List of uniform polyhedra

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Uniform polyhedra and tilings form a well studied group. They are listed here for quick comparison of their properties and varied naming schemes and symbols.

This list includes:

Not included are:

Indexing[edit]

Four numbering schemes for the uniform polyhedra are in common use, distinguished by letters:

  • [C] Coxeter et al., 1954, showed the convex forms as figures 15 through 32; three prismatic forms, figures 33–35; and the nonconvex forms, figures 36–92.
  • [W] Wenninger, 1974, has 119 figures: 1-5 for the Platonic solids, 6-18 for the Archimedean solids, 19-66 for stellated forms including the 4 regular nonconvex polyhedra, and ended with 67-119 for the nonconvex uniform polyhedra.
  • [K] Kaleido, 1993: The 80 figures were grouped by symmetry: 1-5 as representatives of the infinite families of prismatic forms with dihedral symmetry, 6-9 with tetrahedral symmetry, 10-26 with Octahedral symmetry, 46-80 with icosahedral symmetry.
  • [U] Mathematica, 1993, follows the Kaleido series with the 5 prismatic forms moved to last, so that the nonprismatic forms become 1–75.

Table of polyhedra[edit]

The convex forms are listed in order of degree of vertex configurations from 3 faces/vertex and up, and in increasing sides per face. This ordering allows topological similarities to be shown.

Convex forms (3 faces/vertex)[edit]

Name Picture Solid
class
Wythoff
symbol
Vertex figure Bowers-style
acronym
Symmetry
group
W# U# K# Vertices Edges Faces Chi Density Faces by type
Tetrahedron Tetrahedron.png R 3|2 3 Tetrahedron vertfig.png
3.3.3
Tet Td W001 U01 K06 4 6 4 2 1 4{3}
Triangular prism Triangular prism.png P 2 3|2 Triangular prism vertfig.svg
3.4.4
Trip D3h -- -- -- 6 9 5 2 1 2{3}+3{4}
Truncated tetrahedron Truncated tetrahedron.png A 2 3|3 Truncated tetrahedron vertfig.png
3.6.6
Tut Td W006 U02 K07 12 18 8 2 1 4{3}+4{6}
Truncated cube Truncated hexahedron.png A 2 3|4 Truncated cube vertfig.png
3.8.8
Tic Oh W008 U09 K14 24 36 14 2 1 8{3}+6{8}
Truncated dodecahedron Truncated dodecahedron.png A 2 3|5 Truncated dodecahedron vertfig.png
3.10.10
Tid Ih W010 U26 K31 60 90 32 2 1 20{3}+12{10}
Cube Hexahedron.png R 3|2 4 Cube vertfig.png
4.4.4
Cube Oh W003 U06 K11 8 12 6 2 1 6{4}
Pentagonal prism Pentagonal prism.png P 2 5|2 Pentagonal prism vertfig.png
4.4.5
Pip D5h -- U76 K01 10 15 7 2 1 5{4}+2{5}
Hexagonal prism Hexagonal prism.png P 2 6|2 Hexagonal prism vertfig.png
4.4.6
Hip D6h -- -- -- 12 18 8 2 1 6{4}+2{6}
Octagonal prism Octagonal prism.png P 2 8|2 Octagonal prism vertfig.png
4.4.8
Op D8h -- -- -- 16 24 10 2 1 8{4}+2{8}
Decagonal prism Decagonal prism.png P 2 10|2 Decagonal prism vf.png
4.4.10
Dip D10h -- -- -- 20 30 12 2 1 10{4}+2{10}
Dodecagonal prism Dodecagonal prism.png P 2 12|2 Dodecagonal prism vf.png
4.4.12
Twip D12h -- -- -- 24 36 14 2 1 12{4}+2{12}
Truncated octahedron Truncated octahedron.png A 2 4|3 Truncated octahedron vertfig.png
4.6.6
Toe Oh W007 U08 K13 24 36 14 2 1 6{4}+8{6}
Great rhombicuboctahedron Great rhombicuboctahedron.png A 2 3 4| Great rhombicuboctahedron vertfig.png
4.6.8
Girco Oh W015 U11 K16 48 72 26 2 1 12{4}+8{6}+6{8}
Great rhombicosidodecahedron Great rhombicosidodecahedron.png A 2 3 5| Great rhombicosidodecahedron vertfig.png
4.6.10
Grid Ih W016 U28 K33 120 180 62 2 1 30{4}+20{6}+12{10}
Dodecahedron Dodecahedron.png R 3|2 5 Dodecahedron vertfig.png
5.5.5
Doe Ih W005 U23 K28 20 30 12 2 1 12{5}
Truncated icosahedron Truncated icosahedron.png A 2 5|3 Truncated icosahedron vertfig.png
5.6.6
Ti Ih W009 U25 K30 60 90 32 2 1 12{5}+20{6}

Convex forms (4 faces/vertex)[edit]

Name Picture Solid
class
Wythoff
symbol
Vertex figure Bowers-style
acronym
Symmetry
group
W# U# K# Vertices Edges Faces Chi Density Faces by type
Octahedron Octahedron.png R 4|2 3 Octahedron vertfig.png
3.3.3.3
Oct Oh W002 U05 K10 6 12 8 2 1 8{3}
Square antiprism Square antiprism.png P |2 2 4 Square antiprism vertfig.png
3.3.3.4
Squap D4d -- -- -- 8 16 10 2 1 8{3}+2{4}
Pentagonal antiprism Pentagonal antiprism.png P |2 2 5 Pentagonal antiprism vertfig.png
3.3.3.5
Pap D5d -- U77 K02 10 20 12 2 1 10{3}+2{5}
Hexagonal antiprism Hexagonal antiprism.png P |2 2 6 Hexagonal antiprism vertfig.png
3.3.3.6
Hap D6d -- -- -- 12 24 14 2 1 12{3}+2{6}
Octagonal antiprism Octagonal antiprism.png P |2 2 8 Octagonal antiprism vertfig.png
3.3.3.8
Oap D8d -- -- -- 16 32 18 2 1 16{3}+2{8}
Decagonal antiprism Decagonal antiprism.png P |2 2 10 Decagonal antiprism vf.png
3.3.3.10
Dap D10d -- -- -- 20 40 22 2 1 20{3}+2{10}
Dodecagonal antiprism Dodecagonal antiprism.png P |2 2 12 Dodecagonal antiprism vf.png
3.3.3.12
Twap D12d -- -- -- 24 48 26 2 1 24{3}+2{12}
Cuboctahedron Cuboctahedron.png A 2|3 4 Cuboctahedron vertfig.png
3.4.3.4
Co Oh W011 U07 K12 12 24 14 2 1 8{3}+6{4}
Small rhombicuboctahedron Small rhombicuboctahedron.png A 3 4|2 Small rhombicuboctahedron vertfig.png
3.4.4.4
Sirco Oh W013 U10 K15 24 48 26 2 1 8{3}+(6+12){4}
Small rhombicosidodecahedron Small rhombicosidodecahedron.png A 3 5|2 Small rhombicosidodecahedron vertfig.png
3.4.5.4
Srid Ih W014 U27 K32 60 120 62 2 1 20{3}+30{4}+12{5}
Icosidodecahedron Icosidodecahedron.png A 2|3 5 Icosidodecahedron vertfig.png
3.5.3.5
Id Ih W012 U24 K29 30 60 32 2 1 20{3}+12{5}

Convex forms (5 faces/vertex)[edit]

Name Picture Solid
class
Wythoff
symbol
Vertex figure Bowers-style
acronym
Symmetry
group
W# U# K# Vertices Edges Faces Chi Density Faces by type
Icosahedron Icosahedron.png R 5|2 3 Icosahedron vertfig.png
3.3.3.3.3
Ike Ih W004 U22 K27 12 30 20 2 1 20{3}
Snub cube Snub hexahedron.png A |2 3 4 Snub cube vertfig.png
3.3.3.3.4
Snic O W017 U12 K17 24 60 38 2 1 (8+24){3}+6{4}
Snub dodecahedron Snub dodecahedron ccw.png A |2 3 5 Snub dodecahedron vertfig.png
3.3.3.3.5
Snid I W018 U29 K34 60 150 92 2 1 (20+60){3}+12{5}

Nonconvex forms with convex faces[edit]

Name Picture Solid
class
Wythoff
symbol
Vertex figure Bowers-style
acronym
Symmetry
group
W# U# K# Vertices Edges Faces Chi Density Faces by type
Octahemioctahedron Octahemioctahedron.png C+ 3/2 3|3 Octahemioctahedron vertfig.png
6.3/2.6.3
Oho Oh W068 U03 K08 12 24 12 0 4 8{3}+4{6}
Tetrahemihexahedron Tetrahemihexahedron.png C+ 3/2 3|2 Tetrahemihexahedron vertfig.png
4.3/2.4.3
Thah Td W067 U04 K09 6 12 7 1 3 4{3}+3{4}
Cubohemioctahedron Cubohemioctahedron.png C+ 4/3 4|3 Cubohemioctahedron vertfig.png
6.4/3.6.4
Cho Oh W078 U15 K20 12 24 10 -2 4 6{4}+4{6}
Great dodecahedron Great dodecahedron.png R+ 5/2|2 5 Great dodecahedron vertfig.png
(5.5.5.5.5)/2
Gad Ih W021 U35 K40 12 30 12 -6 3 12{5}
Great icosahedron Great icosahedron.png R+ 5/2|2 3 Great icosahedron vertfig.png
(3.3.3.3.3)/2
Gike Ih W041 U53 K58 12 30 20 2 7 20{3}
Great ditrigonal icosidodecahedron Great ditrigonal icosidodecahedron.png C+ 3/2|3 5 Great ditrigonal icosidodecahedron vertfig.png
(5.3.5.3.5.3)/2
Gidtid Ih W087 U47 K52 20 60 32 -8 6 20{3}+12{5}
Small rhombihexahedron Small rhombihexahedron.png C+ 3/2 2 4| Small rhombihexahedron vertfig.png
4.8.4/3.8
Sroh Oh W086 U18 K23 24 48 18 -6 5 12{4}+6{8}
Small cubicuboctahedron Small cubicuboctahedron.png C+ 3/2 4|4 Small cubicuboctahedron vertfig.png
8.3/2.8.4
Socco Oh W069 U13 K18 24 48 20 -4 2 8{3}+6{4}+6{8}
Nonconvex great rhombicuboctahedron Uniform great rhombicuboctahedron.png C+ 3/2 4|2 Uniform great rhombicuboctahedron vertfig.png
4.3/2.4.4
Querco Oh W085 U17 K22 24 48 26 2 5 8{3}+(6+12){4}
Small dodecahemidodecahedron Small dodecahemidodecahedron.png C+ 5/4 5|5 Small dodecahemidodecahedron vertfig.png
10.5/4.10.5
Sidhid Ih W091 U51 K56 30 60 18 -12 6 12{5}+6{10}
Great dodecahemicosahedron Great dodecahemicosahedron.png C+ 5/4 5|3 Great dodecahemicosahedron vertfig.png
6.5/4.6.5
Gidhei Ih W102 U65 K70 30 60 22 -8 10 12{5}+10{6}
Small icosihemidodecahedron Small icosihemidodecahedron.png C+ 3/2 3|5 Small icosihemidodecahedron vertfig.png
10.3/2.10.3
Seihid Ih W089 U49 K54 30 60 26 -4 6 20{3}+6{10}
Small dodecicosahedron Small dodecicosahedron.png C+ 3/2 3 5| Small dodecicosahedron vertfig.png
10.6.10/9.6/5
Siddy Ih W090 U50 K55 60 120 32 -28 6 20{6}+12{10}
Small rhombidodecahedron Small rhombidodecahedron.png C+ 2 5/2 5| Small rhombidodecahedron vertfig.png
10.4.10/9.4/3
Sird Ih W074 U39 K44 60 120 42 -18 3 30{4}+12{10}
Small dodecicosidodecahedron Small dodecicosidodecahedron.png C+ 3/2 5|5 Small dodecicosidodecahedron vertfig.png
10.3/2.10.5
Saddid Ih W072 U33 K38 60 120 44 -16 2 20{3}+12{5}+12{10}
Rhombicosahedron Rhombicosahedron.png C+ 2 5/2 3| Rhombicosahedron vertfig.png
6.4.6/5.4/3
Ri Ih W096 U56 K61 60 120 50 -10 7 30{4}+20{6}
Great icosicosidodecahedron Great icosicosidodecahedron.png C+ 3/2 5|3 Great icosicosidodecahedron vertfig.png
6.3/2.6.5
Giid Ih W088 U48 K53 60 120 52 -8 6 20{3}+12{5}+20{6}

Nonconvex prismatic forms[edit]

Name Picture Solid
class
Wythoff
symbol
Vertex figure Bowers-style
acronym
Symmetry
group
W# U# K# Vertices Edges Faces Chi Density Faces by type
Pentagrammic prism Pentagrammic prism.png P+ 2 5/2|2 Pentagrammic prism vertfig.png
5/2.4.4
Stip D5h -- U78 K03 10 15 7 2 2 5{4}+2{5/2}
Heptagrammic prism (7/3) Heptagrammic prism 7-3.png P+ 2 7/3|2 Septagrammic prism-3-7 vertfig.png
7/3.4.4
Giship D7h -- -- -- 14 21 9 2 3 7{4}+2{7/3}
Heptagrammic prism (7/2) Heptagrammic prism 7-2.png P+ 2 7/2|2 Septagrammic prism vertfig.png
7/2.4.4
Ship D7h -- -- -- 14 21 9 2 2 7{4}+2{7/2}
Octagrammic prism Prism 8-3.png P+ 2 8/3|2 Octagrammic prism vertfig.png
8/3.4.4
Stop D8h -- -- -- 16 24 10 2 3 8{4}+2{8/3}
Pentagrammic antiprism Pentagrammic antiprism.png P+ |2 2 5/2 Pentagrammic antiprism vertfig.png
5/2.3.3.3
Stap D5h -- U79 K04 10 20 12 2 2 10{3}+2{5/2}
Pentagrammic crossed-antiprism Pentagrammic crossed antiprism.png P+ |2 2 5/3 Pentagrammic crossed-antiprism vertfig.png
5/3.3.3.3
Starp D5d -- U80 K05 10 20 12 2 3 10{3}+2{5/2}

Other nonconvex forms with nonconvex faces[edit]

Name Picture Solid
class
Wythoff
symbol
Vertex figure Bowers-style
acronym
Symmetry
group
W# U# K# Vertices Edges Faces Chi Density Faces by type
Small stellated dodecahedron Small stellated dodecahedron.png R+ 5|2 5/2 Small stellated dodecahedron vertfig.png
(5/2)5
Sissid Ih W020 U34 K39 12 30 12 -6 3 12{5/2}
Great stellated dodecahedron Great stellated dodecahedron.png R+ 3|2 5/2 Great stellated dodecahedron vertfig.png
(5/2)3
Gissid Ih W022 U52 K57 20 30 12 2 7 12{5/2}
Ditrigonal dodecadodecahedron Ditrigonal dodecadodecahedron.png S+ 3|5/3 5 Ditrigonal dodecadodecahedron vertfig.png
(5/3.5)3
Ditdid Ih W080 U41 K46 20 60 24 -16 4 12{5}+12{5/2}
Small ditrigonal icosidodecahedron Small ditrigonal icosidodecahedron.png S+ 3|5/2 3 Small ditrigonal icosidodecahedron vertfig.png
(5/2.3)3
Sidtid Ih W070 U30 K35 20 60 32 -8 2 20{3}+12{5/2}
Stellated truncated hexahedron Stellated truncated hexahedron.png S+ 2 3|4/3 Stellated truncated hexahedron vertfig.png
8/3.8/3.3
Quith Oh W092 U19 K24 24 36 14 2 7 8{3}+6{8/3}
Great rhombihexahedron Great rhombihexahedron.png S+ 4/33/2 2| Great rhombihexahedron vertfig.png
4.8/3.4/3.8/5
Groh Oh W103 U21 K26 24 48 18 -6 11 12{4}+6{8/3}
Great cubicuboctahedron Great cubicuboctahedron.png S+ 3 4|4/3 Great cubicuboctahedron vertfig.png
8/3.3.8/3.4
Gocco Oh W077 U14 K19 24 48 20 -4 4 8{3}+6{4}+6{8/3}
Great dodecahemidodecahedron Great dodecahemidodecahedron.png S+ 5/35/2|5/3 Great dodecahemidodecahedron vertfig.png
10/3.5/3.10/3.5/2
Gidhid Ih W107 U70 K75 30 60 18 -12 18 12{5/2}+6{10/3}
Small dodecahemicosahedron Small dodecahemicosahedron.png S+ 5/35/2|3 Small dodecahemicosahedron vertfig.png
6.5/3.6.5/2
Sidhei Ih W100 U62 K67 30 60 22 -8 10 12{5/2}+10{6}
Dodecadodecahedron Dodecadodecahedron.png S+ 2|5/2 5 Dodecadodecahedron vertfig.png
(5/2.5)2
Did Ih W073 U36 K41 30 60 24 -6 3 12{5}+12{5/2}
Great icosihemidodecahedron Great icosihemidodecahedron.png S+ 3/2 3|5/3 Great icosihemidodecahedron vertfig.png
10/3.3/2.10/3.3
Geihid Ih W106 U71 K76 30 60 26 -4 18 20{3}+6{10/3}
Great icosidodecahedron Great icosidodecahedron.png S+ 2|5/2 3 Great icosidodecahedron vertfig.png
(5/2.3)2
Gid Ih W094 U54 K59 30 60 32 2 7 20{3}+12{5/2}
Cubitruncated cuboctahedron Cubitruncated cuboctahedron.png S+ 4/3 3 4| Cubitruncated cuboctahedron vertfig.png
8/3.6.8
Cotco Oh W079 U16 K21 48 72 20 -4 4 8{6}+6{8}+6{8/3}
Great truncated cuboctahedron Great truncated cuboctahedron.png S+ 4/3 2 3| Great truncated cuboctahedron vertfig.png
8/3.4.6
Quitco Oh W093 U20 K25 48 72 26 2 7 12{4}+8{6}+6{8/3}
Truncated great dodecahedron Great truncated dodecahedron.png S+ 2 5/2|5 Truncated great dodecahedron vertfig.png
10.10.5/2
Tigid Ih W075 U37 K42 60 90 24 -6 3 12{5/2}+12{10}
Small stellated truncated dodecahedron Small stellated truncated dodecahedron.png S+ 2 5|5/3 Small stellated truncated dodecahedron vertfig.png
10/3.10/3.5
Quit Sissid Ih W097 U58 K63 60 90 24 -6 9 12{5}+12{10/3}
Great stellated truncated dodecahedron Great stellated truncated dodecahedron.png S+ 2 3|5/3 Great stellated truncated dodecahedron vertfig.png
10/3.10/3.3
Quit Gissid Ih W104 U66 K71 60 90 32 2 13 20{3}+12{10/3}
Truncated great icosahedron Great truncated icosahedron.png S+ 2 5/2|3 Great truncated icosahedron vertfig.png
6.6.5/2
Tiggy Ih W095 U55 K60 60 90 32 2 7 12{5/2}+20{6}
Great dodecicosahedron Great dodecicosahedron.png S+ 5/35/2 3| Great dodecicosahedron vertfig.png
6.10/3.6/5.10/7
Giddy Ih W101 U63 K68 60 120 32 -28 10 20{6}+12{10/3}
Great rhombidodecahedron Great rhombidodecahedron.png S+ 3/25/3 2| Great rhombidodecahedron vertfig.png
4.10/3.4/3.10/7
Gird Ih W109 U73 K78 60 120 42 -18 23 30{4}+12{10/3}
Icosidodecadodecahedron Icosidodecadodecahedron.png S+ 5/3 5|3 Icosidodecadodecahedron vertfig.png
6.5/3.6.5
Ided Ih W083 U44 K49 60 120 44 -16 4 12{5}+12{5/2}+20{6}
Small ditrigonal dodecicosidodecahedron Small ditrigonal dodecicosidodecahedron.png S+ 5/3 3|5 Small ditrigonal dodecicosidodecahedron vertfig.png
10.5/3.10.3
Sidditdid Ih W082 U43 K48 60 120 44 -16 4 20{3}+12{;5/2}+12{10}
Great ditrigonal dodecicosidodecahedron Great ditrigonal dodecicosidodecahedron.png S+ 3 5|5/3 Great ditrigonal dodecicosidodecahedron vertfig.png
10/3.3.10/3.5
Gidditdid Ih W081 U42 K47 60 120 44 -16 4 20{3}+12{5}+12{10/3}
Great dodecicosidodecahedron Great dodecicosidodecahedron.png S+ 5/2 3|5/3 Great dodecicosidodecahedron vertfig.png
10/3.5/2.10/3.3
Gaddid Ih W099 U61 K66 60 120 44 -16 10 20{3}+12{5/2}+12{10/3}
Small icosicosidodecahedron Small icosicosidodecahedron.png S+ 5/2 3|3 Small icosicosidodecahedron vertfig.png
6.5/2.6.3
Siid Ih W071 U31 K36 60 120 52 -8 2 20{3}+12{5/2}+20{6}
Rhombidodecadodecahedron Rhombidodecadodecahedron.png S+ 5/2 5|2 Rhombidodecadodecahedron vertfig.png
4.5/2.4.5
Raded Ih W076 U38 K43 60 120 54 -6 3 30{4}+12{5}+12{5/2}
Nonconvex great rhombicosidodecahedron Uniform great rhombicosidodecahedron.png S+ 5/3 3|2 Uniform great rhombicosidodecahedron vertfig.png
4.5/3.4.3
Qrid Ih W105 U67 K72 60 120 62 2 13 20{3}+30{4}+12{5/2}
Snub dodecadodecahedron Snub dodecadodecahedron.png S+ |2 5/2 5 Snub dodecadodecahedron vertfig.png
3.3.5/2.3.5
Siddid I W111 U40 K45 60 150 84 -6 3 60{3}+12{5}+12{5/2}
Inverted snub dodecadodecahedron Inverted snub dodecadodecahedron.png S+ |5/3 2 5 Inverted snub dodecadodecahedron vertfig.png
3.5/3.3.3.5
Isdid I W114 U60 K65 60 150 84 -6 9 60{3}+12{5}+12{5/2}
Great snub icosidodecahedron Great snub icosidodecahedron.png S+ |2 5/2 3 Great snub icosidodecahedron vertfig.png
3.4.5/2
Gosid I W116 U57 K62 60 150 92 2 7 (20+60){3}+12{5/2}
Great inverted snub icosidodecahedron Great inverted snub icosidodecahedron.png S+ |5/3 2 3 Great inverted snub icosidodecahedron vertfig.png
3.3.5/3
Gisid I W113 U69 K74 60 150 92 2 13 (20+60){3}+12{5/2}
Great retrosnub icosidodecahedron Great retrosnub icosidodecahedron.png S+ |3/25/3 2 Great retrosnub icosidodecahedron vertfig.png
(34.5/2)/2
Girsid I W117 U74 K79 60 150 92 2 23 (20+60){3}+12{5/2}
Great snub dodecicosidodecahedron Great snub dodecicosidodecahedron.png S+ |5/35/2 3 Great snub dodecicosidodecahedron vertfig.png
33.5/3.3.5/2
Gisdid I W115 U64 K69 60 180 104 -16 10 (20+60){3}+(12+12){5/2}
Snub icosidodecadodecahedron Snub icosidodecadodecahedron.png S+ |5/3 3 5 Snub icosidodecadodecahedron vertfig.png
3.3.5.5/3
Sided I W112 U46 K51 60 180 104 -16 4 (20+60){3}+12{5}+12{5/2}
Small snub icosicosidodecahedron Small snub icosicosidodecahedron.png S+ |5/2 3 3 Small snub icosicosidodecahedron vertfig.png
35.5/2
Seside Ih W110 U32 K37 60 180 112 -8 2 (40+60){3}+12{5/2}
Small retrosnub icosicosidodecahedron Small retrosnub icosicosidodecahedron.png S+ |3/23/25/2 Small retrosnub icosicosidodecahedron vertfig.png
(35.5/3)/2
Sirsid Ih W118 U72 K77 60 180 112 -8 22 (40+60){3}+12{5/2}
Great dirhombicosidodecahedron Great dirhombicosidodecahedron.png S+ |3/25/3 3

5/2

Great dirhombicosidodecahedron vertfig.png
(4.5/3.4.3.
4.5/2.4.3/2)/2
Gidrid Ih W119 U75 K80 60 240 124 -56  ?? 40{3}+60{4}+24{5/2}
Icositruncated dodecadodecahedron Icositruncated dodecadodecahedron.png S+ 5/3 3 5| Icositruncated dodecadodecahedron vertfig.png
10/3.6.10
Idtid Ih W084 U45 K50 120 180 44 -16 4 20{6}+12{10}+12{10/3}
Truncated dodecadodecahedron Truncated dodecadodecahedron.png S+ 5/3 2 5| Truncated dodecadodecahedron vertfig.png
10/3.4.10
Quitdid Ih W098 U59 K64 120 180 54 -6 9 30{4}+12{10}+12{10/3}
Great truncated icosidodecahedron Great truncated icosidodecahedron.png S+ 5/3 2 3| Great truncated icosidodecahedron vertfig.png
10/3.4.6
Gaquatid Ih W108 U68 K73 120 180 62 2 13 30{4}+20{6}+12{10/3}

Special case[edit]

Name Picture Solid
class
Wythoff
symbol
Vertex figure Bowers-style
acronym
Symmetry
group
W# U# K# Vertices Edges Faces Chi Density Faces by type
Great disnub dirhombidodecahedron
Skilling's figure
Great disnub dirhombidodecahedron.png S++ | (3/2) 5/3 (3) 5/2 Great disnub dirhombidodecahedron vertfig.png
(5/2.4.3.3.3.4. 5/3.4.3/2.3/2.3/2.4)/2
Gidisdrid Ih -- -- -- 60 240 (*1) 204 24  ?? 120{3}+60{4}+24{5/2}

(*1) : The Great disnub dirhombidodecahedron has 120 edges shared by four faces. If counted as two pairs, then there are a total 360 edges. Because of this edge-degeneracy, it is not always considered a uniform polyhedron.

Column key[edit]

  • Solid classes
    • R = 5 Platonic solids
    • R+= 4 Kepler-Poinsot polyhedra
    • A = 13 Archimedean solids
    • C+= 14 Non-convex polyhedra with only convex faces (all of these uniform polyhedra have faces which intersect each other)
    • S+= 39 Non-convex polyhedra with complex (star) faces
    • S++ = 1 Non-convex polyhedron with complex (star) faces and more than two faces per edge
    • P = Infinite series of Convex Regular Prisms and Antiprisms
    • P+= Infinite series of Non-convex uniform prisms and antiprisms (these all contain complex (star) faces)
  • Bowers style acronym - A unique pronounceable abbreviated name created by mathematician Jonathan Bowers
  • Uniform indexing: U01-U80 (Tetrahedron first, Prisms at 76+)
  • Kaleido software indexing: K01-K80 (Kn = Un-5 for n = 6 to 80) (prisms 1-5, Tetrahedron etc. 6+)
  • Magnus Wenninger Polyhedron Models: W001-W119
    • 1-18 - 5 convex regular and 13 convex semiregular
    • 20-22, 41 - 4 non-convex regular
    • 19-66 Special 48 stellations/compounds (Nonregulars not given on this list)
    • 67-109 - 43 non-convex non-snub uniform
    • 110-119 - 10 non-convex snub uniform
  • Chi: the Euler characteristic, χ. Uniform tilings on the plane correspond to a torus topology, with Euler characteristic of zero.
  • Density: the Density (polytope) represents the number of windings of a polyhedron around its center
  • Note on Vertex figure images:
    • The white polygon lines represent the "vertex figure" polygon. The colored faces are included on the vertex figure images help see their relations. Some of the intersecting faces are drawn visually incorrectly because they are not properly intersected visually to show which portions are in front.

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