Compound of twenty tetrahemihexahedra

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Compound of twenty tetrahemihexahedra
UC19-20 tetrahemihexahedron.png
Type Uniform compound
Index UC19
Polyhedra 20 tetrahemihexahedra
Faces 20+60 triangles, 60 squares
Edges 240
Vertices 60
Symmetry group chiral icosahedral (I)
Subgroup restricting to one constituent 3-fold rotational (C3)

This uniform polyhedron compound is a symmetric arrangement of 20 tetrahemihexahedra. It is chiral with icosahedral symmetry (I).

John Skilling notes, in his enumeration of uniform compounds of uniform polyhedra, that this compound of 20 tetrahemihexahedra is unique in that it cannot be obtained by "adding symmetry to a group in which the basic polyhedron is uniform". Each tetrahemihexahedron in this compound is embedded with symmetry group C3, which does not act transitively over the tetrahemihexahedron's six vertices. However, the compound as a whole can achieve uniformity because two tetrahemihexahedra coincide at each vertex.

Related polyhedra[edit]

This compound shares its edge arrangement with the great dirhombicosidodecahedron, the great disnub dirhombidodecahedron, and the compound of 20 octahedra.

The edges and 20 of the triangular faces occur in one enantiomer of the great snub dodecicosidodecahedron, with the other 60 triangular faces occurring in the other enantiomer.

Convex hull
Great snub dodecicosidodecahedron.png
Great snub dodecicosidodecahedron
Great dirhombicosidodecahedron.png
Great dirhombicosidodecahedron
Great disnub dirhombidodecahedron.png
Great disnub dirhombidodecahedron
UC14-20 octahedra.png
Compound of twenty octahedra
UC19-20 tetrahemihexahedron.png
Compound of twenty tetrahemihexahedra


  • 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 .