Uniform Compounds of Uniform Polyhedra

This list follows the order in the paper "Uniform Compounds of Uniform Polyhedra" by J. Skilling, cited in the references. Many of these 75 compounds (plus duals) have never been made in paper, and their pictures appear nowhere else but here. The notations T, O, I, or iP indicate that the compound has tetrahedral, octahedral, icosahedral, or i-sided prism symmetry, respectively. I haven't finished the prisms yet.

I have also included the duals to many of the uniform compounds. The duals listed are not vertex-uniform, but they are facially uniform. Spinning versions of the compounds with rotational freedom are also avaialble.

Tetrahedra:

• 6 tetrahedra (T, with rotational freedom)
• 12 tetrahedra (O, with rotational freedom)
• 6 tetrahedra (O, rigid)
• 2 tetrahedra (O, stella octangula)
• 5 tetrahedra (I, chiral)
• 10 tetrahedra (I, 5 stella octangulae)

Cubes:

• 6 cubes (O, rotational freedom) (dual: 6 octahedra)
• 3 cubes (O) (dual: 3 octahedra)
• 5 cubes (I) (dual listed below)

Octahedra:

• 4 octahedra (T, rotational freedom) (dual: 4 cubes)
• 8 octahedra (O, rotational freedom) (dual: 8 cubes)
• 4 octahedra (O) (dual: 4 cubes)
• 20 octahedra (I, rotational freedom) (different angle) (dual: 20 cubes, different angle)
• 20 octahedra (I, rigid, 2 per vertex) (dual: 20 cubes)
• 10 octahedra (I) (dual: 10 cubes)
• 10 octahedra (I) (dual: 10 cubes)
• 5 octahedra (I) (dual listed above)

Tetrahemihexahedra:

• 5 tetrahemihexahedra (I, chiral) (one tetrahemihexahedron)
• 20 tetrahemihexahedra (I, chiral)

Prisms/Antiprisms in Prism Symmetry:

• 2k n/m-gonal prisms (knP, rotational freedom)
• k n/m-gonal prisms (knP)
• 2k n/m-gonal antiprisms, m odd, (knP, rotational freedom)
• k n/m-gonal antiprisms, m odd, (knP)
• 2k n/m-gonal antiprisms, m even, (knP, rotational freedom)
• k n/m-gonal antiprisms, m even, (knP)

Prisms/Antiprisms in Non-Prism Symmetry:

• 12 pentagonal antiprisms (I, rotational freedom)
• 6 pentagonal antiprisms (I)
• 12 pentagagrammic inverted antiprisms (I, rotational freedom)
• 6 pentagagrammic inverted antiprisms (I)
• 4 triangular prisms (O, chiral)
• 8 triangular prisms (O)
• 10 triangular prisms (I, chiral)
• 20 triangular prisms (I)
• 6 pentagonal prisms (I, chiral)
• 12 pentagonal prisms (I)
• 6 pentagrammic prisms (I, chiral)
• 12 pentagrammic prisms (I)
• 4 hexagonal prisms (O)
• 10 hexagonal prisms (I)
• 6 decagonal prisms (I)
• 6 decagrammic prisms (I)
• 3 square antiprisms (O, chiral)
• 6 square antiprisms (O)
• 6 pentagrammic antiprisms (I, chiral)
• 12 pentagrammic antiprisms (I)

Icosahedra:

• 2 icosahedra (O) (plus common octahedron) (dual: 2 dodecahedra)
• 5 icosahedra (I) (dual: 5 dodecahedra)

Great Dodecahedra:

• 2 great dodecahedra (O)
• 5 great dodecahedra (I)

Small Stellated Dodecahedra:

• 2 small stellated dodecahedra (O)
• 5 small stellated dodecahedra (I)

Great Icosahedra:

• 2 great icosahedra (O) (dual: 2 great stellated dodecahedra)
• 5 great icosahedra (I) (dual: 5 great stellated dodecahedra)

Truncated Tetrahedra:

• 2 truncated tetrhedra (O) (dual: 2 triakis tetrahedra, one)
• 5 truncated tetrahedra (I, chiral) (dual: 5 triakis tetrahedra)
• 10 truncated tetrahedra (I) (dual: 10 triakis tetrahedra)

Relatives of the 5 Cubes:

• 5 truncated cubes (one) (dual: 5 triakis octahedra, one)
• 5 stellated truncated cubes (one) (dual: 5 great triakis octahedra, one)
• 5 cuboctahedra (one) (dual: 5 rhombic dodecahedra, one)
• 5 cubohemioctahedra (one)
• 5 octahemioctahedra (one)
• 5 rhombicuboctahedra (one) (dual: 5 trapezoidal icositetrahedra, one)
• 5 small rhombicubes (one)
• 5 small cubicuboctahedra (one) (dual: 5 small hexacronic icositetrahedra, one)
• 5 great cubicuboctahedra (one) (dual: 5 great hexacronic icositetrahedra, one)
• 5 great rhombicubes (one)
• 5 great rhombicuboctahedra (one) (dual: 5 great trapezoidal icositetrahedra, one)

Chiral Polyhedron plus its Reflection:

• 2 snub cubes (O) (one) (dual: 2 pentagonal icositetraedra)
• 2 snub dodecahedra (I) (one) (dual: 2 pentagonal hexecontahedra, 2 color version)
• 2 great snub icosidodecahedra (I) (one) (dual: 2 great pentagonal hexecontahedra)
• 2 great inverted snub icosidodecahedra (I) (one) (dual: 2 great inverted pentagonal hexecontahedra)
• 2 great retrosnub icosidodecahedra (I) (one) (dual: 2 great pentagrammic hexecontahedra)
• 2 snub dodecadodecahedra (I) (one) (dual: 2 medial pentagonal hexecontahedra)
• 2 inverted snub dodecadodecahedra (I) (one) (dual: 2 medial inverted pentagonal hexecontahedra)
• 2 snub icosidodecadodecahedra (I) (one) (dual: 2 medial hexagonal hexecontahedra)

Virtual Polyhedra, Copyright 1996, George W. Hart