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

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

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

$\begin{array}{cc}& (\pm 2\sqrt{3}\sin \theta ,\pm ({\tau }^{-1}\sqrt{2}+2\tau \cos \theta ),\pm (\tau \sqrt{2}-2{\tau }^{-1}\cos \theta ))\\ & (\pm (\sqrt{2}-{\tau }^2\cos \theta +{\tau }^{-1}\sqrt{3}\sin \theta ),\pm (\sqrt{2}+(2\tau -1)\cos \theta +\sqrt{3}\sin \theta ),\pm (\sqrt{2}+{\tau }^{-2}\cos \theta -\tau \sqrt{3}\sin \theta ))\\ & (\pm ({\tau }^{-1}\sqrt{2}-\tau \cos \theta -\tau \sqrt{3}\sin \theta ),\pm (\tau \sqrt{2}+{\tau }^{-1}\cos \theta +{\tau }^{-1}\sqrt{3}\sin \theta ),\pm (3\cos \theta -\sqrt{3}\sin \theta ))\\ & (\pm (-{\tau }^{-1}\sqrt{2}+\tau \cos \theta -\tau \sqrt{3}\sin \theta ),\pm (\tau \sqrt{2}+{\tau }^{-1}\cos \theta -{\tau }^{-1}\sqrt{3}\sin \theta ),\pm (3\cos \theta +\sqrt{3}\sin \theta ))\\ & (\pm (-\sqrt{2}+{\tau }^2\cos \theta +{\tau }^{-1}\sqrt{3}\sin \theta ),\pm (\sqrt{2}+(2\tau -1)\cos \theta -\sqrt{3}\sin \theta ),\pm (\sqrt{2}+{\tau }^{-2}\cos \theta +\tau \sqrt{3}\sin \theta ))\end{array}$

where τ = (1 + Template:Radic)/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

• Template:Citation.

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