Compound of twenty octahedra with rotational freedom

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Compound of twenty octahedra with rotational freedom
Type	Uniform compound
Index	UC13
Polyhedra	20 octahedra
Faces	40+120 triangles
Edges	240
Vertices	120
Symmetry group	icosahedral (Ih)
Subgroup restricting to one constituent	6-fold improper rotation (S6)

This uniform polyhedron compound is 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.

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. At a certain intermediate angle the vertices coincide in pairs, yielding the compound of twenty octahedra (without rotational freedom).

[edit] Cartesian coordinates

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

(±2(√3)sinθ, ±(τ⁻¹√2+2τcosθ), ±(τ√2−2τ⁻¹cosθ))

(±(√2−τ²cosθ+τ⁻¹(√3)sinθ), ±(√2+(2τ−1)cosθ+(√3)sinθ), ±(√2+τ⁻²cosθ−τ(√3)sinθ))

(±(τ⁻¹√2−τcosθ−τ(√3)sinθ), ±(τ√2+τ⁻¹cosθ+τ⁻¹(√3)sinθ), ±(3cosθ−(√3)sinθ))

(±(−τ⁻¹√2+τcosθ−τ(√3)sinθ), ±(τ√2+τ⁻¹cosθ−τ⁻¹(√3)sinθ), ±(3cosθ+(√3)sinθ))

(±(−√2+τ²cosθ+τ⁻¹(√3)sinθ), ±(√2+(2τ−1)cosθ−(√3)sinθ), ±(√2+τ⁻²cosθ+τ(√3)sinθ))

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

[edit] References

• John Skilling, Uniform Compounds of Uniform Polyhedra, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 79, pp. 447-457, 1976.