Compound of twenty octahedra with rotational freedom
# Compound of twenty octahedra with rotational freedom

Compound of twenty octahedra with rotational freedom :

| Type | Uniform compound |
| - | - |
| Index | $UC_{13}$ |
| Polyhedra | 20 octahedra |
| Faces | 40+120 triangles |
| Edges | 240 |
| Vertices | 120 |
| Symmetry group | icosahedral $(I_h)$ |
| Subgroup restricting to one constituent | 6-fold improper rotation $(S_6)$ |

The compound of twenty octahedra with rotational freedom is a uniform polyhedron compound. It's composed of a symmetric arrangement of 20 octahedra, considered as triangular antiprisms. It can be constructed by superimposing two copies of the compound of 10 octahedra $UC_{16},$ and for each resulting pair of octahedra, rotating each octahedron in the pair by an equal and opposite angle θ.

When θ is zero or 60 degrees, the octahedra coincide in pairs yielding (two superimposed copies of) the compounds of ten octahedra $UC_{16}$ and $UC_{15}$ respectively. When

    $\theta =2\tan ^{-1}\left({\sqrt {{\frac {1}{3}}\left(13-4{\sqrt {10}}\right)}}\right)\approx 37.76124^{\circ },$

octahedra (from distinct rotational axes) coincide in sets four, yielding the compound of five octahedra. When

    $\theta =2\tan ^{-1}\left({\frac {-4{\sqrt {3}}-2{\sqrt {15}}+{\sqrt {132+60{\sqrt {5}}}}}{4+{\sqrt {2}}+2{\sqrt {5}}+{\sqrt {10}}}}\right)\approx 14.33033^{\circ },$

the vertices coincide in pairs, yielding the compound of twenty octahedra (without rotational freedom).

## Cartesian coordinates

Cartesian coordinates for the vertices of this compound are all the cyclic permutations of

    ${\begin{aligned}&\scriptstyle {\Big\pm 2{\sqrt {3}}\sin \theta ,\,\pm (\tau ^{-1}{\sqrt {2}}+2\tau \cos \theta ),\,\pm (\tau {\sqrt {2}}-2\tau ^{-1}\cos \theta ){\Big )}\\&\scriptstyle {\Big\pm ({\sqrt {2}}-\tau ^{2}\cos \theta +\tau ^{-1}{\sqrt {3}}\sin \theta ),\,\pm ({\sqrt {2}}+(2\tau -1)\cos \theta +{\sqrt {3}}\sin \theta ),\,\pm ({\sqrt {2}}+\tau ^{-2}\cos \theta -\tau {\sqrt {3}}\sin \theta ){\Big )}\\&\scriptstyle {\Big\pm (\tau ^{-1}{\sqrt {2}}-\tau \cos \theta -\tau {\sqrt {3}}\sin \theta ),\,\pm (\tau {\sqrt {2}}+\tau ^{-1}\cos \theta +\tau ^{-1}{\sqrt {3}}\sin \theta ),\,\pm (3\cos \theta -{\sqrt {3}}\sin \theta ){\Big )}\\&\scriptstyle {\Big\pm (-\tau ^{-1}{\sqrt {2}}+\tau \cos \theta -\tau {\sqrt {3}}\sin \theta ),\,\pm (\tau {\sqrt {2}}+\tau ^{-1}\cos \theta -\tau ^{-1}{\sqrt {3}}\sin \theta ),\,\pm (3\cos \theta +{\sqrt {3}}\sin \theta ){\Big )}\\&\scriptstyle {\Big\pm (-{\sqrt {2}}+\tau ^{2}\cos \theta +\tau ^{-1}{\sqrt {3}}\sin \theta ),\,\pm ({\sqrt {2}}+(2\tau -1)\cos \theta -{\sqrt {3}}\sin \theta ),\,\pm ({\sqrt {2}}+\tau ^{-2}\cos \theta +\tau {\sqrt {3}}\sin \theta ){\Big )}\end{aligned}}$

where τ = (1 + √5)/2 is the golden ratio (sometimes written φ).

## Gallery

* Compounds of twenty 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°
