Compound of ten octahedra

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Compounds of ten octahedra
UC15-10 octahedra.pngUC16-10 octahedra.png
Type Uniform compound
Index UC15 and UC16
Polyhedra 10 octahedra
Faces 20+60 triangles
Edges 120
Vertices 60
Symmetry group icosahedral (Ih)
Subgroup restricting to one constituent 3-fold antiprismatic (D3d)

These uniform polyhedron compounds are symmetric arrangements of 10 octahedra, considered as triangular antiprisms, aligned with the axes of three-fold rotational symmetry of an icosahedron. The two compounds differ in the orientation of their octahedra: each compound may be transformed into the other by rotating each octahedron by 60 degrees.

Cartesian coordinates[edit]

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

(0, ±(τ−1√2 + 2sτ), ±(τ√2 − 2sτ−1))
(±(√2 − sτ2), ±(√2 + s(2τ − 1)), ±(√2 + sτ−2))
(±(τ−1√2 − sτ), ±(τ√2 + sτ−1), ±3s)

where τ = (1 + √5)/2 is the golden ratio (sometimes written φ) and s is either +1 or −1. Setting s = −1 gives UC15, while s = +1 gives UC16.

See also[edit]


  • Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79: 447–457, doi:10.1017/S0305004100052440, MR 0397554 .