Compound of ten octahedra

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Compounds of ten octahedra
UC15-10 octahedra.png UC16-10 octahedra.png
Type Uniform compound
IndexUC15 and UC16
Polyhedra10 octahedra
Faces20+60 triangles
Edges120
Vertices60
Symmetry group icosahedral (Ih)
Subgroup restricting to one constituent3-fold antiprismatic (D3d)
3D model of the compound of ten octahedra UC15 First compound of ten octahedra.stl
3D model of the compound of ten octahedra UC15
3D model of the compound of ten octahedra UC16 Second compound of ten octahedra.stl
3D model of the compound of ten octahedra UC16

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.

Contents

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, ±(φ12 + 2sφ), ±(φ2 2sφ1))
(±(2sτφ2), ±(2 + s(2τφ 1)), ±(2 + sφ2))
(±(φ12sφ), ±(φ2 + sφ1), ±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.

See also

References