Editing Polytope compound (section)

== Uniform compounds ==
{{main|Uniform polyhedron compound}}

In 1976 John Skilling published ''Uniform Compounds of Uniform Polyhedra'' which enumerated 75 compounds (including 6 as infinite [[Prism (geometry)|prismatic]] sets of compounds, #20-#25) made from uniform polyhedra with rotational symmetry. (Every vertex is [[vertex-transitive]] and every vertex is transitive with every other vertex.) This list includes the five regular compounds above. [https://archive.today/20070928154042/http://www.interocitors.com/polyhedra/UCs/UniformCompounds.html]

The 75 uniform compounds are listed in the Table below. Most are shown singularly colored by each polyhedron element. Some chiral pairs of face groups are colored by symmetry of the faces within each polyhedron.

* 1-19: Miscellaneous (4,5,6,9,17 are the 5 ''regular compounds'')
{|
|-
|[[Image:UC01-6 tetrahedra.png|100px]]
|[[Image:UC02-12 tetrahedra.png|100px]]
|[[Image:UC03-6 tetrahedra.png|100px]]
|[[Image:UC04-2 tetrahedra.png|100px]]
|[[Image:UC05-5 tetrahedra.png|100px]]
|[[Image:UC06-10 tetrahedra.png|100px]]
|-
|[[Image:UC07-6 cubes.png|100px]]
|[[Image:UC08-3 cubes.png|100px]]
|[[Image:UC09-5 cubes.png|100px]]
|[[Image:UC10-4 octahedra.png|100px]]
|[[Image:UC11-8 octahedra.png|100px]]
|[[Image:UC12-4 octahedra.png|100px]]
|-
|[[Image:UC13-20 octahedra.png|100px]]
|[[Image:UC14-20 octahedra.png|100px]]
|[[Image:UC15-10 octahedra.png|100px]]
|[[Image:UC16-10 octahedra.png|100px]]
|[[Image:UC17-5 octahedra.png|100px]]
|[[Image:UC18-5 tetrahemihexahedron.png|100px]]
|-
|[[Image:UC19-20 tetrahemihexahedron.png|100px]]
|}
* 20-25: Prism symmetry embedded in [[Dihedral symmetry in three dimensions|prism symmetry]],
{|
|[[Image:UC20-2k n-m-gonal prisms.png|100px]]
|[[Image:UC21-k n-m-gonal prisms.png|100px]]
|[[Image:UC22-2k n-m-gonal antiprisms.png|100px]]
|[[Image:UC23-k n-m-gonal antiprisms.png|100px]]
|[[Image:UC24-2k n-m-gonal antiprisms.png|100px]]
|[[Image:UC25-k n-m-gonal antiprisms.png|100px]]
|}
* 26-45: Prism symmetry embedded in [[Octahedral symmetry|octahedral]] or [[icosahedral symmetry]],
{|
|[[Image:UC26-12 pentagonal antiprisms.png|100px]]
|[[Image:UC27-6 pentagonal antiprisms.png|100px]]
|[[Image:UC28-12 pentagrammic crossed antiprisms.png|100px]]
|[[Image:UC29-6 pentagrammic crossed antiprisms.png|100px]]
|[[Image:UC30-4 triangular prisms.png|100px]]
|[[Image:UC31-8 triangular prisms.png|100px]]
|-
|[[Image:UC32-10 triangular prisms.png|100px]]
|[[Image:UC33-20 triangular prisms.png|100px]]
|[[Image:UC34-6 pentagonal prisms.png|100px]]
|[[Image:UC35-12 pentagonal prisms.png|100px]]
|[[Image:UC36-6 pentagrammic prisms.png|100px]]
|[[Image:UC37-12 pentagrammic prisms.png|100px]]
|-
|[[Image:UC38-4 hexagonal prisms.png|100px]]
|[[Image:UC39-10 hexagonal prisms.png|100px]]
|[[Image:UC40-6 decagonal prisms.png|100px]]
|[[Image:UC41-6 decagrammic prisms.png|100px]]
|[[Image:UC42-3 square antiprisms.png|100px]]
|[[Image:UC43-6 square antiprisms.png|100px]]
|-
|[[Image:UC44-6 pentagrammic antiprisms.png|100px]]
|[[Image:UC45-12 pentagrammic antiprisms.png|100px]]
|}
* 46-67: Tetrahedral symmetry embedded in octahedral or icosahedral symmetry,
{|
|[[Image:UC46-2 icosahedra.png|100px]]
|[[Image:UC47-5 icosahedra.png|100px]]
|[[Image:UC48-2 great dodecahedra.png|100px]]
|[[Image:UC49-5 great dodecahedra.png|100px]]
|[[Image:UC50-2 small stellated dodecahedra.png|100px]]
|[[Image:UC51-5 small stellated dodecahedra.png|100px]]
|-
|[[Image:UC52-2 great icosahedra.png|100px]]
|[[Image:UC53-5 great icosahedra.png|100px]]
|[[Image:UC54-2 truncated tetrahedra.png|100px]]
|[[Image:UC55-5 truncated tetrahedra.png|100px]]
|[[Image:UC56-10 truncated tetrahedra.png|100px]]
|[[Image:UC57-5 truncated cubes.png|100px]]
|-
|[[Image:UC58-5 quasitruncated hexahedra.png|100px]]
|[[Image:UC59-5 cuboctahedra.png|100px]]
|[[Image:UC60-5 cubohemioctahedra.png|100px]]
|[[Image:UC61-5 octahemioctahedra.png|100px]]
|[[Image:UC62-5 rhombicuboctahedra.png|100px]]
|[[Image:UC63-5 small rhombihexahedra.png|100px]]
|-
|[[Image:UC64-5 small cubicuboctahedra.png|100px]]
|[[Image:UC65-5 great cubicuboctahedra.png|100px]]
|[[Image:UC66-5 great rhombihexahedra.png|100px]]
|[[Image:UC67-5 great rhombicuboctahedra.png|100px]]
|}
* 68-75: [[enantiomorph]] pairs
{|
|[[Image:UC68-2 snub cubes.png|100px]]
|[[Image:UC69-2 snub dodecahedra.png|100px]]
|[[Image:UC70-2 great snub icosidodecahedra.png|100px]]
|[[Image:UC71-2 great inverted snub icosidodecahedra.png|100px]]
|[[Image:UC72-2 great retrosnub icosidodecahedra.png|100px]]
|[[Image:UC73-2 snub dodecadodecahedra.png|100px]]
|-
|[[Image:UC74-2 inverted snub dodecadodecahedra.png|100px]]
|[[Image:UC75-2 snub icosidodecadodecahedra.png|100px]]
|}