Compound of twenty octahedra with rotational freedom

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 UC16, and for each resulting pair of octahedra, rotating each octahedron in the pair by an equal and opposite angle θ.

Compound of twenty octahedra with rotational freedom
TypeUniform compound
IndexUC13
Polyhedra20 octahedra
Faces40+120 triangles
Edges240
Vertices120
Symmetry groupicosahedral (Ih)
Subgroup restricting to one constituent6-fold improper rotation (S6)

When θ is zero or 60 degrees, the octahedra coincide in pairs yielding (two superimposed copies of) the compounds of ten octahedra UC16 and UC15 respectively. When

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

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

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

References

  • Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79 (3): 447–457, doi:10.1017/S0305004100052440, MR 0397554.


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