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

2 constituent polyhedra incident on each vertex

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

2 constituent polyhedra incident on each vertex

Ih Th
UC10 4 octahedra (8+24){3} 48 24 rotational freedom Th S6
UC11 8 octahedra (16+48){3} 96 48 rotational freedom Oh S6
UC12 4 octahedra (8+24){3} 48 24 Oh D3d
UC13 20 octahedra (40+120){3} 240 120 rotational freedom Ih S6
UC14 20 octahedra (40+120){3} 240 60 2 constituent polyhedra incident on each vertex Ih S6
UC15 10 octahedra (20+60){3} 120 60 Ih D3d
UC16 10 octahedra (20+60){3} 120 60 Ih D3d
UC17 5 octahedra 40{3} 60 30 regular Ih Th
UC18 5 tetrahemihexahedra 20{3}

15{4}

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

60{4}

240 60 2 constituent polyhedra incident on each vertex I C3
UC20 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 n

(n>1)

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

np{4}

3np 2np gcd(p,q)=1, p/q>2 Dnph Dph
UC22 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 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 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 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 12 pentagonal antiprisms 120{3}

24{5}

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

12{5}

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

24{5/2}

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

12{5/2}

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

12{4}

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

24{4}

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

30{4}

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

60{4}

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

12{5}

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

24{5}

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

12{5/2}

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

24{5/2}

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

8{6}

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

20{6}

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

12{10}

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

12{10/3}

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

6{4}

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

12{4}

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

12{5/2}

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

24{5/2}

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

8{6}

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

20{6}

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

40{6}

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

30{8}

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

30{8/3}

180 120 Ih Th
UC59 5 cuboctahedra 40{3}

30{4}

120 60 Ih Th
UC60 5 cubohemioctahedra 30{4}

20{6}

120 60 Ih Th
UC61 5 octahemioctahedra 40{3}

20{6}

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

(30+60){4}

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

30{8}

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

30{4}

30{8}

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

30{4}

30{8/3}

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

30{8/3}

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

(30+60){4}

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

12{4}

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

24{5}

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

24{5/2}

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

24{5/2}

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

24{5/2}

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

24{5}

24{5/2}

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

24{5}

24{5/2}

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

24{5}

24{5/2}

360 120 Ih I

References

  • John Skilling, Uniform Compounds of Uniform Polyhedra, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 79, pp. 447-457, 1976.
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