Compound of four octahedra with rotational freedom explained

bgcolor=#e7dcc3 colspan=2	Compound of four octahedra with rotational freedom
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Type	Uniform compound
Index	UC10
Polyhedra	4 octahedra
Faces	8+24 triangles
Edges	48
Vertices	24
Symmetry group	pyritohedral (Th)
Subgroup restricting to one constituent	6-fold improper rotation (S6)

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) ≈ 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.

References

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