Dynamical systems and fractals Computer graphics experiments in ascal
Experimental mathematics, a child of our ‘Computer Age’, allows us glimpses into
the world of numbers that are breathtaking, not just to mathematicians. Abstract
concepts, until recently known only to specialists - for example Feigenbaum diagrams or
Julia sets - are becoming vivid objects, which even renew the motivation of students.
Beauty and mathematics: they belong together visibly, and not just in the eyes of
mathematicians
Dynamical systems
and fractals
Computer graphics experiments
in Pascal
Published by the Press Syndicate of the University of Cambridge
The Pitt Building, Trumpington Street, Cambridge CB2 1RP
40 West 20th Street, New York, NY lOOllL4211, USA
10 Stamford Road, Oakleigh, Melbourne 3166, Australia
Originally published in German as Computergrafische Experimente mit Pascal: Chaos und
Ordnung in Dynamischen Systemen by Friedr. Vieweg & Sohn, Braunschweig 1986,
second edition 1988, and 0 Friedr. Vieweg & Sohn Verlagsgesellschaft mbH,
Braunschweig 1986, 1988
First published in English 1989
Reprinted 1990 (three times)
English translation 0 Cambridge University Press 1989
Printed in Great Britain at the University Press, Cambridge
Library of Congress cataloguing in publication data available
British Library cataloguing in publication data
Becker, Karl-Heinze
Dynamical systems and fractals
1. Mathematics. Applications of computer graphics
I. Title II. Doffler, Michael III.
Computergrafische Experimente mit Pascal. English
5 lo’.28566
ISBN 0 521 36025 0 hardback
ISBN 0 521 3 6 9 1 0 X paperback
Dynamical systems
and fractals
Computer graphics experiments in Pascal
Karl-Heinz Becker
Michael Diirfler
Translated by Ian Stewart
CAMBRIDGE UNIVERSITY PRESS
Cambridge
New York Port Chester Melbourne Sydney
vi Dynamical Systems and Fractals
7 New Sights - new Insights 179
7.1 Up Hill and Down Dale 186
7.2 Invert It - It’s Worth It! 186
7.3 The World is Round 192
7.4 Inside Story 199
8 Fractal Computer Graphics 203
8.1 All Kinds of Fractal Curves 204
8.2 Landscapes: Trees, Grass, Clouds, Mountains, and Lakes 211
8.3 Graftals 216
8.4 RepetitiveDesigns 224
9 Step by Step into Chaos 231
10 Journey to the Land of Infinite Structures 247
11 Building Blocks for Graphics Experiments 257
11.1 The Fundamental Algorithms 258
11.2 FractalsRevisited 267
11.3 Ready, Steady, Go! 281
11.4 The Loneliness of the Long-distance Reckoner 288
11.5 What You See Is What You Get 303
11.6 A Picture Takes a Trip 319
12 Pascal and the Fig-trees 327
12.1 Some Are More Equal Than Others - Graphics on
Other Systems 328
12.2 MS-DOS and PS/2 Systems 328
12.3 UNIX Systems 337
12.4 Macintosh Systems 347
12.5 Atari Systems 361
12.6 Apple II Systems 366
12.7 ‘Kermit Here’ - Communications 374
13 Appendices 379
13.1 Data for Selected Computer Graphics 380
13.2 Figure Index 383
13.3 Program Index 388
13.4 Bibliography 391
13.5 Acknowledgements 393
Index 395
Contents
Foreword
New Directions in Computer Graphics : Experimental Mathematics vii
Preface to the German Edition xi
1 Researchers Discover Chaos
1.1 Chaos and Dynamical Systems - What Are They? 3
1.2 Computer Graphics Experiments and Art 6
2 Between Order and Chaos: Feigenbaum Diagrams 17
2.1 First Experiments 18
2.1.1 It’s Prettier with Graphics 27
2.1.2 GraphicalIteration 34
2.2 Fig-trees Forever 37
2.2.1 Bifurcation Scenario - the Magic Number ‘Delta 46
2.2.2 Attractors and Frontiers 48
2.2.3 FeigenbaumLandscapes 51
2.3 Chaos - Two Sides to the Same Coin 53
3 Strange Attractors 55
3.1 The Strange Attractor 56
3.2 The Henon Attractor 62
3.3 The Lorenz Attractor 64
4 Greetings from Sir Isaac 71
4.1 Newton’s Method 72
4.2 Complex Is Not Complicated 81
4.3 Carl Friedrich Gauss meets Isaac Newton 86
5 Complex Frontiers 91
5.1 Julia and His Boundaries 92
5.2 Simple Formulas give Interesting Boundaries 108
6 Encounter with the Gingerbread Man 127
6.1 A Superstar with Frills 128
6.2 Tomogram of the Gingerbread Man 145
6.3 Fig-tree and Gingerbread Man 159
6.4 Metamorphoses 167
. ..
Vlll Dynamical Systems and Fmctals
members to carry out far more complicated mathematical experiments. Complex
dynamical systems are studied here; in particular mathematical models of changing or
self-modifying systems that arise from physics, chemistry, or biology (planetary orbits,
chemical reactions, or population development). In 1983 one of the Institute’s research
groups concerned itself with so-called Julia sets. The bizarre beauty of these objects
lent wings to fantasy, and suddenly was born the idea of displaying the resulting pictures
as a public exhibition.
Such a step down from the ‘ivory tower’ of science, is of course not easy.
Nevertheless, the stone began to roll. The action group ‘Bremen and its University’, as
well as the generous support of Bremen Savings Bank, ultimately made it possible: in
January 1984 the exhibition Harmony in Chaos and Cosmos opened in the large bank
lobby. After the hectic preparation for the exhibition, and the last-minute completion of a
programme catalogue, we now thought we could dot the i’s and cross the last t’s. But
something different happened: ever louder became the cry to present the results of our
experiments outside Bremen, too. And so, within a few months, the almost completely
new exhibition Morphology of Complex Boundan’es took shape. Its journey through
many universities and German institutes began in the Max Planck Institute for
Biophysical Chemistry (Gottingen) and the Max Planck Institute for Mathematics (in
Bonn Savings Bank).
An avalanche had broken loose. The boundaries within which we were able to
present our experiments and the theory of dynamical systems became ever wider. Even
in (for us) completely unaccustomed media, such as the magazine Gw on ZDF
television, word was spread. Finally, even the Goethe Institute opted for a world-wide
exhibition of our computer graphics. So we began a third time (which is everyone’s
right, as they say in Bremen), equipped with fairly extensive experience. Graphics,
which had become for us a bit too brightly coloured, were worked over once more.
Naturally, the results of our latest experiments were added as well. The premiere was
celebrated in May 1985 in the ‘BGttcherstrasse Gallery’. The exhibition SchSnheit im
Chaos/Frontiers of Chaos has been travelling throughout the world ever since, and is
constantly booked. Mostly, it is shown in natural science museums.
It’s no wonder that every day we receive many enquiries about computer graphics,
exhibition catalogues (which by the way were all sold out) and even programming
instructions for the experiments. Naturally, one can’t answer all enquiries personally. But
what are books for? The Beauty of Fractals, that is to say the book about the exhibition,
became a prizewinner and the greatest success of the scientific publishing company
Springer-Verlag. Experts can enlighten themselves over the technical details in The
Science of Fractal Images, and with The Game of FractaJ Images lucky Macintosh II
owners, even without any further knowledge, can boot up their computers and go on a
journey of discovery at once. But what about all the many home computer fans, who
themselves like to program, and thus would like simple, but exact. information? The
book lying in front of you by Karl-Heinz Becker and Michael DGrfler fills a gap that has
Foreword
New Directions in Computer Graphics:
Experimental Mathematics
As a mathematician one is accustomed to many things. Hardly any other academics
encounter as much prejudice as we do. To most people, mathematics is the most
colourless of all school subjects - incomprehensible, boring, or just terribly dry. And
presumably, we mathematicians must be the same, or at least somewhat strange. We deal
with a subject that (as everyone knows) is actually complete. Can there still be anything
left to find out? And if yes, then surely it must be totally uninteresting, or even
superfluous.
Thus it is for us quite unaccustomed that our work should so suddenly be
confronted with so much public interest. In a way, a star has risen on the horizon of
scientific knowledge, that everyone sees in their path.
Experimental mathematics, a child of our ‘Computer Age’, allows us glimpses into
the world of numbers that are breathtaking, not just to mathematicians. Abstract
concepts, until recently known only to specialists - for example Feigenbaum diagrams or
Julia sets - are becoming vivid objects, which even renew the motivation of students.
Beauty and mathematics: they belong together visibly, and not just in the eyes of
mathematicians.
Experimental mathematics: that sounds almost like a self-contradiction!
Mathematics is supposed to be founded on purely abstract, logically provable
relationships. Experiments seem to have no place here. But in reality, mathematicians, by
nature, have always experimented: with pencil and paper, or whatever equivalent was
available. Even the relationship a%@=~?, well-known to all school pupils, for the
sides of a right-angled triangle, didn’t just fall into Pythagoras’ lap out of the blue. The
proof of this equation came after knowledge of many examples. The working out of
examples is a‘typical part of mathematical work. Intuition develops from examples.
Conjectures are formed, and perhaps afterwards a provable relationship is discerned.
But it may also demonstrate that a conjecture was wrong: a single counter-example
suffices.
Computers and computer graphics have lent a new quality to the working out of
examples. The enormous calculating power of modem computers makes it possible to
study problems that could never be assaulted with pencil and paper. This results in
gigantic data sets, which describe the results of the particular calculation. Computer
graphics enable us to handle these data sets: they become visible. And so, we are
currently gaining insights into mathematical structures of such infinite complexity that we
could not even have dreamed of it until recently.
Some years ago the Institute for Dynamical Systems of the University of Bremen
was able to begin the installation of an extensive computer laboratory, enabling its
Foreword ix
too long been open.
The two authors of this book became aware of our experiments in 1984, and
through our exhibitions have taken wing with their own experiments. After didactic
preparation they now provide, in this book, a quasi-experimental introduction to our field
of research. A veritable kaleidoscope is laid out: dynamical systems are introduced,
bifurcation diagrams are computed, chaos is produced, Julia sets unfold, and over it all
looms the ‘Gingerbread Man’ (the nickname for the Mandelbrot set). For all of these,
there are innumerable experiments, some of which enable us to create fantastic computer
graphics for ourselves. Naturally, a lot of mathematical theory lies behind it all, and is
needed to understand the problems in full detail. But in order to experiment oneself (even
if in perhaps not quite as streetwise a fashion as a mathematician) the theory is luckily not
essential. And so every home computer fan can easily enjoy the astonishing results of
his or her experiments. But perhaps one or the other of these will let themselves get
really curious. Now that person can be helped, for that is why it exists: the study of
mathematics.
But next, our research group wishes you lots of fun studying this book, and great
success in your own experiments. And please, be patient: a home computer is no
‘express train’ (or, more accurately, no supercomputer). Consequently some of the
experiments may tax the ‘little ones’ quite nicely. Sometimes, we also have the same
problems in our computer laboratory. But we console ourselves: as always, next year
there will be a newer, faster, and simultaneously cheaper computer. Maybe even for
Christmas... but please with colour graphics, because then the fun really starts.
Research Group in Complex Dynamics
University of Bremen Hartmut Jikgens
Xii Dynamical Systems and Fractals
hardly any insight would be possible without the use of computer systems and graphical
data processing.
This book divides into two main parts. In the first part (Chapters 1 -lo), the reader
is introduced to interesting problems and sometimes a solution in the form of a program
fragment. A large number of exercises lead to individual experimental work and
independent study. The fist part closes with a survey of ‘possible’ applications of this
new theory.
In the second part (from Chapter 11 onwards) the modular concept of our program
fragments is introduced in connection with selected problem solutions. In particular,
readers who have never before worked with Pascal will find in Chapter 11 - and indeed
throughout the entire book - a great number of program fragments, with whose aid
independent computer experimentation can be carried out. Chapter 12 provides reference
programs and special tips for dealing with graphics in different operating systems and
programming languages. The contents apply to MS-DOS systems with Turbo Pascal
and UNIX 4.2 BSD systems, with hints on Berkeley Pascal and C. Further example
programs, which show how the graphics routines fit together, are given for Macintosh
systems (Turbo Pascal, Lightspeed Pascal, Lightspeed C), the Atari (ST Pascal Plus), the
Apple IIe (UCSD Pascal), and the Apple IIGS (TML Pascal).
We are grateful to the Bremen research group and the Vieweg Company for
extensive advice and assistance. And, not least, to our readers. Your letters and hints
have convinced us to rewrite the fist edition so much that the result is virtually a new
book - which, we hope, is more beautiful, better, more detailed, and has many new ideas
for computer graphics experiments.
Bremen Karl-Heinz Becker Michael Dbffler
Preface to the German Edition
Today the ‘theory of complex dynamical systems’ is often referred to as a revolution,
illuminating all of science. Computer-graphical methods and experiments today define
the methodology of a new branch of mathematics: ‘experimental mathematics’. Its content
is above all the theory of complex dynamical systems. ‘Experimental’ here refers
primarily to computers and computer graphics. In contrast to the experiments are
‘mathematical cross-connections’, analysed with the aid of computers, whose examples
were discovered using computer-graphical methods. The mysterious structure of these
computer graphics conceals secrets which still remain unknown, and lie at the frontiers of
thought in several areas of science. If what we now know amounts to a revolution, then
we must expect further revolutions to occur.
. The groundwork must therefore be prepared, and
. people must be found who can communicate the new knowledge.
We believe that the current favourable research situation has been created by the growing
power and cheapness of computers. More and more they are being used as research
tools. But science’s achievement has always been to do what can be done. Here we
should mention the name of Benoi§t B. Mandelbrot, a scientific outsider who worked for
many years to develop the fundamental mathematical concept of a fractal and to bring it to
life.
Other research teams have developed special graphical techniques. At the
University of Bremen fruitful interaction of mathematicians and physicists has led to
results which have been presented to a wide public. In this context the unprecedented
popular writings of the group working under Professors Heinz-Otto Peitgen and Peter
H. Richter must be mentioned. They brought computer graphics to an interested public
in many fantastic exhibitions. The questions formulated were explained non-technically
in the accompanying programmes and exhibition catalogues and were thus made
accessible to laymen. They recognised a further challenge, to emerge from the ‘Ivory
Tower’ of science, so that scientific reports and congresses were arranged not only in the
university. More broadly, the research group presented its results in the magazine Geo,
on ZDF television programmes, and in worldwide exhibitions arranged by the Goethe
Institute. We know of no other instance where the bridge from the foremost frontier of
research to a wide lay public has been built in such a short time. In our own way we
hope to extend that effort in this book. We hope, while dealing with the discoveries of
the research group, to open for many readers the path to their own experiments. Perhaps
in this way we can lead them towards a deeper understanding of the problems connected
with mathematical feedback.
Our book is intended for everyone who has a computer system at their disposal and
who enjoys experimenting with computer graphics. The necessary mathematical formulas
are so simple that they can easily be understood or used in simple ways. The reader will
rapidly be brought into contact with a frontier of today’s scientific research, in which
2 Dynamical Systems and Fractals
The story which today so fascinates researchers, and which is associated with chaos
theory and experimental mathematics, came to our attention around 1983 in Bremen. A t
that time a research group in dynamical systems under the leadership of Professors
Peitgen and Richter was founded at Bremen University. This starting-point led to a
collaboration lasting many years with members of the Computer Graphics Laboratory at
the University of Utah in the USA.
Equipped with a variety of research expertise, the research group began to install its
own computer graphics laboratory. In January and February of 1984 they made their
results public. These results were startling and caused a great sensation. For what they
exhibited was beautiful, coloured computer graphics reminiscent of artistic paintings. The
first exhibition, Harmony in Chaos and Cosmos, was followed by the exhibition
Moqhology of Complex Frontiers. With the next exhibition the results became
internationally known. In 1985 and 1986, under the title Frontiers of Chaos and with
assistance from the Goethe Institute, this third exhibition was shown in the UK and the
USA. Since then the computer graphics have appeared in many magazines and on
television, a witches’ brew of computer-graphic simulations of dynamical systems.
What is so stimulating about it?
Why did these pictures cause so great a sensation?
We think that these new directions in research are fascinating on several grounds. It
seems that we are observing a ’ celestial conjunction’ - a conjunction as brilliant as that
which occurs when Jupiter and Saturn pass close together in the sky, something that
happens only once a century. Similar events have happened from time to time in the
history of science. When new theories overturn or change previous knowledge, we.
speak of a paradigm change. 1
The implications of such a paradigm change are influenced by science and society.
We think that may also be the case here. At any rate, from the scientific viewpoint, this
much is clear:
. A new theory, the so-called chaos theory, has shattered the scientific world-
view. We will discuss it shortly.
. New techniques are changing the traditional methods of work of mathematics and
lead to the concept of experimental mathematics.
For centuries mathematicians have stuck to their traditional tools and methods such
as paper, pen, and simple calculating machines, so that the typical means of progress in
mathematics have been proofs and logical deductions. Now for the first time some
mathematicians are working like engineers and physicists. The mathematical problem
under investigation is planned and carried out like an experiment. The experimental
apparatus for this investigatory mathematics is the computer. Without it, research in this
field would be impossible. The mathematical processes that we wish to understand are
‘Paradigm = ‘example’. By a paradigm we mean a basic Point of view, a fundamental unstated
assumption, a dogma, through which scientists direct their investigations.
1 Researchers Discover Chaos
4 Dynamical Systems and Ftactals
Why does the computer - the very incarnation of exactitude - find its limitations
here?
Let us take a look at how meteorologists, with the aid of computers, make their
predictions. The assumptions of the meteorologist are based on the causality principle.
This states that equal causes produce equal effects - which nobody would seriously
doubt. Therefore the knowledge of all weather data must make an exact prediction
possible. Of course this cannot be achieved in practice, because we cannot set up
measuring stations for collecting weather data in an arbitrarily large number of places.
For this reason the meteorologists appeal to the strong causality principle, which holds
that similar causes produce similar effects. In recent decades theoretical models for the
changes in weather have been derived from this assumption.
Data: Parameters:
Air-pressure Time of year
Temperature Vegetation
Cloud-cover Snow
Wind-direction Sunshine
Wind-speed
Innut Mathematical formulas
- I 1 1 represZt~ ~~atkZaviour i piii5
Situation
at 12.00
for 06.00
Figure 1.1-i Feedback cycle of weather research.
Such models, in the form of complicated mathematical equations, are calculated with
the aid of the computer and used for weather prediction. In practice weather data from the
worldwide network of measuring stations, such as pressure, temperature, wind direction,
and many other quantities, are entered into the computer system, which calculates the
resulting weather with the aid of the underlying model. For example, in principle the
method for predicting weather 6 hours ahead is illustrated in Figure 1.1-l. The 24-
hour forecast can easily be obtained, by feeding the data for the l&hour computation
back into the model. In other words, the computer system generates output data with the
aid of the weather forecasting program. The data thus obtained are fed back in again as
input data. They produce new output data, which can again be treated as input data. The
data are thus repeatedly fed back into the program.
Discovering Chaos 3
visual&d in the form of computer graphics. From the graphics we draw conclusions
about the mathematics. The outcome is changed and improved, the experiment carried out
with the new data. And the cycle starts anew.
. Two previously separate disciplines, mathematics and computer graphics, are
growing together to create something qualitatively new.
Even here a further connection with the experimental method of the physicist can be seen.
In physics, bubble-chambers and semiconductor detectors are instruments for visualising
the microscopically small processes of nuclear physics. Thus these processes become
representable and accessible to experience. Computer graphics, in the area of dynamical
systems, are similar to bubble-chamber photographs, making dynamical processes
visible.
Above all, this direction of research seems to us to have social significance:
. The ‘ivory tower’ of science is becoming transparent.
In this connection you must realise that the research group is interdisciplinary.
Mathematicians and physicists work together, to uncover the mysteries of this new
discipline. In our experience it has seldom previously been the case that scientists have
emerged from their own ‘closed’ realm of thought, and made their research results known
to a broad lay public. That occurs typically here.
l These computer graphics, the results of mathematical research, are very surprising
and have once more raised the question of what ‘art’ really is.
Are these computer graphics to become a symbol of our ‘hi-tech’ age?
b For the first time in the history of science the distance between the
utmost frontiers of research, and what can be understood by the ‘man
in the street’, has become vanishingly small.
Normally the distance between mathematical research, and what is taught in schools, is
almost infinitely large. But here the concerns of a part of today’s mathematical research
can be made transparent. That has not been possible for a long time.
Anyone can join in the main events of this new research area, and come to a basic
understanding of mathematics. The central figure in the theory of dynamical systems, the
Mandelbrot set - the so-called ‘Gingerbread Man’ - was discovered only in 1980.
Today, virtually anyone who owns a computer can generate this computer graphic for
themselves, and investigate how its hidden structures unravel.
1 .l Chaos and Dynamical Systems - What Are They?
An old farmer’s saying runs like this: ‘When the cock crows on the dungheap, the
weather will either change, or stay as it is.’ Everyone can be 100 per cent correct with
this weather forecast. We obtain a success rate of 60 per cent if we use the rule that
tomorrow’s weather will be the same as today’s. Despite satellite photos, worldwide
measuring networks for weather data, and supercomputers, the success rate of
computer-generated predictions stands no higher than 80 per cent.
Why is it not better?
6 Dynamical Systems and Fractals
chemistry and mathematics, and also in economic areas.
The research area of dynamical systems theory is manifestly interdisciplinary. The
theory that causes this excitement is still quite young and - initially - so simple
mathematically that anyone who has a computer system and can carry out elementary
programming tasks can appreciate its startling results.
Possible Parameters
Initial Specification of a I, R e s u l t
Value process
Feedback
Figure 1.1-2 General feedback scheme.
The aim of chaos research is to understand in general how the transition from order
to chaos takes place.
An important possibility for investigating the sensitivity of chaotic systems is to
represent their behaviour by computer graphics. Above all, graphical representation of
the results and independent experimentation has considerable aesthetic appeal, and is
exciting.
In the following chapters we will introduce you to such experiments with different
dynamical systems and their graphical representation. At the same time we will give you
- a bit at a time - a vivid introduction to the conceptual world of this new research area.
1.2 Computer Graphics Experiments and Art
In their work, scientists distinguish two important phases. In the ideal case they
alternate between experimental and theoretical phases. When scientists carry out an
experiment, they pose a particular question to Nature. As a rule they offer a definite
point of departure: this might be a chemical substance or a piece of technical apparatus,
with which the experiment should be performed. They look for theoretical interpretations
of the answers, which they mostly obtain by making measurements with their
instruments.
For mathematicians, this procedure is relatively new. In their case the apparatus or
Discovering Chaos 5
One might imagine that the results thus obtained become ever more accurate. The
opposite can often be the case. The computed weather forecast, which for several days
has matched the weather very well, can on the following day lead to a catastrophically
false prognosis. Even if the ‘model system weather’ gets into a ‘harmonious’ relation to
the predictions, it can sometimes appear to behave ‘chaotically’. The stability of the
computed weather forecast is severely over-estimated, if the weather can change in
unpredictable ways. For meteorologists, no more stability or order is detectable in such
behaviour. The model system ‘weather’ breaks down in apparent disorder, in ‘chaos’.
This phenomenon of unpredictablity is characteristic of complex systems. In the
transition from ‘harmony’ (predictability) into ‘chaos’ (unpredictability) is concealed the
secret for understanding both concepts.
The concepts ‘chaos’ and ‘chaos theory’ are ambiguous. At the moment we agree
to speak of chaos only when ‘predictability breaks down’. As with the weather (whose
correct prediction we classify as an ‘ordered’ result), we describe the meteorologists -
often unfairly - as ‘chaotic’, when yet again they get it wrong.
Such concepts as ‘order’ and ‘chaos’ must remain unclear at the start of our
investigation. To understand them we will soon carry out our own experiments. For
this purpose we must clarify the many-sided concept of a dynamical system.
In general by a system we understand a collection of elements and their effects on
each other. That seems rather abstract. But in fact we are surrounded by systems.
The weather, a wood, the global economy, a crowd of people in a football stadium,
biological populations such as the totality of all fish in a pond, a nuclear power station:
these are all systems, whose ‘behaviour’ can change very rapidly. The elements of the
dynamical system ‘football stadium’, for example, are people: their relations with each
other can be very different and of a multifaceted kind.
Real systems signal their presence through three factors:
l They are dynamic, that is, subject to lasting changes.
l They are complex, that is, depend on many parameters.
. They are iterative, that is, the laws that govern their behaviour can be
described by feedback.
Today nobody can completely describe the interactions of such a system through
mathematical formulas, nor predict the behaviour of people in a football stadium.
Despite this, scientists try to investigate the regularities that form the basis of such
dynamical systems. In particular one exercise is to find simple mathematical models, with
whose help one can simulate the behaviour of such a system.
We can represent this in schematic form as in Figure 1.1-2.
Of course in a system such as the weather, the transition from order to chaos is hard
to predict. The cause of ‘chaotic’ behaviour is based on the fact that negligible changes to
quantities that are coupled by feedback can produce unexpected chaotic effects. This is an
apparently astonishing phenomenon, which scientists of many disciplines have studied
with great excitement. It applies in particular to a range of problems that might bring into
question recognised theories or stimulate new formulations, in biology, physics,
Dynamical Systems and l%ctah
Figure 1.2-2 Vulcan’s Eye.
Discovering Chaos 7
measuring instrument is a computer. The questions are presented as formulas,
representing a series of steps in an investigation. The results of measurement are
numbers, which must be interpreted. To be able to grasp this multitude of numbers, they
must be represented clearly. Often graphical methods are used to achieve this. Bar-
charts and pie-charts, as well as coordinate systems with curves, are widespread
examples. In most cases not only is a picture ‘worth a thousand words’: the picture is
perhaps the only way to show the precise state of affairs.
Over the last few years experimental mathematics has become an exciting area, not
just for professional researchers, but for the interested layman. With the availability of
efficient personal computers, anyone can explore the new territory for himself.
The results of such computer graphics experiments are not just very attractive
visually - in general they have never been produced by anyone else before.
In this book we will provide programs to make the different questions from this
area of mathematics accessible. At first we will give the programs at full length; but later
- following the building-block principle - we shall give only the new parts that have not
occurred repeatedly.
Before we clarify the connection between experimental mathematics and computer
graphics, we will show you some of these computer graphics. Soon you will be
producing these, or similar, graphics for yourself. Whether they can be described as
computer art you must decide for yourself.
Figure 1.2-l Rough Diamond.
10 Dynamical Systems and Fractals
Figure 1.2-4 Tornado Convention.2
2Tbis picture was christened by Prof. K. Kenkel of Dartmouth College.
