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

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When equal shares are best 7 Figure 2.2. What is the best price for two fields, of a total area of 40 hectares temptation to draw moral lessons for politicians at this point.) A parable will serve to illustrate this principle in action. A farmer, attracted by certain European subsidies, seeks to purchase two fields with a total area of 40 hectares. The prices for fields depend on the field size as shown in the diagram of figure 2.2. What is he to do Playing with the different possibilities quickly convinces him that he should buy two fields of 20 hectares each. The combination of ten and 30 hectares is more expensive; its price is twice that indicated by the open circle on the diagram, which lies above the price of a 20 hectare field. What property of the price structure forces him to choose equal-sized fields It is the upward curvature of figure 2.2, which we may call convexity. Packing a single disc A packing problem may be posed for a single disc of radius Ê by requiring it to fit into a polygonal boundary—a sheep-pen—which is as small as possible. The sheep-pen has Ò sides: what shape should it be The answer is: a regular polygon (that is, totally symmetric with equal edges and equal angles between edges). To demonstrate this we note that the area of a polygon can be divided into sectors, as shown in figure 2.3(a). Each sector corresponds to an angle and the angles must add up to ¾. We need to share this total among Ò angles. But the area of each sector must be greater or equal than that shown in figure 2.3(b) (isosceles triangle) and this is a convex function of .
8 Loose change and tight packing Figure 2.3. (a) A polygon with a disc inside can be divided into triangular sectors with angles . (b) The isosceles triangle that touches the disc is the sector that minimizes the area for a given . The area of such a sector is a convex function of , therefore a division in equal sectors is best. So equal angles are best, and the strategy of equal shares results in Ò such isosceles triangles. Packing many discs We are ready to complete the proof of the disc-packing problem in two dimensions: what is the most dense arrangement of equal discs, infinite in number The answer will be the triangular close packing of figure 2.1. In his book, which covers many such problems 1 , Fejes Tóth attributes the ½ Fejes Tóth L 1953 Lagerungen in der Ebene auf der Kugel und im Raum (Die Grundlehren der Math. Wiss. 65) (Berlin: Springer).
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 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 48 and 49: 4.2 The programme of Thomas Hales T
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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