title: Compound of twenty tetrahemihexahedra
text: # Compound of twenty tetrahemihexahedra
  
  Compound of twenty tetrahemihexahedra :
  
  | Type | Uniform compound |
  | - | - |
  | Index | $UC_{19}$ |
  | 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 $(C_3)$ |
  
  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 $C_3,$ 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
  
  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 |
  