Compound of twenty octahedra with rotational freedom

Compound of twenty octahedra with rotational freedom
Type	Uniform compound
Index	UC13
Polyhedra	20 octahedra
Faces	40+120 triangles
Edges	240
Vertices	120
Symmetry group	icosahedral (Ih)
Subgroup restricting to one constituent	6-fold improper rotation (S6)

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 UC₁₆, 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 UC₁₆ and UC₁₅ respectively. When

${\displaystyle \theta =2\tan ^{-1}\left({\sqrt {{\frac {1}{3}}\left(13-4{\sqrt {10}}\right)}}\right)\approx 37.76124^{\circ },}$

octahedra (from distinct rotational axes) coincide in sets four, yielding the compound of five octahedra. When

${\displaystyle \theta =2\tan ^{-1}\left({\frac {-4{\sqrt {3}}-2{\sqrt {15}}+{\sqrt {132+60{\sqrt {5}}}}}{4+{\sqrt {2}}+2{\sqrt {5}}+{\sqrt {10}}}}\right)\approx 14.33033^{\circ },}$

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

${\displaystyle {\begin{aligned}&\scriptstyle {\Big (}\pm 2{\sqrt {3}}\sin \theta ,\,\pm (\tau ^{-1}{\sqrt {2}}+2\tau \cos \theta ),\,\pm (\tau {\sqrt {2}}-2\tau ^{-1}\cos \theta ){\Big )}\\&\scriptstyle {\Big (}\pm ({\sqrt {2}}-\tau ^{2}\cos \theta +\tau ^{-1}{\sqrt {3}}\sin \theta ),\,\pm ({\sqrt {2}}+(2\tau -1)\cos \theta +{\sqrt {3}}\sin \theta ),\,\pm ({\sqrt {2}}+\tau ^{-2}\cos \theta -\tau {\sqrt {3}}\sin \theta ){\Big )}\\&\scriptstyle {\Big (}\pm (\tau ^{-1}{\sqrt {2}}-\tau \cos \theta -\tau {\sqrt {3}}\sin \theta ),\,\pm (\tau {\sqrt {2}}+\tau ^{-1}\cos \theta +\tau ^{-1}{\sqrt {3}}\sin \theta ),\,\pm (3\cos \theta -{\sqrt {3}}\sin \theta ){\Big )}\\&\scriptstyle {\Big (}\pm (-\tau ^{-1}{\sqrt {2}}+\tau \cos \theta -\tau {\sqrt {3}}\sin \theta ),\,\pm (\tau {\sqrt {2}}+\tau ^{-1}\cos \theta -\tau ^{-1}{\sqrt {3}}\sin \theta ),\,\pm (3\cos \theta +{\sqrt {3}}\sin \theta ){\Big )}\\&\scriptstyle {\Big (}\pm (-{\sqrt {2}}+\tau ^{2}\cos \theta +\tau ^{-1}{\sqrt {3}}\sin \theta ),\,\pm ({\sqrt {2}}+(2\tau -1)\cos \theta -{\sqrt {3}}\sin \theta ),\,\pm ({\sqrt {2}}+\tau ^{-2}\cos \theta +\tau {\sqrt {3}}\sin \theta ){\Big )}\end{aligned}}}$ 

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°

References

• Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79 (3): 447–457, doi:10.1017/S0305004100052440, MR 0397554.