title: Compound of four octahedra with rotational freedom
text: # Compound of four octahedra with rotational freedom
  
  Compound of four octahedra with rotational freedom :
  
  | Type | Uniform compound |
  | - | - |
  | Index | $UC_{10}$ |
  | Polyhedra | 4 octahedra |
  | Faces | 8+24 triangles |
  | Edges | 48 |
  | Vertices | 24 |
  | Symmetry group | pyritohedral $(T_h)$ |
  | Subgroup restricting to one constituent | 6-fold improper rotation $(S_6)$ |
  
  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
  
      $\theta =2\tan ^{-1}\left({\sqrt {15}}-2{\sqrt {3}}\right)\approx 44.47751^{\circ },$
  
  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.
  
  ## 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°
  