Compound of twenty tetrahemihexahedra

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

Compound of twenty tetrahemihexahedra
TypeUniform compound
IndexUC19
Polyhedra20 tetrahemihexahedra
Faces20+60 triangles, 60 squares
Edges240
Vertices60
Symmetry groupchiral icosahedral (I)
Subgroup restricting to one constituent3-fold rotational (C3)

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.

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
(Nonuniform rhombicosidodecahedron)
Great snub dodecicosidodecahedron
Great dirhombicosidodecahedron

Great disnub dirhombidodecahedron
Compound of twenty octahedra
Compound of twenty tetrahemihexahedra
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See also

References
  • 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.


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