Compound of ten octahedra
# Compound of ten octahedra

Compounds of ten octahedra :

| Type | Uniform compound |
| - | - |
| Index | $UC_{15}$ and $UC_{16}$ |
| Polyhedra | 10 octahedra |
| Faces | 20+60 triangles |
| Edges | 120 |
| Vertices | 60 |
| Symmetry group | icosahedral $(I_h)$ |
| Subgroup restricting to one constituent | 3-fold antiprismatic $(D_{3d})$ |

The compounds of ten octahedra $UC_{15}$ and $UC_{16}$ 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 − $sτ^2),$ ±(√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 $UC_{15},$ while s = +1 gives $UC_{16}.$

