Aste T., Weaire D. Pursuit of perfect packing (IOP 2000)(147s).pdf

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4.2 The programme of Thomas Hales The programme of Thomas Hales 37 Mathematicians often speak of a ‘programme’ 1 upon whose construction they are engaged, and do not mean a computer program. Rather, it is a tactical plan designed to achieve some objective. Like mountaineers who wish to conquer Everest, they define in advance the route and various camps which must be established along the way. It is in this spirit that Thomas Hales has attacked the Kepler problem. He had established base camp when we set out to write this book. He had proved a number of intermediate results and felt that the summit could be reached. Hales’ programme was based on reducing the Kepler problem to certain local statements about packing. Earlier we pointed to the impossibility of packing regular tetrahedra (chapter 3), and saw the Kepler problem as one of ‘frustration’, for this reason. But this does not mean that the global packing problem cannot be reduced to a local one, in some more subtle sense. Hales considers a saturated packing, which is an assembly of non-intersecting spheres where no further non-intersecting spheres can be added. He uses shells: local configurations made of a sphere and its surrounding neighbours. He calculates the local density and a score, this quantity is associated with the empty and occupied volume in the local packing around the sphere. The programme of Hales was to prove that all the possible local configurations have scores lower or equal than 8.0, that is, the one associated with Kepler’s dense packing. This is enough to prove Kepler’s conjecture. Hales started the implementation of this programme around 1992. He soon proved that a large class of local packings score less than eight, but there remained a few local configurations for which this proof was extremely tricky. In spring 1998 one could read on his home-page on the Internet: ‘When asked how long all this will take, I leave myself a year or two. But I hope these pages convince you that the end is in sight!’. He estimated that would take until year 2000 to finish the proof. The main problem in this kind of proof is to find a good way to partition a packing into local configurations. This is a key issue: how does one properly define ‘shells’ in the packing All the major breakthroughs in the history of the Kepler conjecture (including the Hsiang attempt) are associated with different ways of partitioning space. In particular, there are two natural ways to divide the space around a given sphere in a packing. The first is the Voronoï decomposition (chapter 2). The Voronoï cell is a polyhedron, the interior of which consists of all points of the space which are closer to the centre of the given sphere than to any other. This was the kind of decomposition adopted by Fejes Tóth to reduce the Kepler problem to the ‘determination of the minimum of a function of a finite number of variables’. But this method meets with difficulties for some local configurations, such as when a sphere is surrounded by 12 spheres with centres on the vertices of a regular ½ Only east of the Atlantic are there extra letters which maintain a useful distinction between these two spellings.
38 Proof positive Figure 4.1. The assembly of 12 spheres around a central one in a pentahedral prism configuration. icosahedron. In this case the local packing fraction associated with the Voronoi cell is ¼ which is bigger than the value of the Kepler packing ( ¼¼ ) and its score is bigger than eight. The second natural way of dividing space is the Delaunay decomposition. Here space is divided in Delaunay simplexes which are tetrahedra with vertices on the centres of the neighbouring spheres chosen in a way that no other spheres in the packing have centres within the circumsphere of a Delaunay simplex. The local configuration considered is now the union of the Delaunay tetrahedra with a common vertex in the centre of a given sphere (this is called the Delaunay star). This was the kind of decomposition first adopted by Hales in his programme. For instance, the Delaunay decomposition succeeds in the icosahedral case, giving a score of 7.999 98. But there exists at least one local configuration with a higher score. This is an assembly of 13 spheres around a central one, which is obtained by taking 12 spheres centred at the vertices of an icosahedron and distorting the arrangement by pressing the 13th sphere into one of the faces. This configuration scores 8.34 and has local packing fraction ¼½. Another nasty configuration which has a score dangerously close to eight is the ‘pentahedral prism’. This is an assembly of 12 spheres around a central one, shown in figure 4.1. The Voronoï and Delaunay decompositions can be mixed in infinitely many ways. This is what Hales attempted by decomposing space in ‘Q-systems’ and associated stars. This decomposition was successful to establish the score of the pentahedral prism at 7.9997. However, new and nasty configurations remained to be considered....
Page 1: The Pursuit of Perfect Packing
Page 4 and 5: c○ IOP Publishing Ltd 2000 All ri
Page 7 and 8: Contents Preface xi 1 How many swee
Page 9: Contents ix 11 Soccer balls, golf b
Page 12 and 13: Chapter 1 How many sweets in the ja
Page 14 and 15: How many sweets in the jar 3 Figure
Page 16 and 17: Chapter 2 Loose change and tight pa
Page 18 and 19: When equal shares are best 7 Figure
Page 20 and 21: When equal shares are best 9 Figure
Page 22 and 23: Regular and semi-regular packings 1
Page 24 and 25: Disordered, quasi-ordered and fract
Page 26 and 27: Disordered, quasi-ordered and fract
Page 28 and 29: The Voronoï construction 17 Figure
Page 30 and 31: The Voronoï construction 19 analys
Page 32 and 33: Balls in bags 21 (a) (b) (c) Figure
Page 34 and 35: A new way of looking 23 Figure 3.3.
Page 36 and 37: Osborne Reynolds: a footprint on th
Page 38 and 39: Ordered loose packings 27 flux inde
Page 40 and 41: The Kepler Conjecture 29 Figure 3.5
Page 42 and 43: The Kepler Conjecture 31 Figure 3.7
Page 44 and 45: Marvellous clarity: the life of Kep
Page 46 and 47: Chapter 4 Proof positive 4.1 News f
Page 50 and 51: At last 39 4.3 At last On 10 August
Page 52 and 53: At last 41 > - when were you first
Page 54 and 55: The problem of proof 43 tists, who
Page 56 and 57: Chapter 5 Peas and pips 5.1 Vegetab
Page 58 and 59: Stephen Hales 47 Figure 5.2. Stephe
Page 60 and 61: The improbable seed 49 Figure 5.3.
Page 62 and 63: Biological cells, lead shot and soa
Page 64 and 65: Biological cells, lead shot and soa
Page 66 and 67: The honeycomb problem 55 Figure 6.1
Page 68 and 69: What the bees do not know 57 Figure
Page 70 and 71: Chapter 7 Toils and troubles with b
Page 72 and 73: A blind man in the kingdom of the s
Page 74 and 75: Proving Plateau 63 Figure 7.4. Diff
Page 76 and 77: Minimal surfaces Foam and ether 65
Page 78 and 79: Foam and ether 67 Figure 7.6. Kelvi
Page 80 and 81: The Kelvin cell 69 (a) (b) Figure 7
Page 82 and 83: Simulated soap 71 was a great labou
Page 84 and 85: A discovery in Dublin 73 Figure 7.9
Page 86 and 87: Chapter 8 The architecture of the w
Page 88 and 89: Atoms and molecules: begging the qu
Page 90 and 91: Atoms as points 79 (a) (b) (c) Figu
Page 92 and 93: Playing hardball 81 Figure 8.5. Som
Page 94 and 95: Crystalline packings 83 the means t
Page 96 and 97: Tetrahedral packing 85 (a) (b) (c)
Page 98 and 99:
Quasicrystals 87 restricted to pent
Page 100 and 101:
Amorphous solids 89 struction expla
Page 102 and 103:
Chapter 9 Apollonius and concrete 9
Page 104 and 105:
Apollonian packing 93 coined by Ben
Page 106 and 107:
Packing fraction in granular aggreg
Page 108 and 109:
Chapter 10 The Giant’s Causeway 1
Page 110 and 111:
The first official report 99 Figure
Page 112 and 113:
Mallett’s model 101 Figure 10.3.
Page 114 and 115:
Chapter 11 Soccer balls, golf balls
Page 116 and 117:
Buckyballs 105 We know of no physic
Page 118 and 119:
Buckminster Fuller 107 chemical sta
Page 120 and 121:
The Tammes problem 109 Figure 11.4.
Page 122 and 123:
Helical packings 111 Figure 11.5. C
Page 124 and 125:
Chapter 12 Packings and kisses in h
Page 126 and 127:
Packing in many dimensions 115 Tabl
Page 128 and 129:
More kisses 117 Table 12.2. Known l
Page 130 and 131:
Chapter 13 Odds and ends 13.1 Parki
Page 132 and 133:
Filling boxes 121 volume tend to be
Page 134 and 135:
Packing pentagons 123 Figure 13.2.
Page 136 and 137:
The Malfatti problem 125 Figure 13.
Page 138 and 139:
Order from shaking 127 energy—bet
Page 140 and 141:
Segregation 129 Figure 13.7. Why ar
Page 142 and 143:
Turning down the heat: simulated an
Page 144 and 145:
Chapter 14 Conclusion We stated at
Page 146 and 147:
Index 135 Faraday, M, 60 Fejes Tót
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