Compound of six tetrahedra with rotational freedom

Compound of six tetrahedra with rotational freedom
Type	Uniform compound
Index	UC1
Polyhedra	6 tetrahedra
Faces	24 triangles
Edges	36
Vertices	24
Symmetry group	tetrahedral (Td)
Subgroup restricting to one constituent	4-fold improper rotation (S4)

The compound of six tetrahedra with rotational freedom is a uniform polyhedron compound made of a symmetric arrangement of 6 tetrahedra, considered as antiprisms. It can be constructed by superimposing six tetrahedra within a cube, and then rotating them in pairs about the three axes that pass through the centres of two opposite cubic faces. Each tetrahedron is rotated by an equal (and opposite, within a pair) angle θ. Equivalently, a tetrahedron may be inscribed within each cube in the compound of six cubes with rotational freedom, in such a way as to preserve tetrahedral symmetry.

When θ = 0, all six tetrahedra coincide. When θ is 45 degrees, the more symmetric compound of six tetrahedra (without rotational freedom) arises. Otherwise, the compound of six tetrahedra with rotational freedom shares the same vertex arrangement as a nonuniform truncated octahedron in a way related to a beveled tetrahedron.

Gallery

• Compounds of six tetrahedra with rotational freedom
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  Tetrahedron.stl θ = 0°
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  Compound of six tetrahedra with rotational freedom (5°).stl θ = 5°
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  Compound of six tetrahedra with rotational freedom (10°).stl θ = 10°
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  Compound of six tetrahedra with rotational freedom (15°).stl θ = 15°
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  Compound of six tetrahedra with rotational freedom (20°).stl θ = 20°
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  Compound of six tetrahedra with rotational freedom (25°).stl θ = 25°
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  Compound of six tetrahedra with rotational freedom (30°).stl θ = 30°
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  Compound of six tetrahedra with rotational freedom (35°).stl θ = 35°
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  Compound of six tetrahedra with rotational freedom (40°).stl θ = 40°
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  Compound of six tetrahedra.stl θ = 45°

References

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