Table Of ContentThe Science of Fractal Images
Heinz-Otto Peitgen Dietmar Saupe
Editors
The Science
of Fractal Images
Michael F. Barnsley Robert L. Devaney Benoit B. Mandelbrot
Heinz-Otto Peitgen Dietmar Saupe Richard F. Voss
With Contributions by
Yuval Fisher Michael McGuire
With 142 Illustrations in 277 Parts and 39 Color Plates
Springer-Verlag New York Berlin Heidelberg
London Paris Tokyo
Heinz-Otto Peitgen
Institut fUr Oynamische Systeme, Universitat Bremen, 0-2800 Bremen 33, Federal Republic
of Germany, and
Department of Mathematics, University of California, Santa Cruz, CA 95064, USA
Oietmar Saupe
Institut fUr Oynamische Systeme, Universitat Bremen, 0-2800 Bremen 33, Federal Republic
of Germany
The cover picture shows a fractal combining the two main topics of this book: deterministic and random
fractals. The deterministic part is given in the form of the potential surface in a neighborhood of the Mandelbrot
set. The sky is generated using random fractals. The image is produced by H. Jurgens, H.-O. Peitgen and
D. Saupe.
The back cover images are: Black Forest in Winter (top left, M. Bamsley, F. Jacquin, A. Malassenet, A.
Sloan, L. Reuter); Distance estimates at boundary of Mandelbrot set (top right, H. Jurgens, H.-O. Peitgen,
D. Saupe); Floating island of Gulliver's Travels (center left, R. Voss); Foggy fractally cratered landscape
(center right, R. Voss); Fractal landscaping (bottom, B. Mandelbrot, K. Musgrave).
Library of Congress Cataloging-in-Publication Data
The Science of fractal images: edited by Heinz-Otto Peitgen
and Dietmar Saupe ; contributions by Michael F. Bamsley ... let al.l.
p. cm.
Based on notes for the course Fractals-introduction, basics, and
perspectives given by Michael F. Bamsley, and others, as part of
the SIGGRAPH '87 (Anaheim, Calif.) course program.
Bibliography: p.
Includes index.
I. Fractals. I. Peitgen, Heinz-Otto, 1945- II. Saupe, Dietmar,
1954- III. Bamsley, M. F. (Michael Fielding), 1946-
QA614.86.S35 1988
5 I 6--dc 19 88-12683
ISBN-13: 978-1-4612-8349-2 e-ISBN-13: 978-1-4612-3784-6
001: 10.1007/978-1-4612-3784-6
© 1988 by Springer-Verlag New York Inc; the copyright of all artwork and images remains with the individual
authors.
Softcover reprint of the hardcover 1st edition 1988
All rights reserved. This work may not be translated or copied in whole or in part without the written
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brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information
storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology
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are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks
and Merchandise Marks Act, may accordingly be used freely by anyone.
This book was prepared with IiITEX on Macintosh computers and reproduced by Springer-Verlag from
camera-ready copy supplied by the editors.
TEX is a trademark of the American Mathematical Society. Macintosh is a trademark of Applt; Computer,
Inc.
9 8 765
Preface
This book is based on notes for the course Fractals:lntroduction, Basics and
Perspectives given by MichaelF. Barnsley, RobertL. Devaney, Heinz-Otto Peit
gen, Dietmar Saupe and Richard F. Voss. The course was chaired by Heinz-Otto
Peitgen and was part of the SIGGRAPH '87 (Anaheim, California) course pro
gram. Though the five chapters of this book have emerged from those courses
we have tried to make this book a coherent and uniformly styled presentation
as much as possible. It is the first book which discusses fractals solely from the
point of view of computer graphics. Though fundamental concepts and algo
rithms are not introduced and discussed in mathematical rigor we have made a
serious attempt to justify and motivate wherever it appeared to be desirable. Ba
sic algorithms are typically presented in pseudo-code or a description so close
to code that a reader who is familiar with elementary computer graphics should
find no problem to get started.
Mandelbrot's fractal geometry provides both a description and a mathemat
ical model for many of the seemingly complex forms and patterns in nature and
the sciences. Fractals have blossomed enormously in the past few years and
have helped reconnect pure mathematics research with both natural sciences
and computing. Computer graphics has played an essential role both in its de
velopment and rapidly growing popularity. Conversely, fractal geometry now
plays an important role in the rendering, modelling and animation of natural
phenomena and fantastic shapes in computer graphics.
We are proud and grateful that Benoit B. Mandelbrot agreed to write a de
tailed foreword for our book. In these beautiful notes the Father of Fractals
shares with us some of the computer graphical history of fractals.
The five chapters of our book cover:
• an introduction to the basic axioms of fractals and their applications in the
natural sciences,
• a survey of random fractals together with many pseudo codes for selected
algorithms,
• an introduction into fantastic fractals, such as the Mandelbrot set, Julia
sets and various chaotic attractors, together with a detailed discussion of
algorithms,
• fractal modelling of real world objects.
V
Chapters 1 and 2 are devoted to random fractals. While Chapter 1 also
gives an introduction to the basic concepts and the scientific potential of frac
tals, Chapter 2 is essentially devoted to algorithms and their mathematical back
ground. Chapters 3,4 and 5 deal with deterministic fractals and develop a dy
namical systems point of view. The first part of Chapter 3 serves as an intro
duction to Chapters 4 and 5, and also describes some links to the recent chaos
theory.
The Appendix of our book has four parts. In Appendix A Benoit B. Mandel
brot contributes some of his brand new ideas to create random fractals which are
directed towards the simulation of landscapes, including mountains and rivers.
In Appendix B we present a collection of magnificent photographs created and
introduced by Michael Mc Guire, who works in the tradition of Ansel Adams.
The other two appendices were added at the last minute. In Appendix C Diet
mar Saupe provides a short introduction to rewriting systems, which are used
for the modelling of branching patterns of plants and the drawing of classic frac
tal curves. These are topics which are otherwise not covered in this book but
certainly have their place in the computer graphics of fractals. The final Appen
dix D by Yuval Fisher from Cornell University shares with us the fundamentals
of a new algorithm for the Mandelbrot set which is very efficient and therefore
has potential to become popular for PC based experiments.
Almost throughout the book we provide selected pseudo codes for the most
fundamental algorithms being developed and discussed, some of them for be
ginning and some others for advanced readers. These codes are intended to
illustrate the methods and to help with a first implementation, therefore they are
not optimized for speed.
The center of the book displays 39 color plates which exemplify the potential
of the algorithms discussed in the book. They are referred to in the text as Plate
followed by a single number N. Color plate captions are found on the pages
immediately preceding and following the color work. There we also describe
the front and back cover images of the book. All black and white figures are
listed as Figure N.M. Here N refers to the chapter number and M is a running
number within the chapter.
After our first publication in the Scientific American, August 1985, the Man
delbrot set has become one of the brightest stars of amateur mathematics. Since
then we have received numerous mailings from enthusiasts around the world.
VI
We have reproduced some of the most beautiful experiments (using MSetDEM( ) ,
see Chapter 4) on pages 20 and 306. These were suggested by David Brooks
and Daniel N. Kalikow, Framingham, Massachusetts.
Bremen, March 1988
Heinz-Otto Peitgen and Dietmar Saupe
Acknowledgements
Michael F. Barnsley acknowledges collaboration with Laurie Reuter and Alan
D. Sloan, both from Georgia Institute of Technology. Robert L. Devaney thanks
Chris Frazier from the University of Bremen for putting his black and white
figures in final form. Chris Frazier also produced Figures 0.2, 5.1, 5.2 and 5.17.
Heinz-Otto Peitgen acknowledges collaboration with Hartmut Jurgens and Diet
mar Saupe, University of Bremen, and thanks Chris Frazier for some of the
black and white images. Dietmar Saupe thanks Richard F. Voss for sharing his
expertise on random fractals. Besides the authors several people contributed to
the color plates of this book, which is gratefully acknowledged. A detailed list
of credits is given in the Color Plate Section.
Michael F. Bamsley
Robert L. Devaney
Yuval Fisher
Benoit B. Mandelbrot
Michael McGuire
Heinz-Otto Peitgen
Dietmar Saupe
Richard F. Voss
VII
Michael F. Barnsley. *1946 in Folkestone (England). Ph. D. in Theoretical
Chemistry, University of Wisconsin (Madison) 1972. Professor of Mathematics,
Georgia Institute of Technology, Atlanta, since 1983. Formerly at University
of Wisconsin (Madison), University of Bradford (England), Centre d'Etudes
Nucleaires de Saclay (Paris). Founding officer of Iterated Systems, Inc.
Robert L. Devaney. * 1948 in Lawrence, Mass. (USA). Ph. D. at the University
of California, Berkeley, 1973. Professor of Mathematics, Boston University
1980. Formerly at Northwestern University and Tufts University. Research in
terests: complex dynamics, Hamiltonian systems.
M.F. Barnsley R.L. Devaney
Yuval Fisher. *1962 in Israel. 1984 B.S. in Mathematics and Physics, University
of California, Irvine. 1986 M.S. in Computer Science, Cornell University. 1988
Ph. D. in Mathematics (expected), Cornell University.
Benoit B. Mandelbrot. *1924 in Warsaw (Poland). Moved to Paris in 1936, to
USA in 1958. Diploma 1947, Ecole Poly technique, D. Sc. 1952, Paris, Dr.
Sc. (h. c.) Syracuse, Laurentian, Boston, SUNY. 1974 I.B.M. Fellow at Thomas
1. Watson Research Center and 1987 Abraham Robinson Adjunct Professor of
Mathematical Science, Yale University. Barnard Medal 1985,. Franklin Medal
1986. Member of the American Academy of Arts and Sciences and of the U.S.
National Academy of Sciences.
Y. Fisher B.B. Mandelbrot
Michael McGuire. *1945 in Ballarat, Victoria (Australia). Ph. D. in Physics,
University of Washington (Seattle), 1974. He has worked in the field of atomic
frequency standards at the University of Mainz, Germany, NASA, Goddard
Space Flight Center, and Hewlett Packard Laboratories.
Heinz-Otto Peitgen. * 1945 in Bruch (Germany). Dr. rer. nat. 1973, Habilitation
1976, both from the University of Bonn. Research on nonlinear analysis and
dynamical systems. 1977 Professor of Mathematics at the University of Bremen
and since 1985 also Professor of Mathematics at the University of California
at Santa Cruz. Visiting Professor in Belgium, Italy, Mexico and USA.
M. McGuire H.-O. Peitgen
Dietmar Saupe. *1954 in Bremen (Germany). Dr. rer. nat. 1982 at the University
of Bremen. Visiting Assistant Professor of Mathematics at the University of
California, Santa Cruz, 1985-87 and since 1987 at the University of Bremen.
There he is a researcher at the Dynamical Systems Graphics Laboratory. Research
interests: mathematical computer graphics and experimental mathematics.
Richard F. Voss. * 1948 in St. Paul, Minnesota (USA). 1970 B. S. in Physics
from M.LT. 1975 Ph. D. in Physics from U. C. Berkeley. 1975-present: Re
search Staff Member at the LB.M. Thomas 1. Watson Research Laboratory in
Yorktown Heights, NY. Research in condensed matter physics.
D. Saupe R.F. Voss
VIII
Contents
Preface V
Foreword: People and events behind the "Science of Fractal Images" 1
Benoit B.Mandelbrot
0.1 The prehistory of some fractals-to-be: Poincare, Fricke, Klein
and Escher . . . . . . . . . . . . . . 2
0.2 Fractals at IBM. . . . . . . . . . . 5
0.3 The fractal mountains by R.E Voss . 6
0.4 Old films . . . . . . . . . . . . . . 8
0.5 Star Trek II . . . . . . . . . . . . . 8
0.6 Midpoint displacement in Greek geometry: The Archimedes
construction for the parabola . 11
0.7 Fractal clouds ......................... 12
0.8 Fractal trees .... . . . . . . . . . . . . . . . . . . . . .. 13
0.9 Iteration, yesterday's dry mathematics and today's weird and
wonderful new fractal shapes, and the "Geometry Supercom-
puter Project" . . . . . . . . . . . . . . . . . . . . . . 14
0.10 Devaney, Bamsley and the Bremen "Beauty of Fractals" . 17
1 Fractals in nature: From characterization to simulation 21
Richard F.Voss
1.1 Visual introduction to fractals: Coastlines, mountains and clouds 22
1.1.1 Mathematical monsters: The fractal heritage . . . . 25
1.1.2 Fractals and self-similarity . . . . . . . . . . . . . 25
1.1.3 An early monster: The von Koch snowflake curve . 26
1.1.4 Self-similarity and dimension 28
1.1.5 Statistical self-similarity . . 30
1.1.6 Mandelbrot landscapes . . . . 30
1.1.7 Fractally distributed craters . 31
1.1.8 Fractal planet: Brownian motion on a sphere 33
1.1.9 Fractal flakes and clouds . . . . . . . . . . . 33
1.2 Fractals in nature: A brief survey from aggregation to music. 35
1.2.1 Fractals at large scales . . . . . . . . . . . 36
1.2.2 Fractals at small scales: Condensing matter 36
IX
1.2.3 Scaling randomness in time: -jp-noises 39
1.2.4 Fractal music. . . . . . . . . . . . . . 40
1.3 Mathematical models: Fractional Brownian motion 42
1.3.1 Self-affinity ....... . . . . . . . . . 44
1.3.2 Zerosets . . . . . . . . . . . . . . . . . . 45
1.3.3 Self-affinity in higher dimensions: Mandelbrot land-
scapes and clouds .... . . . . . . . . . . . . . .. 45
1.3.4 Spectral densities for fBm and the spectral exponent f3 47
1.4 Algorithms: Approximating fBm on a finite grid . 47
1.4.1 Brownian motion as independent cuts 48
1.4.2 Fast Fourier Transform filtering 49
1.4.3 Random midpoint displacement . . . 51
1.4.4 Successive random additions. . . . . 54
1.4.5 Weierstrass-Mandelbrot random fractal function. 55
1.5 Laputa: A concluding tale . . . . . 57
1.6 Mathematical details and formalism . . . . 58
1.6.1 Fractional Brownian motion . . . . 58
1.6.2 Exact and statistical self-similarity . 59
1.6.3 Measuring the fractal dimension D . 61
1.6.4 Self-affinity ..... . . . . . . . 62
1.6.5 The relation of D to H for self-affine fractional Brow-
nian motion . . . . . . . . . 63
1.6.6 Trails of fBm. . . . . . . . . . . . . . . . . . . . . . 64
1.6.7 Self-affinity in E dimensions . . . . . . . . . . . . . . 64
1.6.8 Spectral densities for fBm and the spectral exponent f3 65
1.6.9 Measuring fractal dimensions: Mandelbrot measures 66
1.6.10 Lacunarity . . . . . . . . . . . . . . . . . . . 67
¥ t:
1.6.11 Random cuts with H Campbell's theorem 69
1.6.12 FFT filtering in 2 and 3 dimensions ..... . 69
2 Algorithms for random fractals 71
Dietmar Saupe
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . 71
2.2 First case study: One-dimensional Brownian motion . 74
2.2.1 Definitions.............. 74
2.2.2 Integrating white noise . . . . . . . . . 75
2.2.3 Generating Gaussian random numbers . 76
2.2.4 Random midpoint displacement method 78
2.2.5 Independent jumps . . . . . . . . . . . 80
2.3 Fractional Brownian motion: Approximation by spatial methods 82
2.3.1 Definitions........... 82
2.3.2 Midpoint displacement methods 84
2.3.3 Displacing interpolated points . 87
x
