Polyhedra Coordinates

Here are the 62 (x, y, z) coordinates for the 120 Polyhedron.

Recall that the 120 Polyhedron's vertices are also the vertices for:

and, of course, the 120 Polyhedron itself.

These coordinates are given in terms of the Golden Mean (a.k.a. Golden Ratio) which is here symbolized by the letter "p".

"p^2" and "p^3" are the 2nd and 3rd powers of the Golden Mean.

Also below, I give the vertex, edge and face maps for each of the polyhedra mentioned above.

With this information, it is easy to program a computer (CAD system) to display, manipulate and obtain numeric information about these polyhedra.



Coordinates

Vertex Type X Y Z
1 A 0 0 2p^2
2 B p^2 0 p^3
3 A p p^2 p^3
4 C 0 p p^3
5 A -p p^2 p^3
6 B -p^2 0 p^3
7 A -p -p^2 p^3
8 C 0 -p p^3
9 A p -p^2 p^3
10 A p^3 p p^2
11 C p^2 p^2 p^2
12 B 0 p^3 p^2
13 C -p^2 p^2 p^2
14 A -p^3 p p^2
15 A -p^3 -p p^2
16 C -p^2 -p^2 p^2
17 B 0 -p^3 p^2
18 C p^2 -p^2 p^2
19 A p^3 -p p^2
20 C p^3 0 p
21 A p^2 p^3 p
22 A -p^2 p^3 p
23 C -p^3 0 p
24 A -p^2 -p^3 p
25 A p^2 -p^3 p
Vertex Type X Y Z
26 A 2p^2 0 0
27 B p^3 p^2 0
28 C p p^3 0
29 A 0 2p^2 0
30 C -p p^3 0
31 B -p^3 p^2 0
32 A -2p^2 0 0
33 B -p^3 -p^2 0
34 C -p -p^3 0
35 A 0 -2p^2 0
36 C p -p^3 0
37 B p^3 -p^2 0
Vertex Type X Y Z
38 C p^3 0 -p
39 A p^2 p^3 -p
40 A -p^2 p^3 -p
41 C -p^3 0 -p
42 A -p^2 -p^3 -p
43 A p^2 -p^3 -p
44 A p^3 p -p^2
45 C p^2 p^2 -p^2
46 B 0 p^3 -p^2
47 C -p^2 p^2 -p^2
48 A -p^3 p -p^2
49 A -p^3 -p -p^2
50 C -p^2 -p^2 -p^2
51 B 0 -p^3 -p^2
52 C p^2 -p^2 -p^2
53 A p^3 -p -p^2
54 B p^2 0 -p^3
55 A p p^2 -p^3
56 C 0 p -p^3
57 A -p p^2 -p^3
58 B -p^2 0 -p^3
59 A -p -p^2 -p^3
60 C 0 -p -p^3
61 A p -p^2 -p^3
62 A 0 0 -2p^2


Maps for Constructing the Polyhedra

10 Tetrahedra

Tetrahedron 1

Vertices { 4, 34, 38, 47}
Edge Map { 4, 34}, { 4, 38}, { 4, 47}, {34, 38}, {34, 47}, {38, 47}
Face Map { 4, 34, 47}, { 4, 38, 34}, { 4, 47, 38}, {34, 38, 47}

Tetrahedron 2

Vertices {18, 23, 28, 60}
Edge Map {18, 23}, {18, 28}, {18, 60}, {23, 28}, {23, 60}, {28, 60}
Face Map {18, 23, 28}, {18, 23, 60}, {18, 28, 60}, {23, 28, 60}

Tetrahedron 3

Vertices { 4, 36, 41, 45}
Edge Map { 4, 36}, { 4, 41}, { 4, 45}, {36, 41}, {36, 45}, {41, 45}
Face Map { 4, 41, 36}, { 4, 36, 45}, { 4, 45, 41}, {36, 41, 45}

Tetrahedron 4

Vertices {16, 20, 30, 60}
Edge Map {16, 20}, {16, 30}, {16, 60}, {20, 30}, {20, 60}, {30, 60}
Face Map {16, 20, 30}, {16, 20, 60}, {16, 30, 60}, {20, 30, 60}

Tetrahedron 5

Vertices { 8, 28, 41, 52}
Edge Map { 8, 28}, { 8, 41}, { 8, 52}, {28, 41}, {28, 52}, {41, 52}
Face Map { 8, 28, 41}, { 8, 28, 52}, { 8, 41, 52}, {28, 41, 52}

Tetrahedron 6

Vertices {13, 20, 34, 56}
Edge Map {13, 20}, {13, 34}, {13, 56}, {20, 34}, {20, 56}, {34, 56}
Face Map {13, 20, 34}, {13, 20, 56}, {13, 34, 56}, {20, 34, 56}

Tetrahedron 7

Vertices { 8, 30, 38, 50}
Edge Map { 8, 30}, { 8, 38}, { 8, 50}, {30, 38}, {30, 50}, {38, 50}
Face Map { 8, 30, 38}, { 8, 30, 50}, { 8, 38, 50}, {30, 38, 50}

Tetrahedron 8

Vertices {11, 23, 36, 56}
Edge Map {11, 23}, {11, 36}, {11, 56}, {23, 36}, {23, 56}, {36, 56}
Face Map {11, 23, 36}, {11, 23, 56}, {11, 36, 56}, {23, 36, 56}

Tetrahedron 9

Vertices {11, 16, 47, 52}
Edge Map {11, 16}, {11, 47}, {11, 52}, {16, 47}, {16, 52}, {47, 52}
Face Map {47, 16, 11}, {52, 47, 11}, {16, 52, 11}, {47, 52, 16}

Tetrahedron 10

Vertices {13, 18, 45, 50}
Edge Map {13, 18}, {13, 45}, {13, 50}, {18, 45}, {18, 50}, {45, 50}
Face Map {13, 18, 45}, {13, 18, 50}, {13, 45, 50}, {18, 45, 50}


5 Cubes

Note: For the face maps, the Cube's square faces are divided into triangles. It is a common practice in graphics applications to divide polygons into triangles.

Cube 1

Vertices { 4, 18, 23, 28, 34, 38, 47, 60}
Edge Map { 4, 18}, {18, 38}, {38, 28}, {28, 4},
{ 4, 23}, {18, 34}, {28, 47}, {38, 60},
{23, 34}, {34, 60}, {60, 47}, {47, 23}
Face Map {( 4, 18, 38), (38, 28, 4)}, {( 4, 23, 18), (18, 23, 34)},
{( 4, 28, 47), ( 4, 47, 23)}, {(28, 38, 60), (28, 60, 47)},
{(23, 47, 34), (47, 60, 34)}, {(38, 18, 60), (18, 34, 60)}

Cube 2

Vertices { 4, 16, 20, 30, 36, 41, 45, 60}
Edge Map { 4, 16}, {16, 36}, {36, 20}, {20, 4},
{ 4, 30}, {16, 41}, {20, 45}, {36, 60},
{30, 41}, {41, 60}, {60, 45}, {45, 30}
Face Map {( 4, 16, 20), (16, 36, 20)}, {( 4, 20, 45), ( 4, 45, 30)},
{( 4, 30, 41), ( 4, 41, 16)}, {(45, 41, 30), (45, 60, 41)},
{(20, 36, 60), (20, 60, 45)}, {(36, 16, 41), (36, 41, 60}}

Cube 3

Vertices { 8, 13, 20, 28, 34, 41, 52, 56}
Edge Map { 8, 13}, {13, 28}, {28, 20}, {20, 8},
{ 8, 34}, {13, 41}, {28, 56}, {20, 52},
{34, 41}, {41, 56}, {56, 52}, {52, 34}
Face Map {( 8, 13, 20), (13, 28, 20)}, {( 8, 20, 52), ( 8, 52, 34)},
{( 8, 34, 41), ( 8, 41, 13)}, {(20, 28, 56), (20, 56, 52)},
{(52, 56, 41), (52, 41, 34)}, {(28, 13, 41), (28, 41, 56)}

Cube 4

Vertices { 8, 11, 23, 30, 36, 38, 50, 56}
Edge Map { 8, 11}, {11, 30}, {30, 23}, {23, 8},
{ 8, 36}, {11, 38}, {23, 50}, {30, 56},
{36, 38}, {38, 56}, {56, 50}, {50, 36}
Face Map {( 8, 11, 23), (11, 30, 23)}, {( 8, 23, 50), ( 8, 50, 36)},
{( 8, 36, 38), ( 8, 38, 11)}, {(23, 30, 56), (23, 56, 50)},
{(50, 56, 38), (50, 38, 36)}, {(30, 11, 56), (11, 38, 56)}

Cube 5

Vertices {11, 13, 16, 18, 45, 47, 50, 52}
Edge Map {11, 13}, {13, 16}, {16, 18}, {18, 11},
{11, 45}, {13, 47}, {16, 50}, {18, 52},
{45, 47}, {47, 50}, {50, 52}, {52, 45}
Face Map {(11, 13, 16), (11, 16, 18)}, {(11, 45, 47), (11, 47, 13)},
{(13, 47, 50), (13, 50, 16)}, {(16, 50, 52), (16, 52, 18)},
{(18, 52, 45), (18, 45, 11)}, {(45, 50, 52), (45, 47, 50)}


5 Octahedra

Octahedron 1

Vertices { 7, 10, 22, 43, 49, 55}
Edge Map { 7, 10}, { 7, 22}, { 7, 43}, { 7, 49},
{10, 22}, {10, 43}, {22, 49}, {43, 49},
{10, 55}, {22, 55}, {43, 55}, {49, 55}
Face Map { 7, 10, 43}, { 7, 22, 10}, { 7, 43, 49}, { 7, 49, 22},
{55, 10, 43}, {55, 22, 10}, {55, 43, 49}, {55, 49, 22}

Octahedron 2

Vertices { 9, 14, 21, 42, 53, 57}
Edge Map { 9, 14}, { 9, 21}, { 9, 42}, { 9, 53},
{14, 21}, {14, 42}, {21, 53}, {42, 53},
{14, 57}, {21, 57}, {42, 57}, {53, 57}
Face Map { 9, 14, 21}, { 9, 21, 53}, { 9, 42, 14}, { 9, 53, 42},
{57, 14, 21}, {57, 21, 53}, {57, 42, 14}, {57, 53, 42}

Octahedron 3

Vertices { 3, 15, 25, 40, 44, 59}
Edge Map { 3, 15}, { 3, 25}, { 3, 40}, { 3, 44},
{15, 25}, {15, 40}, {40, 44}, {25, 44},
{25, 59}, {15, 59}, {40, 59}, {44, 59}
Face Map { 3, 15, 25}, { 3, 25, 44}, { 3, 40, 15}, { 3, 44, 40},
{59, 25, 15}, {59, 15, 40}, {59, 40, 44}, {59, 44, 25}

Octahedron 4

Vertices { 5, 19, 24, 39, 48, 61}
Edge Map { 5, 19}, { 5, 24}, { 5, 39}, { 5, 48},
{19, 24}, {19, 39}, {24, 48}, {39, 48},
{19, 61}, {24, 61}, {39, 61}, {48, 61}
Face Map { 5, 19, 39}, { 5, 24, 19}, { 5, 39, 48}, { 5, 48, 24},
{61, 19, 39}, {61, 24, 19}, {61, 39, 48}, {61, 48, 24}

Octahedron 5

Vertices { 1, 26, 29, 32, 35, 62}
Edge Map { 1, 26}, { 1, 29}, { 1, 32}, { 1, 35},
{26, 29}, {29, 32}, {32, 35}, {35, 26},
{62, 26}, {62, 29}, {62, 32}, {62, 35}
Face Map { 1, 26, 29}, { 1, 29, 32}, { 1, 32, 35}, { 1, 35, 26},
{62, 29, 26}, {62, 32, 29}, {62, 35, 32}, {62, 26, 35}


5 Rhombic Dodecahedra

Note: For the face maps, the rhombic Dodecahedron's diamond faces are divided into triangles. It is a common practice in graphics applications to divide polygons into triangles.

Rhombic Dodecahedron 1

Vertices { 4, 7, 10, 18, 22, 23, 28, 34, 38, 43, 47, 49, 55, 60},
Edge Map { 7, 4}, { 7, 18}, { 7, 23}, { 7, 34},
{10, 4}, {10, 18}, {10, 28}, {10, 38},
{22, 4}, {22, 23}, {22, 28}, {22, 47},
{43, 18}, {43, 34}, {43, 38}, {43, 60},
{49, 23}, {49, 34}, {49, 47}, {49, 60},
{55, 28}, {55, 38}, {55, 47}, {55, 60}
Face Map {( 7, 4, 18), (10, 4, 18)}, {( 7, 18, 34), (43, 18, 34)},
{( 7, 34, 23), (49, 34, 23)}, {( 7, 4, 23), (22, 4, 23)},
{(22, 4, 28), (10, 4, 28)}, {(18, 10, 43), (38, 10, 43)},
{(34, 49, 43), (60, 49, 43)}, {(23, 49, 22), (47, 49, 22)},
{(55, 38, 60), (43, 38, 60)}, {(55, 60, 47), (49, 60, 47)},
{(55, 47, 28), (22, 47, 28)}, {(55, 28, 38), (10, 28, 38)}

Rhombic Dodecahedron 2

Vertices { 4, 9, 14, 16, 20, 21, 30, 36, 41, 42, 45, 53, 57, 60},
Edge Map { 9, 4}, { 9, 16}, { 9, 20}, { 9, 36},
{14, 4}, {14, 16}, {14, 30}, {14, 41},
{21, 4}, {21, 20}, {21, 30}, {21, 45},
{42, 16}, {42, 36}, {42, 41}, {42, 60},
{53, 20}, {53, 36}, {53, 45}, {53, 60},
{57, 30}, {57, 41}, {57, 45}, {57, 60}
Face Map {( 9, 4, 16), (14, 4, 16)}, {( 9, 16, 36), (42, 16, 36)},
{( 9, 36, 20), (53, 36, 20)}, {( 9, 20, 4), (21, 20, 4)},
{(14, 4, 30), (21, 4, 30)}, {(16, 42, 14), (41, 42, 14)},
{(36, 42, 53), (60, 42, 53)}, {(20, 21, 53), (45, 21, 53)},
{(42, 41, 60), (57, 41, 60)}, {(53, 45, 60), (57, 45, 60)},
{(21, 30, 45), (57, 30, 45)}, {(14, 41, 30), (57, 41, 30)}

Rhombic Dodecahedron 3

Vertices { 3, 8, 13, 15, 20, 25, 28, 34, 40, 41, 44, 52, 56, 59},
Edge Map { 3, 8}, { 3, 13}, { 3, 20}, { 3, 28},
{15, 8}, {15, 13}, {15, 34}, {15, 41},
{25, 8}, {25, 20}, {25, 34}, {25, 52},
{40, 13}, {40, 28}, {40, 41}, {40, 56},
{44, 20}, {44, 28}, {44, 52}, {44, 56},
{59, 34}, {59, 41}, {59, 52}, {59, 56}
Face Map {( 3, 8, 13), (15, 8, 13)}, {( 3, 13, 28), (40, 13, 28)},
{( 3, 28, 20), (44, 28, 20)}, {( 3, 20, 8), (25, 20, 8)},
{( 8, 25, 15), (34, 25, 15)}, {(13, 15, 40), (41, 15, 40)},
{(28, 44, 40), (56, 44, 40)}, {(20, 44, 25), (52, 44, 25)},
{(15, 41, 34), (59, 41, 34)}, {(40, 56, 41), (59, 56, 41)},
{(44, 56, 52), (59, 56, 52)}, {(25, 34, 52), (59, 34, 52)}

Rhombic Dodecahedron 4

Vertices { 5, 8, 11, 19, 23, 24, 30, 36, 38, 39, 48, 50, 56, 61},
Edge Map { 5, 8}, { 5, 11}, { 5, 23}, { 5, 30},
{19, 8}, {19, 11}, {19, 36}, {19, 38},
{24, 8}, {24, 23}, {24, 36}, {24, 50},
{39, 11}, {39, 30}, {39, 38}, {39, 56},
{48, 23}, {48, 30}, {48, 50}, {48, 56},
{61, 36}, {61, 38}, {61, 50}, {61, 56}
Face Map {( 5, 8, 11), (19, 8, 11)}, {( 5, 11, 30), (39, 11, 30)},
{( 5, 30, 23), (48, 30, 23)}, {( 5, 23, 8), (24, 23, 8)},
{( 8, 19, 24), (36, 19, 24)}, {(11, 19, 39), (38, 19, 39)},
{(30, 39, 48), (56, 39, 48)}, {(23, 48, 24), (50, 48, 24)},
{(19, 38, 36), (61, 38, 36)}, {(39, 38, 56), (61, 38, 56)},
{(48, 50, 56), (61, 50, 56)}, {(24, 36, 50), (61, 36, 50)}

Rhombic Dodecahedron 5

Vertices { 1, 11, 13, 16, 18, 26, 29, 32, 35, 45, 47, 50, 52, 62},
Edge Map { 1, 11}, { 1, 13}, { 1, 16}, { 1, 18},
{26, 18}, {26, 11}, {26, 45}, {26, 52},
{29, 11}, {29, 13}, {29, 47}, {29, 45},
{32, 13}, {32, 16}, {32, 50}, {32, 47},
{35, 16}, {35, 18}, {35, 52}, {35, 50},
{62, 45}, {62, 47}, {62, 50}, {62, 52}
Face Map {( 1, 11, 29), (29, 13, 1)}, {( 1, 13, 32), (32, 16, 1)},
{( 1, 16, 35), (35, 18, 1)}, {( 1, 18, 26), (26, 11, 1)},
{(26, 45, 29), (29, 11, 26)}, {(29, 47, 32), (32, 13, 29)},
{(32, 50, 35), (35, 16, 32)}, {(35, 52, 26), (26, 18, 35)},
{(62, 45, 26), (26, 52, 62)}, {(62, 47, 29), (29, 45, 62)},
{(62, 50, 32), (32, 47, 62)}, {(62, 52, 35), (35, 50, 62)}


Regular Dodecahedron

Note: For the face maps, the Dodecahedron's pentagon faces are divided into triangles. It is a common practice in graphics applications to divide polygons into triangles.

Vertices { 4, 8, 11, 13, 16, 18, 20, 23, 28, 30,
34, 36, 38, 41, 45, 47, 50, 52, 56, 60},
Edge Map { 4, 8}, { 4, 11}, { 4, 13}, { 8, 16}, { 8, 18}, {11, 20},
{11, 28}, {13, 30}, {13, 23}, {16, 23}, {16, 34}, {18, 36},
{18, 20}, {20, 38}, {23, 41}, {28, 30}, {28, 45}, {30, 47},
{34, 50}, {34, 36}, {36, 52}, {38, 45}, {38, 52}, {41, 47},
{41, 50}, {45, 56}, {47, 56}, {50, 60}, {52, 60}, {56, 60}
Face Map {( 4, 8, 11), (11, 8, 18), (11, 18, 20)},
{( 4, 13, 23), ( 4, 23, 8), ( 8, 23, 16)},
{( 4, 11, 28), ( 4, 28, 30), ( 4, 30, 13)},
{( 8, 16, 34), ( 8, 34, 18), (18, 34, 36)},
{(11, 20, 28), (20, 45, 28), (20, 38, 45)},
{(13, 30, 23), (23, 30, 41), (41, 30, 47)},
{(16, 23, 34), (34, 23, 50), (50, 23, 41)},
{(18, 36, 52), (18, 52, 38), (18, 38, 20)},
{(28, 45, 56), (28, 56, 47), (28, 47, 30)},
{(34, 50, 60), (34, 60, 36), (36, 60, 52)},
{(38, 52, 60), (38, 60, 56), (38, 56, 45)},
{(41, 47, 56), (41, 56, 60), (41, 60, 50}


Icosahedron

Vertices { 2, 6, 12, 17, 27, 31, 33, 37, 46, 51, 54, 58}
Edge Map { 2, 6}, { 2, 12}, { 2, 17}, { 2, 37}, { 2, 27}, { 6, 12},
{ 6, 17}, { 6, 31}, { 6, 33}, {12, 27}, {12, 46}, {12, 31},
{17, 33}, {17, 51}, {17, 37}, {27, 37}, {27, 54}, {27, 46},
{31, 46}, {31, 58}, {31, 33}, {33, 58}, {33, 51}, {37, 51},
{37, 54}, {46, 54}, {46, 58}, {51, 54}, {51, 58}, {54, 58}
Face Map { 2, 6, 17}, { 2, 12, 6}, { 2, 17, 37}, { 2, 37, 27},
{ 2, 27, 12}, {37, 54, 27}, {27, 54, 46}, {27, 46, 12},
{12, 46, 31}, {12, 31, 6}, { 6, 31, 33}, { 6, 33, 17},
{17, 33, 51}, {17, 51, 37}, {37, 51, 54}, {58, 54, 51},
{58, 46, 54}, {58, 31, 46}, {58, 33, 31}, {58, 51, 33}


Rhombic Triacontahedra

Note: For the face maps, the rhombic Triacontahedron's diamond faces are divided into triangles. It is a common practice in graphics applications to divide polygons into triangles.

Rhombic Triacontahedron

Vertices { 2, 4, 6, 8, 11, 12, 13, 16, 17, 18, 20, 23,
27, 28, 30, 31, 33, 34, 36, 37, 38, 41, 45, 46,
47, 50, 51, 52, 54, 56, 58, 60}
Edge Map { 2, 4}, { 4, 6}, { 6, 8}, { 8, 2}, { 2, 11}, {11, 12}, { 4, 12},
{12, 13}, {13, 6}, { 6, 23}, { 6, 16}, {16, 17}, {17, 8}, {17, 18},
{ 2, 18}, { 2, 20}, {20, 27}, {27, 28}, {12, 28}, {12, 30}, {13, 31},
{23, 31}, {23, 33}, {33, 16}, {18, 37}, {37, 20}, {11, 27}, {54, 56},
{56, 58}, {58, 60}, {60, 54}, {54, 45}, {45, 46}, {46, 56}, {58, 47},
{58, 41}, {58, 50}, {60, 51}, {52, 54}, {54, 38}, {38, 27}, {27, 45},
{46, 47}, {47, 31}, {31, 41}, {41, 33}, {33, 50}, {50, 51}, {51, 52},
{52, 37}, {37, 38}, {28, 46}, {30, 46}, {30, 31}, {17, 36}, {36, 51},
{51, 34}, {34, 17}, {36, 37}, {33, 34}
Face Map {( 2, 4, 6), ( 6, 8, 2)}, {( 2, 11, 4), ( 4, 11, 12)},
{( 4, 12, 13), ( 4, 13, 6)}, {( 6, 16, 8), ( 8, 16, 17)},
{( 8, 17, 18), ( 8, 18, 2)}, {( 2, 18, 37), ( 2, 37, 20)},
{( 2, 20, 27), ( 2, 27, 11)}, {(11, 27, 28), (11, 28, 12)},
{( 6, 13, 31), ( 6, 31, 23)}, {( 6, 23, 33), ( 6, 33, 16)},
{(54, 60, 58), (58, 56, 54)}, {(54, 56, 45), (45, 56, 46)},
{(56, 58, 47), (47, 46, 56)}, {(47, 58, 41), (41, 31, 47)},
{(58, 50, 33), (33, 41, 58)}, {(58, 60, 51), (51, 50, 58)},
{(60, 54, 52), (52, 51, 60)}, {(54, 38, 37), (37, 52, 54)},
{(45, 27, 38), (38, 54, 45)}, {(20, 37, 38), (38, 27, 20)},
{(23, 31, 41), (41, 33, 23)}, {(12, 28, 46), (46, 30, 12)},
{(12, 30, 31), (31, 13, 12)}, {(31, 30, 46), (46, 47, 31)},
{(28, 27, 45), (45, 46, 28)}, {(17, 34, 51), (51, 36, 17)},
{(18, 17, 36), (36, 37, 18)}, {(37, 36, 51), (51, 52, 37)},
{(17, 16, 33), (33, 34, 17)}, {(34, 33, 50), (50, 51, 34)}


120 Polyhedron

Vertices { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42,
43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56,
57, 58, 59, 60, 61, 62}
Edge Map { 1, 2}, { 1, 4}, { 1, 6}, { 1, 8}, { 2, 3}, { 2, 4}, { 2, 8},
{ 2, 9}, { 2, 10}, { 2, 11}, { 2, 18}, { 2, 19}, { 2, 20}, { 3, 4},
{ 3, 11}, { 3, 12}, { 4, 5}, { 4, 6}, { 4, 12}, { 5, 6}, { 5, 12},
{ 5, 13}, { 6, 7}, { 6, 8}, { 6, 13}, { 6, 14}, { 6, 15}, { 6, 16},
{ 6, 23}, { 7, 8}, { 7, 16}, { 7, 17}, { 8, 9}, { 8, 17}, { 9, 17},
{ 9, 18}, {10, 11}, {10, 20}, {10, 27}, {11, 12}, {11, 21}, {11, 27},
{12, 13}, {12, 21}, {12, 28}, {12, 29}, {12, 22}, {12, 30}, {13, 14},
{13, 22}, {13, 31}, {14, 23}, {14, 31}, {15, 16}, {15, 23}, {15, 33},
{16, 17}, {16, 24}, {16, 33}, {17, 18}, {17, 24}, {17, 25}, {17, 34},
{17, 35}, {17, 36}, {18, 19}, {18, 25}, {18, 37}, {19, 20}, {19, 37},
{20, 26}, {20, 27}, {20, 37}, {21, 27}, {21, 28}, {22, 30}, {22, 31},
{23, 31}, {23, 32}, {23, 33}, {24, 33}, {24, 34}, {25, 36}, {25, 37},
{26, 27}, {26, 37}, {26, 38}, {27, 28}, {27, 38}, {27, 39}, {27, 44},
{27, 45}, {28, 29}, {28, 39}, {28, 46}, {29, 30}, {29, 46}, {30, 31},
{30, 40}, {30, 46}, {31, 32}, {31, 40}, {31, 41}, {31, 47}, {31, 48},
{32, 33}, {32, 41}, {33, 34}, {33, 41}, {33, 42}, {33, 49}, {33, 50},
{34, 35}, {34, 42}, {34, 51}, {35, 36}, {35, 51}, {36, 37}, {36, 43},
{36, 51}, {37, 38}, {37, 43}, {37, 52}, {37, 53}, {38, 44}, {38, 53},
{38, 54}, {39, 45}, {39, 46}, {40, 46}, {40, 47}, {41, 48}, {41, 49},
{41, 58}, {42, 50}, {42, 51}, {43, 51}, {43, 52}, {44, 45}, {44, 54},
{45, 54}, {45, 55}, {45, 46}, {46, 47}, {46, 55}, {46, 56}, {46, 57},
{47, 48}, {47, 57}, {47, 58}, {48, 58}, {49, 50}, {49, 58}, {50, 51},
{50, 58}, {50, 59}, {51, 52}, {51, 59}, {51, 60}, {51, 61}, {52, 53},
{52, 61}, {52, 54}, {53, 54}, {54, 55}, {54, 56}, {54, 60}, {54, 61},
{54, 62}, {55, 56}, {56, 57}, {56, 58}, {56, 62}, {57, 58}, {58, 59},
{58, 60}, {58, 62}, {59, 60}, {60, 61}, {60, 62}
Face Map { 1, 2, 4}, { 2, 3, 4}, { 2, 20, 10}, { 2, 10, 11}, { 2, 11, 3},
{ 3, 11, 12}, { 3, 12, 4}, {20, 26, 27}, {20, 27, 10}, {10, 27, 11},
{11, 27, 21}, {11, 21, 12}, {21, 27, 28}, {12, 21, 28}, {12, 28, 29},
{ 1, 4, 6}, { 4, 12, 5}, { 4, 5, 6}, { 5, 12, 13}, { 5, 13, 6},
{ 6, 13, 14}, { 6, 14, 23}, {12, 29, 30}, {12, 30, 22}, {12, 22, 13},
{13, 22, 31}, {22, 30, 31}, {13, 31, 14}, {14, 31, 23}, {23, 31, 32},
{ 1, 6, 8}, { 6, 23, 15}, { 6, 15, 16}, { 6, 16, 7}, { 6, 7, 8},
{ 8, 7, 17}, { 7, 16, 17}, {23, 32, 33}, {15, 23, 33}, {16, 15, 33},
{24, 16, 33}, {34, 24, 33}, {17, 16, 24}, {17, 24, 34}, {17, 34, 35},
{ 1, 8, 2}, { 8, 17, 9}, { 8, 9, 2}, { 9, 17, 18}, { 9, 18, 2},
{ 2, 18, 19}, { 2, 19, 20}, {17, 35, 36}, {17, 36, 25}, {17, 25, 18},
{18, 25, 37}, {25, 36, 37}, {19, 18, 37}, {20, 19, 37}, {20, 37, 26},
{27, 26, 38}, {27, 38, 44}, {27, 44, 45}, {27, 45, 39}, {27, 39, 28},
{28, 39, 46}, {28, 46, 29}, {39, 45, 46}, {38, 54, 44}, {55, 45, 54},
{45, 44, 54}, {45, 55, 46}, {46, 55, 56}, {55, 54, 56}, {56, 54, 62},
{30, 29, 46}, {30, 46, 40}, {31, 30, 40}, {40, 46, 47}, {31, 40, 47},
{31, 47, 48}, {31, 48, 41}, {31, 41, 32}, {46, 56, 57}, {47, 46, 57},
{47, 57, 58}, {48, 47, 58}, {41, 48, 58}, {57, 56, 58}, {58, 56, 62},
{33, 32, 41}, {33, 41, 49}, {33, 49, 50}, {33, 50, 42}, {33, 42, 34},
{34, 42, 51}, {42, 50, 51}, {35, 34, 51}, {49, 41, 58}, {50, 49, 58},
{50, 58, 59}, {51, 50, 59}, {51, 59, 60}, {59, 58, 60}, {60, 58, 62},
{36, 35, 51}, {36, 51, 43}, {37, 36, 43}, {43, 51, 52}, {37, 43, 52},
{37, 52, 53}, {37, 53, 38}, {37, 38, 26}, {51, 60, 61}, {52, 51, 61},
{52, 61, 54}, {53, 52, 54}, {38, 53, 54}, {54, 61, 60}, {54, 60, 62}

Copyright November 2001, Robert W. Gray

rwgray@rwgrayprojects.com