- Page 1 and 2: The Louis and Jeanette Brooks Engin
- Page 3 and 4: Copyright 2001 by Jean Le Mée
- Page 5 and 6: Dodecahedron ......................
- Page 7 and 8: Acknowledgements Special thanks are
- Page 9 and 10: viii The realization of the project
- Page 11 and 12: Introduction: 1 Ad- Quadratum Const
- Page 13 and 14: Generation of the Platonic Forms: 1
- Page 15: …which can be more briefly put as
- Page 19 and 20: Designate CH by x, then cos ˆ C i
- Page 21 and 22: Dodecahedron: 11 For the dodecahedr
- Page 23 and 24: with a 10 5 13 5 5R a 2 10 55R
- Page 25 and 26: 15 A simple way of constructing eac
- Page 27 and 28: Similarly, cosU ˆ O S 1 ,and ther
- Page 29 and 30: C ˆ D ˆ O i 2 The cube dihedra
- Page 31 and 32: 21 As indicated for the cube (fig.
- Page 33 and 34: 23 or MO 5 2 3 R so that if we (ar
- Page 35 and 36: Tetrahedron: For the tetrahedron, a
- Page 37 and 38: a 6 6 R 2 6 6 27 R 12 6 r 3 3
- Page 39 and 40: 29 r 0. 794 R The more complex for
- Page 41 and 42: 31 Now, consider triangles AA HO a
- Page 43 and 44: Dˆ i cos 2 33 We can therefor
- Page 45 and 46: 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
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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
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fig. 67 115 fig. 69 fig. 6 fig. 68
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the structure of all regular polyhe
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119 Ad Quadratum method and the Gen
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121 fig. 74
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123 Ad Quadratum and the Pythagorea
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125 fig. 78
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OT y 1 x 127 y So that, if OT is a
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129 fruitful paradigm. For more tha
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5 3 1 (unisson) 5 3 30 3 20 2 (fi
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133 fig. 79
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135 The difference between a whole
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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