Compound of five truncated cubes

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)

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.

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.