Compounds of ten octahedraTypeUniform compoundIndexUC15 and UC16Polyhedra10 octahedraFaces20+60 trianglesEdges120Vertices60Symmetry groupicosahedral (Ih)Subgroup restricting to one constituent3-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 Cartesian coordinates for the vertices of this compound are all the cyclic permutations of
(0, ±(φ−1 + 2sφ), ±(φ − 2sφ−1)) (±( − sτφ2), ±( + s(2τφ − 1)), ±( + sφ−2)) (±(φ−1 − sφ), ±(φ + sφ−1), ±3s)
where φ = (1 + )/2 is the golden ratio and s is either +1 or −1. Setting s = −1 gives UC15, while s = +1 gives UC16.
See also Compound of three octahedra Compound of four octahedra Compound of five octahedra Compound of twenty octahedra
References .