Computers, Pattern, Chaos and Beauty [NOOK Book]


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.
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Computers, Pattern, Chaos and Beauty

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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.
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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 (
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Product Details

  • ISBN-13: 9780486151618
  • Publisher: Dover Publications
  • Publication date: 6/14/2012
  • Sold by: Barnes & Noble
  • Format: eBook
  • Pages: 416
  • Sales rank: 1,224,456
  • File size: 60 MB
  • Note: This product may take a few minutes to download.

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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Table of Contents

Chapter 1. Computers and Creativity
1.1 Objectives
1.2 Lateral Use of Computer Software Tools
1.3 Reading List for Chapter 1
Chapter 2. Hidden Words
2.1 "Digits, Symbols, Pictures"
2.2 Computers and Art
2.3 Computer Graphics: Past and Present
2.4 Computers: Past and Present
2.5 The Human Brain vs. the Computer Brain
2.5.1 The Human Brain
2.5.2 Human Computers
2.6 New Applications of Calculating: A Sampling and Digression
2.7 One Final Word
2.8 Reading List for Chapter 2
Chapter 3. Fourier Transforms (The Prisms of Science)
3.1 Fourier Analysis: A Digression and Review
3.2 Singing Computers
3.3 "Bach, Beethoven, The Beatles"
3.4 Breathing Proteins
3.5 Cancer Genes (DNA Waveforms)
3.6 Reading List for Chapter 3
Chapter 4. Unusual Graphic Representations
4.1 Acoustics
4.2 New Ways of Displaying Data
4.3 Snowflakes from Sound: Art and Science
4.4 Medicine: Cardiology and SDPs
4.5 Another Dot-Display Used in Molecular Biophysics
4.6 Autocorrelation Cartoon-Faces for Speech
4.7 Cartoon Faces in Education
4.7.1 Cognitive Association of Coordinates with Facial Features
4.7.2 Target-Pictures for Children
4.7.3 Learning by Means of Analogy
4.8 Educational Aid for the Presentation of Statistical Concepts
4.9 Commercial and Military Air Traffic Control
4.10 Faces and Cancer Genes
4.11 Back to Acoustics: Phase Vectorgrams
4.12 Fractal Characterization of Speech Waveform Graphs
4.12.1 Scale Invariance
4.12.2 Fractal Characterization of Speech
4.12.3 Classification of Shapes
4.12.4 Speech Fractal Dimension
4.12.5 Catalog of Other Acoustic Sounds
4.13 A Monte Carlo Approach to Fractal Dimension
4.14 "Other Speech Graphics, Wigner Distributions, FM Synthesis"
4.14.1 Wigner Distribution
4.14.2 FM Synthesis of Speech
4.15 Molecular Genetics: DNA Vectorgram
4.15.1 Background
4.15.2 DNA Vectorgram
4.16 Reading List for Chapter 4
Chapter 5. Image Processing of the Shroud of Turin
5.1 Image Processing
5.2 The Shroud
5.3 A Sampling of Results of Past Scientific Examination
5.4 Past Image Processing Studies
5.5 Present Study
5.6 Pseudo-Gray Transformation via Look-up Table (LUT)
5.7 Processing via Pseudocolor LUT
5.8 Reading List for Chapter 5
Chapter 6. Physics: Charged Curves
6.1 Recipe for Accelerating the Iterative Process
6.2 Reading List for Chapter 6
Chapter 7. Summary of Part II
Chapter 8. Genesis Equations (or Biological Feedback Forms)
8.1 Feedback
8.2 "A Computer Program "Bug"
8.3 Genesis Equations
8.3.1 The Leeuwenhoek Program
8.3.2 Brief Sampling of Past Work in Natural Object Creation
8.3.3 Biological Forms Generated from Mathematical Feedback Loops
8.3.4 The Habitation of Abstract Geometric Space
8.3.5 Creation and Search in the Manifold Dimensions of Space
8.4 Reading List for Chapter 8
Chapter 9. More Beauty from Complex Variables
9.1 Turbulent Complex Curls
9.2 Transcendental Functions in the µ Plane
9.2.1 "What is "cosh"?"
9.2.2 Review of Some Definitions of Terms Used in This Chapter
9.2.3 Graphics Gallery and Observations
9.2.4 Recipes for Cosh Pictures
9.2.5 Maps
9.3 How to Design Textures Using Recursive Composite Functions
9.3.1 Secrets and Tricks
9.3.2 Chaos and Pattern
9.3.3 Where Do We Go From Here?
9.3.4 Recipe
9.3.5 "Networks, Repeller-Towers"
9.4 Reading List for Chapter 9
Chapter 10. Mathematical Chaos
10.1 What is Chaos?
10.2 Pattern Formation and Chaos in Networks
10.2.1 Networks
10.2.2 Drawing the Network
10.2.3 Some Examples
10.2.4 Summary
10.2.5 Recipes for Generating Networks
10.2.6 Automatic Computer Generation of Networks
10.3 "Graphics, Bifurcation, Order and Chaos"
10.3.1 Recipe for Bifurcation Plot Generation
10.4 Image Processing Techniques and Deterministic Chaos
10.5 A Note on Rendering 3-D Strange Attractors
10.5.1 Recipe
10.6 Quaternion Images
10.6.1 Quaternion Julia Sets
10.7 A Note on Rendering Chaotic Repeller Distance-Towers
10.8 Reading List for Chapter 10
Chapter 11. Number Theory
11.1 Introduction
11.2 Exotic Symmetries from Large Pascal's Triangles
11.2.1 Computation
11.2.2 Nomenclature
11.2.3 Gaskets
11.2.4 Size Changes of Internal Triangles
11.2.5 Symmetries
11.2.6 Plotting a Complete PT
11.2.7 Perfect Numbers
11.2.8 Practical Importance
11.2.9 Future Work
11.3 Patterns in Hailstone (3n+1) Numbers
11.4 Is There a Double Smoothly Undulating Integer?
11.5 Reading List For Chapter 11
Chapter 12. Synthesizing Nature
12.1 Wild Monopodial Tendril Plant Growth
12.1.1 Spirals in Nature
12.1.2 Graphics Gallery of Plant Forms
12.1.3 Natural Growth
12.1.4 Recipe for Picture Computation
12.2 How to Design with Directed Random Walks
12.2.1 Directed Random Walk
12.2.2 The Algorithm
12.2.3 Recipe for Transition Probability Plants
12.3 "A Sampling of Spirals in Science, Nature, and Art"
12.3.1 "Traditional Mathematical Spirals (Clothoid, Littus, et al.)"
12.3.2 Spirals Made by Humans
12.3.3 Strange Spirals
12.3.4 Galaxies
12.3.5 Seashells
12.3.6 Fuzzy Spirals
12.3.7 Spiral-Like Forms from Differential Equations
12.3.8 Trigonometric Iteration (Bushy Spirals)
12.3.9 Concluding Remarks on Spirals
12.4 A Vacation on Mars
12.5 Reading List for Chapter 12
Chapter 13. Synthesizing Ornamental Textures
13.1 Self-decorating Eggs
13.1.1 The Algorithm
13.1.2 Graphics Gallery
13.1.3 Summary of Egg Tiling Patterns
13.2 Self-Decorating Surfaces (Beauty from Noise)
13.2.1 Recipe
13.3 Biometric Art from Retinal Vascular Patterns
13.4 Reading List for Chapter 13
Chapter 14. Dynamical Systems
14.1 Time-discrete Phase Plans Associated with a Cyclic System
14.1.1 Recipe
14.2 Cyles and Centers in Time-Discrete Dynamical Systems
14.2.1 A Review of Some Definitions of Terms
14.3 Reading List for Chapter 14
Chapter 15. Numerical Approximation Methods
15.1 Halley Maps for a Trigonometric and Rational Function
15.1.1 Overrelaxation and Chaos
15.1.2 Buridan's Ass
15.2 "Beauty, Symmetry and Chaos in Chebyshev's Paradise"
15.2.1 Chebyshev Rings
15.3 Recipe for Chebyshev's Paradise
15.4 Reading List for Chapter 15
Chapter 16. Tesselation Automata Derived from a Single Defect
16.1 Method and Observations
16.1.1 TA Type 1
16.1.2 TA Type 2. Time Dependence of Rules
16.1.3 TA Type 3. Contest Between Defects
16.1.4 TA Type 4. Defects in a Centered Rectangular Lattice
16.1.5 TA Type 5. Large Local Neighborhood
16.2 Recipe for Tesselation Automata
16.3 Reading List for Chapter 16
Chapter 17. Summary of Part III and Conclusion of Book
Appendix A. Color Plates
Appendix B. Additional Recipes
B.1 Chaos from bits
B.2 Dynamical Systems
B.3 Cartoon Faces for Multidimensional Data Portrayal
B.4 Symmetrized Random-Dot Patterns
B.4.1 Recipe
B.4.2 Sqrt(r) Dot-Patterns
B.4.3 Cartoon Faces for Multidimensional Data Portrayal
B.5 Burnt Paper
B.6 Picturing Randomness with Truchet Tiles
B.7 Circles Which Kiss: A Note on Osculatory Packing
B.8 Markov Aggregation on a Sticky Circle
Appendix C. Suggestions for Future Experiments
Appendix D. Descriptions for Chapter Frontpiece Figures
References (General)
Recommended Reading
"Interesting and Unusual Newsletters, Distributors, Columns, Misc."
Recommended Journals
References by Topic and Section
Part I - Introduction
Part II - Representing Nature
"Part III - Pattern, Symmetry, and Beauty"
Additional Reading
Works of the Author
A Final Illustration of Chaos
About the Author
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