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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
- Page 47 and 48: Cube: Octahedron: Dodecahedron: r R
- Page 49 and 50: Comparison with Orthoscheme Approac
- Page 51 and 52: 1R O R ri R 41 cos csc 3 3 cos
- Page 53 and 54: 43 They are not what is generally t
- Page 55 and 56: 45 Kepler (1571-1630) discovered in
- Page 57 and 58: 47 We shall now determine the diame
- Page 59 and 60: 49 The stellation process is one wh
- Page 61 and 62: 51 i. Case of the Dodecahedron The
- Page 63 and 64: 53 These dipyramids in turn can be
- Page 65 and 66: 55 If the faces of the pyramids on
- Page 67 and 68: 57 - the great dodecahedron by addi
- Page 69 and 70: 59 and therefore through the vertic
- Page 71 and 72: R e a the well-known relation in
- Page 73 and 74: from Since ABCD is a golden rectang
- Page 75 and 76: We have: Noting that cos Di 2 65 R
- Page 77 and 78: 67 Fig. 46A shows such a pentagonal
- Page 79 and 80: The inner geometry of the GD will b
- Page 81 and 82: 71 2 2 2 Now H sin And sin 1c
- Page 83 and 84: 73 GSD. Since the apexes of the pyr
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- Page 87 and 88: where R S is the circumsphere of th
- Page 89 and 90: And since We verify that as establi
- Page 91 and 92: 81 The intersection of these pyrami
- Page 93 and 94: Now, as has just been established,
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- Page 97: We have tanw 87 2 2 1 1 4 1 5 Wi
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- Page 103 and 104: 93 For the dodecahedron, the golde
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- Page 107 and 108: Alternative Methods of Generating t
- Page 109 and 110: 99 As Fernand Hallyn 25 points our,
- Page 111 and 112: 101 held by their authors. From the
- Page 113 and 114: 103 By joining each such vertex to
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- Page 117 and 118: 107 For the cube, select a pole on
- Page 119 and 120: 109 fig. 63A fig. 63B fig. 63C
- Page 121 and 122: 111 r 2 2 2.828 (We consider the
- Page 123 and 124: 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
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- Page 133 and 134: 123 Ad Quadratum and the Pythagorea
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- 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
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- Page 145 and 146: 135 The difference between a whole
- Page 147 and 148: 137 In either case, consonance is l
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139 The interval between the notes
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We can also write: MN ML 141 1 3 L
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143 Ad Quadratum and the Just Inton
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145 from nothing. It is indivisible
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AA' OA1 2 1 BB' 2 = PP' OP 1 147
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149 fig. 87
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151
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153 Ad Quadratum, the Millennium Sp
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155 own positions in it. Kepler’s
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157 The Millennium Sphere and the L
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159 fig. 89 fig. 90
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161 fig. 91
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This determines the envelope of des
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Remarks TABLE 1 TRIGONOMETRIC PROPE
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TABLE 3 REGULAR CONVEX POLYHEDRA ME
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R G r R G R S R S r 169 From r to
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3. Octahedron: fig. 3Ap 4. Dodecahe
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173 One can also reason that there
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Replacing in (1) by (2) or, rearran
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and from (6) we see that 177 But 3
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179 Alternatively, from (8) we obta
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181 fig. 56
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183 and finally R a 4 Therefore a