The -I 7 Golden Ratio and Fibonacci Numbers
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Golden Ratio u~~ Fibonacci Numbers Richard A, Dunlap Dai~~u~ie ~ni~ersity Canada r World Scientific NewJersey London Singapore HongKong
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA oflce: Suite 202,1060 Main Street, River Edge, NJ 07661 UKuflee: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Dunlap, R. A. The golden ratio and Fibonacci numbers I R. A. Dunlap. p. cw. Includes biblio~phi~ references (pp. 153-155) and index. ISBN 9810232640 (alk. paper) 1. Golden section. 2. Fibonacci numbers. I. Title. QA466.D86 1997 512:12--dc21 97-28158 CIP British Library Cataloguing-in-Publication Data A catdope record for this book is available from the British Library. First published 1997 Reprinted 1998,1999,2003 Copyright Q 1997 by World Scientific Publishing Co. Pte. Ltd. A~~rig~fs reserved. Thisbook, orpa~st~ereo~ may notbe repro~uce~inanyformor~any mea~, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permissionfrom the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore.
PREFACE The golden ratio and Fibonacci numbers have numerous applications which range from the description of plant growth and the crystallographic structure of certain solids to the development of computer algorithms for searching data bases. Although much has been written about these numbers, the present book will h0-y IYI the gap between those sources which take a philosophical or even mystical approach and the formal mathematical texts. I have tried to stress not only fundamental properties of these numbers but their application to diverse fields of mathematics, computer science, physics and biology. I believe that this is the k t book to take this approach since the application of models involving the golden ratio to the description of incommensurate structures and quasicrystals in the 1970 s and 1980 s. This book will, hopefully, be of intern to the general reader with an interest in mathematics and its application to the physical and biological sciences. It may also be mitable supplementary reading for an introductory university come in number theory, geometty or general mathematics. Finally, the present volume should be suf iciently infomathe to provide a general introduction to the golden ratio and Fibonacci numbers for those researchers and graduate students who are working in fields where these numbers have found applications. Formal mathematics has been kept to a minimum, although readers should have a general knowledge of algebra, geometry and trigonometry at the high school or first year university level. My own intern in the golden ratio and related topics developed from my involvement in research on the physical properties of incommensurate solids and quasiaystals. Over the years I have benefited greatly from discussions with colleagues in this field and many of the ideas presented in this book have been derived fiom these discussions. Without their involvement in my research in solid state physics, this book would not have been written. For their comments and ideas which eventually led to the pmnt volume I would like to acknowledge Derek Lawther, Srinivas Veeturi, Dhiren Bahadur, Mike McHenry, Bob O Handley and Bob March. I would also like to thank V
vi The Golden Ratio and Fibonacci Numbers Ewa Dunlap, Rene Codombe, Jerry MacKay and Jody O Brien for their advice and assistance during the preparation of the manuscript. Halifii, Nova Scoria June 1997 RA. I)uNLAp
CONTENTS PREFACE CHAPTER 1: INTRODUCTION 1 CHAPTER 2: BASIC PROPERTIES OF THE GOLDEN RATIO 7 CHAPTER 3: GEOMETRIC PROBLEMS IN TWO DIMENSIONS 15 CHAPTER 4: GEOMETRIC PROBLEMS IN THREE DIMENSIONS 23 CHAPTER 5: FIBONACCI NUMBERS 35 CHAPTER 6: LUCAS NUMEIERS AND GENERALIZED FIBONACCI NUMBERS 51 CHAPl'ER 7: CONTINUED FRACTIONS AND RATIONAL APPROXIMANTS 63 CHAPTER 8: GENERALIZED FIBONACCI REPRESENTATION THEOREMS 71 CHAPTER 9: OPTIMAL SPACING AND SEARCH ALGORITHMS 79 CHAPTER 10: COMMENSURATE AND INCOMMENSURATE PROJECTIONS 87 CHAFTER 11: PENROSE TILINGS 97 CHAFTER 12: QUASICRYSTALLOGRAPHY 111 CHAPTER 13 : BIOLOGICAL APPLICATIONS 123 APPENDIX I: CONSTRUCTION OF THE REGULAR PENTAGON 137 APPENDIX 11: THE FIRST 100 FIBONACCI AND LUCAS NUMBERS 139 APPENDIX 111: RELATIONSHIPS INVOLVING THE GOLDEN RATIO AND GENERALIZED FIBONACCI NUMBERS 143 REFERENCES 153 INDEX 157 V vii