Ad Quadratum Construction and Study of the Regular Polyhedra

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TABLE 2 TRIGONOMETRIC PROPERTIES OF THE HALF-INTERNAL ANGLE FORM sin cos tan Remarks Tetrahedron Ti sin Ti 2 cos Ci 2 2 tan Ti 2 1 3 0.5773 cos Ti 2 2 3 0.8165 sin Ti 2 2 54o 74 sin Ci 2 cos Ti 2 1.4142 tan Ti 2 2 tan Ci 2 sin 4 cos 4 tan Ci 2 tan Ci 2 sin Oi 2 tan Ci 2 cos Ci 2 1 3 sin Ci 2 Cube Ci 2 35o 27 cos Oi 2 1 2 0.7071 2 3 0.8165 0.5773 sin Oi 2 cos Oi 2 Octahedron Oi 45o 2 4 Oi 2 1 tan Oi 2 1 2 0.7071 cos Oi 2 1 2 0.7071 sin Oi 2 tan Ci 2 166 1 2 tan Di 2 2 3 3 sin 3 10 3 21sin 10 4 sin 2 cos 10 5 cos Di 2 cos Di 2 1 2 cos Di 2 2 sin Di 2 2 3 1 5 2 3 cos Di 2 1 3 sin Di 2 Dodecahedron Di 2 20o 90 tan Di 2 sin Di 2 tan Di 2 sin Di 2 3 5 2 0.3819 1 3 3 sin 10 2 3 3 sin Di 2 tan Di 2 0.9342 5 1 2 3 3 I i 2 I i 2 sin 5 2 5 5 sin Ii 2 cos 1 2 0.3568 sin Ii 2 1 1 I i tan 2 3 2sin 10 2cos 5 cos I i 2 sin Ii 2 1 tan Ii 2 1 2 cos I i 2 Icosahedron Ii 2 31o 72 3 cos 10 2 sin 5 2 5 5 sin tan Ii 2 tan Di 2 1 0.6180 1 5 25 5 2 5 5 10 1 2 2 5 5 cos I i 2 1 tan Di 2 tan Ii 2 0.8506 0.5257 1 tan Ii 2 2sin 3 10
TABLE 3 REGULAR CONVEX POLYHEDRA METRICS Tetrahedron Cube Octahedron Dodecahedron Icosahedron 6 3 2 R R aT R ac R ao 4 2 2 ad 3 15 4 3 ad 2 R i a 25 5 4 ai 2 2 2 2 3 aT 6R ac 3 3 R ad ao 2R 3 5 1R ai 3 2 3 1R 3 10 55R 5 2 2 R aT 2sin Ti 2 R ac 2sin Ci 2 R ao 2sin Oi 2 R ad 2sin Di 2 R ai 2sin I i 2 R 1 3 a o a T 2 ac ad aT ac 2 167 1 ac ac I i ac tan 2 d a ai ac 3 sin I i 2 3 2 2 a c a T 2 2 2 0. 4787R A i 2 2 a 3 A o 2 d 0. 8760R 3 R 4 2 rd 0. 794R 3 R Di I i cos cos R 3 2 2 2 4 R 3 2 2 aT 2 3 3R 4 3 (face area) rT 3 R (insphere radius) T A A Ac ac o i r R I i 2 0. 794R Di cos cos 2 r o R 3 3 r c 3 3 vi 1. 463 3R 3 vd 1. 606 3R 3 3R 4 3 v o 3 3R 8 9 v c 8 vt 3R 27 (volume) 3 3 4 1. 463 vd vs 60. 495% 3 3 4 1. 606 d % 235 . 12 3 2 9 t v % 133 . 55 3 o % 705 . 36 3 2 3 c v s s v % 406 . 66 v v s v v s v (vs =circumsphere volume)
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The Louis and Jeanette Brooks Engin
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Copyright 2001 by Jean Le Mée
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Dodecahedron ......................
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Acknowledgements Special thanks are
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viii The realization of the project
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Introduction: 1 Ad- Quadratum Const
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Generation of the Platonic Forms: 1
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…which can be more briefly put as
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7 Tetrahedron: Cube: Octahedron: Do
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Designate CH by x, then cos ˆ C i
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Dodecahedron: 11 For the dodecahedr
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with a 10 5 13 5 5R a 2 10 55R
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15 A simple way of constructing eac
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Similarly, cosU ˆ O S 1 ,and ther
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C ˆ D ˆ O i 2 The cube dihedra
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21 As indicated for the cube (fig.
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23 or MO 5 2 3 R so that if we (ar
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Tetrahedron: For the tetrahedron, a
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a 6 6 R 2 6 6 27 R 12 6 r 3 3
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29 r 0. 794 R The more complex for
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31 Now, consider triangles AA HO a
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Dˆ i cos 2 33 We can therefor
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35 Ratio of Intersphere to Circumsp
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Cube: Octahedron: Dodecahedron: r R
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Comparison with Orthoscheme Approac
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1R O R ri R 41 cos csc 3 3 cos
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43 They are not what is generally t
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45 Kepler (1571-1630) discovered in
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47 We shall now determine the diame
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49 The stellation process is one wh
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51 i. Case of the Dodecahedron The
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53 These dipyramids in turn can be
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55 If the faces of the pyramids on
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57 - the great dodecahedron by addi
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59 and therefore through the vertic
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R e a the well-known relation in
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from Since ABCD is a golden rectang
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We have: Noting that cos Di 2 65 R
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67 Fig. 46A shows such a pentagonal
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The inner geometry of the GD will b
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71 2 2 2 Now H sin And sin 1c
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73 GSD. Since the apexes of the pyr
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The circumsphere radius of the enve
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where R S is the circumsphere of th
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And since We verify that as establi
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81 The intersection of these pyrami
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Now, as has just been established,
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85 fig. 45
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We have tanw 87 2 2 1 1 4 1 5 Wi
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89 The structure of this scaled G.I
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the three mutually perpendicular go
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93 For the dodecahedron, the golde
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95 so that 2R cos ˆ D i 2 For the
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Alternative Methods of Generating t
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99 As Fernand Hallyn 25 points our,
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101 held by their authors. From the
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103 By joining each such vertex to
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105 fig. 61
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107 For the cube, select a pole on
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109 fig. 63A fig. 63B fig. 63C
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111 r 2 2 2.828 (We consider the
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113 fig. 64 fig. 65 fig. 66
Page 125 and 126: fig. 67 115 fig. 69 fig. 6 fig. 68
Page 127 and 128: the structure of all regular polyhe
Page 129 and 130: 119 Ad Quadratum method and the Gen
Page 131 and 132: 121 fig. 74
Page 133 and 134: 123 Ad Quadratum and the Pythagorea
Page 135 and 136: 125 fig. 78
Page 137 and 138: OT y 1 x 127 y So that, if OT is a
Page 139 and 140: 129 fruitful paradigm. For more tha
Page 141 and 142: 5 3 1 (unisson) 5 3 30 3 20 2 (fi
Page 143 and 144: 133 fig. 79
Page 145 and 146: 135 The difference between a whole
Page 147 and 148: 137 In either case, consonance is l
Page 149 and 150: 139 The interval between the notes
Page 151 and 152: We can also write: MN ML 141 1 3 L
Page 153 and 154: 143 Ad Quadratum and the Just Inton
Page 155 and 156: 145 from nothing. It is indivisible
Page 157 and 158: AA' OA1 2 1 BB' 2 = PP' OP 1 147
Page 159 and 160: 149 fig. 87
Page 161 and 162: 151
Page 163 and 164: 153 Ad Quadratum, the Millennium Sp
Page 165 and 166: 155 own positions in it. Kepler’s
Page 167 and 168: 157 The Millennium Sphere and the L
Page 169 and 170: 159 fig. 89 fig. 90
Page 171 and 172: 161 fig. 91
Page 173 and 174: This determines the envelope of des
Page 175: Remarks TABLE 1 TRIGONOMETRIC PROPE
Page 179 and 180: R G r R G R S R S r 169 From r to
Page 181 and 182: 3. Octahedron: fig. 3Ap 4. Dodecahe
Page 183 and 184: 173 One can also reason that there
Page 185 and 186: Replacing in (1) by (2) or, rearran
Page 187 and 188: and from (6) we see that 177 But 3
Page 189 and 190: 179 Alternatively, from (8) we obta
Page 191 and 192: 181 fig. 56
Page 193 and 194: 183 and finally R a 4 Therefore a
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Ad Quadratum Construction and Study of the Regular Polyhedra

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