THE PURSUIT OF PERFECT PACKING SECOND EDITION THE PURSUIT OF PERFECT PACKING SECOND EDITION TOMASO ASTE AUSTRALIAN NATIONAL UNIVERSITY CANBERRA, AUSTRALIA DENIS WEAIRE TRINITY COLLEGE DUBLIN, IRELAND Taylor & Francis Taylor Si Francis Croup ® New York London Taylor & Francis is an imprint of the Taylor & Francis Group, an informa business CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2008 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4200-6817-7 (Hardcover) This book contains information obtained from authentic and highly regarded sources Reason - able efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. 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QA166.7.A78 2008 511’.6--dc22 2007044016 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Dedicated to Colette and Tiziana Contents Preface to First Edition ..........................................xi Preface to Second Edition .....................................xiii 1 How Many Sweets in the Jar?...................................1 2 Loose Change and Tight Packing ...............................5 2.1 A Teasing But Tractable Problem ............................5 2.2 A Handful of Coins..........................................5 2.3 Order and Disorder .........................................8 3 Hard Problems with Hard Spheres ........................... 11 3.1 The Greengrocer’s Dilemma ................................11 3.2 Ordered Close Packing—The Kepler Problem...............11 3.3 The Kepler Conjecture .....................................14 3.4 Marvelous Clarity, Neurotic Anxiety: The Life of Kepler......16 3.5 Progress by Leaps and Bounds? ............................ 18 4 Proof Positive? ................................................21 4.1 News from the Western Front ..............................21 4.2 The Programme of Thomas Hales ..........................23 4.3 At Last? ....................................................25 4.4 Polishing Off the Programme ...............................29 4.5 The Acceptance of Proof ...................................30 4.6 The Flyspeck Project .......................................32 4.7 Who Cares? ................................................33 4.8 The Power of Thought .....................................33 5 Disordered Packings ..........................................35 5.1 Balls in Bags...............................................35 5.2 A New Way of Looking ....................................37 vii viii Contents 5.3 How Many Balls in the Bag? ...............................38 5.4 Is the Bernal Close Packing Well Defined? ..................39 5.5 Bernal’s Long-Running Ball Game ..........................39 5.6 Tomography Takes Over ...................................40 6 Sands and Grains..............................................43 6.1 Granular Materials .........................................43 6.2 Osborne Reynolds: A Footprint in the Sand .................44 6.3 Major Bagnold’s Desert Drive...............................47 6.4 Dunes .....................................................49 6.5 Order from Shaking ........................................51 6.6 Segregation ................................................53 6.7 Granular Temperature......................................55 7 Divide and Conquer: Tiling Space.............................57 7.1 Packing and Tiling .........................................57 7.2 The Vorono¨ı Construction ..................................58 7.3 The Dual Construction of Delaunay ........................ 60 7.4 Applications ...............................................60 7.5 Vertices in Tilings ..........................................61 7.6 Three Dimensions .........................................61 7.7 Regular and Semiregular Packings ..........................62 8 Peas and Pips..................................................67 8.1 Vegetable Staticks ..........................................67 8.2 Stephen Hales .............................................69 8.3 Pomegranate Pips ..........................................70 8.4 Biological Cells, Lead Shot, Rubber Balls, and Soap Bubbles: Plusca ¸ Change .........................71 9 Enthusiastic Admiration: The Honeycomb ...................75 9.1 The Honeycomb Problem ..................................75 9.2 What the Bees Do Not Know...............................78 10 A Search for Structure.........................................81 10.1 A Voice in the Wilderness..................................81 10.2 The Two-Dimensional Soap Froth ..........................82 10.3 The Rules of the Game ....................................83 10.4 In a Cambridge Garden ....................................85 11 Toils and Troubles with Bubbles ..............................87 11.1 Playing with Bubbles .....................................87 11.2 A Blind Man in the Kingdom of the Sighted ...............88 11.3 Proving Plateau...........................................91 11.4 Foam and Ether ..........................................93 Contents ix 11.5 The Kelvin Cell ...........................................97 11.6 Most Beautiful and Regular ...............................98 11.7 The Twinkling of an Eye..................................99 11.8 Simulated Soap..........................................100 11.9 A Discovery in Dublin ...................................101 11.10 Crystals of Small Bubbles ................................104 12 Bubbles in Beijing............................................107 12.1 An Olympian Vision .....................................107 12.2 Fun and Fit for Purpose? .................................107 12.3 A Flexible Friend? .......................................108 13 The Architecture of the World of Atoms.....................111 13.1 Molecular Tactics ........................................111 13.2 Atoms and Molecules: Begging the Question .............113 13.3 Atoms as Points .........................................115 13.4 Playing Hardball.........................................116 13.5 Modern Crystallography .................................120 13.6 Crystalline Packings .....................................121 13.7 Packing Tetrahedra .....................................123 13.8 Changed Utterly: Quasicrystals ...........................125 13.9 Amorphous Solids .......................................128 13.10 Crystal Nonsense ........................................129 14 Apollonius and Concrete ....................................131 14.1 Mixing Concrete.........................................131 14.2 Apollonian Packing......................................132 14.3 Packing Fraction and Fractal Dimension ..................136 14.4 Packing Fraction in Granular Aggregates .................136 15 Packings and Kisses in High Dimensions ...................139 15.1 Packing in Many Dimensions ............................139 15.2 A Kissing Competition ...................................143 15.3 Kissing the Neighbors in Higher Dimensions .............143 15.4 Will Disorder Win in the End? ...........................144 16 The Sweets in the Jar, the Pebbles on the Beach... ..........147 16.1 Those Sweets ...........................................147 16.2 Hey, What Shape Do You Want Your Ice Cubes?..........148 16.3 Another Walk on the Beach..............................148 17 The Giant’s Causeway ........................................151 17.1 Worth Seeing? ...........................................151 17.2 Idealization Oversteps Again.............................152