title: Compound of twenty octahedra with rotational freedom
text: # Compound of twenty octahedra with rotational freedom
  
  Compound of twenty octahedra with rotational freedom :
  
  | Type | Uniform compound |
  | - | - |
  | Index | $UC_{13}$ |
  | Polyhedra | 20 octahedra |
  | Faces | 40+120 triangles |
  | Edges | 240 |
  | Vertices | 120 |
  | Symmetry group | icosahedral $(I_h)$ |
  | Subgroup restricting to one constituent | 6-fold improper rotation $(S_6)$ |
  
  The compound of twenty octahedra with rotational freedom is a uniform polyhedron compound. It's composed of a symmetric arrangement of 20 octahedra, considered as triangular antiprisms. It can be constructed by superimposing two copies of the compound of 10 octahedra $UC_{16},$ 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 $UC_{16}$ and $UC_{15}$ respectively. When
  
      $\theta =2\tan ^{-1}\left({\sqrt {{\frac {1}{3}}\left(13-4{\sqrt {10}}\right)}}\right)\approx 37.76124^{\circ },$
  
  octahedra (from distinct rotational axes) coincide in sets four, yielding the compound of five octahedra. When
  
      $\theta =2\tan ^{-1}\left({\frac {-4{\sqrt {3}}-2{\sqrt {15}}+{\sqrt {132+60{\sqrt {5}}}}}{4+{\sqrt {2}}+2{\sqrt {5}}+{\sqrt {10}}}}\right)\approx 14.33033^{\circ },$
  
  the vertices coincide in pairs, yielding the compound of twenty octahedra (without rotational freedom).
  
  ## Cartesian coordinates
  
  Cartesian coordinates for the vertices of this compound are all the cyclic permutations of
  
      ${\begin{aligned}&\scriptstyle {\Big\pm 2{\sqrt {3}}\sin \theta ,\,\pm (\tau ^{-1}{\sqrt {2}}+2\tau \cos \theta ),\,\pm (\tau {\sqrt {2}}-2\tau ^{-1}\cos \theta ){\Big )}\\&\scriptstyle {\Big\pm ({\sqrt {2}}-\tau ^{2}\cos \theta +\tau ^{-1}{\sqrt {3}}\sin \theta ),\,\pm ({\sqrt {2}}+(2\tau -1)\cos \theta +{\sqrt {3}}\sin \theta ),\,\pm ({\sqrt {2}}+\tau ^{-2}\cos \theta -\tau {\sqrt {3}}\sin \theta ){\Big )}\\&\scriptstyle {\Big\pm (\tau ^{-1}{\sqrt {2}}-\tau \cos \theta -\tau {\sqrt {3}}\sin \theta ),\,\pm (\tau {\sqrt {2}}+\tau ^{-1}\cos \theta +\tau ^{-1}{\sqrt {3}}\sin \theta ),\,\pm (3\cos \theta -{\sqrt {3}}\sin \theta ){\Big )}\\&\scriptstyle {\Big\pm (-\tau ^{-1}{\sqrt {2}}+\tau \cos \theta -\tau {\sqrt {3}}\sin \theta ),\,\pm (\tau {\sqrt {2}}+\tau ^{-1}\cos \theta -\tau ^{-1}{\sqrt {3}}\sin \theta ),\,\pm (3\cos \theta +{\sqrt {3}}\sin \theta ){\Big )}\\&\scriptstyle {\Big\pm (-{\sqrt {2}}+\tau ^{2}\cos \theta +\tau ^{-1}{\sqrt {3}}\sin \theta ),\,\pm ({\sqrt {2}}+(2\tau -1)\cos \theta -{\sqrt {3}}\sin \theta ),\,\pm ({\sqrt {2}}+\tau ^{-2}\cos \theta +\tau {\sqrt {3}}\sin \theta ){\Big )}\end{aligned}}$
  
  where τ = (1 + √5)/2 is the golden ratio (sometimes written φ).
  
  ## Gallery
  
  * Compounds of twenty 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°
  