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Compound of five cuboctahedra

Polyhedral compound From Wikipedia, the free encyclopedia

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

Compound of five cuboctahedra
Thumb
TypeUniform compound
IndexUC59
Polyhedra5 cuboctahedra
Faces40 triangles, 30 squares
Edges120
Vertices60
Symmetry groupicosahedral (Ih)
Subgroup restricting to one constituentpyritohedral (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.


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