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 uniform polyhedra[edit]

Name
Bowers-style acronym
Picture Wythoff
symbol
Vertex
figure
Sym. W# U# K# Vert. Ed. Fa. Χ Den. Faces by type
Tetrahedron
tet
Tetrahedron.png Tetrahedron vertfig.png
3.3.3
3 | 2 3 Td W001 U01 K06 4 6 4 2 1 4{3}
Triangular prism
trip
Triangular prism.png Triangular prism vertfig.png
3.4.4
2 3 | 2 D3h -- U76a K01a 6 9 5 2 1 2{3}
+3{4}
Truncated tetrahedron
tut
Truncated tetrahedron.png Truncated tetrahedron vertfig.png
3.6.6
2 3 | 3 Td W006 U02 K07 12 18 8 2 1 4{3}
+4{6}
Truncated cube
tic
Truncated hexahedron.png Truncated cube vertfig.png
3.8.8
2 3 | 4 Oh W008 U09 K14 24 36 14 2 1 8{3}
+6{8}
Truncated dodecahedron
tid
Truncated dodecahedron.png Truncated dodecahedron vertfig.png
3.10.10
2 3 | 5 Ih W010 U26 K31 60 90 32 2 1 20{3}
+12{10}
Cube
cube
Hexahedron.png Cube vertfig.png
4.4.4
3 | 2 4 Oh W003 U06 K11 8 12 6 2 1 6{4}
Pentagonal prism
pip
Pentagonal prism.png Pentagonal prism vertfig.png
4.4.5
2 5 | 2 D5h -- U76b K01b 10 15 7 2 1 5{4}
+2{5}
Hexagonal prism
hip
Hexagonal prism.png Hexagonal prism vertfig.png
4.4.6
2 6 | 2 D6h -- U76c K01c 12 18 8 2 1 6{4}
+2{6}
Octagonal prism
op
Octagonal prism.png Octagonal prism vertfig.png
4.4.8
2 8 | 2 D8h -- U76e K01e 16 24 10 2 1 8{4}
+2{8}
Decagonal prism
dip
Decagonal prism.png Decagonal prism vf.png
4.4.10
2 10 | 2 D10h -- U76g K01g 20 30 12 2 1 10{4}
+2{10}
Dodecagonal prism
twip
Dodecagonal prism.png Dodecagonal prism vf.png
4.4.12
2 12 | 2 D12h -- U76i K01i 24 36 14 2 1 12{4}
+2{12}
Truncated octahedron
toe
Truncated octahedron.png Truncated octahedron vertfig.png
4.6.6
2 4 | 3 Oh W007 U08 K13 24 36 14 2 1 6{4}
+8{6}
Truncated cuboctahedron
Girco
Great rhombicuboctahedron.png Great rhombicuboctahedron vertfig.png
4.6.8
2 3 4 | Oh W015 U11 K16 48 72 26 2 1 12{4}
+8{6}
+6{8}
Truncated icosidodecahedron
grid
Great rhombicosidodecahedron.png Great rhombicosidodecahedron vertfig.png
4.6.10
2 3 5 | Ih W016 U28 K33 120 180 62 2 1 30{4}
+20{6}
+12{10}
Dodecahedron
doe
Dodecahedron.png Dodecahedron vertfig.png
5.5.5
3 | 2 5 Ih W005 U23 K28 20 30 12 2 1 12{5}
Truncated icosahedron
ti
Truncated icosahedron.png Truncated icosahedron vertfig.png
5.6.6
2 5 | 3 Ih W009 U25 K30 60 90 32 2 1 12{5}
+20{6}
Octahedron
Oct
Octahedron.png Octahedron vertfig.png
3.3.3.3
4 | 2 3 Oh W002 U05 K10 6 12 8 2 1 8{3}
Square antiprism
Squap
Square antiprism.png Square antiprism vertfig.png
3.3.3.4
| 2 2 4 D4d -- U77a K02a 8 16 10 2 1 8{3}
+2{4}
Pentagonal antiprism
Pap
Pentagonal antiprism.png Pentagonal antiprism vertfig.png
3.3.3.5
| 2 2 5 D5d -- U77b K02b 10 20 12 2 1 10{3}
+2{5}
Hexagonal antiprism
Hap
Hexagonal antiprism.png Hexagonal antiprism vertfig.png
3.3.3.6
| 2 2 6 D6d -- U77c K02c 12 24 14 2 1 12{3}
+2{6}
Octagonal antiprism
Oap
Octagonal antiprism.png Octagonal antiprism vertfig.png
3.3.3.8
| 2 2 8 D8d -- U77e K02e 16 32 18 2 1 16{3}
+2{8}
Decagonal antiprism
Dap
Decagonal antiprism.png Decagonal antiprism vf.png
3.3.3.10
| 2 2 10 D10d -- U77g K02g 20 40 22 2 1 20{3}
+2{10}
Dodecagonal antiprism
Twap
Dodecagonal antiprism.png Dodecagonal antiprism vf.png
3.3.3.12
| 2 2 12 D12d -- U77i K02i 24 48 26 2 1 24{3}
+2{12}
Cuboctahedron
Co
Cuboctahedron.png Cuboctahedron vertfig.png
3.4.3.4
2 | 3 4 Oh W011 U07 K12 12 24 14 2 1 8{3}
+6{4}
Rhombicuboctahedron
Sirco
Small rhombicuboctahedron.png Small rhombicuboctahedron vertfig.png
3.4.4.4
3 4 | 2 Oh W013 U10 K15 24 48 26 2 1 8{3}
+(6+12){4}
Rhombicosidodecahedron
Srid
Small rhombicosidodecahedron.png Small rhombicosidodecahedron vertfig.png
3.4.5.4
3 5 | 2 Ih W014 U27 K32 60 120 62 2 1 20{3}
+30{4}
+12{5}
Icosidodecahedron
Id
Icosidodecahedron.png Icosidodecahedron vertfig.png
3.5.3.5
2 | 3 5 Ih W012 U24 K29 30 60 32 2 1 20{3}
+12{5}
Icosahedron
ike
Icosahedron.png Icosahedron vertfig.png
3.3.3.3.3
5 | 2 3 Ih W004 U22 K27 12 30 20 2 1 20{3}
Snub cube
snic
Snub hexahedron.png Snub cube vertfig.png
3.3.3.3.4
| 2 3 4 O W017 U12 K17 24 60 38 2 1 (8+24){3}
+6{4}
Snub dodecahedron
snid
Snub dodecahedron ccw.png Snub dodecahedron vertfig.png
3.3.3.3.5
| 2 3 5 I W018 U29 K34 60 150 92 2 1 (20+60){3}
+12{5}

Uniform star polyhedra[edit]

Name Image Wyth
sym
Vert.
fig
Sym. W# U# K# Vert. Ed. Faces Chi Dens. Faces by type
Octahemioctahedron Octahemioctahedron.png 3/2 3 | 3 Octahemioctahedron vertfig.png
6.3/2.6.3
Oh W068 U03 K08 12 24 12 0 4 8{3}+4{6}
Tetrahemihexahedron Tetrahemihexahedron.png 3/2 3 | 2 Tetrahemihexahedron vertfig.png
4.3/2.4.3
Td W067 U04 K09 6 12 7 1 3 4{3}+3{4}
Cubohemioctahedron Cubohemioctahedron.png 4/3 4 | 3 Cubohemioctahedron vertfig.png
6.4/3.6.4
Oh W078 U15 K20 12 24 10 -2 4 6{4}+4{6}
Great
dodecahedron
Great dodecahedron.png 5/2 | 2 5 Great dodecahedron vertfig.png
(5.5.5.5.5)/2
Ih W021 U35 K40 12 30 12 -6 3 12{5}
Great
icosahedron
Great icosahedron.png 5/2 | 2 3 Great icosahedron vertfig.png
(3.3.3.3.3)/2
Ih W041 U53 K58 12 30 20 2 7 20{3}
Great
ditrigonal
icosidodecahedron
Great ditrigonal icosidodecahedron.png 3/2 | 3 5 Great ditrigonal icosidodecahedron vertfig.png
(5.3.5.3.5.3)/2
Ih W087 U47 K52 20 60 32 -8 6 20{3}+12{5}
Small
rhombihexahedron
Small rhombihexahedron.png 3/2 2 4 | Small rhombihexahedron vertfig.png
4.8.4/3.8
Oh W086 U18 K23 24 48 18 -6 5 12{4}+6{8}
Small
cubicuboctahedron
Small cubicuboctahedron.png 3/2 4 | 4 Small cubicuboctahedron vertfig.png
8.3/2.8.4
Oh W069 U13 K18 24 48 20 -4 2 8{3}+6{4}+6{8}
great
rhombicuboctahedron
Uniform great rhombicuboctahedron.png 3/2 4 | 2 Uniform great rhombicuboctahedron vertfig.png
4.3/2.4.4
Oh W085 U17 K22 24 48 26 2 5 8{3}+(6+12){4}
Small dodecahemi-
dodecahedron
Small dodecahemidodecahedron.png 5/4 5 | 5 Small dodecahemidodecahedron vertfig.png
10.5/4.10.5
Ih W091 U51 K56 30 60 18 -12 6 12{5}+6{10}
Great dodecahem-
icosahedron
Great dodecahemicosahedron.png 5/4 5 | 3 Great dodecahemicosahedron vertfig.png
6.5/4.6.5
Ih W102 U65 K70 30 60 22 -8 10 12{5}+10{6}
Small icosihemi-
dodecahedron
Small icosihemidodecahedron.png 3/2 3 | 5 Small icosihemidodecahedron vertfig.png
10.3/2.10.3
Ih W089 U49 K54 30 60 26 -4 6 20{3}+6{10}
Small
dodecicosahedron
Small dodecicosahedron.png 3/2 3 5 | Small dodecicosahedron vertfig.png
10.6.10/9.6/5
Ih W090 U50 K55 60 120 32 -28 6 20{6}+12{10}
Small
rhombidodecahedron
Small rhombidodecahedron.png 2 5/2 5 | Small rhombidodecahedron vertfig.png
10.4.10/9.4/3
Ih W074 U39 K44 60 120 42 -18 3 30{4}+12{10}
Small dodecicosi-
dodecahedron
Small dodecicosidodecahedron.png 3/2 5 | 5 Small dodecicosidodecahedron vertfig.png
10.3/2.10.5
Ih W072 U33 K38 60 120 44 -16 2 20{3}+12{5}+12{10}
Rhombicosahedron Rhombicosahedron.png 2 5/2 3 | Rhombicosahedron vertfig.png
6.4.6/5.4/3
Ih W096 U56 K61 60 120 50 -10 7 30{4}+20{6}
Great
icosicosi-
dodecahedron
Great icosicosidodecahedron.png 3/2 5 | 3 Great icosicosidodecahedron vertfig.png
6.3/2.6.5
Ih W088 U48 K53 60 120 52 -8 6 20{3}+12{5}+20{6}
Pentagrammic
prism
Pentagrammic prism.png 2 5/2 | 2 Pentagrammic prism vertfig.png
5/2.4.4
D5h -- U78 K03 10 15 7 2 2 5{4}+2{5/2}
Heptagrammic
prism (7/3)
Heptagrammic prism 7-3.png 2 7/3 | 2 Septagrammic prism-3-7 vertfig.png
7/3.4.4
D7h -- -- -- 14 21 9 2 3 7{4}+2{7/3}
Heptagrammic
prism (7/2)
Heptagrammic prism 7-2.png 2 7/2 | 2 Septagrammic prism vertfig.png
7/2.4.4
D7h -- -- -- 14 21 9 2 2 7{4}+2{7/2}
Octagrammic
prism
Prism 8-3.png 2 8/3 | 2 Octagrammic prism vertfig.png
8/3.4.4
D8h -- -- -- 16 24 10 2 3 8{4}+2{8/3}
Pentagrammic antiprism Pentagrammic antiprism.png | 2 2 5/2 Pentagrammic antiprism vertfig.png
5/2.3.3.3
D5h -- U79 K04 10 20 12 2 2 10{3}+2{5/2}
Pentagrammic
crossed-antiprism
Pentagrammic crossed antiprism.png | 2 2 5/3 Pentagrammic crossed-antiprism vertfig.png
5/3.3.3.3
D5d -- U80 K05 10 20 12 2 3 10{3}+2{5/2}
Small
stellated
dodecahedron
Small stellated dodecahedron.png 5 | 2 5/2 Small stellated dodecahedron vertfig.png
(5/2)5
Ih W020 U34 K39 12 30 12 -6 3 12{5/2}
Great
stellated
dodecahedron
Great stellated dodecahedron.png 3 | 2 5/2 Great stellated dodecahedron vertfig.png
(5/2)3
Ih W022 U52 K57 20 30 12 2 7 12{5/2}
Ditrigonal
dodeca-
dodecahedron
Ditrigonal dodecadodecahedron.png 3 | 5/3 5 Ditrigonal dodecadodecahedron vertfig.png
(5/3.5)3
Ih W080 U41 K46 20 60 24 -16 4 12{5}+12{5/2}
Small
ditrigonal
icosidodecahedron
Small ditrigonal icosidodecahedron.png 3 | 5/2 3 Small ditrigonal icosidodecahedron vertfig.png
(5/2.3)3
Ih W070 U30 K35 20 60 32 -8 2 20{3}+12{5/2}
Stellated
truncated
hexahedron
Stellated truncated hexahedron.png 2 3 | 4/3 Stellated truncated hexahedron vertfig.png
8/3.8/3.3
Oh W092 U19 K24 24 36 14 2 7 8{3}+6{8/3}
Great
rhombihexahedron
Great rhombihexahedron.png 4/33/2 2 | Great rhombihexahedron vertfig.png
4.8/3.4/3.8/5
Oh W103 U21 K26 24 48 18 -6 11 12{4}+6{8/3}
Great
cubicuboctahedron
Great cubicuboctahedron.png 3 4 | 4/3 Great cubicuboctahedron vertfig.png
8/3.3.8/3.4
Oh W077 U14 K19 24 48 20 -4 4 8{3}+6{4}+6{8/3}
Great dodecahemi-
dodecahedron
Great dodecahemidodecahedron.png 5/35/2 | 5/3 Great dodecahemidodecahedron vertfig.png
10/3.5/3.10/3.5/2
Ih W107 U70 K75 30 60 18 -12 18 12{5/2}+6{10/3}
Small dodecahemi-
cosahedron
Small dodecahemicosahedron.png 5/35/2 | 3 Small dodecahemicosahedron vertfig.png
6.5/3.6.5/2
Ih W100 U62 K67 30 60 22 -8 10 12{5/2}+10{6}
Dodeca-
dodecahedron
Dodecadodecahedron.png 2 | 5/2 5 Dodecadodecahedron vertfig.png
(5/2.5)2
Ih W073 U36 K41 30 60 24 -6 3 12{5}+12{5/2}
Great icosihemi-
dodecahedron
Great icosihemidodecahedron.png 3/2 3 | 5/3 Great icosihemidodecahedron vertfig.png
10/3.3/2.10/3.3
Ih W106 U71 K76 30 60 26 -4 18 20{3}+6{10/3}
Great
icosidodecahedron
Great icosidodecahedron.png 2 | 5/2 3 Great icosidodecahedron vertfig.png
(5/2.3)2
Ih W094 U54 K59 30 60 32 2 7 20{3}+12{5/2}
Cubitruncated
cuboctahedron
Cubitruncated cuboctahedron.png 4/3 3 4 | Cubitruncated cuboctahedron vertfig.png
8/3.6.8
Oh W079 U16 K21 48 72 20 -4 4 8{6}+6{8}+6{8/3}
Great
truncated
cuboctahedron
Great truncated cuboctahedron.png 4/3 2 3 | Great truncated cuboctahedron vertfig.png
8/3.4.6
Oh W093 U20 K25 48 72 26 2 7 12{4}+8{6}+6{8/3}
Truncated
great
dodecahedron
Great truncated dodecahedron.png 2 5/2 | 5 Truncated great dodecahedron vertfig.png
10.10.5/2
Ih W075 U37 K42 60 90 24 -6 3 12{5/2}+12{10}
Small stellated
truncated
dodecahedron
Small stellated truncated dodecahedron.png 2 5 | 5/3 Small stellated truncated dodecahedron vertfig.png
10/3.10/3.5
Ih W097 U58 K63 60 90 24 -6 9 12{5}+12{10/3}
Great stellated
truncated
dodecahedron
Great stellated truncated dodecahedron.png 2 3 | 5/3 Great stellated truncated dodecahedron vertfig.png
10/3.10/3.3
Ih W104 U66 K71 60 90 32 2 13 20{3}+12{10/3}
Truncated
great
icosahedron
Great truncated icosahedron.png 2 5/2 | 3 Great truncated icosahedron vertfig.png
6.6.5/2
Ih W095 U55 K60 60 90 32 2 7 12{5/2}+20{6}
Great
dodecicosahedron
Great dodecicosahedron.png 5/35/2 3 | Great dodecicosahedron vertfig.png
6.10/3.6/5.10/7
Ih W101 U63 K68 60 120 32 -28 10 20{6}+12{10/3}
Great
rhombidodecahedron
Great rhombidodecahedron.png 3/25/3 2 | Great rhombidodecahedron vertfig.png
4.10/3.4/3.10/7
Ih W109 U73 K78 60 120 42 -18 23 30{4}+12{10/3}
Icosidodeca-
dodecahedron
Icosidodecadodecahedron.png 5/3 5 | 3 Icosidodecadodecahedron vertfig.png
6.5/3.6.5
Ih W083 U44 K49 60 120 44 -16 4 12{5}+12{5/2}+20{6}
Small ditrigonal
dodecicosi-
dodecahedron
Small ditrigonal dodecicosidodecahedron.png 5/3 3 | 5 Small ditrigonal dodecicosidodecahedron vertfig.png
10.5/3.10.3
Ih W082 U43 K48 60 120 44 -16 4 20{3}+12{;5/2}+12{10}
Great ditrigonal
dodecicosi-
dodecahedron
Great ditrigonal dodecicosidodecahedron.png 3 5 | 5/3 Great ditrigonal dodecicosidodecahedron vertfig.png
10/3.3.10/3.5
Ih W081 U42 K47 60 120 44 -16 4 20{3}+12{5}+12{10/3}
Great
dodecicosi-
dodecahedron
Great dodecicosidodecahedron.png 5/2 3 | 5/3 Great dodecicosidodecahedron vertfig.png
10/3.5/2.10/3.3
Ih W099 U61 K66 60 120 44 -16 10 20{3}+12{5/2}+12{10/3}
Small icosicosi-
dodecahedron
Small icosicosidodecahedron.png 5/2 3 | 3 Small icosicosidodecahedron vertfig.png
6.5/2.6.3
Ih W071 U31 K36 60 120 52 -8 2 20{3}+12{5/2}+20{6}
Rhombidodeca-
dodecahedron
Rhombidodecadodecahedron.png 5/2 5 | 2 Rhombidodecadodecahedron vertfig.png
4.5/2.4.5
Ih W076 U38 K43 60 120 54 -6 3 30{4}+12{5}+12{5/2}
Great
rhombicosi-
dodecahedron
Uniform great rhombicosidodecahedron.png 5/3 3 | 2 Uniform great rhombicosidodecahedron vertfig.png
4.5/3.4.3
Ih W105 U67 K72 60 120 62 2 13 20{3}+30{4}+12{5/2}
Snub dodeca-
dodecahedron
Snub dodecadodecahedron.png | 2 5/2 5 Snub dodecadodecahedron vertfig.png
3.3.5/2.3.5
I W111 U40 K45 60 150 84 -6 3 60{3}+12{5}+12{5/2}
Inverted
snub dodeca-
dodecahedron
Inverted snub dodecadodecahedron.png | 5/3 2 5 Inverted snub dodecadodecahedron vertfig.png
3.5/3.3.3.5
I W114 U60 K65 60 150 84 -6 9 60{3}+12{5}+12{5/2}
Great
snub
icosidodecahedron
Great snub icosidodecahedron.png | 2 5/2 3 Great snub icosidodecahedron vertfig.png
3.4.5/2
I W116 U57 K62 60 150 92 2 7 (20+60){3}+12{5/2}
Great
inverted
snub
icosidodecahedron
Great inverted snub icosidodecahedron.png | 5/3 2 3 Great inverted snub icosidodecahedron vertfig.png
3.3.5/3
I W113 U69 K74 60 150 92 2 13 (20+60){3}+12{5/2}
Great
retrosnub
icosidodecahedron
Great retrosnub icosidodecahedron.png | 3/25/3 2 Great retrosnub icosidodecahedron vertfig.png
(34.5/2)/2
I W117 U74 K79 60 150 92 2 23 (20+60){3}+12{5/2}
Great
snub
dodecicosi-
dodecahedron
Great snub dodecicosidodecahedron.png | 5/35/2 3 Great snub dodecicosidodecahedron vertfig.png
33.5/3.3.5/2
I W115 U64 K69 60 180 104 -16 10 (20+60){3}+(12+12){5/2}
Snub
icosidodeca-
dodecahedron
Snub icosidodecadodecahedron.png | 5/3 3 5 Snub icosidodecadodecahedron vertfig.png
3.3.5.5/3
I W112 U46 K51 60 180 104 -16 4 (20+60){3}+12{5}+12{5/2}
Small snub icos-
icosidodecahedron
Small snub icosicosidodecahedron.png | 5/2 3 3 Small snub icosicosidodecahedron vertfig.png
35.5/2
Ih W110 U32 K37 60 180 112 -8 2 (40+60){3}+12{5/2}
Small retrosnub
icosicosi-
dodecahedron
Small retrosnub icosicosidodecahedron.png | 3/23/25/2 Small retrosnub icosicosidodecahedron vertfig.png
(35.5/3)/2
Ih W118 U72 K77 60 180 112 -8 22 (40+60){3}+12{5/2}
Great
dirhombicosi-
dodecahedron
Great dirhombicosidodecahedron.png | 3/25/3 3

5/2

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

Special case[edit]

Name
Bowers-style acronym
Picture Wythoff
symbol
Vertex figure Symmetry
group
W# U# K# Vertices Edges Faces Chi Density Faces by type
Great disnub dirhombidodecahedron
Skilling's figure
gidisdrid
Great disnub dirhombidodecahedron.png | (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
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]

  • 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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