Compound of six tetrahedra with rotational freedom

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Compound of six tetrahedra with rotational freedom
UC01-6 tetrahedra.png
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)

This uniform polyhedron compound is 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.


  • Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79 (03): 447–457, doi:10.1017/S0305004100052440, MR 0397554 .