Ad Quadratum Construction and Study of the Regular Polyhedra

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146 relations can be expressed in simple arithmetical ratios….1:1, 1:2, 2:3, 3:4.” 45 “Beautiful things please by proportions…equality is not found only in sounds for the ear and in bodily movements, but also in visible forms…” says Augustine. 46 Centuries later, Leonardo da Vinci would write 47 : “I give the degrees of the objects seen by the eye as the musician does the notes heard by the ear.” …thereby introducing perspective as an analogy to musical harmony, and Kepler would add 48 : “The mathematics of the senses, if they are recognized, excite an intellectual mathematics, previously present to the inner man, such that there actively lies within the soul what beforehand was hidden beneath the veil of potentiality.” This visible form of harmony manifested through modules is evident for instance in the simple arrangements of a row of evenly spaced columns or trees (or the ties of a railroad) perceived as a harmonic series (see fig. 85). Here a constant frequency on the ground, mathematically expressed as an arithmetic series (the equal spacing between trees is additive) is transformed through the eye or the pinhole of a camera oscura by the effect of perspective (projection on the retina or the plate of the camera) into an harmonic series (each term, i.e., length of the column, tree or tie appears to be the harmonic means of its two adjoining terms) i.e., the frequency appears to get compressed more and more as the distance between items seems to keep on decreasing. Referring to fig. 85, we assume for simplicity’s sake unit distance between columns (or trees or ties), i.e., O’A’=A’B’ etc., columns two units tall, i.e., AA’=2O’A’ etc., and A observation point O (pinhole) at midheight, i.e., OO A . Now consider triangles 2 OAA’ and OPP’; OBB’ and OPBP' B ; OCC’ and OPC 'P'C ; ODD’ and OPD'P D ; they are respectively similar. We can therefore write in turn: 45 Von Simson, Otto, The Gothic Cathedral. Princeton/Bollingen, 1974. p. 21. 46 De Musica VI, xiii, 38. 47 The Notebooks of Leonardo da Vinci. Jean Paul Richter; 102 Vol. I. Dover, NY, 1970 48 Kepler, J. Gesammelte Werke, W. von Dyck et. Al. Ed. Munich, Beck, 1938. p. 226, Vol. 6. Quoted in Hallyn, op. cit. p. 171.
AA' OA1 2 1 BB' 2 = PP' OP 1 147 PP' 21 OB1 2 = PBP B PB P B 2 OP1 1 2 OP1 1 CC' OC1 2 3 = PC P C PC P C 2 OP1 1 3 DD' OD1 2 4 = PD P D PD P D 2 OP1 1 4 Now, projections on plane PP’ are inverted and can be considered as projection on the plate of a camera oscura or the retina (assuming it flat). Different focal distances can be fig. 85 fig. 86
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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
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: 145 from nothing. It is indivisible
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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