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Uniform polyhedron compound
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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 uniform 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.

CompoundBowersacronymPicturePolyhedralcountPolyhedral typeFacesEdgesVerticesNotesSymmetry groupSubgrouprestrictingto oneconstituent
UC01sis6tetrahedra24{3}3624Rotational freedomTdS4
UC02dis12tetrahedra48{3}7248Rotational freedomOhS4
UC03snu6tetrahedra24{3}3624OhD2d
UC04so2tetrahedra8{3}128RegularOhTd
UC05ki5tetrahedra20{3}3020RegularIT
UC06e10tetrahedra40{3}6020Regular

2 polyhedra per vertex

IhT
UC07risdoh6cubes(12+24){4}7248Rotational freedomOhC4h
UC08rah3cubes(6+12){4}3624OhD4h
UC09rhom5cubes30{4}6020Regular

2 polyhedra per vertex

IhTh
UC10dissit4octahedra(8+24){3}4824Rotational freedomThS6
UC11daso8octahedra(16+48){3}9648Rotational freedomOhS6
UC12sno4octahedra(8+24){3}4824OhD3d
UC13addasi20octahedra(40+120){3}240120Rotational freedomIhS6
UC14dasi20octahedra(40+120){3}240602 polyhedra per vertexIhS6
UC15gissi10octahedra(20+60){3}12060IhD3d
UC16si10octahedra(20+60){3}12060IhD3d
UC17se5octahedra40{3}6030RegularIhTh
UC18hirki5tetrahemihexahedra20{3}

15{4}

6030IT
UC19sapisseri20tetrahemihexahedra(20+60){3}

60{4}

240602 polyhedra per vertexIC3
UC20-2n

(2n ≥ 2)

p/q-gonal prisms4n{p/q}

2np{4}

6np4npRotational freedomDnphCph
UC21-n

(n ≥ 2)

p/q-gonal prisms2n{p/q}

np{4}

3np2npDnphDph
UC22-2n

(2n ≥ 2)

(q odd)

p/q-gonal antiprisms

(q odd)

4n{p/q} (if ⁠p/q⁠ ≠ 2)

4np{3}

8np4npRotational freedomDnpd (if n odd)

Dnph (if n even)

S2p
UC23-n

(n ≥ 2)

p/q-gonal antiprisms

(q odd)

2n{p/q} (if ⁠p/q⁠ ≠ 2)

2np{3}

4np2npDnpd (if n odd)

Dnph (if n even)

Dpd
UC24-2n

(2n ≥ 2)

p/q-gonal antiprisms

(q even)

4n{p/q} (if ⁠p/q⁠ ≠ 2)

4np{3}

8np4npRotational freedomDnphCph
UC25-n

(n ≥ 2)

p/q-gonal antiprisms

(q even)

2n{p/q} (if ⁠p/q⁠ ≠ 2)

2np{3}

4np2npDnphDph
UC26gadsid12pentagonal antiprisms120{3}

24{5}

240120Rotational freedomIhS10
UC27gassid6pentagonal antiprisms60{3}

12{5}

12060IhD5d
UC28gidasid12pentagrammic crossed antiprisms120{3}

24{5/2}

240120Rotational freedomIhS10
UC29gissed6pentagrammic crossed antiprisms60{3}

125

12060IhD5d
UC30ro4triangular prisms8{3}

12{4}

3624OD3
UC31dro8triangular prisms16{3}

24{4}

7248OhD3
UC32kri10triangular prisms20{3}

30{4}

9060ID3
UC33dri20triangular prisms40{3}

60{4}

180602 polyhedra per vertexIhD3
UC34kred6pentagonal prisms30{4}

12{5}

9060ID5
UC35dird12pentagonal prisms60{4}

24{5}

180602 polyhedra per vertexIhD5
UC36gikrid6pentagrammic prisms30{4}

12{5/2}

9060ID5
UC37giddird12pentagrammic prisms60{4}

24{5/2}

180602 polyhedra per vertexIhD5
UC38griso4hexagonal prisms24{4}

8{6}

7248OhD3d
UC39rosi10hexagonal prisms60{4}

20{6}

180120IhD3d
UC40rassid6decagonal prisms60{4}

12{10}

180120IhD5d
UC41grassid6decagrammic prisms60{4}

12{10/3}

180120IhD5d
UC42gassic3square antiprisms24{3}

6{4}

4824OD4
UC43gidsac6square antiprisms48{3}

12{4}

9648OhD4
UC44sassid6pentagrammic antiprisms60{3}

12{5/2}

12060ID5
UC45sadsid12pentagrammic antiprisms120{3}

24{5/2}

240120IhD5
UC46siddo2icosahedra(16+24){3}6024OhTh
UC47sne5icosahedra(40+60){3}15060IhTh
UC48presipsido2great dodecahedra24{5}6024OhTh
UC49presipsi5great dodecahedra60{5}15060IhTh
UC50passipsido2small stellated dodecahedra24{5/2}6024OhTh
UC51passipsi5small stellated dodecahedra60{5/2}15060IhTh
UC52sirsido2great icosahedra(16+24){3}6024OhTh
UC53sirsei5great icosahedra(40+60){3}15060IhTh
UC54tisso2truncated tetrahedra8{3}

8{6}

3624OhTd
UC55taki5truncated tetrahedra20{3}

20{6}

9060IT
UC56te10truncated tetrahedra40{3}

40{6}

180120IhT
UC57tar5truncated cubes40{3}

30{8}

180120IhTh
UC58quitar5stellated truncated hexahedra40{3}

30{8/3}

180120IhTh
UC59arie5cuboctahedra40{3}

30{4}

12060IhTh
UC60gari5cubohemioctahedra30{4}

20{6}

12060IhTh
UC61iddei5octahemioctahedra40{3}

20{6}

12060IhTh
UC62rasseri5rhombicuboctahedra40{3}

(30+60){4}

240120IhTh
UC63rasher5small rhombihexahedra60{4}

30{8}

240120IhTh
UC64rahrie5small cubicuboctahedra40{3}

30{4}

30{8}

240120IhTh
UC65raquahri5great cubicuboctahedra40{3}

30{4}

30{8/3}

240120IhTh
UC66rasquahr5great rhombihexahedra60{4}

30{8/3}

240120IhTh
UC67rosaqri5nonconvex great rhombicuboctahedra40{3}

(30+60){4}

240120IhTh
UC68disco2snub cubes(16+48){3}

12{4}

12048OhO
UC69dissid2snub dodecahedra(40+120){3}

24{5}

300120IhI
UC70giddasid2great snub icosidodecahedra(40+120){3}

24{5/2}

300120IhI
UC71gidsid2great inverted snub icosidodecahedra(40+120){3}

24{5/2}

300120IhI
UC72gidrissid2great retrosnub icosidodecahedra(40+120){3}

24{5/2}

300120IhI
UC73disdid2snub dodecadodecahedra120{3}

24{5}

24{5/2}

300120IhI
UC74idisdid2inverted snub dodecadodecahedra120{3}

24{5}

24{5/2}

300120IhI
UC75desided2snub icosidodecadodecahedra(40+120){3}

24{5}

24{5/2}

360120IhI