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 (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.

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

2 polyhedra per vertex

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

2 polyhedra per vertex

Ih Th
UC10 dissit 4 octahedra (8+24){3} 48 24 Rotational freedom Th S6
UC11 daso 8 octahedra (16+48){3} 96 48 Rotational freedom Oh S6
UC12 sno 4 octahedra (8+24){3} 48 24 Oh D3d
UC13 addasi 20 octahedra (40+120){3} 240 120 Rotational freedom Ih S6
UC14 dasi 20 octahedra (40+120){3} 240 60 2 polyhedra per vertex Ih S6
UC15 gissi 10 octahedra (20+60){3} 120 60 Ih D3d
UC16 si 10 octahedra (20+60){3} 120 60 Ih D3d
UC17 se 5 octahedra 40{3} 60 30 Regular Ih Th
UC18 hirki 5 tetrahemihexahedra 20{3}

15{4}

60 30 I T
UC19 sapisseri 20 tetrahemihexahedra (20+60){3}

60{4}

240 60 2 polyhedra per vertex I C3
UC20 - 2n

(2n ≥ 2)

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

2np{4}

6np 4np Rotational freedom Dnph Cph
UC21 - n

(n ≥ 2)

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

np{4}

3np 2np Dnph Dph
UC22 - 2n

(2n ≥ 2)

(q odd)

p/q-gonal antiprisms

(q odd)

4n{p/q} (if ⁠p/q⁠ ≠ 2)

4np{3}

8np 4np Rotational freedom Dnpd (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}

4np 2np Dnpd (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}

8np 4np Rotational freedom Dnph Cph
UC25 - n

(n ≥ 2)

p/q-gonal antiprisms

(q even)

2n{p/q} (if ⁠p/q⁠ ≠ 2)

2np{3}

4np 2np Dnph Dph
UC26 gadsid 12 pentagonal antiprisms 120{3}

24{5}

240 120 Rotational freedom Ih S10
UC27 gassid 6 pentagonal antiprisms 60{3}

12{5}

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

24{5/2}

240 120 Rotational freedom Ih S10
UC29 gissed 6 pentagrammic crossed antiprisms 60{3}

125

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

12{4}

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

24{4}

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

30{4}

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

60{4}

180 60 2 polyhedra per vertex Ih D3
UC34 kred 6 pentagonal prisms 30{4}

12{5}

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

24{5}

180 60 2 polyhedra per vertex Ih D5
UC36 gikrid 6 pentagrammic prisms 30{4}

12{5/2}

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

24{5/2}

180 60 2 polyhedra per vertex Ih D5
UC38 griso 4 hexagonal prisms 24{4}

8{6}

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

20{6}

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

12{10}

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

12{10/3}

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

6{4}

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

12{4}

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

12{5/2}

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

24{5/2}

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

8{6}

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

20{6}

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

40{6}

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

30{8}

180 120 Ih Th
UC58 quitar 5 stellated truncated hexahedra 40{3}

30{8/3}

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

30{4}

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

20{6}

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

20{6}

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

(30+60){4}

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

30{8}

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

30{4}

30{8}

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

30{4}

30{8/3}

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

30{8/3}

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

(30+60){4}

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

12{4}

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

24{5}

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

24{5/2}

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

24{5/2}

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

24{5/2}

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

24{5}

24{5/2}

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

24{5}

24{5/2}

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

24{5}

24{5/2}

360 120 Ih I

References

  • Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79 (3): 447–457, doi:10.1017/S0305004100052440, MR 0397554.