesting than any

Concluding remarks and the future of polynomiography

In this paper I have described some general techniques for creating polynomiographic images. However, even with polynomiography software at one's disposal, from the user's point of view it is essential to have available a very detailed and systematic manual and/or a book. The creation of such publications is one of my future goals. Indeed, different users will benefit from different such publications.

It is believed that through various software programs polynomiography could grow not only into

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