Ad Quadratum Construction and Study of the Regular Polyhedra

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70 fig. 52
71 2 2 2 Now H sin And sin 1cos 2 2 1 4 1 2 3 H a 1 3 2 and sin ˆ G J 2 H G ˆ sin 2 a 1 2 a 1 3 2 G ˆ sin 0.85065 2 G ˆ 2 58o28 so that G ˆ 116 o 56 1 3 Such is the dihedral angle of the pentagonal planes, of course the same as the dihedral angle of the convex dodecahedron already calculated. Naturally, a complete ruler and compass descriptive geometry construction is possible starting from the top view of a dimple to find the point view of (or any other edge such as or , and therefore the true size of G ˆ . The construction is shown on fig. 52. c. The Great Stellated Dodecahedron: GSD (Kepler) If the edges of the convex icosahedron enveloping the SSD and GD are extended, they form triangular based pyramids on each of the 20 faces of this icosahedron (fig. 53). The geometry of these pyramids, as in the case of the SSD, determines the geometry of the
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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
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
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: The inner geometry of the GD will b
Page 83 and 84: 73 GSD. Since the apexes of the pyr
Page 85 and 86: The circumsphere radius of the enve
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,
Page 95 and 96: 85 fig. 45
Page 97 and 98: We have tanw 87 2 2 1 1 4 1 5 Wi
Page 99 and 100: 89 The structure of this scaled G.I
Page 101 and 102: the three mutually perpendicular go
Page 103 and 104: 93 For the dodecahedron, the golde
Page 105 and 106: 95 so that 2R cos ˆ D i 2 For the
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
Page 115 and 116: 105 fig. 61
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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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
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Ad Quadratum Construction and Study of the Regular Polyhedra

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