# 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 (UC$_{20}$ and UC$_{21}$) exist only when ⁠p/q⁠ > 2, and when p and q are coprime. The prismatic compounds of {p/q}-gonal antiprisms (UC$_{22}$, UC$_{23}$, UC$_{24}$ and UC$_{25}$) 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.
| Compound | Bowers acronym | Polyhedral count | Polyhedral type | Faces | Edges | Vertices | Notes | Symmetry group | Subgroup restricting to one constituent |
| UC$_{01}$ | sis | 6 | tetrahedra | 24{3} | 36 | 24 | Rotational freedom | T$_d$ | S$_4$ |
| UC$_{02}$ | dis | 12 | tetrahedra | 48{3} | 72 | 48 | Rotational freedom | O$_h$ | S$_4$ |
| UC$_{03}$ | snu | 6 | tetrahedra | 24{3} | 36 | 24 | | O$_h$ | D$_{2d}$ |
| UC$_{04}$ | so | 2 | tetrahedra | 8{3} | 12 | 8 | Regular | O$_h$ | T$_d$ |
| UC$_{05}$ | ki | 5 | tetrahedra | 20{3} | 30 | 20 | Regular | I | T |
| UC$_{06}$ | e | 10 | tetrahedra | 40{3} | 60 | 20 | Regular 2 polyhedra per vertex | I$_h$ | T |
| UC$_{07}$ | risdoh | 6 | cubes | (12+24){4} | 72 | 48 | Rotational freedom | O$_h$ | C$_{4h}$ |
| UC$_{08}$ | rah | 3 | cubes | (6+12){4} | 36 | 24 | | O$_h$ | D$_{4h}$ |
| UC$_{09}$ | rhom | 5 | cubes | 30{4} | 60 | 20 | Regular 2 polyhedra per vertex | I$_h$ | T$_h$ |
| UC$_{10}$ | dissit | 4 | octahedra | (8+24){3} | 48 | 24 | Rotational freedom | T$_h$ | S$_6$ |
| UC$_{11}$ | daso | 8 | octahedra | (16+48){3} | 96 | 48 | Rotational freedom | O$_h$ | S$_6$ |
| UC$_{12}$ | sno | 4 | octahedra | (8+24){3} | 48 | 24 | | O$_h$ | D$_{3d}$ |
| UC$_{13}$ | addasi | 20 | octahedra | (40+120){3} | 240 | 120 | Rotational freedom | I$_h$ | S$_6$ |
| UC$_{14}$ | dasi | 20 | octahedra | (40+120){3} | 240 | 60 | 2 polyhedra per vertex | I$_h$ | S$_6$ |
| UC$_{15}$ | gissi | 10 | octahedra | (20+60){3} | 120 | 60 | | I$_h$ | D$_{3d}$ |
| UC$_{16}$ | si | 10 | octahedra | (20+60){3} | 120 | 60 | | I$_h$ | D$_{3d}$ |
| UC$_{17}$ | se | 5 | octahedra | 40{3} | 60 | 30 | Regular | I$_h$ | T$_h$ |
| UC$_{18}$ | hirki | 5 | tetrahemihexahedra | 20{3} 15{4} | 60 | 30 | | I | T |
| UC$_{19}$ | sapisseri | 20 | tetrahemihexahedra | (20+60){3} 60{4} | 240 | 60 | 2 polyhedra per vertex | I | C$_3$ |
| UC$_{20}$ | - | 2n (2n ≥ 2) | p/q-gonal prisms | 4n{p/q} 2np{4} | 6np | 4np | Rotational freedom | D$_{nph}$ | C$_{ph}$ |
| UC$_{21}$ | - | n (n ≥ 2) | p/q-gonal prisms | 2n{p/q} np{4} | 3np | 2np | | D$_{nph}$ | D$_{ph}$ |
| UC$_{22}$ | - | 2n (2n ≥ 2) (q odd) | p/q-gonal antiprisms (q odd) | 4n{p/q} (if ⁠p/q⁠ ≠ 2) 4np{3} | 8np | 4np | Rotational freedom | D$_{npd}$ (if n odd) D$_{nph}$ (if n even) | S$_{2p}$ |
| UC$_{23}$ | - | n (n ≥ 2) | p/q-gonal antiprisms (q odd) | 2n{p/q} (if ⁠p/q⁠ ≠ 2) 2np{3} | 4np | 2np | | D$_{npd}$ (if n odd) D$_{nph}$ (if n even) | D$_{pd}$ |
| UC$_{24}$ | - | 2n (2n ≥ 2) | p/q-gonal antiprisms (q even) | 4n{p/q} (if ⁠p/q⁠ ≠ 2) 4np{3} | 8np | 4np | Rotational freedom | D$_{nph}$ | C$_{ph}$ |
| UC$_{25}$ | - | n (n ≥ 2) | p/q-gonal antiprisms (q even) | 2n{p/q} (if ⁠p/q⁠ ≠ 2) 2np{3} | 4np | 2np | | D$_{nph}$ | D$_{ph}$ |
| UC$_{26}$ | gadsid | 12 | pentagonal antiprisms | 120{3} 24{5} | 240 | 120 | Rotational freedom | I$_h$ | S$_{10}$ |
| UC$_{27}$ | gassid | 6 | pentagonal antiprisms | 60{3} 12{5} | 120 | 60 | | I$_h$ | D$_{5d}$ |
| UC$_{28}$ | gidasid | 12 | pentagrammic crossed antiprisms | 120{3} 24{5/2} | 240 | 120 | Rotational freedom | I$_h$ | S$_{10}$ |
| UC$_{29}$ | gissed | 6 | pentagrammic crossed antiprisms | 60{3} 125 | 120 | 60 | | I$_h$ | D$_{5d}$ |
| UC$_{30}$ | ro | 4 | triangular prisms | 8{3} 12{4} | 36 | 24 | | O | D$_3$ |
| UC$_{31}$ | dro | 8 | triangular prisms | 16{3} 24{4} | 72 | 48 | | O$_h$ | D$_3$ |
| UC$_{32}$ | kri | 10 | triangular prisms | 20{3} 30{4} | 90 | 60 | | I | D$_3$ |
| UC$_{33}$ | dri | 20 | triangular prisms | 40{3} 60{4} | 180 | 60 | 2 polyhedra per vertex | I$_h$ | D$_3$ |
| UC$_{34}$ | kred | 6 | pentagonal prisms | 30{4} 12{5} | 90 | 60 | | I | D$_5$ |
| UC$_{35}$ | dird | 12 | pentagonal prisms | 60{4} 24{5} | 180 | 60 | 2 polyhedra per vertex | I$_h$ | D$_5$ |
| UC$_{36}$ | gikrid | 6 | pentagrammic prisms | 30{4} 12{5/2} | 90 | 60 | | I | D$_5$ |
| UC$_{37}$ | giddird | 12 | pentagrammic prisms | 60{4} 24{5/2} | 180 | 60 | 2 polyhedra per vertex | I$_h$ | D$_5$ |
| UC$_{38}$ | griso | 4 | hexagonal prisms | 24{4} 8{6} | 72 | 48 | | O$_h$ | D$_{3d}$ |
| UC$_{39}$ | rosi | 10 | hexagonal prisms | 60{4} 20{6} | 180 | 120 | | I$_h$ | D$_{3d}$ |
| UC$_{40}$ | rassid | 6 | decagonal prisms | 60{4} 12{10} | 180 | 120 | | I$_h$ | D$_{5d}$ |
| UC$_{41}$ | grassid | 6 | decagrammic prisms | 60{4} 12{10/3} | 180 | 120 | | I$_h$ | D$_{5d}$ |
| UC$_{42}$ | gassic | 3 | square antiprisms | 24{3} 6{4} | 48 | 24 | | O | D$_4$ |
| UC$_{43}$ | gidsac | 6 | square antiprisms | 48{3} 12{4} | 96 | 48 | | O$_h$ | D$_4$ |
| UC$_{44}$ | sassid | 6 | pentagrammic antiprisms | 60{3} 12{5/2} | 120 | 60 | | I | D$_5$ |
| UC$_{45}$ | sadsid | 12 | pentagrammic antiprisms | 120{3} 24{5/2} | 240 | 120 | | I$_h$ | D$_5$ |
| UC$_{46}$ | siddo | 2 | icosahedra | (16+24){3} | 60 | 24 | | O$_h$ | T$_h$ |
| UC$_{47}$ | sne | 5 | icosahedra | (40+60){3} | 150 | 60 | | I$_h$ | T$_h$ |
| UC$_{48}$ | presipsido | 2 | great dodecahedra | 24{5} | 60 | 24 | | O$_h$ | T$_h$ |
| UC$_{49}$ | presipsi | 5 | great dodecahedra | 60{5} | 150 | 60 | | I$_h$ | T$_h$ |
| UC$_{50}$ | passipsido | 2 | small stellated dodecahedra | 24{5/2} | 60 | 24 | | O$_h$ | T$_h$ |
| UC$_{51}$ | passipsi | 5 | small stellated dodecahedra | 60{5/2} | 150 | 60 | | I$_h$ | T$_h$ |
| UC$_{52}$ | sirsido | 2 | great icosahedra | (16+24){3} | 60 | 24 | | O$_h$ | T$_h$ |
| UC$_{53}$ | sirsei | 5 | great icosahedra | (40+60){3} | 150 | 60 | | I$_h$ | T$_h$ |
| UC$_{54}$ | tisso | 2 | truncated tetrahedra | 8{3} 8{6} | 36 | 24 | | O$_h$ | T$_d$ |
| UC$_{55}$ | taki | 5 | truncated tetrahedra | 20{3} 20{6} | 90 | 60 | | I | T |
| UC$_{56}$ | te | 10 | truncated tetrahedra | 40{3} 40{6} | 180 | 120 | | I$_h$ | T |
| UC$_{57}$ | tar | 5 | truncated cubes | 40{3} 30{8} | 180 | 120 | | I$_h$ | T$_h$ |
| UC$_{58}$ | quitar | 5 | stellated truncated hexahedra | 40{3} 30{8/3} | 180 | 120 | | I$_h$ | T$_h$ |
| UC$_{59}$ | arie | 5 | cuboctahedra | 40{3} 30{4} | 120 | 60 | | I$_h$ | T$_h$ |
| UC$_{60}$ | gari | 5 | cubohemioctahedra | 30{4} 20{6} | 120 | 60 | | I$_h$ | T$_h$ |
| UC$_{61}$ | iddei | 5 | octahemioctahedra | 40{3} 20{6} | 120 | 60 | | I$_h$ | T$_h$ |
| UC$_{62}$ | rasseri | 5 | rhombicuboctahedra | 40{3} (30+60){4} | 240 | 120 | | I$_h$ | T$_h$ |
| UC$_{63}$ | rasher | 5 | small rhombihexahedra | 60{4} 30{8} | 240 | 120 | | I$_h$ | T$_h$ |
| UC$_{64}$ | rahrie | 5 | small cubicuboctahedra | 40{3} 30{4} 30{8} | 240 | 120 | | I$_h$ | T$_h$ |
| UC$_{65}$ | raquahri | 5 | great cubicuboctahedra | 40{3} 30{4} 30{8/3} | 240 | 120 | | I$_h$ | T$_h$ |
| UC$_{66}$ | rasquahr | 5 | great rhombihexahedra | 60{4} 30{8/3} | 240 | 120 | | I$_h$ | T$_h$ |
| UC$_{67}$ | rosaqri | 5 | nonconvex great rhombicuboctahedra | 40{3} (30+60){4} | 240 | 120 | | I$_h$ | T$_h$ |
| UC$_{68}$ | disco | 2 | snub cubes | (16+48){3} 12{4} | 120 | 48 | | O$_h$ | O |
| UC$_{69}$ | dissid | 2 | snub dodecahedra | (40+120){3} 24{5} | 300 | 120 | | I$_h$ | I |
| UC$_{70}$ | giddasid | 2 | great snub icosidodecahedra | (40+120){3} 24{5/2} | 300 | 120 | | I$_h$ | I |
| UC$_{71}$ | gidsid | 2 | great inverted snub icosidodecahedra | (40+120){3} 24{5/2} | 300 | 120 | | I$_h$ | I |
| UC$_{72}$ | gidrissid | 2 | great retrosnub icosidodecahedra | (40+120){3} 24{5/2} | 300 | 120 | | I$_h$ | I |
| UC$_{73}$ | disdid | 2 | snub dodecadodecahedra | 120{3} 24{5} 24{5/2} | 300 | 120 | | I$_h$ | I |
| UC$_{74}$ | idisdid | 2 | inverted snub dodecadodecahedra | 120{3} 24{5} 24{5/2} | 300 | 120 | | I$_h$ | I |
| UC$_{75}$ | desided | 2 | snub icosidodecadodecahedra | (40+120){3} 24{5} 24{5/2} | 360 | 120 | | I$_h$ | I |