Polyhedra Overview

The (x, y, z) coordinates for all of the vertices can be put into the form
n φ^m
where n = -2, -1, 0, 1, 2 and m = -1, 0, 1 and φ is the Golden ratio.

Vertex	Type	X	Y	Z
1	A	0	0	2
2	B	1	0	φ
3	A	φ^-1	1	φ
4	C	0	φ^-1	φ
5	A	-φ^-1	1	φ
6	B	-1	0	φ
7	A	-φ^-1	-1	φ
8	C	0	-φ^-1	φ
9	A	φ^-1	-1	φ
10	A	φ	φ^-1	1
11	C	1	1	1
12	B	0	φ	1
13	C	-1	1	1
14	A	-φ	φ^-1	1
15	A	-φ	-φ^-1	1
16	C	-1	-1	1
17	B	0	-φ	1
18	C	1	-1	1
19	A	φ	-φ^-1	1
20	C	φ	0	φ^-1
21	A	1	φ	φ^-1
22	A	-1	φ	φ^-1
23	C	-φ	0	φ^-1
24	A	-1	-φ	φ^-1
25	A	1	-φ	φ^-1

Vertex	Type	X	Y	Z
26	A	2	0	0
27	B	φ	1	0
28	C	φ^-1	φ	0
29	A	0	2	0
30	C	-φ^-1	φ	0
31	B	-φ	1	0
32	A	-2	0	0
33	B	-φ	-1	0
34	C	-φ^-1	-φ	0
35	A	0	-2	0
36	C	φ^-1	-φ	0
37	B	φ	-1	0

Vertex	Type	X	Y	Z
38	C	φ	0	-φ^-1
39	A	1	φ	-φ^-1
40	A	-1	φ	-φ^-1
41	C	-φ	0	-φ^-1
42	A	-1	-φ	-φ^-1
43	A	1	-φ	-φ^-1
44	A	φ	φ^-1	-1
45	C	1	1	-1
46	B	0	φ	-1
47	C	-1	1	-1
48	A	-φ	φ^-1	-1
49	A	-φ	-φ^-1	-1
50	C	-1	-1	-1
51	B	0	-φ	-1
52	C	1	-1	-1
53	A	φ	-φ^-1	-1
54	B	1	0	-φ
55	A	φ^-1	1	-φ
56	C	0	φ^-1	-φ
57	A	-φ^-1	1	-φ
58	B	-1	0	-φ
59	A	-φ^-1	-1	-φ
60	C	0	-φ^-1	-φ
61	A	φ^-1	-1	-φ
62	A	0	0	-2

Note the z-components are 2, φ, 1, φ^-1, 0, -φ^-1, -1, -φ, -2.
This defines 9 layers or planes (in this orientation of the 120 Polyhedron).

The Cube's 4 vertex-to-opposite-vertex axes were used
for the Cube and Octahedron's rotations.

What are the rotation angles?

For an arbitrary (x, y, z) coordinate, if rotated by these angles
what is the resulting coordinate (x', y', z')?

Type "A" =	Octahedron vertex
Type "B" =	Icosahedron vertex
Type "C" =	Tetrahedron, Cube, Dodecahedron vertex

The 120 Polyhedron