See [MathWorld] for more information about the icosahedron (and other polyhedra).
vertices: 12
faces: 20
edges: 30
genus: 0
characteristic polynomial:
(-5 + x)(-5 + x2)3 (1 + x)5
dual's characteristic polynomial:
(-3 + x)(-1 + x)5 x4 (2 + x)4 (-5 + x2)3
chromatic number: 3
lattice group:
A(5)
After taking two neighboured faces and rotating them as a collective unit (generalization of Rubik-like move).
vertices: 12
faces: 20
edges: 30
genus: 0
characteristic polynomial:
(1 + x) (2 + x) (-1 - 2x + x2) (-7 - 4x + 2x2 + x3) (22 + 43x + 9x2 - 14x3 - 3x4 + x5)
dual's characteristic polynomial:
(-3 + x) (-1 + x)3 x2 (1 + x) (2 + x)2 (-5 + x2) (2 - 5x2 + x4) (-2 + 18x - 5x2 - 9x3 + x4 + x5)
chromatic number: 3
lattice group (lossy):
Z(2) 29 × A(30)