Uniform polyhedron compound

In geometry, a uniform polyhedron compound is a polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra, in an arrangement that is also uniform, i.e. the symmetry group of the compound acts transitively on the compound's vertices.

The uniform polyhedron compounds were first enumerated by John Skilling in 1976, with a proof that the enumeration is complete. The following table lists them according to his numbering.

The prismatic compounds of ${p/q}-$gonal prisms (UC20 and UC21) exist only when $p⁄q > 2$, and when $p$ and $q$ are coprime. The prismatic compounds of ${p/q}-$gonal antiprisms (UC22, UC23, UC24 and UC25) exist only when $p⁄q > 3⁄2$, and when $p$ and $q$ are coprime. Furthermore, when $p⁄q = 2$, the antiprisms degenerate into tetrahedra with digonal bases.