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RANDOMNICITY Rules and Randomness in the Realm of the Infinite Copyright © 2008 by Imperial College Press All Rights Reserved Rules and Randomness in the Realm of the Infinite P558tp.indd 1 8/20/08 4:20:43 PM This page intentionally left blank Rules and Randomness in the Realm of the Infinite Anastasios A. Tsonis Department of Mathematical Sciences, University of Wisconsin, Milwaukee Imperial College Press ICP P558tp.indd 2 8/20/08 4:20:43 PM Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. On the cover: Jackson Pollock’s Autumn Rhythm © Pollock–Krasner Foundation Artists Rights Society (ARS), New York RANDOMNICITY Rules and Randomness in the Realm of the Infinite Copyright © 2008 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from 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. ISBN-13 978-1-84816-197-9 ISBN-10 1-84816-197-2 ISBN-13 978-1-84816-205-1 (pbk) ISBN-10 1-84816-205-7 (pbk) Typeset by Stallion Press Email: [email protected] Printed in Singapore. ChianYang - Randomnicity.pmd 1 12/10/2008, 7:44 PM August 19, 2008 15:21 9in x 6in B-634 fm To My brother Takis, my other half. v August 19, 2008 15:21 9in x 6in B-634 fm This page intentionally left blank August 19, 2008 15:21 9in x 6in B-634 fm Preface It is dawn and the battlefield is waiting. It is sometime in the 12th century B.C. and a critical moment in the Trojan War must be decided. Paris seduced and ran away with Helen, the wife of the king of Sparta; and now Menelaus, the king and a unified Greek army have invaded the Trojan land and are asking for revenge. The war has been dragging on for years and Troy is not falling. In fact, it appears that the Trojans, led by Hector, are gaining the upper hand. Somebody from the Greek army has to step in and fight man- to-man with Hector. Who will it be? The decision will be left to chance. Each of the volunteer marks his own lot, then the lots are put in a helmet and are shaken. A lot is drawn from the helmet and identifies the soldier who will fight Hector. It is Ajax. In Homer’s Iliad and in many other early epics, such decisions were often left to chance. The concept of randomness appears to have been an integral part of the actions and feelings of early cultures. One might wonder why this would ever happen. After all, early in human history, people believed that the gods controlled every little detail (determinism) and therefore nothing was left to chance. This, however, is not a paradox. Randomness in early civilizations emerges as part of God. It is controlled only by God, thus eliminating human intervention and allows the will of God to apply. Thus, randomness cannot be separated from God (determinism). Randomness and determinism are thus established early in the human mind as being inter- connected and associated with something bigger like God, who is boundless and everywhere at any time. This sentiment is clearly reflected in Aristotle’s Physics Book II, 4. “Others there are who believe that chance is a cause but that it is inscrutable to human intelligence as being a divine thing and full of mystery.” Later, in his Metaphysics, Book XI, 8, Aristotle extended his logical arguments about chance to relate “the divine thing” to reason and nature. vii August 19, 2008 15:21 9in x 6in B-634 fm viii Randomnicity “Since nothing accidental is prior to the essential neither are accidental causes prior. If, then, luck or spontaneity is a cause of the material universe, reason and nature are causes before it.” This cyclic, counterintuitive association and interplay of randomness and rules is a fascinating mathematical issue and is the main reason for writing this book. In doing so I made an effort to present and communicate to the reader the mathematical and scientific ideas step by step, using examples as simple as possible. Parts I and II open with a hypothetical story where the main characters engage in a fictitious conversation. This blend of fiction and scientific facts is designed to familiarize the reader with the mathematical and physical concepts that follow. In some cases, for the more mathemati- cally inclined reader, notes on some of the details have been included at the end of the book. The sequence of chapters has been designed so that what follows is understood from what has been presented up to that point. I hope that you will enjoy the book. Anastasios Tsonis Milwaukee, Wisconsin [email protected] August 19, 2008 15:21 9in x 6in B-634 fm Acknowledgements This year (2008) Professor Edward Lorenz passed away. I would like to remember him here as the humblest of all the great minds I have met and for his inspiration and help whenever I needed it. Also this year my father- in-law Michal Koryzna passed away. I would like to remember him here as well for all the (good) randomness he injected into my life. I would like to thank Benoit Mandelbrot, Wolfram Research Inc., Brian Hayes, Hendrik Lenstra, B. de Smit, Winton Clitheroe, Heinz- Otto Peitgen, Dietmar Saupe, J.P. Lewis, Richard Voss, Michael Barnsley, Kenneth Libbrecht, Eugene Stanley, David B. Searls, and Steven Strogatz for allowing me free of charge usage of their figures or images. I am thankful to M.C. Escher Company, Holland, to Pollock-Krasner Foundation, New York, and to Metropolitan Museum, New York for giving me permission to publish artwork of these two great painters. I am indebted to Leila de Carvalho, John Casti, Douglas Hofstadter, Chris Essex, John Lambris, and Jose Luis Millan for critically reading the manuscript and offering criticism. ix August 19, 2008 15:21 9in x 6in B-634 fm This page intentionally left blank August 19, 2008 15:21 9in x 6in B-634 fm Contents Preface vii Acknowledgements ix Part I: The Three Parts of Everything 1 1. Gödel Visits Escher’s Studio 3 2. Gödel’s Theorem 10 3. Slippery Road 19 4. Fuller than Full 27 5. Cantor Would Have Been Pleased 34 Part II: Sources of Randomness 39 6. A Private Lesson a Long, Long Time Ago 41 7. The Five Faces of Order 47 8. Randomness of the First Kind 55 9. Randomness of the Second Kind 59 10. Randomness of the Third Kind 65 Part III: Randomness in the Universe 69 11. From Aristotle to Einstein 71 12. The World According to an Electron 76 13. Chaos 86 xi August 19, 2008 15:21 9in x 6in B-634 fm xii Randomnicity 14. The Supreme Law 91 15. Randomness of the Fourth Kind? 98 16. Connections 102 17. Allowed Behaviors 107 Part IV: The Emergence of Real World 109 18. The Fractal Character of Nature 111 19. Physics Plus Randomness 129 20. Stochastic Processes 135 21. Self-Organization 138 22. The Blueprint of Life 140 23. Human Products and Social Phenomena 144 24. The Principle of Minimum Energy Consumption 159 Part V: The Role of Randomness 161 25. Efficiency, Paradoxes and Beauty 163 In Closing... 173 Notes and References 175 Index 189 August 4, 2008 10:37 9in x 6in B-634 ch01 PART I The Three Parts of Everything Is randomness an intrinsic property of mathematics? 1 August 4, 2008 10:37 9in x 6in B-634 ch01 This page intentionally left blank 2 August 4, 2008 10:37 9in x 6in B-634 ch01 Chapter 1 Gödel Visits Escher’s Studio “I paint things as I think of them, not as I see them.” Picasso In a rare break from his everyday occupation of proving theorems, Gödel decides to visit Escher’s studio. He has heard rumors that this fellow is experimenting with painting paradoxical pictures. He arrives at the studio at about 11:00 am. It is quite a gloomy and foggy day in Baarn but this does not seem to bother Gödel. He rings the bell and very soon a serious, well- dressed bearded man opens the door to a small, frail-looking man wearing heavy rim glasses. “Good morning sir, my name is Dr. Kurt Gödel. I am a mathematician and I would like, if you don’t mind, to look at your work. I am not an art connoisseur. In fact I am not sure I understand modern art, but I understand you are on to something different. Is this a bad time?” “Not at all. Please come in,” replies Escher showing Gödel the way in. Once inside the studio Escher discretely lets his visitor wander around. Gödel, his hands clasped behind his back, is silently moving around looking at the pictures, some of which are framed and hanging on the walls and some are laying flat on large tables. “If you have any questions please call me,” Escher says after a while.
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