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 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.

CompoundBowers
acronym		PicturePolyhedral
count		 Polyhedral type FacesEdgesVerticesNotes Symmetry group Subgroup
restricting
to one
constituent
UC01 sis UC01-6 tetrahedra.png 6 tetrahedra 24{3}3624Rotational freedom Td S4
UC02 dis UC02-12 tetrahedra.png 12 tetrahedra 48{3}7248Rotational freedom Oh S4
UC03 snu UC03-6 tetrahedra.png 6 tetrahedra 24{3}3624 Oh D2d
UC04 so UC04-2 tetrahedra.png 2 tetrahedra 8{3}128Regular Oh Td
UC05 ki UC05-5 tetrahedra.png 5 tetrahedra 20{3}3020Regular I T
UC06 e UC06-10 tetrahedra.png 10 tetrahedra 40{3}6020Regular

2 polyhedra per vertex

 Ih T
UC07 risdoh UC07-6 cubes.png 6 cubes (12+24){4}7248Rotational freedom Oh C4h
UC08 rah UC08-3 cubes.png 3 cubes (6+12){4}3624 Oh D4h
UC09 rhom UC09-5 cubes.png 5 cubes 30{4}6020Regular

2 polyhedra per vertex

 Ih Th
UC10 dissit UC10-4 octahedra.png 4 octahedra (8+24){3}4824Rotational freedom Th S6
UC11 daso UC11-8 octahedra.png 8 octahedra (16+48){3}9648Rotational freedom Oh S6
UC12 sno UC12-4 octahedra.png 4 octahedra (8+24){3}4824 Oh D3d
UC13 addasi UC13-20 octahedra.png 20 octahedra (40+120){3}240120Rotational freedom Ih S6
UC14 dasi UC14-20 octahedra.png 20 octahedra (40+120){3}240602 polyhedra per vertex Ih S6
UC15 gissi UC15-10 octahedra.png 10 octahedra (20+60){3}12060 Ih D3d
UC16 si UC16-10 octahedra.png 10 octahedra (20+60){3}12060 Ih D3d
UC17 se UC17-5 octahedra.png 5 octahedra 40{3}6030Regular Ih Th
UC18 hirki UC18-5 tetrahemihexahedron.png 5 tetrahemihexahedra 20{3}

15{4}

6030 I T
UC19 sapisseri UC19-20 tetrahemihexahedron.png 20 tetrahemihexahedra (20+60){3}

60{4}

240602 polyhedra per vertex I C3
UC20 - UC20-2k n-m-gonal prisms.png 2n

(2n ≥ 2)

p/q-gonal prisms 4n{p/q}

2np{4}

6np4npRotational freedom Dnph Cph
UC21 - UC21-k n-m-gonal prisms.png n

(n ≥ 2)

p/q-gonal prisms 2n{p/q}

np{4}

3np2np Dnph Dph
UC22 - UC22-2k n-m-gonal antiprisms.png 2n

(2n ≥ 2)

(q odd)

p/q-gonal antiprisms

(q odd)

4n{p/q} (if p/q ≠ 2)

4np{3}

8np4npRotational freedom Dnpd (if n odd)

Dnph (if n even)

 S2p
UC23 - UC23-k n-m-gonal antiprisms.png n

(n ≥ 2)

p/q-gonal antiprisms

(q odd)

2n{p/q} (if p/q ≠ 2)

2np{3}

4np2np Dnpd (if n odd)

Dnph (if n even)

 Dpd
UC24 - UC24-2k n-m-gonal antiprisms.png 2n

(2n ≥ 2)

p/q-gonal antiprisms

(q even)

4n{p/q} (if p/q ≠ 2)

4np{3}

8np4npRotational freedom Dnph Cph
UC25 - UC25-k n-m-gonal antiprisms.png n

(n ≥ 2)

p/q-gonal antiprisms

(q even)

2n{p/q} (if p/q ≠ 2)

2np{3}

4np2np Dnph Dph
UC26 gadsid UC26-12 pentagonal antiprisms.png 12 pentagonal antiprisms 120{3}

24{5}

240120Rotational freedom Ih S10
UC27gassid UC27-6 pentagonal antiprisms.png 6 pentagonal antiprisms 60{3}

12{5}

12060 Ih D5d
UC28gidasid UC28-12 pentagrammic crossed antiprisms.png 12 pentagrammic crossed antiprisms 120{3}

24{5/2}

240120Rotational freedom Ih S10
UC29 gissed UC29-6 pentagrammic crossed antiprisms.png 6 pentagrammic crossed antiprisms 60{3}

125

12060 Ih D5d
UC30 ro UC30-4 triangular prisms.png 4 triangular prisms 8{3}

12{4}

3624 O D3
UC31 dro UC31-8 triangular prisms.png 8 triangular prisms 16{3}

24{4}

7248 Oh D3
UC32 kri UC32-10 triangular prisms.png 10 triangular prisms 20{3}

30{4}

9060 I D3
UC33 dri UC33-20 triangular prisms.png 20 triangular prisms 40{3}

60{4}

180602 polyhedra per vertex Ih D3
UC34 kred UC34-6 pentagonal prisms.png 6 pentagonal prisms 30{4}

12{5}

9060 I D5
UC35 dird UC35-12 pentagonal prisms.png 12 pentagonal prisms 60{4}

24{5}

180602 polyhedra per vertex Ih D5
UC36 gikrid UC36-6 pentagrammic prisms.png 6 pentagrammic prisms 30{4}

12{5/2}

9060 I D5
UC37 giddird UC37-12 pentagrammic prisms.png 12 pentagrammic prisms 60{4}

24{5/2}

180602 polyhedra per vertex Ih D5
UC38 griso UC38-4 hexagonal prisms.png 4 hexagonal prisms 24{4}

8{6}

7248 Oh D3d
UC39 rosi UC39-10 hexagonal prisms.png 10 hexagonal prisms 60{4}

20{6}

180120 Ih D3d
UC40 rassid UC40-6 decagonal prisms.png 6 decagonal prisms 60{4}

12{10}

180120 Ih D5d
UC41 grassid UC41-6 decagrammic prisms.png 6 decagrammic prisms 60{4}

12{10/3}

180120 Ih D5d
UC42gassic UC42-3 square antiprisms.png 3 square antiprisms 24{3}

6{4}

4824 O D4
UC43gidsac UC43-6 square antiprisms.png 6 square antiprisms 48{3}

12{4}

9648 Oh D4
UC44sassid UC44-6 pentagrammic antiprisms.png 6 pentagrammic antiprisms 60{3}

12{5/2}

12060 I D5
UC45sadsid UC45-12 pentagrammic antiprisms.png 12 pentagrammic antiprisms 120{3}

24{5/2}

240120 Ih D5
UC46 siddo UC46-2 icosahedra.png 2 icosahedra (16+24){3}6024 Oh Th
UC47 sne UC47-5 icosahedra.png 5 icosahedra (40+60){3}15060 Ih Th
UC48 presipsido UC48-2 great dodecahedra.png 2 great dodecahedra 24{5}6024 Oh Th
UC49 presipsi UC49-5 great dodecahedra.png 5 great dodecahedra 60{5}15060 Ih Th
UC50 passipsido UC50-2 small stellated dodecahedra.png 2 small stellated dodecahedra 24{5/2}6024 Oh Th
UC51 passipsi UC51-5 small stellated dodecahedra.png 5 small stellated dodecahedra 60{5/2}15060 Ih Th
UC52 sirsido UC52-2 great icosahedra.png 2 great icosahedra (16+24){3}6024 Oh Th
UC53 sirsei UC53-5 great icosahedra.png 5 great icosahedra (40+60){3}15060 Ih Th
UC54 tisso UC54-2 truncated tetrahedra.png 2 truncated tetrahedra 8{3}

8{6}

3624 Oh Td
UC55 taki UC55-5 truncated tetrahedra.png 5 truncated tetrahedra 20{3}

20{6}

9060 I T
UC56 te UC56-10 truncated tetrahedra.png 10 truncated tetrahedra 40{3}

40{6}

180120 Ih T
UC57 tar UC57-5 truncated cubes.png 5 truncated cubes 40{3}

30{8}

180120 Ih Th
UC58 quitar UC58-5 quasitruncated hexahedra.png 5 stellated truncated hexahedra 40{3}

30{8/3}

180120 Ih Th
UC59 arie UC59-5 cuboctahedra.png 5 cuboctahedra 40{3}

30{4}

12060 Ih Th
UC60 gari UC60-5 cubohemioctahedra.png 5 cubohemioctahedra 30{4}

20{6}

12060 Ih Th
UC61 iddei UC61-5 octahemioctahedra.png 5 octahemioctahedra 40{3}

20{6}

12060 Ih Th
UC62 rasseri UC62-5 rhombicuboctahedra.png 5 rhombicuboctahedra 40{3}

(30+60){4}

240120 Ih Th
UC63 rasher UC63-5 small rhombihexahedra.png 5 small rhombihexahedra 60{4}

30{8}

240120 Ih Th
UC64 rahrie UC64-5 small cubicuboctahedra.png 5 small cubicuboctahedra 40{3}

30{4}

30{8}

240120 Ih Th
UC65 raquahri UC65-5 great cubicuboctahedra.png 5 great cubicuboctahedra 40{3}

30{4}

30{8/3}

240120 Ih Th
UC66 rasquahr UC66-5 great rhombihexahedra.png 5 great rhombihexahedra 60{4}

30{8/3}

240120 Ih Th
UC67 rosaqri UC67-5 great rhombicuboctahedra.png 5 nonconvex great rhombicuboctahedra 40{3}

(30+60){4}

240120 Ih Th
UC68 disco UC68-2 snub cubes.png 2 snub cubes (16+48){3}

12{4}

12048 Oh O
UC69 dissid UC69-2 snub dodecahedra.png 2 snub dodecahedra (40+120){3}

24{5}

300120 Ih I
UC70 giddasid UC70-2 great snub icosidodecahedra.png 2 great snub icosidodecahedra (40+120){3}

24{5/2}

300120 Ih I
UC71 gidsid UC71-2 great inverted snub icosidodecahedra.png 2 great inverted snub icosidodecahedra (40+120){3}

24{5/2}

300120 Ih I
UC72 gidrissid UC72-2 great retrosnub icosidodecahedra.png 2 great retrosnub icosidodecahedra (40+120){3}

24{5/2}

300120 Ih I
UC73 disdid UC73-2 snub dodecadodecahedra.png 2 snub dodecadodecahedra 120{3}

24{5}

24{5/2}

300120 Ih I
UC74 idisdid UC74-2 inverted snub dodecadodecahedra.png 2 inverted snub dodecadodecahedra 120{3}

24{5}

24{5/2}

300120 Ih I
UC75 desided UC75-2 snub icosidodecadodecahedra.png 2 snub icosidodecadodecahedra (40+120){3}

24{5}

24{5/2}

360120 Ih I

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