Uniform polyhedron compound

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

Template:Short description

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}-Script error: No such module "Check for unknown parameters".gonal prisms (UC20 and UC21) exist only when Template:Sfrac > 2Script error: No such module "Check for unknown parameters"., and when Template:Mvar and Template:Mvar are coprime. The uniform prismatic compounds of {p/q}-Script error: No such module "Check for unknown parameters".gonal antiprisms (UC22, UC23, UC24 and UC25) exist only when Template:Sfrac > Template:SfracScript error: No such module "Check for unknown parameters"., and when Template:Mvar and Template:Mvar are coprime. Furthermore, when Template:Sfrac = 2Script error: No such module "Check for unknown parameters"., 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 File:UC01-6 tetrahedra.png 6 tetrahedra 24{3} 36 24 Rotational freedom Td S4
UC02 dis File:UC02-12 tetrahedra.png 12 tetrahedra 48{3} 72 48 Rotational freedom Oh S4
UC03 snu File:UC03-6 tetrahedra.png 6 tetrahedra 24{3} 36 24 Oh D2d
UC04 so File:UC04-2 tetrahedra.png 2 tetrahedra 8{3} 12 8 Regular Oh Td
UC05 ki File:UC05-5 tetrahedra.png 5 tetrahedra 20{3} 30 20 Regular I T
UC06 e File:UC06-10 tetrahedra.png 10 tetrahedra 40{3} 60 20 Regular

2 polyhedra per vertex

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

2 polyhedra per vertex

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

15{4}

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

60{4}

240 60 2 polyhedra per vertex I C3
UC20 - File:UC20-2k n-m-gonal prisms.png 2n

(2n ≥ 2)

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

2np{4}

6np 4np Rotational freedom Dnph Cph
UC21 - File:UC21-k n-m-gonal prisms.png n

(n ≥ 2)

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

np{4}

3np 2np Dnph Dph
UC22 - File:UC22-2k n-m-gonal antiprisms.png 2n

(2n ≥ 2)

(q odd)

p/q-gonal antiprisms

(q odd)

4n{p/q} (if Template:Sfrac ≠ 2)

4np{3}

8np 4np Rotational freedom Dnpd (if n odd)

Dnph (if n even)

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

(n ≥ 2)

p/q-gonal antiprisms

(q odd)

2n{p/q} (if Template:Sfrac ≠ 2)

2np{3}

4np 2np Dnpd (if n odd)

Dnph (if n even)

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

(2n ≥ 2)

p/q-gonal antiprisms

(q even)

4n{p/q} (if Template:Sfrac ≠ 2)

4np{3}

8np 4np Rotational freedom Dnph Cph
UC25 - File:UC25-k n-m-gonal antiprisms.png n

(n ≥ 2)

p/q-gonal antiprisms

(q even)

2n{p/q} (if Template:Sfrac ≠ 2)

2np{3}

4np 2np Dnph Dph
UC26 gadsid File:UC26-12 pentagonal antiprisms.png 12 pentagonal antiprisms 120{3}

24{5}

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

12{5}

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

24{5/2}

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

125

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

12{4}

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

24{4}

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

30{4}

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

60{4}

180 60 2 polyhedra per vertex Ih D3
UC34 kred File:UC34-6 pentagonal prisms.png 6 pentagonal prisms 30{4}

12{5}

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

24{5}

180 60 2 polyhedra per vertex Ih D5
UC36 gikrid File:UC36-6 pentagrammic prisms.png 6 pentagrammic prisms 30{4}

12{5/2}

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

24{5/2}

180 60 2 polyhedra per vertex Ih D5
UC38 griso File:UC38-4 hexagonal prisms.png 4 hexagonal prisms 24{4}

8{6}

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

20{6}

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

12{10}

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

12{10/3}

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

6{4}

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

12{4}

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

12{5/2}

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

24{5/2}

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

8{6}

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

20{6}

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

40{6}

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

30{8}

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

30{8/3}

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

30{4}

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

20{6}

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

20{6}

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

(30+60){4}

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

30{8}

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

30{4}

30{8}

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

30{4}

30{8/3}

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

30{8/3}

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

(30+60){4}

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

12{4}

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

24{5}

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

24{5/2}

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

24{5/2}

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

24{5/2}

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

24{5}

24{5/2}

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

24{5}

24{5/2}

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

24{5}

24{5/2}

360 120 Ih I

References

  • Script error: No such module "citation/CS1"..

External links

Retrieved from "http://debianws.lexgopc.com/wiki143/index.php?title=Uniform_polyhedron_compound&oldid=3980628"