Ad Quadratum Construction and Study of the Regular Polyhedra

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and cos ˆ O i 2 Dodecahedron: As previously established, Icosahedron: 1 1 10 2 2 ri R ri R 36 2 2 D ˆ i cos 2 1 5 with 2 1 5 2 3 15 3 6 Making use of previous results, we can write: ri I ˆ i cos R 2 Ratio of Insphere to Intersphere Radii: Finally we can also establish the ratio of the insphere radius r to the intersphere radius ri, r for each polyhedron, ri Tetrahedron: since r ri r R ri R r cosC ˆ i R 1 ri T ˆ i 3 ; cos 3 R 2 3
Cube: Octahedron: Dodecahedron: r R r ri r ri cosC ˆ i cos ˆ T i 2 1 3 3 3 37 1 3 0.577 3 3 r 3 cosC ˆ i R 3 ri C ˆ i 2 ; cos 3 R 2 3 cos 3 cosC ˆ i cos ˆ C i 2 3 3 2 3 3 1 3 2 2 2 2 0.707 r 3 cosC ˆ i R 3 ri O ˆ i 2 1 ; cos 3 R 2 2 2 r ri 3 3 2 2 D ˆ i 2 cos I ˆ i 1 2 3 2 0.816 3 ; ri R D ˆ i cos 2
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 and 16: …which can be more briefly put as
Page 17 and 18: 7 Tetrahedron: Cube: Octahedron: Do
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: 35 Ratio of Intersphere to Circumsp
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
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
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 and 176:
Remarks TABLE 1 TRIGONOMETRIC PROPE
Page 177 and 178:
TABLE 3 REGULAR CONVEX POLYHEDRA ME
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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