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 prisms (UC20 and UC21) exist only when , and when and are coprime. The uniform prismatic compounds of antiprisms (UC22, UC23, UC24 and UC25) exist only when , and when and are coprime. Furthermore, when , the antiprisms degenerate into tetrahedra with digonal bases.

CompoundBowersacronymPicturePolyhedralcountPolyhedral typeFacesEdgesVerticesNotesSymmetry groupSubgrouprestrictingto oneconstituentUC01sis6tetrahedra24{3}3624Rotational freedomTdS4UC02dis12tetrahedra48{3}7248Rotational freedomOhS4UC03snu6tetrahedra24{3}3624OhD2dUC04so2tetrahedra8{3}128RegularOhTdUC05ki5tetrahedra20{3}3020RegularITUC06e10tetrahedra40{3}6020Regular

2 polyhedra per vertexIhTUC07risdoh6cubes(12+24){4}7248Rotational freedomOhC4hUC08rah3cubes(6+12){4}3624OhD4hUC09rhom5cubes30{4}6020Regular

2 polyhedra per vertexIhThUC10dissit4octahedra(8+24){3}4824Rotational freedomThS6UC11daso8octahedra(16+48){3}9648Rotational freedomOhS6UC12sno4octahedra(8+24){3}4824OhD3dUC13addasi20octahedra(40+120){3}240120Rotational freedomIhS6UC14dasi20octahedra(40+120){3}240602 polyhedra per vertexIhS6UC15gissi10octahedra(20+60){3}12060IhD3dUC16si10octahedra(20+60){3}12060IhD3dUC17se5octahedra40{3}6030RegularIhThUC18hirki5tetrahemihexahedra20{3}

15{4}6030ITUC19sapisseri20tetrahemihexahedra(20+60){3}

60{4}240602 polyhedra per vertexIC3UC20 -2n

(2n ≥ 2)p/q-gonal prisms4n{p/q}

2np{4}6np4npRotational freedomDnphCphUC21 -n

(n ≥ 2)p/q-gonal prisms2n{p/q}

np{4}3np2npDnphDphUC22 -2n

(2n ≥ 2)

(q odd)p/q-gonal antiprisms

(q odd)4n{p/q} (if ≠ 2)

4np{3}8np4npRotational freedomDnpd (if n odd)

Dnph (if n even)S2pUC23 -n

(n ≥ 2)p/q-gonal antiprisms

(q odd)2n{p/q} (if ≠ 2)

2np{3}4np2npDnpd (if n odd)

Dnph (if n even)DpdUC24 -2n

(2n ≥ 2)p/q-gonal antiprisms

(q even)4n{p/q} (if ≠ 2)

4np{3}8np4npRotational freedomDnphCphUC25 -n

(n ≥ 2)p/q-gonal antiprisms

(q even)2n{p/q} (if ≠ 2)

2np{3}4np2npDnphDphUC26gadsid12pentagonal antiprisms120{3}

24{5}240120Rotational freedomIhS10UC27gassid6pentagonal antiprisms60{3}

12{5}12060IhD5dUC28gidasid12pentagrammic crossed antiprisms120{3}

24{5/2}240120Rotational freedomIhS10UC29gissed6pentagrammic crossed antiprisms60{3}

1212060IhD5dUC30ro4triangular prisms8{3}

12{4}3624OD3UC31dro8triangular prisms16{3}

24{4}7248OhD3UC32kri10triangular prisms20{3}

30{4}9060ID3UC33dri20triangular prisms40{3}

60{4}180602 polyhedra per vertexIhD3UC34kred6pentagonal prisms30{4}

12{5}9060ID5UC35dird12pentagonal prisms60{4}

24{5}180602 polyhedra per vertexIhD5UC36gikrid6pentagrammic prisms30{4}

12{5/2}9060ID5UC37giddird12pentagrammic prisms60{4}

24{5/2}180602 polyhedra per vertexIhD5UC38griso4hexagonal prisms24{4}

8{6}7248OhD3dUC39rosi10hexagonal prisms60{4}

20{6}180120IhD3dUC40rassid6decagonal prisms60{4}

12{10}180120IhD5dUC41grassid6decagrammic prisms60{4}

12{10/3}180120IhD5dUC42gassic3square antiprisms24{3}

6{4}4824OD4UC43gidsac6square antiprisms48{3}

12{4}9648OhD4UC44sassid6pentagrammic antiprisms60{3}

12{5/2}12060ID5UC45sadsid12pentagrammic antiprisms120{3}

24{5/2}240120IhD5UC46siddo2icosahedra(16+24){3}6024OhThUC47sne5icosahedra(40+60){3}15060IhThUC48presipsido2great dodecahedra24{5}6024OhThUC49presipsi5great dodecahedra60{5}15060IhThUC50passipsido2small stellated dodecahedra24{5/2}6024OhThUC51passipsi5small stellated dodecahedra60{5/2}15060IhThUC52sirsido2great icosahedra(16+24){3}6024OhThUC53sirsei5great icosahedra(40+60){3}15060IhThUC54tisso2truncated tetrahedra8{3}

8{6}3624OhTdUC55taki5truncated tetrahedra20{3}

20{6}9060ITUC56te10truncated tetrahedra40{3}

40{6}180120IhTUC57tar5truncated cubes40{3}

30{8}180120IhThUC58quitar5stellated truncated hexahedra40{3}

30{8/3}180120IhThUC59arie5cuboctahedra40{3}

30{4}12060IhThUC60gari5cubohemioctahedra30{4}

20{6}12060IhThUC61iddei5octahemioctahedra40{3}

20{6}12060IhThUC62rasseri5rhombicuboctahedra40{3}

(30+60){4}240120IhThUC63rasher5small rhombihexahedra60{4}

30{8}240120IhThUC64rahrie5small cubicuboctahedra40{3}

30{4}

30{8}240120IhThUC65raquahri5great cubicuboctahedra40{3}

30{4}

30{8/3}240120IhThUC66rasquahr5great rhombihexahedra60{4}

30{8/3}240120IhThUC67rosaqri5nonconvex great rhombicuboctahedra40{3}

(30+60){4}240120IhThUC68disco2snub cubes(16+48){3}

12{4}12048OhOUC69dissid2snub dodecahedra(40+120){3}

24{5}300120IhIUC70giddasid2great snub icosidodecahedra(40+120){3}

24{5/2}300120IhIUC71gidsid2great inverted snub icosidodecahedra(40+120){3}

24{5/2}300120IhIUC72gidrissid2great retrosnub icosidodecahedra(40+120){3}

24{5/2}300120IhIUC73disdid2snub dodecadodecahedra120{3}

24{5}

24{5/2}300120IhIUC74idisdid2inverted snub dodecadodecahedra120{3}

24{5}

24{5/2}300120IhIUC75desided2snub icosidodecadodecahedra(40+120){3}

24{5}

24{5/2}360120IhI

References .

External links http://www.interocitors.com/polyhedra/UCs/ShortNames.html - Bowers style acronyms for uniform polyhedron compounds