Compound of five truncated cubes
Polyhedral compound
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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.
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