Compound of five cuboctahedra

In geometry, this uniform polyhedron compound is a composition of 5 cuboctahedra. It has icosahedral symmetry I_h. It could also be called the anticosicosahedron.

Compound of five cuboctahedra
Type	Uniform compound
Index	UC59
Polyhedra	5 cuboctahedra
Faces	40 triangles, 30 squares
Edges	120
Vertices	60
Symmetry group	icosahedral (Ih)
Subgroup restricting to one constituent	pyritohedral (Th)

Cartesian coordinates

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

(±2, 0, ±2)

(±τ, ±τ⁻¹, ±(2τ−1))

(±1, ±τ⁻², ±τ²)

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

Construction

The compound of 5 cuboctahedra could be made by the rectification of the compound of five cubes or compound of five octahedra. It could also be formed by the expansion of the compound of five or ten tetrahedra.

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.
• McCooey, Robert. "Uniform Polyhedron Compounds". Hedron Dude. Retrieved 24 June 2025.