Uniform polyhedron compound

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A uniform polyhedron compound is a polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra, in an arrangement that is also uniform: 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.

Compound Bowers
acronym
Picture Polyhedral
count
Polyhedral type Faces Edges Vertices Notes Symmetry group Subgroup
restricting
to one
constituent
UC01 sis UC01-6 tetrahedra.png 6 tetrahedra 24{3} 36 24 rotational freedom Td S4
UC02 dis UC02-12 tetrahedra.png 12 tetrahedra 48{3} 72 48 rotational freedom Oh S4
UC03 snu UC03-6 tetrahedra.png 6 tetrahedra 24{3} 36 24 Oh D2d
UC04 so UC04-2 tetrahedra.png 2 tetrahedra 8{3} 12 8 regular Oh Td
UC05 ki UC05-5 tetrahedra.png 5 tetrahedra 20{3} 30 20 regular I T
UC06 e UC06-10 tetrahedra.png 10 tetrahedra 40{3} 60 20 regular

2 constituent polyhedra incident on each vertex

Ih T
UC07 risdoh UC07-6 cubes.png 6 cubes (12+24){4} 72 48 rotational freedom Oh C4h
UC08 rah UC08-3 cubes.png 3 cubes (6+12){4} 36 24 Oh D4h
UC09 rhom UC09-5 cubes.png 5 cubes 30{4} 60 20 regular

2 constituent polyhedra incident on each vertex

Ih Th
UC10 dissit UC10-4 octahedra.png 4 octahedra (8+24){3} 48 24 rotational freedom Th S6
UC11 daso UC11-8 octahedra.png 8 octahedra (16+48){3} 96 48 rotational freedom Oh S6
UC12 sno UC12-4 octahedra.png 4 octahedra (8+24){3} 48 24 Oh D3d
UC13 addasi UC13-20 octahedra.png 20 octahedra (40+120){3} 240 120 rotational freedom Ih S6
UC14 dasi UC14-20 octahedra.png 20 octahedra (40+120){3} 240 60 2 constituent polyhedra incident on each vertex Ih S6
UC15 gissi UC15-10 octahedra.png 10 octahedra (20+60){3} 120 60 Ih D3d
UC16 si UC16-10 octahedra.png 10 octahedra (20+60){3} 120 60 Ih D3d
UC17 se UC17-5 octahedra.png 5 octahedra 40{3} 60 30 regular Ih Th
UC18 hirki UC18-5 tetrahemihexahedron.png 5 tetrahemihexahedra 20{3}

15{4}

60 30 I T
UC19 sapisseri UC19-20 tetrahemihexahedron.png 20 tetrahemihexahedra (20+60){3}

60{4}

240 60 2 constituent polyhedra incident on each vertex I C3
UC20 - UC20-2k n-m-gonal prisms.png 2n

(n>0)

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

2np{4}

6np 4np rotational freedom

gcd(p,q)=1, p/q>2

Dnph Cph
UC21 - UC21-k n-m-gonal prisms.png n

(n>1)

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

np{4}

3np 2np gcd(p,q)=1, p/q>2 Dnph Dph
UC22 - UC22-2k n-m-gonal antiprisms.png 2n

(n>0)

p/q-gonal antiprisms (tetrahedra if p/q=2)

(q odd)

4n{p/q} (unless p/q=2)

4np{3}

8np 4np rotational freedom

gcd(p,q)=1, p/q>3/2

Dnpd (if n odd)

Dnph (if n even)

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

(n>1)

p/q-gonal antiprisms (tetrahedra if p/q=2)

(q odd)

2n{p/q} (unless p/q=2)

2np{3}

4np 2np gcd(p,q)=1, p/q>3/2 Dnpd (if n odd)

Dnph (if n even)

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

(n>0)

p/q-gonal antiprisms

(q even)

4n{p/q}

4np{3}

8np 4np rotational freedom

gcd(p,q)=1, p/q>3/2

Dnph Cph
UC25 - UC25-k n-m-gonal antiprisms.png n

(n>1)

p/q-gonal antiprisms

(q even)

2n{p/q}

2np{3}

4np 2np gcd(p,q)=1, p/q>3/2 Dnph Dph
UC26 gadsid UC26-12 pentagonal antiprisms.png 12 pentagonal antiprisms 120{3}

24{5}

240 120 rotational freedom Ih S10
UC27 gassid UC27-6 pentagonal antiprisms.png 6 pentagonal antiprisms 60{3}

12{5}

120 60 Ih D5d
UC28 gidasid UC28-12 pentagrammic crossed antiprisms.png 12 pentagrammic crossed antiprisms 120{3}

24{5/2}

240 120 rotational freedom Ih S10
UC29 gissed UC29-6 pentagrammic crossed antiprisms.png 6 pentagrammic crossed antiprisms 60{3}

12{5/2}

120 60 Ih D5d
UC30 ro UC30-4 triangular prisms.png 4 triangular prisms 8{3}

12{4}

36 24 O D3
UC31 dro UC31-8 triangular prisms.png 8 triangular prisms 16{3}

24{4}

72 48 Oh D3
UC32 kri UC32-10 triangular prisms.png 10 triangular prisms 20{3}

30{4}

90 60 I D3
UC33 dri UC33-20 triangular prisms.png 20 triangular prisms 40{3}

60{4}

180 60 2 constituent polyhedra incident on each vertex Ih D3
UC34 red UC34-6 pentagonal prisms.png 6 pentagonal prisms 30{4}

12{5}

90 60 I D5
UC35 dird UC35-12 pentagonal prisms.png 12 pentagonal prisms 60{4}

24{5}

180 60 2 constituent polyhedra incident on each vertex Ih D5
UC36 gikrid UC36-6 pentagrammic prisms.png 6 pentagrammic prisms 30{4}

12{5/2}

90 60 I D5
UC37 giddird UC37-12 pentagrammic prisms.png 12 pentagrammic prisms 60{4}

24{5/2}

180 60 2 constituent polyhedra incident on each vertex Ih D5
UC38 griso UC38-4 hexagonal prisms.png 4 hexagonal prisms 24{4}

8{6}

72 48 Oh D3d
UC39 rosi UC39-10 hexagonal prisms.png 10 hexagonal prisms 60{4}

20{6}

180 120 Ih D3d
UC40 rassid UC40-6 decagonal prisms.png 6 decagonal prisms 60{4}

12{10}

180 120 Ih D5d
UC41 grassid UC41-6 decagrammic prisms.png 6 decagrammic prisms 60{4}

12{10/3}

180 120 Ih D5d
UC42 gassic UC42-3 square antiprisms.png 3 square antiprisms 24{3}

6{4}

48 24 O D4
UC43 gidsac UC43-6 square antiprisms.png 6 square antiprisms 48{3}

12{4}

96 48 Oh D4
UC44 sassid UC44-6 pentagrammic antiprisms.png 6 pentagrammic antiprisms 60{3}

12{5/2}

120 60 I D5
UC45 sadsid UC45-12 pentagrammic antiprisms.png 12 pentagrammic antiprisms 120{3}

24{5/2}

240 120 Ih D5
UC46 siddo UC46-2 icosahedra.png 2 icosahedra (16+24){3} 60 24 Oh Th
UC47 sne UC47-5 icosahedra.png 5 icosahedra (40+60){3} 150 60 Ih Th
UC48 presipsido UC48-2 great dodecahedra.png 2 great dodecahedra 24{5} 60 24 Oh Th
UC49 presipsi UC49-5 great dodecahedra.png 5 great dodecahedra 60{5} 150 60 Ih Th
UC50 passipsido UC50-2 small stellated dodecahedra.png 2 small stellated dodecahedra 24{5/2} 60 24 Oh Th
UC51 passipsi UC51-5 small stellated dodecahedra.png 5 small stellated dodecahedra 60{5/2} 150 60 Ih Th
UC52 sirsido UC52-2 great icosahedra.png 2 great icosahedra (16+24){3} 60 24 Oh Th
UC53 sirsei UC53-5 great icosahedra.png 5 great icosahedra (40+60){3} 150 60 Ih Th
UC54 tisso UC54-2 truncated tetrahedra.png 2 truncated tetrahedra 8{3}

8{6}

36 24 Oh Td
UC55 taki UC55-5 truncated tetrahedra.png 5 truncated tetrahedra 20{3}

20{6}

90 60 I T
UC56 te UC56-10 truncated tetrahedra.png 10 truncated tetrahedra 40{3}

40{6}

180 120 Ih T
UC57 tar UC57-5 truncated cubes.png 5 truncated cubes 40{3}

30{8}

180 120 Ih Th
UC58 quitar UC58-5 quasitruncated hexahedra.png 5 stellated truncated hexahedra 40{3}

30{8/3}

180 120 Ih Th
UC59 arie UC59-5 cuboctahedra.png 5 cuboctahedra 40{3}

30{4}

120 60 Ih Th
UC60 gari UC60-5 cubohemioctahedra.png 5 cubohemioctahedra 30{4}

20{6}

120 60 Ih Th
UC61 iddei UC61-5 octahemioctahedra.png 5 octahemioctahedra 40{3}

20{6}

120 60 Ih Th
UC62 rasseri UC62-5 rhombicuboctahedra.png 5 rhombicuboctahedra 40{3}

(30+60){4}

240 120 Ih Th
UC63 rasher UC63-5 small rhombihexahedra.png 5 small rhombihexahedra 60{4}

30{8}

240 120 Ih Th
UC64 rahrie UC64-5 small cubicuboctahedra.png 5 small cubicuboctahedra 40{3}

30{4}

30{8}

240 120 Ih Th
UC65 raquahri UC65-5 great cubicuboctahedra.png 5 great cubicuboctahedra 40{3}

30{4}

30{8/3}

240 120 Ih Th
UC66 rasquahr UC66-5 great rhombihexahedra.png 5 great rhombihexahedra 60{4}

30{8/3}

240 120 Ih Th
UC67 rosaqri UC67-5 great rhombicuboctahedra.png 5 nonconvex great rhombicuboctahedra 40{3}

(30+60){4}

240 120 Ih Th
UC68 disco UC68-2 snub cubes.png 2 snub cubes (16+48){3}

12{4}

120 48 Oh O
UC69 dissid UC69-2 snub dodecahedra.png 2 snub dodecahedra (40+120){3}

24{5}

300 120 Ih I
UC70 giddasid UC70-2 great snub icosidodecahedra.png 2 great snub icosidodecahedra (40+120){3}

24{5/2}

300 120 Ih I
UC71 gidsid UC71-2 great inverted snub icosidodecahedra.png 2 great inverted snub icosidodecahedra (40+120){3}

24{5/2}

300 120 Ih I
UC72 gidrissid UC72-2 great retrosnub icosidodecahedra.png 2 great retrosnub icosidodecahedra (40+120){3}

24{5/2}

300 120 Ih I
UC73 disdid UC73-2 snub dodecadodecahedra.png 2 snub dodecadodecahedra 120{3}

24{5}

24{5/2}

300 120 Ih I
UC74 idisdid UC74-2 inverted snub dodecadodecahedra.png 2 inverted snub dodecadodecahedra 120{3}

24{5}

24{5/2}

300 120 Ih I
UC75 desided UC75-2 snub icosidodecadodecahedra.png 2 snub icosidodecadodecahedra (40+120){3}

24{5}

24{5/2}

360 120 Ih I

References[edit]

  • Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society 79: 447–457, doi:10.1017/S0305004100052440, MR 0397554 .

External links[edit]