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 |
[edit] References
- John Skilling, Uniform Compounds of Uniform Polyhedra, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 79, pp. 447-457, 1976.
[edit] External links
- http://www.interocitors.com/polyhedra/UCs/ShortNames.html - Bowers style acronyms for uniform polyhedron compounds