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Computers, Pattern, Chaos and Beauty

Computers, Pattern, Chaos and Beauty

by Clifford A. Pickover

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Combining fractal theory with computer art, this book introduces a creative use of computers. It describes graphic methods for detecting patterns in complicated data and illustrates simple techniques for visualizing chaotic behavior. "Beautiful." — Martin Gardner, Scientific American. Over 275 illustrations, 29 in color.


Combining fractal theory with computer art, this book introduces a creative use of computers. It describes graphic methods for detecting patterns in complicated data and illustrates simple techniques for visualizing chaotic behavior. "Beautiful." — Martin Gardner, Scientific American. Over 275 illustrations, 29 in color.

Editorial Reviews

Publishers Weekly - Publisher's Weekly
Computer graphics reveal hidden relationships in complex systems, make confusing data understandable and provide scientists and mathematicians with a tool for discovery and problem-solving. Featuring 200 black-and-white computer images and eight pages in color, this sourcebook includes programming exercises and mathematical recreations. Though most of the narrative requires advanced mathematical understanding, the general reader will find the computer artwork intriguing. The sounds of human speech yield snowflake-like patterns; Art Nouveau-ish images emerge out of mathematical relationships; structural changes in biomolecules produce graphics resembling galaxies and whirlpools. Diligent readers will gain an appreciation of how computer imaging helps scientists simulate plant tendril growth, analyze the Shroud of Turin, unravel the structure of cancer genes and investigate spiral patterns in DNA and galaxies' arms. Pickover is an editor at Computers and Graphics. (May)
It is in the field of advanced computer graphics that Pickover finds themes from high-level mathematics and chaos theory to philosophy, art and aesthetics all interweaving in a synthesis of science and art, illustrated in eerily phantasmagoric images, eight pages of them in color. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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Read an Excerpt

Computers, Pattern, Chaos and Beauty

Graphics from an Unseen World

By Clifford A. Pickover

Dover Publications, Inc.

Copyright © 2001 Clifford A. Pickover
All rights reserved.
ISBN: 978-0-486-15161-8


Computers and Creativity

"The heavens call to you, and circle about you, displaying to you their eternal splendors, and your eye gazes only to earth."


Imagine a world with no shadows, no sun.

Imagine computing, with no mystery, no creativity, no human dreamer. The beauty and importance of computers lie mainly in their usefulness as a tool for reasoning, creating and discovering. Computers are one of our most important tools for reasoning beyond our own intuition. In order to show the eclectic nature of computer "territory," this book contains a collage of topics which have in common their highly visual nature, and each can be effectively explored using a computer.

Imagery is the heart of much of the work described in this book. To help understand what is around us, we need eyes to see it. Computers with graphics can be used to produce visual representations with a myriad of perspectives. These perspectives are demonstrated by the subjects presented in this book. The applications are varied and include fields as diverse as speech synthesis, molecular biology, mathematics, and art. Yet it is hoped that they all combine to illustrate the wonder in "lateral thinking" with computers (defined in Sect. 1.2 "Lateral Use of Computer Software Tools" on page 4).

1.1 Objectives

Where possible, the material is organized by subject area. The purpose of this book is:

1. to present several novel graphical ways of representing complicated data,

2. to show the role of the aesthetics in mathematics and to suggest how computer graphics gives an appreciation of the complexity and beauty underlying apparently simple processes,

3. to show, in general, the beauty, adventure and potential importance of creative thinking using computers,

4. to show how the computer can be used as an instrument for simulation and discovery.

1.2 Lateral Use of Computer Software Tools

"He calmly rode on, leaving it to his horse's discretion to go which way it pleased, firmly believing that in this consisted the very essence of adventures. "

Cervantes, Don Quixote

"Lateral thinking" is a term discussed by writer/philosopher, Robert Pirsig (author of Zen and the Art of Motorcycle Maintenance). As he explains it, lateral thinking is reasoning in a direction not naturally pointed to by a scientific discipline. It is reasoning in a direction unexpected from the actual goal one is working toward (see also de Bono, 1975). In this book, the term "lateral thinking" is used in an extended way to indicate not only action motivated by unexpected results, but also the deliberate drift of thinking in new directions to discover what can be learned. It is also used to indicate the application of a single computer software tool to several unrelated fields.

Let's list a few examples of the lateral use of computer software tools. These examples will be discussed in greater detail later in the book. To give some personal history and examples: while creating analysis tools for speech synthesis research (Chapter 3), the author drifted laterally and examined their application to the study of the breathing motions of proteins. This naturally led to other biological molecules such as genes. In this application, the sequence of bases in a human bladder cancer gene is treated as if it were a speech waveform in order to gain a new perspective. These studies presented traditional graphics and analysis in new applications in an effort to visualize complex data.

This idea of novel ways for making complicated data understandable led to the application of Chernoff faces (cartoon faces whose facial coordinates depend on the input data). These faces can be applied to a range of sounds, mathematical equations, and genetic sequences. The faces rely on the feature-integration abilities of the human brain to condense a vast amount of data.

Does there exist an optimal representation for visual characterization and detection of significant information in data? This question, along with the face research, further stimulated my interest in the human visual system. Part of Chapter 4 discusses the use of a perceptual illusion, achieved with patterns of dots, to the characterization of subunit relationships in proteins. These patterns, called "Moire interference patterns," resemble galaxies and whirlpools. The interference patterns led to another question concerning vision and data characterization: Can symmetry operators, like the mirrors in a child's kaleidoscope, help us to understand data? To answer this question, another dot-based tool was developed; this representation is comprised of snowflake-like patterns of colored dots and is used to characterize sounds.

Intriguing even as an art form, these dot patterns may be a way of visually fingerprinting natural and synthetic sounds and of allowing researchers to detect patterns not easily captured with traditional analyses.

A short quote from Robert Pirsig can apply to the joy computer programmers, artists, and scientists often experience when experimenting on a computer:

"It's the sides of the mountain which sustain life, not the top. Here's where things grow."

1.3 Reading List for Chapter 1

Two interesting books on the topics of creativity and lateral thinking are De Bono (1970) and Pirsig (1975).


Hidden Worlds

"If we wish to understand the nature of the Universe we have an inner hidden advantage: we are ourselves little portions of the universe and so carry the answer within us."

Jacques Boivin, The Heart Single Field Theory

2.1 Digits, Symbols, Pictures

We live in a civilization where numbers play a role in virtually all facets of human endeavor. Even in our daily lives we encounter multidigit zip-codes, social security numbers, credit card numbers, and phone numbers. In many ways the requirements for ordinary living are a great deal more complicated than ever before. Digits ... digits ... digits.... It all seems so dry sometimes. And yet, when one gazes at a page in a scientific journal and sees a set of complicated-looking equations, such as those chosen from pages of scientific texts (Figure 2.1), a sense of satisfaction is generated: the human mind, when aided by numbers and symbols, is capable of expressing and understanding concepts of great complexity. Ever since "visionary" mathematical and physical relations trickled like rain onto the rooftop of 20th century man, we have begun to realize that some descriptions of nature lie beyond our traditional, unaided ways of thinking.

The expression of complicated relations and equations is one magnificent step — insight gained from these relations is another. Today, computers with graphics can be used to produce representations of data from a number of perspectives and to characterize natural phenomena with increasing clarity and usefulness. "Mathematicians couldn't solve it until they could see it!" a caption in a popular scientific magazine recently exclaimed when describing work done on curved mathematical surfaces (Science Digest, January, 1986, p. 49). In addition, cellular automata and fractals — classes of simple mathematical systems with exotic behavior — are beginning to show promise as models for a variety of physical processes (see "Genesis Equations" on page 104 and "Tesselation Automata Derived from a Single Defect" on page 295). Though the rules governing the creation of these systems are simple, the patterns they produce are complicated and some-times seem almost random, like a turbulent fluid flow or the output of a cryptographic system.

Today, in almost all branches of the scientific world, computer graphics is helping to provide incito and to reveal hidden relationships in complicated systems. Figure 2.2 is just one example of the use of graphics to represent the behavior of mathematical functions. Notice the complexity of the behavior exhibited by the function used to create Figure 2.2 — behavior mathematicians could not fully appreciate before computers could display it.

Like computer models of a host of natural phenomena such as vortices, fluid flow, and other chaotic (irregular) systems, pictures such as these reveal an unpredictable, exciting and visually attractive universe.

2.2 Computers and Art

"Salvador Dali once exploded a bomb filled with nails against a copper plate, producing a striking but random pattern. Many other artists have also utilized explosives in their work, but the results have generally been unpredictable."

Febr. 1989, Scientific American

Not only can computers and graphics be used in counting and measuring, but they also are of enormous help in producing visual art (Figure 2.3). (See the


Reading List at the end of the chapter for more information on computer art.) The break between artistic and scientific pursuits is often apparent today. Whereas the earlier thinkers pursued science and art in the light of guiding principles such as harmony and proportion, today some hold the view that science stifles the artistic spirit. Nevertheless, the computer is capable of creating images of captivating beauty and power. Techniques such as animation, color and shading all help to create fantastic effects (Figure 2.4).

In much of the work in this book, beauty, science and art are intertwined, and — judging from the response from readers — this contributes to the fascination of these approaches for both scientists and laypeople. From an artistic standpoint, mathematical equations provide a vast and deep reservoir from which artists can draw. New algorithms ("recipes"), such as those outlined in this book, interact with such traditional elements as form, shading and color to produce futuristic images. The mathematical recipes function as the artist's assistant, quickly taking care of much of the repetitive and sometimes tedious detail. By becoming familiar with advanced computer graphics, the computer artist may change our perception of art.

2.3 Computer Graphics: Past and Present

"Computers are useless. They can only give you answers."

Pablo Picasso

In the beginning of the modern computer age, computer graphics consisted of the multitude of Abe Lincolns, Mona Lisas, and Charlie Brown cartoons spewed forth from crude character line-printers in campuses and laboratories. Better hardware led to better images. In the 1970s we saw an increasing amount of computer animation, computer generated-commercials and films — and Pacman. Today, in science, computer graphics is used to reveal a variety of subtle patterns in nature and mathematics. The field of computer graphics is very important in: 1) revealing hidden correlations and unexpected relationships (and as an adjunct to numerical analysis), 2) simulating nature, and 3) providing a source of general scientific intuition. Naturally, these three uses overlap. Pseudo-color, animation, three-dimensional figures, and a variety of shading schemes are among the techniques used to reveal relations not easily visible in more traditional data representations.

2.4 Computers: Past and Present

Taking a step back: how long ago did computing really begin? Probably, the first calculating machine to help expand the mind of man was the abacus. The abacus is a manually operated storage device which aids a human calculator. It consists of beads and rods and originated in the Orient more than 5,000 years ago. Archeologists have since found geared calculators, dated back to 80 BC, in the sea off northwestern Crete. Since then, other primitive calculating machines have evolved, with a variety of esoteric sounding names, including: Napier's bones (consisting of sticks of bones or ivory), Pascal's arithmetic machine (utilizing a mechanical gear system), Leibniz' Stepped Reckoner, and Babbage's analytical engine (which used punched cards) (see Gardner, 1986, for more detail).

Continuing with more history: the Atanasoff-Berry computer, made in 1939, (Mackintosh, 1988), and the 1500 vacuum tube Colossus, were the first program-mable electronic machines. The Colossus first ran in 1943 in order to break a German coding machine named Enigma. The first computer able to store programs was the Manchester University Mark I. It ran its first program in 1948. Later, the transistor and the integrated circuit enabled micro-miniaturization and led to the modern computer.

In 1988, one of the world's most powerful and fastest computers is the liquid-cooled Cray2 produced by Cray Research. It performs 250 million floating point arithmetic operations per second — much more expensive than the abacus or Napier's bones, but also much faster!

2.5 The Human Brain vs. the Computer Brain

2.5.1 The Human Brain

While it's clear that the computer "brain" is vastly superior to man's brain in certain tasks, for perspective it is useful to mention some of the lesser known capacities of the human mind-machine.

The human brain weighs about three pounds and is made of roughly 10 billion neurons, each neuron receiving connections from perhaps 100 other neurons and connecting to still 100 more (Figure 2.5). The web of interconnections is so complex that the whole cortex can be thought of as one entity of integrated activity. Many neurobiologists believe that memory, learning, emotions, creativity, imagination — all the unique elements of human character — will ultimately be shown to reside in the precise patterns of synaptic interconnections in the human brain. The importance of the brain's system of pathways has led some scientists to hypothesize an equation for consciousness itself: C = f1(n)f2(s) (Rose, 1976). Consciousness C is represented on the cellular level by a function of neural cell number, n, and connectivity s. It has been shown that small systems of neurons (i.e., under 10,000 neurons), such as those in simple invertebrates, are capable of learning and memory. In 1987, computer models of neural networks helped researchers begin to untangle the complexities of biological processes such as vision.

2.5.2 Human Computers

We know that the human brain is capable of profound and important functions such as creativity and imagination, but often little is said of its computing and storage capabilities. In some instances, the human memory can be great. For example, in 1974, one individual recited 16,000 pages of Buddhist texts without error. Later, a 23-year old Indian man recited π from memory to 31,811 places in about 3 hours. (Note that in 1987 a NEC SX-2 supercomputer calculated π to more than 134 million digits. In 1989, the Chudnovsky brothers, two Columbia University mathematicians, computed over one billion digits of π using a Cray 2 and an IBM 3090-VF computer.)

As an example of computational capabilities of the human brain, Willem Klein in 1981 was able to extract the 13th root of a 100-digit number in about one minute. In addition, there are the autistic savants — people who can perform mental feats at a level far beyond the capacity of a normal person but whose overall IQ is very low.


Excerpted from Computers, Pattern, Chaos and Beauty by Clifford A. Pickover. Copyright © 2001 Clifford A. Pickover. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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