Compound of ten octahedra

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.

For UC15, the convex hull of this compound is a nonuniform rhombicosidodecahedron. For UC16, the convex hull would be a nonuniform truncated icosahedron.

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

(0, ±(φ⁻¹√2 + 2sφ), ±(φ√2 − 2sφ⁻¹))

(±(√2 − sτφ²), ±(√2 + s(2τφ − 1)), ±(√2 + sφ⁻²))

(±(φ⁻¹√2 − sφ), ±(φ√2 + sφ⁻¹), ±3s)

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

• Compound of three octahedra
• Compound of four octahedra
• Compound of five octahedra
• Compound of twenty octahedra

• 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.