Compound of five truncated cubes

This uniform polyhedron compound is a composition of 5 truncated cubes, formed by truncating each of the cubes in the compound of 5 cubes. It is also called the truncated rhombihedron.

Compound of five truncated cubes
Type	Uniform compound
Index	UC57
Polyhedra	5 truncated cubes
Faces	40 triangles, 30 octagons
Edges	180
Vertices	120
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+√2), ±√2, ±(2+√2))

(±τ, ±(τ⁻¹+τ⁻¹√2), ±(2τ−1+τ√2))

(±1, ±(τ⁻²−τ⁻¹√2), ±(τ²+τ√2))

(±(1+√2), ±(−τ⁻²−√2), ±(τ²+√2))

(±(τ+τ√2), ±(−τ⁻¹), ±(2τ−1+τ⁻¹√2))

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, Bibcode:1976MPCPS..79..447S, doi:10.1017/S0305004100052440, MR 0397554, S2CID 123279687.
• McCooey, Robert. "Uniform Polyhedron Compounds". Hedron Dude. Retrieved 24 June 2025.