Compound of five icosahedra

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Compound of five icosahedra
UC47-5 icosahedra.png
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)

This uniform polyhedron compound is a composition of 5 icosahedra. It has icosahedral symmetry Ih.

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.

Cartesian coordinates[edit]

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).


  • 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 .