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Polyhedral compound
The compound of twenty octahedra with rotational freedom is a
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
octahedra (from distinct rotational axes) coincide in sets four, yielding the compound of five octahedra. When
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
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, .