Appendix D: Lists of Illustrations, Photos, Tables

Appendix D: Lists of Illustrations, Photos, Tables

1Illustrations

2 Figure no.page

3 Fig. 2 1 "Triangling" versus "squaring" 23

4 Fig. 2 2 "Cubing" versus "tetrahedroning" 24

5 Fig. 3 3 6 connections between 4 events, defining a tetrahedral system 30

6 Fig. 4 4 3 triangles, 4 triangles around a vertex 42

7 Fig. 4 5 5 triangles around each vertex form an icosahedron 43

8 Fig. 4 6 3 squares around each vertex form a cube 44

9 Fig. 4 7 3 pentagons per vertex form a pentagonal dodecahedron 44

10 Fig. 4 8 3 hexagons around each vertex form a flat honeycomb 45

11 Fig. 4 9 The 5 regular polyhedra 46

12 Fig. 4 10 "Dualing" polyhedra 50

13 Fig. 4 11 Degenerate truncation of a tetrahedron. 51

14 Fig. 4 12 Stellated cube and octahedron. 52

15 Fig. 4 13 Truncation of duals 53

16 Fig. 4 14 Icosidodecahedron. 54

17 Fig. 4 15 Degenerate stellation of cube 55

18 Fig. 4 16 Cuboctahedron and rhombic dodecahedron are dual polyhedra 56

19 Fig. 4 17 Symmetry of icosahedron and of the letters M, S, and R. 57

20 Fig. 5 18 Square necklace, collapsed 60

21 Fig. 5 19 Leverage 61

22 Fig. 5 20 Inscribed tetrahedron stabilizes cube. 65

23 Fig. 5 21 "Dimpling" 69

24 Fig. 5 22 Dimpling: ¡ of octahedron caves in to nest inside other half. 69

25 Fig. 6 23 Cartesian vs. spherical coordinates. 79

26 Fig. 6 24 "4D": Tetrahedron / octahedron embody four distinct directions 80

27 Fig. 6 25 A cone: sliced and flattened 83

28 Fig. 6 26 Nets of icosahedron and octahedron 83

29 Fig. 6 27 4-frequency icosahedron 86

30 Fig. 6 28 Angle types 87

31 Fig. 7 29 Stable, metastable, and neutral equilibrium 89

32 Fig. 7 30 Entropy: closed-container experiment 92

33 Fig. 7 31 "Close-packed cookies": triangular vs. square pattern 93

34 Fig. 7 32 In search of equivalent radial and circumferential vectors. 95

35 Fig. 7 33 Vector equilibrium. 97

36 Fig. 7 34 (a) cuboctahedron; (b) twist cuboctahedron. 97

37 Fig. 7 35 Eight radiating tetrahedra alternate with six ¡-octahedra. 99

38 Fig. 7 36 Four hexagonal cross-sections of the VE. 101

39 Fig. 7 37 Motion of partially restrained disc in a plane 102

40 Fig. 7 38 Minimum of 12 spokes needed for stability 103

41 Fig. 7 39 Ball restrained in space 105

42 Fig. 8 40 Closepacked spheres form array of equally spaced points in space 109

43 Fig. 8 41 Closepacking of three pennies 111

44 Fig. 8 42 Six spheres closepacked around one 111

45 Fig. 8 43 Cubic packing (top) versus hexagonal packing (bottom) 112

46 Fig. 8 44 Hexagonal (left) vs cubic (right): spheres 12-around-1 113

47 Fig. 8 45 Five oranges stacked to form ¡-octahedron. 115

48 Fig. 8 46 Tetrahedral and octahedral closest sphere packing clusters. 116

49 Fig. 8 47 Triangular numbers. 117

50 Fig. 8 48 "Four balls lock" 119

51 Fig. 8 49 Frequency in closepacking of spheres 120

52 Fig. 8 50 Tetrahedral frequency (v) progression: 1v, 2v, 3v 121

53 Fig. 8 51 "Yes-no-no-yes-no-no" nest layers 122

54 Fig. 8 52 Vector equilibrium progression: 1v, 2v, 3v 123

55 Fig. 8 53 Removing nuclear sphere from 1-frequency VE creates icosahedron 126

56 Fig. 8 54 Square pattern of spheres compressed into rhomboid shape 127

57 Fig. 8 55 Pyramids formed by triangular-number layers + one ball in central nest 130

58 Fig. 8 56 Interprecessing: two closepacked pairs of spheres creates a tetrahedron 131

59 Fig. 8 57 Tetrahedron as chef’s hats 132

60 Fig. 8 58 Chef’s hats back to tetrahedron 132

61 Fig. 8 59 Two 1/8-octahedra create minimum stable cube 133

62 Fig. 9 60 Unit-vector matrix within closepacked sphere cluster 135

63 Fig. 9 61 Alternating tetrahedra and octahedra 136

64 Fig. 9 62 Isotropic vector matrix and Octet Truss 137

65 Fig. 9 63 Octahedral cavities between adjacent VEs 138

66 Fig. 9 64 60ř framework vs 90ř framework 139

67 Fig. 9 65 Tetrahedra and octahedra team up to fill space 140

68 Fig. 9 66 Octahedral "egg-crate" 142

69 Fig. 9 67 Oct / Tet supplementary angles 143

70 Fig. 9 68 3-frequency (3v) truncations 144

71 Fig. 9 69 Central nodes of tetrahedron and octahedron. 145

72 Fig. 9 70 Relationship of cube and IVM. 146

73 Fig. 9 71 Degenerate stellation of the octahedron 147

74 Fig. 10 72 Self-similarity in subdivision of triangles and tetrahedra. 155

75 Fig. 10 73 2-frequency (2v) cube consists of 8 unit cubes 156

76 Fig. 10 74 2-frequency (2v) tetrahedron has altitude of 2 tetrahedra 157

77 Fig. 10 75 "Unwrap" a hidden central octahedron 160

78 Fig. 10 76 Derivation of an octant: 1/8-octahedron. 161

79 Fig. 10 77 One octant altitude is equal to ¡ the altitude of the tetrahedron 161

80 Fig. 10 78 Four octants added to one tetrahedron produce one cube 162

81 Fig. 10 79 Subdivision of tetrahedron into 4 shallow pyramids 163

82 Fig. 10 80 2v tetrahedron with octahedron core 165

83 Fig. 10 81 2v octahedron with cubocahedron (VE) core 166

84 Fig. 10 82 Hierarchical nest within 4v tetrahedron 167

85 Fig. 11 83 "Jitterbug" transformation of VE 171

86 Fig. 11 84 Further jitterbugging. 173

87 Fig. 11 85 Octahedron within icosahedron 177

88 Fig. 11 86 Icosahedron within octahedron 177

89 Fig. 11 87 S module. 179

90 Fig. 11 88 Golden- ratio relationship: Icosa / Rhombic dodecahedron 180

91 Fig. 11 89 Four independent axes of rotation 181

92 Fig. 11 90 Tetrahedra "bonding" models of phases of matter 185

93 Fig. 12 91 Plane tesselations: Regular and semiregular tilings of the plane. 188

94 Fig. 12 92 Octahedron at junction of 8 cubes 190

95 Fig. 12 93 Space-filling rhombic dodecahedron via degenerate stellation of cube 194

96 Fig. 12 94 Space-filling tetrakaidecahedron by truncating 3v octahedron 196

97 Fig. 12 96 Volume cross-sections of 4v tetrahedron 200

98 Fig. 13 97 Development of A module. 203

99 Fig. 13 98 Modules for intertransformability 204

100 Fig. 13 99 Planar nets 207

101 Fig. 13 100 LCD of IVM: "Mite" 209

102 Fig. 13 102 The LCD of both cube and rhombic dodecahedron is the Mite. 212

103 Fig. 13 103 Each Mite is one octant of the coupler. 213

104 Fig. 13 104 "Coupler" 214

105 Fig. 13 105 Different rearrangements of 8 Mites in the coupler 216

106 Fig. 14 106 A lesser-circle path is always the long way around 219

107 Fig. 14 107 Spherical polyhedra 221

108 Fig. 14 108 Different sets of axes of symmetry 223

109 Fig. 14 109 The 6 great circles of the octahedron 224

110 Fig. 14 110 The 4 great-circle edges of the VE 224

111 Fig. 14 111 The 25 great circles of the VE 225

112 Fig. 14 112 The 6 great circles of the icosahedron 226

113 Fig. 14 113 The 31 great circles of the icosahedron 227

114 Fig. 14 114 Maximum of 120 equivalent domains: LCD 228

115 Fig. 14 115 LCD unit 229

116 Fig. 14 116 Revealed in the 31-great-circle pattern 230

117 Fig. 14 117 Revealed in 25 great-circle pattern 232

118 Fig. 14 118 Making bowties: relating central and surface angles 233

119 Fig. 14 119 4 bowties create the 4 great circles of the VE 234

120 Fig. 14 120 Reflective bounces in local holding-pattern figure-8 loop 236

121 Fig. 14 121 6 bowties create the 6 great circles of the cube or octahedron 237

122 Fig. 14 122 The 3-great-circle model requires 6 foldable circles 238

123 Fig. 14 123 6 pentagonal bowties create the 6 great circles of the icosahedron 239

124 Fig. 14 124 15-great-circle model, created by doubling the edges 240

125 Fig. 15 125 Reed sphere of six interwoven great circles 247

126 Fig. 15 126 31-great circle subdivision of icosahedron LCD 248

127 Fig. 15 127 4-frequency (4v) triangles superimposed on icosahedron 249

128 Fig. 15 128 Transformation from planar to spherical 251

129 Fig. 15 129 4v icosahedron: transformation of Fig. 15-3 251

130 Fig. 15 130 5/8 truncation of 4v icosaas basis of geodesic dome. 255

131 Fig. 15 131 Geodesic pattern possibilities 256

132 Fig. 15 132 LCD in higher frequencies 257

133 Fig. 15 133 Compression: neutral axis, girth, and spherical limit case 262

134 Fig. 15 134 Wheel with compression spokes 265

135 Fig. 15 135 Tensegrity icosahedron and its Platonic (planar) counterpart. 269

136 Fig. 16 136 DymaxionTM Map 281

137 Fig. 16 137 Hanging bookshelf, U.S. Patent 4,377,114 (1983). 282

138 Photos

139 Photo no.page

140 Photo 11 1 Complex of jitterbugs 182

141 Photo 15 2 Tensegrity icosahedron and tensegrity tetrahedron. 267

142 Photo 15 3 3v tensegrity icosahedron with 90 struts. 268

143 Photo 15 4 Six-strut "expanded octahedron" tensegrity. 270

144 Tables

145 Table no.page

146 Table 4 1 Euler’s Law (V + F = E + 2) 47

147 Table 5 2 "Structural quanta":
volume per edge for the 3 prime structural systems. 67

148 Table 6 3 "Structural quanta":
polyhedra edges in even multiples of the tetrahedron’s 6 85

149 Table 7 4 VE: equal radial and circumferential vectors and angles 98

150 Table 10 5 Volume Ratios 153

151 Table 12 6 Octahedron-tetrahedron ratio in space filling 197