Compound of four octahedra with rotational freedom
| Compound of four octahedra with rotational freedom | |
|---|---|
| |
| 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]\displaystyle{ \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 = 5°
θ = 10°
θ = 15°
θ = 20°
θ = 25°
θ = 30°
θ = 35°
θ = 40°
θ = 45°
θ = 50°
θ = 55°
θ = 60°
References
- Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society 79 (3): 447–457, doi:10.1017/S0305004100052440.
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