Compound of five icosahedra

Compound of five icosahedra
Type	Uniform compound
Index	UC47
Polyhedra	5 icosahedra
Faces	40+60 Triangles
Edges	150
Vertices	60
Symmetry group	icosahedral (Ih)
Subgroup restricting to one constituent	pyritohedral (Th)

3D model of a compound of five icosahedra

The compound of five icosahedra is uniform polyhedron compound. It's composed of 5 icosahedra, rotated around a common axis. It has icosahedral symmetry I_h.

The triangles in this compound decompose into two orbits under action of the symmetry group: 40 of the triangles lie in coplanar pairs in icosahedral planes, while the other 60 lie in unique planes.

The compound of five icosahedra shares the same vertex arrangement of a nonuniform rhombicosidodecahedron.

Cartesian coordinates

Cartesian coordinates for the vertices of this compound are all the cyclic permutations of

(0, ±2, ±2τ)

(±τ⁻¹, ±1, ±(1+τ²))

(±τ, ±τ², ±(2τ−1))

where τ = (1+√5)/2 is the golden ratio (sometimes written φ).

References

• Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society , 79 (3): 447–457, doi:10.1017/S0305004100052440, MR   0397554 .