Ad Quadratum Construction and Study of the Regular Polyhedra

More documents

Recommendations

Info

174 1 5 3 4 a d A more analytical method can of course be established. Referring to fig. 45, it will be observed that the intersphere, radius r i (as described on p. 35), passes through points P and L. OP OL r i Furthermore CL a 2 and PC is the edge view of a pentagonal face of the dodecahedron show- ing the height of the pentagon in true length (see figs. 6 Ap. and 7 Ap.) PC h Point C is on the circumsphere so that OC R PC ri a 2 h 2 2 ri (ri a 2 ) 2 (1) fig. 6Ap As can be seen from fig. 66 (p. 113), CD on fig. 7 Ap. is the pentagon diagonal fig. 7Ap so that 1 5 CD a a 2 PC 2 a 2 2 a 2 4 = a 2 ( 2 1 4 )
Replacing in (1) by (2) or, rearranging: 175 PC h a 2 1 4 2 ri a 3 4 h a 2 5 2 5 (2) (ri a 2 )2 a2 (2 5 5) 4 2 ri a 2 ri a2 ( 52) o (3) 4 Solution to this quadratic will yield r i f (a) which, introduced into (4), below, will in turn yield the sought-for relation between R and a. 2 2 a R ri 4 (4) Let ( 5 2) Then r a 2 i 2 ri a2 o 4 ri 1 a 2 2 a2 4 ri a 1 1 4 4 2 a 4 4
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 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 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: 173 One can also reason that there
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
show all

Ad Quadratum Construction and Study of the Regular Polyhedra

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?