Fractals and Chaos. A.J. Crilly R.A. Earnshaw H. Jones Editors. With 146 Figures in 173 Parts, 57 in Color

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1 Fractals and Chaos

2 A.J. Crilly R.A. Earnshaw H. Jones Editors Fractals and Chaos With 146 Figures in 173 Parts, 57 in Color Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona

3 A.J. Crilly Middlesex Polytechnic Barnet, Herts EN4 OPT United Kingdom R.A. Earnshaw University of Leeds Leeds LS2 9JT United Kingdom H. Jones Middlesex Polytechnic London NIl 2NQ United Kingdom Cover illustration: Fractal planet based on the rescale-and-add method. For more details see Plate 25 in the colour insert (image by D. Saupe, University of Bremen). Library of Congress Cataloging-in-Publication Data Fractals and chaos / A.J. Crilly, R.A. Earnshaw, H. Jones, editors. p. cm. 1. Fractals. 2. Chaotic behaviour in systems. I. Crilly, A.J. II. Earnshaw, Rae A., III. Jones, Huw, QA F '.74-dc Printed on acid-free by Springer-Verlag New York Inc. Softcover reprint of the hardcover 1st edition 1991 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for 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 now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc. in this publication, even if the former 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. Typeset by Nancy A. Rogers using 'lex. Photocomposition on a Chelgraph IBX ISBN-13: e-isbn-13: DOl: /

4 Contents About the Editors vii Preface ix Introduction 1 Part 1 Chapter 1 Jones: Fractals Before Mandelbrot Fractals 7 Chapter 2 Reeve: Mandelbrot, Julia Sets and Nonlinear Mappings Chapter 3 Batty: Cities as Fractals: Simulating Growth and Form Chapter 4 Kaandorp: Modelling Growth Forms of Sponges with Fractal Techniques 71 Chapter 5 Saupe: Random Fractals in Image Synthesis Chapter 6 Horn: IFSs and the Interactive Design of Tiling Structures Chapter 7 Bressloff and Stark: Neural Networks, Learning Automata and Iterated Function Systems 145

5 vi Contents Part 2 Chaos Chapter 8 Crilly: The Roots of Chaos-A Brief Guide 193 Chapter 9 LansdoW'Tl,,: Chaos, Design and Creativity 211 Chapter 10 Novak: Relativistic Particles in a Magnetic Field 225 Chapter 11 Mullin: Chaos in Physical Systems 237 Chapter 12 Darbyshire and Price: Phase Portraits from Chaotic Time Series 247 Chapter 13 Pottinger: Data Visualisation Techniques for Nonlinear Systems 259 Index 269

6 About the Editors Fractals and Chaos A.J. Crilly Tony Crilly began his education in Sydney, Australia and later obtained undergraduate and masters degrees in mathematics from the University of Hull in the United Kingdom. He received his Ph.D. in the history of mathematics in 1981 from the Council for National Academic Awards. He has served as both secretary and treasurer of the British Society for the History of Mathematics and is currently a committee member. He is a Fellow of the Institute of Mathematics and its Applications ttnti a member of the British Computer Society Documentation and Displays Group. He has taught in the United States at the University of Michigan and recently spent two years teaching at the newly established City Polytechnic of Hong Kong. While in Hong Kong, he helped set up a department of Applied Mathematics and designed courses in engineering mathematics and discrete mathematics. His present interests lie in the geometry of computer graphics and in mathematical education. He is currently Principal Lecturer in the Faculty of Engineering, Science and Mathematics at Middlesex Polytechnic. He is married with four children and lives in St. Albans, Hertfordshire. R.A. Earnshaw Rae Earnshaw is Head of Computer Graphics at the University of Leeds, with interests in graphics algorithms, integrated graphics and text, display technology, CAD/CAM and human-computer interface issues. He gained his Ph.D. in computer science at the University of Leeds. He has been a Visiting Professor at lit, Chicago, USA, Northwestern Polytechnical University, China and George Washington University, Washington DC, USA. He was a Co-Chair of the BCS/ ACM International Summer Institute on 'State of the Art in Computer Graphics' held in Sterling, Scotland in 1986; in Exeter, England in 1988; and in Edinburgh, Scotland in Dr. Earnshaw is also a Director of the NATO ASIon 'Theoretical Foundations of Computer Graphics and CAD' held in Italy in 1987.

7 viii About the Editors H. Jones Huw Jones was brought up in South Wales and graduated from University College Swansea with a B.S. in Applied Mathematics in The following year he obtained a Diploma in Education from the same institution and, after a short period as a schoolmaster, has spent the rest of his working life as a lecturer in higher education in London. During this period he obtained his Master of Science in Statistics from BruneI University, became a Fellow of the Royal Statistical Society, a member of the European Association for Computer Graphics and a member of the British Computer Society's Computer Graphics and Displays Group Committee. He is currently a Principal Lecturer specialising in Computer Graphics in the Faculty of Engineering, Science and Mathematics at Middlesex Polytechnic, where he is head of the Master of Science in Computer Graphics course. He is married to Judy, a mathematics teacher, and has a son, Rhodri, and a daughter, Ceri.

8 Preface This volume is based upon the presentations made at an international conference in London on the subject of 'Fractals and Chaos'. The objective of the conference was to bring together some of the leading practitioners and exponents in the overlapping fields of fractal geometry and chaos theory, with a view to exploring some of the relationships between the two domains. Based on this initial conference and subsequent exchanges between the editors and the authors, revised and updated papers were produced. These papers are contained in the present volume. We thank all those who contributed to this effort by way of planning and organisation, and also all those who helped in the production of this volume. In particular, we wish to express our appreciation to Gerhard Rossbach, Computer Science Editor, Craig Van Dyck, Production Director, and Nancy A. Rogers, who did the typesetting. A.J. Crilly R.A. Earnshaw H. Jones 1 March 1990

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