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)

3D model of the compound of ten octahedra UC₁₅

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.

Cartesian coordinates

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

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

References

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