Английская Википедия:Compound of four octahedra with rotational freedom

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Compound of four octahedra with rotational freedom
Файл:UC10-4 octahedra.png
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

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

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 = 0°

• θ = 5°

  θ = 5°

• θ = 10°

  θ = 10°

• θ = 15°

  θ = 15°

• θ = 20°

  θ = 20°

• θ = 25°

  θ = 25°

• θ = 30°

  θ = 30°

• θ = 35°

  θ = 35°

• θ = 40°

  θ = 40°

• θ = 45°

  θ = 45°

• θ = 50°

  θ = 50°

• θ = 55°

  θ = 55°

• θ = 60°

  θ = 60°

References

• Шаблон:Citation.

Шаблон:Polyhedron-stub