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qalle2qalle2
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math moved to readme
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‎plato.scad

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1-
// The four non-built-in Platonic solids for OpenSCAD (regular tetrahedron,
2-
// octahedron, dodecahedron and icosahedron).
3-
// By Kalle (http://qalle.net).
1+
// The non-built-in Platonic solids.
2+
// See the readme file for the math.
43

54
module tetrahedron() {
65
// regular tetrahedron (centered at origin, edge length 1)
7-
// http://en.wikipedia.org/wiki/Equilateral_triangle
8-
// http://en.wikipedia.org/wiki/Tetrahedron#Regular_tetrahedron
96

10-
a = 1 / 2; // edge length / 2
11-
b = sqrt(3) / 6; // radius of incircle of faces
12-
c = sqrt(3) / 3; // radius of circumcircle of faces
13-
d = sqrt(6) / 12; // radius of insphere
14-
e = sqrt(6) / 4; // radius of circumsphere
7+
he = 1 / 2; // half the edge length
8+
fi = sqrt(3) / 6; // faces - incircle radius
9+
fc = sqrt(3) / 3; // faces - circumcircle radius
10+
i = sqrt(6) / 12; // insphere radius
11+
c = sqrt(6) / 4; // circumsphere radius
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1613
polyhedron(
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// vertices
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[
19-
[ 0, 0, e], // 0: top
20-
[ 0, c, -d], // 1: bottom front
21-
[-a, -b, -d], // 2: bottom rear left
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[ a, -b, -d], // 3: bottom rear right
16+
[ 0, 0, c], // 0: top
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[ 0, fc, -i], // 1: bottom front
18+
[-he, -fi, -i], // 2: bottom rear left
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[ he, -fi, -i], // 3: bottom rear right
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],
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// faces
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[
@@ -33,20 +30,19 @@ module tetrahedron() {
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3431
module octahedron() {
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// regular octahedron (centered at origin, edge length 1)
36-
// http://en.wikipedia.org/wiki/Octahedron#Regular_octahedron
3733

38-
a = 1 / 2; // edge length / 2
39-
b = sqrt(2) / 2; // radius of circumsphere
34+
he = 1 / 2; // half the edge length
35+
c = sqrt(2) / 2; // circumsphere radius
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polyhedron(
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// vertices
4339
[
44-
[ 0, 0, b], // 0: top
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[-a, a, 0], // 1: front left
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[ a, a, 0], // 2: front right
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[ a, -a, 0], // 3: rear right
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[-a, -a, 0], // 4: rear left
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[ 0, 0, -b], // 5: bottom
40+
[ 0, 0, c], // 0: top
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[-he, he, 0], // 1: front left
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[ he, he, 0], // 2: front right
43+
[ he, -he, 0], // 3: rear right
44+
[-he, -he, 0], // 4: rear left
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[ 0, 0, -c], // 5: bottom
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],
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// faces
5248
[
@@ -64,45 +60,37 @@ module octahedron() {
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6561
module dodecahedron() {
6662
// regular dodecahedron (centered at origin, edge length 1)
67-
// http://en.wikipedia.org/wiki/Regular_dodecahedron
68-
//
69-
// phi = (1 + sqrt(5)) / 2
70-
// phi^2 = phi + 1
71-
//
72-
// if edge = 2/phi, coordinates of vertices are:
73-
// (+-1, +-1, +-1)
74-
// circular permutations of (0, +-phi, +-1/phi)
75-
// to get coordinates with edge length 1, multiply them by phi/2:
76-
// (+-phi/2, +-phi/2, +-phi/2)
77-
// circular permutations of (0, +-(phi+1)/2, +-1/2)
7863

79-
a = (1 + sqrt(5)) / 4; // phi / 2
80-
b = (3 + sqrt(5)) / 4; // (phi + 1) / 2
81-
c = 1 / 2;
64+
// coordinates of the "cube"
65+
c = (1 + sqrt(5)) / 4; // phi / 2
66+
// coordinates of the "rectangular cuboid"
67+
r1 = 0;
68+
r2 = (3 + sqrt(5)) / 4; // (phi + 1) / 2
69+
r3 = 1 / 2;
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polyhedron(
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// vertices
8573
[
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[ 0, b, c], // 0: front top
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[ 0, b, -c], // 1: front bottom
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[ 0, -b, c], // 2: rear top
89-
[ 0, -b, -c], // 3: rear bottom
90-
[ c, 0, b], // 4: top right
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[ c, 0, -b], // 5: bottom right
92-
[-c, 0, b], // 6: top left
93-
[-c, 0, -b], // 7: bottom left
94-
[ a, a, a], // 8: top front right
95-
[ a, a, -a], // 9: bottom front right
96-
[ a, -a, a], // 10: top rear right
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[ a, -a, -a], // 11: bottom rear right
98-
[-a, a, a], // 12: top front left
99-
[-a, a, -a], // 13: bottom front left
100-
[-a, -a, a], // 14: top rear left
101-
[-a, -a, -a], // 15: bottom rear left
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[ b, c, 0], // 16: right front
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[ b, -c, 0], // 17: right rear
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[-b, c, 0], // 18: left front
105-
[-b, -c, 0], // 19: left rear
74+
[ r1, r2, r3], // 0: front top
75+
[ r1, r2, -r3], // 1: front bottom
76+
[ r1, -r2, r3], // 2: rear top
77+
[ r1, -r2, -r3], // 3: rear bottom
78+
[ r3, r1, r2], // 4: top right
79+
[ r3, r1, -r2], // 5: bottom right
80+
[-r3, r1, r2], // 6: top left
81+
[-r3, r1, -r2], // 7: bottom left
82+
[ c, c, c], // 8: top front right
83+
[ c, c, -c], // 9: bottom front right
84+
[ c, -c, c], // 10: top rear right
85+
[ c, -c, -c], // 11: bottom rear right
86+
[ -c, c, c], // 12: top front left
87+
[ -c, c, -c], // 13: bottom front left
88+
[ -c, -c, c], // 14: top rear left
89+
[ -c, -c, -c], // 15: bottom rear left
90+
[ r2, r3, r1], // 16: right front
91+
[ r2, -r3, r1], // 17: right rear
92+
[-r2, r3, r1], // 18: left front
93+
[-r2, -r3, r1], // 19: left rear
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],
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// faces
10896
[
@@ -124,29 +112,26 @@ module dodecahedron() {
124112

125113
module icosahedron() {
126114
// regular icosahedron (centered at origin, edge length 1)
127-
// http://en.wikipedia.org/wiki/Regular_icosahedron
128115

129-
// coordinates of vertices:
130-
// circular permutations of 0, +-1/2, +-phi/2
131-
132-
a = 1 / 2;
133-
b = (1 + sqrt(5)) / 4; // phi/2
116+
c1 = 0; // coordinate 1
117+
c2 = 1 / 2; // coordinate 2
118+
c3 = (1 + sqrt(5)) / 4; // coordinate 3; phi / 2
134119

135120
polyhedron(
136121
// vertices
137122
[
138-
[ b, 0, a], // 0: right top
139-
[ b, 0, -a], // 1: right bottom
140-
[-b, 0, a], // 2: left top
141-
[-b, 0, -a], // 3: left bottom
142-
[ a, b, 0], // 4: front right
143-
[ a, -b, 0], // 5: rear right
144-
[-a, b, 0], // 6: front left
145-
[-a, -b, 0], // 7: rear left
146-
[ 0, a, b], // 8: top front
147-
[ 0, a, -b], // 9: bottom front
148-
[ 0, -a, b], // 10: top rear
149-
[ 0, -a, -b], // 11: bottom rear
123+
[ c3, c1, c2], // 0: right top
124+
[ c3, c1, -c2], // 1: right bottom
125+
[-c3, c1, c2], // 2: left top
126+
[-c3, c1, -c2], // 3: left bottom
127+
[ c2, c3, c1], // 4: front right
128+
[ c2, -c3, c1], // 5: rear right
129+
[-c2, c3, c1], // 6: front left
130+
[-c2, -c3, c1], // 7: rear left
131+
[ c1, c2, c3], // 8: top front
132+
[ c1, c2, -c3], // 9: bottom front
133+
[ c1, -c2, c3], // 10: top rear
134+
[ c1, -c2, -c3], // 11: bottom rear
150135
],
151136
// faces
152137
[

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