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Uniform polyhedron compound ： ウィキペディア英語版
Uniform polyhedron compound
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.
|72
|48
|rotational freedom
|''O''h
|''S''4
|-
|UC03
|snu
|50px
|6
|tetrahedra
|24
|36
|24
|
|''O''h
|''D''2d
|-
|UC04
|so
|50px
|2
|tetrahedra
|8
|12
|8
|regular
|''O''h
|''T''d
|-
|UC05
|ki
|50px
|5
|tetrahedra
|20
|30
|20
|regular
|''I''
|''T''
|-
|UC06
|e
|50px
|10
|tetrahedra
|40
|60
|20
|regular
2 constituent polyhedra incident on each vertex
|''I''h
|''T''
|-
|UC07
|risdoh
|50px
|6
|cubes
|(12+24)
|72
|48
|rotational freedom
|''O''h
|''C''4h
|-
|UC08
|rah
|50px
|3
|cubes
|(6+12)
|36
|24
|
|''O''h
|''D''4h
|-
|UC09
|rhom
|50px
|5
|cubes
|30
|60
|20
|regular
2 constituent polyhedra incident on each vertex
|''I''h
|''T''h
|-
|UC10
|dissit
|50px
|4
|octahedra
|(8+24)
|48
|24
|rotational freedom
|''T''h
|''S''6
|-
|UC11
|daso
|50px
|8
|octahedra
|(16+48)
|96
|48
|rotational freedom
|''O''h
|''S''6
|-
|UC12
|sno
|50px
|4
|octahedra
|(8+24)
|48
|24
|
|''O''h
|''D''3d
|-
|UC13
|addasi
|50px
|20
|octahedra
|(40+120)
|240
|120
|rotational freedom
|''I''h
|''S''6
|-
|UC14
|dasi
|50px
|20
|octahedra
|(40+120)
|240
|60
|2 constituent polyhedra incident on each vertex
|''I''h
|''S''6
|-
|UC15
|gissi
|50px
|10
|octahedra
|(20+60)
|120
|60
|
|''I''h
|''D''3d
|-
|UC16
|si
|50px
|10
|octahedra
|(20+60)
|120
|60
|
|''I''h
|''D''3d
|-
|UC17
|se
|50px
|5
|octahedra
|40
|60
|30
|regular
|''I''h
|''T''h
|-
|UC18
|hirki
|50px
|5
|tetrahemihexahedra
|20
15
|60
|30
|
|''I''
|''T''
|-
|UC19
|sapisseri
|50px
|20
|tetrahemihexahedra
|(20+60)
60
|240
|60
|2 constituent polyhedra incident on each vertex
|''I''
|''C''3
|-
|UC20
| -
|50px
|2''n''
(''n''>0)
|''p''/''q''-gonal prisms
|4''n''
2''np''
|6''np''
|4''np''
|rotational freedom
gcd(''p'',''q'')=1, ''p''/''q''>2
|''D''''np''h
|''C''''p''h
|-
|UC21
| -
|50px
|''n''
(''n''>1)
|''p''/''q''-gonal prisms
|2''n''
''np''
|3''np''
|2''np''
|gcd(''p'',''q'')=1, ''p''/''q''>2
|''D''''np''h
|''D''''p''h
|-
|UC22
| -
|50px
|2''n''
(''n''>0)
|''p''/''q''-gonal antiprisms (tetrahedra if ''p''/''q''=2)
(''q'' odd)
|4''n'' (unless ''p''/''q''=2)
4''np''
|8''np''
|4''np''
|rotational freedom
gcd(''p'',''q'')=1, ''p''/''q''>3/2
|''D''''np''d (if ''n'' odd)
''D''''np''h (if ''n'' even)
|''S''2''p''
|-
|UC23
| -
|50px
|''n''
(''n''>1)
|''p''/''q''-gonal antiprisms (tetrahedra if ''p''/''q''=2)
(''q'' odd)
|2''n'' (unless ''p''/''q''=2)
2''np''
|4''np''
|2''np''
|gcd(''p'',''q'')=1, ''p''/''q''>3/2
|''D''''np''d (if ''n'' odd)
''D''''np''h (if ''n'' even)
|''D''''p''d
|-
|UC24
| -
|50px
|2''n''
(''n''>0)
|''p''/''q''-gonal antiprisms
(''q'' even)
|4''n''
4''np''
|8''np''
|4''np''
|rotational freedom
gcd(''p'',''q'')=1, ''p''/''q''>3/2
|''D''''np''h
|''C''''p''h
|-
|UC25
| -
|50px
|''n''
(''n''>1)
|''p''/''q''-gonal antiprisms
(''q'' even)
|2''n''
2''np''
|4''np''
|2''np''
|gcd(''p'',''q'')=1, ''p''/''q''>3/2
|''D''''np''h
|''D''''p''h
|-
|UC26
|gadsid
|50px
|12
|pentagonal antiprisms
|120
24
|240
|120
|rotational freedom
|''I''h
|''S''10
|-
|UC27
|gassid
|50px
|6
|pentagonal antiprisms
|60
12
|120
|60
|
|''I''h
|''D''5d
|-
|UC28
|gidasid
|50px
|12
|pentagrammic crossed antiprisms
|120
24
|240
|120
|rotational freedom
|''I''h
|''S''10
|-
|UC29
|gissed
|50px
|6
|pentagrammic crossed antiprisms
|60
12
|120
|60
|
|''I''h
|''D''5d
|-
|UC30
|ro
|50px
|4
|triangular prisms
|8
12
|36
|24
|
|''O''
|''D''3
|-
|UC31
|dro
|50px
|8
|triangular prisms
|16
24
|72
|48
|
|''O''h
|''D''3
|-
|UC32
|kri
|50px
|10
|triangular prisms
|20
30
|90
|60
|
|''I''
|''D''3
|-
|UC33
|dri
|50px
|20
|triangular prisms
|40
60
|180
|60
|2 constituent polyhedra incident on each vertex
|''I''h
|''D''3
|-
|UC34
|red
|50px
|6
|pentagonal prisms
|30
12
|90
|60
|
|''I''
|''D''5
|-
|UC35
|dird
|50px
|12
|pentagonal prisms
|60
24
|180
|60
|2 constituent polyhedra incident on each vertex
|''I''h
|''D''5
|-
|UC36
|gikrid
|50px
|6
|pentagrammic prisms
|30
12
|90
|60
|
|''I''
|''D''5
|-
|UC37
|giddird
|50px
|12
|pentagrammic prisms
|60
24
|180
|60
|2 constituent polyhedra incident on each vertex
|''I''h
|''D''5
|-
|UC38
|griso
|50px
|4
|hexagonal prisms
|24
8
|72
|48
|
|''O''h
|''D''3d
|-
|UC39
|rosi
|50px
|10
|hexagonal prisms
|60
20
|180
|120
|
|''I''h
|''D''3d
|-
|UC40
|rassid
|50px
|6
|decagonal prisms
|60
12
|180
|120
|
|''I''h
|''D''5d
|-
|UC41
|grassid
|50px
|6
|decagrammic prisms
|60
12
|180
|120
|
|''I''h
|''D''5d
|-
|UC42
|gassic
|50px
|3
|square antiprisms
|24
6
|48
|24
|
|''O''
|''D''4
|-
|UC43
|gidsac
|50px
|6
|square antiprisms
|48
12
|96
|48
|
|''O''h
|''D''4
|-
|UC44
|sassid
|50px
|6
|pentagrammic antiprisms
|60
12
|120
|60
|
|''I''
|''D''5
|-
|UC45
|sadsid
|50px
|12
|pentagrammic antiprisms
|120
24
|240
|120
|
|''I''h
|''D''5
|-
|UC46
|siddo
|50px
|2
|icosahedra
|(16+24)
|60
|24
|
|''O''h
|''T''h
|-
|UC47
|sne
|50px
|5
|icosahedra
|(40+60)
|150
|60
|
|''I''h
|''T''h
|-
|UC48
|presipsido
|50px
|2
|great dodecahedra
|24
|60
|24
|
|''O''h
|''T''h
|-
|UC49
|presipsi
|50px
|5
|great dodecahedra
|60
|150
|60
|
|''I''h
|''T''h
|-
|UC50
|passipsido
|50px
|2
|small stellated dodecahedra
|24
|60
|24
|
|''O''h
|''T''h
|-
|UC51
|passipsi
|50px
|5
|small stellated dodecahedra
|60
|150
|60
|
|''I''h
|''T''h
|-
|UC52
|sirsido
|50px
|2
|great icosahedra
|(16+24)
|60
|24
|
|''O''h
|''T''h
|-
|UC53
|sirsei
|50px
|5
|great icosahedra
|(40+60)
|150
|60
|
|''I''h
|''T''h
|-
|UC54
|tisso
|50px
|2
|truncated tetrahedra
|8
8
|36
|24
|
|''O''h
|''T''d
|-
|UC55
|taki
|50px
|5
|truncated tetrahedra
|20
20
|90
|60
|
|''I''
|''T''
|-
|UC56
|te
|50px
|10
|truncated tetrahedra
|40
40
|180
|120
|
|''I''h
|''T''
|-
|UC57
|tar
|50px
|5
|truncated cubes
|40
30
|180
|120
|
|''I''h
|''T''h
|-
|UC58
|quitar
|50px
|5
|stellated truncated hexahedra
|40
30
|180
|120
|
|''I''h
|''T''h
|-
|UC59
|arie
|50px
|5
|cuboctahedra
|40
30
|120
|60
|
|''I''h
|''T''h
|-
|UC60
|gari
|50px
|5
|cubohemioctahedra
|30
20
|120
|60
|
|''I''h
|''T''h
|-
|UC61
|iddei
|50px
|5
|octahemioctahedra
|40
20
|120
|60
|
|''I''h
|''T''h
|-
|UC62
|rasseri
|50px
|5
|rhombicuboctahedra
|40
(30+60)
|240
|120
|
|''I''h
|''T''h
|-
|UC63
|rasher
|50px
|5
|small rhombihexahedra
|60
30
|240
|120
|
|''I''h
|''T''h
|-
|UC64
|rahrie
|50px
|5
|small cubicuboctahedra
|40
30
30
|240
|120
|
|''I''h
|''T''h
|-
|UC65
|raquahri
|50px
|5
|great cubicuboctahedra
|40
30
30
|240
|120
|
|''I''h
|''T''h
|-
|UC66
|rasquahr
|50px
|5
|great rhombihexahedra
|60
30
|240
|120
|
|''I''h
|''T''h
|-
|UC67
|rosaqri
|50px
|5
|nonconvex great rhombicuboctahedra
|40
(30+60)
|240
|120
|
|''I''h
|''T''h
|-
|UC68
|disco
|50px
|2
|snub cubes
|(16+48)
12
|120
|48
|
|''O''h
|''O''
|-
|UC69
|dissid
|50px
|2
|snub dodecahedra
|(40+120)
24
|300
|120
|
|''I''h
|''I''
|-
|UC70
|giddasid
|50px
|2
|great snub icosidodecahedra
|(40+120)
24
|300
|120
|
|''I''h
|''I''
|-
|UC71
|gidsid
|50px
|2
|great inverted snub icosidodecahedra
|(40+120)
24
|300
|120
|
|''I''h
|''I''
|-
|UC72
|gidrissid
|50px
|2
|great retrosnub icosidodecahedra
|(40+120)
24
|300
|120
|
|''I''h
|''I''
|-
|UC73
|disdid
|50px
|2
|snub dodecadodecahedra
|120
24
24
|300
|120
|
|''I''h
|''I''
|-
|UC74
|idisdid
|50px
|2
|inverted snub dodecadodecahedra
|120
24
24
|300
|120
|
|''I''h
|''I''
|-
|UC75
|desided
|50px
|2
|snub icosidodecadodecahedra
|(40+120)
24
24
|360
|120
|
|''I''h
|''I''
|}
== References ==

*.

抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
■ウィキペディアで「Uniform polyhedron compound」の詳細全文を読む



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