Ad Quadratum Construction and Study of the Regular Polyhedra

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170 Circumradius and Edge Relationships The relationships given on page 7 are derived here for each of the five regular convex polyhedra. 1. Cube: fig. 1Ap 2. Tetrahedron: fig. 2Ap AB = a AC = a 2 BC = 2R = (a 2) 2 a 2 a 3 R = a 2 and a 3 2 3 3 R The tetrahedron having the same circumsphere as the cube and therefore the same circumradius is ABCD. Let the edge of the tetrahedron be aT . a T = a C 2 where a C = cube edge. Since R is the same as that of the cube R a C 2 Replacing a C by a C a T 2 a T 3 2 2
3. Octahedron: fig. 3Ap 4. Dodecahedron: 171 R a T 6 4 a 2 3 6R AO R AB a a 2R and R a 2 2 a 6 4 fig. 4Ap Euclid constructed the dodecahedron on the basis of the cube as detailed by Heath (op. cit. p. 253). Indeed, consideration of a dodecahedron model (fig. 5 Ap.) readily shows that a dodecahedron can be seen as a cube on the faces of which a roof-like structure would be erected, the edges of the cube being the diagonals of the pentagonal faces of the dodecahedron. Further examination will show that, at each vertex, two such diagonals converge to form, with those of the other two pentagonal faces constituting that vertex, two sets of three mutually perpendicular lines determining the corners of the two cubes that can be traced on the dodecahedron model. One such set is AB, CB, DB. To prove that they are mutually perpendicular, ie., to show that ABDEHCGH is a cube, we note that ABDE must be a square since it has: 4 equal sides (diagonals of the pentagonal faces of the dodecahedron) 4 equal angles by symmetry furthermore in a quadrilateral figure the sum of the angles is 2 , each angle is therefore equal to fig. 5Ap 2 The same will obviously obtain for the other faces such as BCHD etc., making ABDEHCGH a cube. The dodecahedron will include five such cubes. This results immediately from the fact that there are five diagonals in a pentagonal face. The total number of diagonals in the whole dodecahedron is therefore 5 12 = 60 and since each cube has 12 edges, the number of cubes must be 60 5. 12
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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
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fig. 67 115 fig. 69 fig. 6 fig. 68
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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: R G r R G R S R S r 169 From r to
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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