Compound of ten octahedra

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Compounds of ten octahedra
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)

The compounds of ten octahedra UC15 and UC16 are two uniform polyhedron compounds. They are composed of a symmetric arrangement 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.

3D model of the compound of ten octahedra UC15
3D model of the compound of ten octahedra UC16

Cartesian coordinates

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

(0, ±(τ−12 + 2sτ), ±(τ2 − 2sτ−1))
(±(2sτ2), ±(2 + s(2τ − 1)), ±(2 + sτ−2))
(±(τ−12sτ), ±(τ2 + 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

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References

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