Compound of four octahedra with rotational freedom
# Compound of four octahedra with rotational freedom

Compound of four octahedra with rotational freedom :

| Type | Uniform compound |
| - | - |
| Index | $UC_{10}$ |
| Polyhedra | 4 octahedra |
| Faces | 8+24 triangles |
| Edges | 48 |
| Vertices | 24 |
| Symmetry group | pyritohedral $(T_h)$ |
| Subgroup restricting to one constituent | 6-fold improper rotation $(S_6)$ |

The compound of four octahedra with rotational freedom is a uniform polyhedron compound. It consists in a symmetric arrangement of 4 octahedra, considered as triangular antiprisms. It can be constructed by superimposing four identical octahedra, and then rotating each by an equal angle θ about a separate axis passing through the centres of two opposite octahedral faces, in such a way as to preserve pyritohedral symmetry.

Superimposing this compound with a second copy, in which the octahedra have been rotated by the same angle θ in the opposite direction, yields the compound of eight octahedra with rotational freedom.

When θ = 0, all four octahedra coincide. When θ is 60 degrees, the more symmetric compound of four octahedra (without rotational freedom) arises. In another notable case (pictured), when

    $\theta =2\tan ^{-1}\left({\sqrt {15}}-2{\sqrt {3}}\right)\approx 44.47751^{\circ },$

24 of the triangles form coplanar pairs, and the compound assumes the form of the compound of five octahedra with one of the octahedra removed.

## Gallery

* Compounds of four octahedra with rotational freedom
* θ = 0°
* θ = 5°
* θ = 10°
* θ = 15°
* θ = 20°
* θ = 25°
* θ = 30°
* θ = 35°
* θ = 40°
* θ = 45°
* θ = 50°
* θ = 55°
* θ = 60°
