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In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry.

Uniform polyhedra can be divided between convex forms with convex regular polygon faces and star forms. Star forms have either regular star polygon faces or vertex figures or both.

This list includes these:

It was proven in 1970 that there are only 75 uniform polyhedra other than the infinite families of prisms and antiprisms. John Skilling discovered an overlooked degenerate example, by relaxing the condition that only two faces may meet at an edge. This is a degenerate uniform polyhedron rather than a uniform polyhedron, because some pairs of edges coincide.

Not included are:

Contents

IndexingEdit

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.

Names of polyhedra by number of sidesEdit

The generic geometric names for the most common polyhedra. The 5 regular polyhedra are called a tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron with 4, 6, 8, 12, and 20 sides respectively.

Table of polyhedraEdit

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 polyhedraEdit

Name Picture Vertex
type
Wythoff
symbol
Sym. C# W# U# K# Vert. Edges Faces Faces by type
Tetrahedron    
3.3.3
3 | 2 3 Td C15 W001 U01 K06 4 6 4 4{3}
Triangular prism    
3.4.4
2 3 | 2 D3h C33a -- U76a K01a 6 9 5 2{3}
+3{4}
Truncated tetrahedron    
3.6.6
2 3 | 3 Td C16 W006 U02 K07 12 18 8 4{3}
+4{6}
Truncated cube    
3.8.8
2 3 | 4 Oh C21 W008 U09 K14 24 36 14 8{3}
+6{8}
Truncated dodecahedron    
3.10.10
2 3 | 5 Ih C29 W010 U26 K31 60 90 32 20{3}
+12{10}
Cube    
4.4.4
3 | 2 4 Oh C18 W003 U06 K11 8 12 6 6{4}
Pentagonal prism    
4.4.5
2 5 | 2 D5h C33b -- U76b K01b 10 15 7 5{4}
+2{5}
Hexagonal prism    
4.4.6
2 6 | 2 D6h C33c -- U76c K01c 12 18 8 6{4}
+2{6}
Octagonal prism    
4.4.8
2 8 | 2 D8h C33e -- U76e K01e 16 24 10 8{4}
+2{8}
Decagonal prism    
4.4.10
2 10 | 2 D10h C33g -- U76g K01g 20 30 12 10{4}
+2{10}
Dodecagonal prism    
4.4.12
2 12 | 2 D12h C33i -- U76i K01i 24 36 14 12{4}
+2{12}
Truncated octahedron    
4.6.6
2 4 | 3 Oh C20 W007 U08 K13 24 36 14 6{4}
+8{6}
Truncated cuboctahedron    
4.6.8
2 3 4 | Oh C23 W015 U11 K16 48 72 26 12{4}
+8{6}
+6{8}
Truncated icosidodecahedron    
4.6.10
2 3 5 | Ih C31 W016 U28 K33 120 180 62 30{4}
+20{6}
+12{10}
Dodecahedron    
5.5.5
3 | 2 5 Ih C26 W005 U23 K28 20 30 12 12{5}
Truncated icosahedron    
5.6.6
2 5 | 3 Ih C27 W009 U25 K30 60 90 32 12{5}
+20{6}
Octahedron    
3.3.3.3
4 | 2 3 Oh C17 W002 U05 K10 6 12 8 8{3}
Square antiprism    
3.3.3.4
| 2 2 4 D4d C34a -- U77a K02a 8 16 10 8{3}
+2{4}
Pentagonal antiprism    
3.3.3.5
| 2 2 5 D5d C34b -- U77b K02b 10 20 12 10{3}
+2{5}
Hexagonal antiprism    
3.3.3.6
| 2 2 6 D6d C34c -- U77c K02c 12 24 14 12{3}
+2{6}
Octagonal antiprism    
3.3.3.8
| 2 2 8 D8d C34e -- U77e K02e 16 32 18 16{3}
+2{8}
Decagonal antiprism    
3.3.3.10
| 2 2 10 D10d C34g -- U77g K02g 20 40 22 20{3}
+2{10}
Dodecagonal antiprism    
3.3.3.12
| 2 2 12 D12d C34i -- U77i K02i 24 48 26 24{3}
+2{12}
Cuboctahedron    
3.4.3.4
2 | 3 4 Oh C19 W011 U07 K12 12 24 14 8{3}
+6{4}
Rhombicuboctahedron    
3.4.4.4
3 4 | 2 Oh C22 W013 U10 K15 24 48 26 8{3}
+(6+12){4}
Rhombicosidodecahedron    
3.4.5.4
3 5 | 2 Ih C30 W014 U27 K32 60 120 62 20{3}
+30{4}
+12{5}
Icosidodecahedron    
3.5.3.5
2 | 3 5 Ih C28 W012 U24 K29 30 60 32 20{3}
+12{5}
Icosahedron    
3.3.3.3.3
5 | 2 3 Ih C25 W004 U22 K27 12 30 20 20{3}
Snub cube    
3.3.3.3.4
| 2 3 4 O C24 W017 U12 K17 24 60 38 (8+24){3}
+6{4}
Snub dodecahedron    
3.3.3.3.5
| 2 3 5 I C32 W018 U29 K34 60 150 92 (20+60){3}
+12{5}

Uniform star polyhedraEdit

Name Image Wyth
sym
Vert.
fig
Sym. C# W# U# K# Vert. Edges Faces Chi Orient
able?
Dens. Faces by type
Octahemioctahedron   3/2 3 | 3  
6.3/2.6.3
Oh C37 W068 U03 K08 12 24 12 0 Yes   8{3}+4{6}
Tetrahemihexahedron   3/2 3 | 2  
4.3/2.4.3
Td C36 W067 U04 K09 6 12 7 1 No   4{3}+3{4}
Cubohemioctahedron   4/3 4 | 3  
6.4/3.6.4
Oh C51 W078 U15 K20 12 24 10 -2 No   6{4}+4{6}
Great
dodecahedron
  5/2 | 2 5  
(5.5.5.5.5)/2
Ih C44 W021 U35 K40 12 30 12 -6 Yes 3 12{5}
Great
icosahedron
  5/2 | 2 3  
(3.3.3.3.3)/2
Ih C69 W041 U53 K58 12 30 20 2 Yes 7 20{3}
Great
ditrigonal
icosidodecahedron
  3/2 | 3 5  
(5.3.5.3.5.3)/2
Ih C61 W087 U47 K52 20 60 32 -8 Yes 6 20{3}+12{5}
Small
rhombihexahedron
  2 4 (3/2 4/2) |  
4.8.4/3.8/7
Oh C60 W086 U18 K23 24 48 18 -6 No   12{4}+6{8}
Small
cubicuboctahedron
  3/2 4 | 4  
8.3/2.8.4
Oh C38 W069 U13 K18 24 48 20 -4 Yes 2 8{3}+6{4}+6{8}
Great
rhombicuboctahedron
  3/2 4 | 2  
4.3/2.4.4
Oh C59 W085 U17 K22 24 48 26 2 Yes 5 8{3}+(6+12){4}
Small dodecahemi-
dodecahedron
  5/4 5 | 5  
10.5/4.10.5
Ih C65 W091 U51 K56 30 60 18 -12 No   12{5}+6{10}
Great dodecahem-
icosahedron
  5/4 5 | 3  
6.5/4.6.5
Ih C81 W102 U65 K70 30 60 22 -8 No   12{5}+10{6}
Small icosihemi-
dodecahedron
  3/2 3 | 5  
10.3/2.10.3
Ih C63 W089 U49 K54 30 60 26 -4 No   20{3}+6{10}
Small
dodecicosahedron
  3 5 (3/2 5/4) |  
10.6.10/9.6/5
Ih C64 W090 U50 K55 60 120 32 -28 No   20{6}+12{10}
Small
rhombidodecahedron
  2 5 (3/2 5/2) |  
10.4.10/9.4/3
Ih C46 W074 U39 K44 60 120 42 -18 No   30{4}+12{10}
Small dodecicosi-
dodecahedron
  3/2 5 | 5  
10.3/2.10.5
Ih C42 W072 U33 K38 60 120 44 -16 Yes 2 20{3}+12{5}+12{10}
Rhombicosahedron   2 3 (5/4 5/2) |  
6.4.6/5.4/3
Ih C72 W096 U56 K61 60 120 50 -10 No   30{4}+20{6}
Great
icosicosi-
dodecahedron
  3/2 5 | 3  
6.3/2.6.5
Ih C62 W088 U48 K53 60 120 52 -8 Yes 6 20{3}+12{5}+20{6}
Pentagrammic
prism
  2 5/2 | 2  
5/2.4.4
D5h C33b -- U78a K03a 10 15 7 2 Yes 2 5{4}+2{5/2}
Heptagrammic
prism (7/2)
  2 7/2 | 2  
7/2.4.4
D7h C33d -- U78b K03b 14 21 9 2 Yes 2 7{4}+2{7/2}
Heptagrammic
prism (7/3)
  2 7/3 | 2  
7/3.4.4
D7h C33d -- U78c K03c 14 21 9 2 Yes 3 7{4}+2{7/3}
Octagrammic
prism
  2 8/3 | 2  
8/3.4.4
D8h C33e -- U78d K03d 16 24 10 2 Yes 3 8{4}+2{8/3}
Pentagrammic antiprism   | 2 2 5/2  
5/2.3.3.3
D5h C34b -- U79a K04a 10 20 12 2 Yes 2 10{3}+2{5/2}
Pentagrammic
crossed-antiprism
  | 2 2 5/3  
5/3.3.3.3
D5d C35a -- U80a K05a 10 20 12 2 Yes 3 10{3}+2{5/2}
Heptagrammic
antiprism (7/2)
  | 2 2 7/2  
7/2.3.3.3
D7h C34d -- U79b K04b 14 28 16 2 Yes 3 14{3}+2{7/2}
Heptagrammic
antiprism (7/3)
  | 2 2 7/3  
7/3.3.3.3
D7d C34d -- U79c K04c 14 28 16 2 Yes 3 14{3}+2{7/3}
Heptagrammic
crossed-antiprism
  | 2 2 7/4  
7/4.3.3.3
D7h C35b -- U80b K05b 14 28 16 2 Yes 4 14{3}+2{7/3}
Octagrammic
antiprism
  | 2 2 8/3  
8/3.3.3.3
D8d C34e -- U79d K04d 16 32 18 2 Yes 3 16{3}+2{8/3}
Octagrammic
crossed-antiprism
  | 2 2 8/5  
8/5.3.3.3
D8d C35c -- U80c K05c 16 32 18 2 Yes 5 16{3}+2{8/3}
Small
stellated
dodecahedron
  5 | 2 5/2  
(5/2)5
Ih C43 W020 U34 K39 12 30 12 -6 Yes 3 12{5/2}
Great
stellated
dodecahedron
  3 | 2 5/2  
(5/2)3
Ih C68 W022 U52 K57 20 30 12 2 Yes 7 12{5/2}
Ditrigonal
dodeca-
dodecahedron
  3 | 5/3 5  
(5/3.5)3
Ih C53 W080 U41 K46 20 60 24 -16 Yes 4 12{5}+12{5/2}
Small
ditrigonal
icosidodecahedron
  3 | 5/2 3  
(5/2.3)3
Ih C39 W070 U30 K35 20 60 32 -8 Yes 2 20{3}+12{5/2}
Stellated
truncated
hexahedron
  2 3 | 4/3  
8/3.8/3.3
Oh C66 W092 U19 K24 24 36 14 2 Yes 7 8{3}+6{8/3}
Great
rhombihexahedron
  2 4/3 (3/2 4/2) |  
4.8/3.4/3.8/5
Oh C82 W103 U21 K26 24 48 18 -6 No   12{4}+6{8/3}
Great
cubicuboctahedron
  3 4 | 4/3  
8/3.3.8/3.4
Oh C50 W077 U14 K19 24 48 20 -4 Yes 4 8{3}+6{4}+6{8/3}
Great dodecahemi-
dodecahedron
  5/35/2 | 5/3  
10/3.5/3.10/3.5/2
Ih C86 W107 U70 K75 30 60 18 -12 No   12{5/2}+6{10/3}
Small dodecahemi-
cosahedron
  5/35/2 | 3  
6.5/3.6.5/2
Ih C78 W100 U62 K67 30 60 22 -8 No   12{5/2}+10{6}
Dodeca-
dodecahedron
  2 | 5/2 5  
(5/2.5)2
Ih C45 W073 U36 K41 30 60 24 -6 Yes 3 12{5}+12{5/2}
Great icosihemi-
dodecahedron
  3/2 3 | 5/3  
10/3.3/2.10/3.3
Ih C85 W106 U71 K76 30 60 26 -4 No   20{3}+6{10/3}
Great
icosidodecahedron
  2 | 5/2 3  
(5/2.3)2
Ih C70 W094 U54 K59 30 60 32 2 Yes 7 20{3}+12{5/2}
Cubitruncated
cuboctahedron
  4/3 3 4 |  
8/3.6.8
Oh C52 W079 U16 K21 48 72 20 -4 Yes 4 8{6}+6{8}+6{8/3}
Great
truncated
cuboctahedron
  4/3 2 3 |  
8/3.4.6/5
Oh C67 W093 U20 K25 48 72 26 2 Yes 1 12{4}+8{6}+6{8/3}
Truncated
great
dodecahedron
  2 5/2 | 5  
10.10.5/2
Ih C47 W075 U37 K42 60 90 24 -6 Yes 3 12{5/2}+12{10}
Small stellated
truncated
dodecahedron
  2 5 | 5/3  
10/3.10/3.5
Ih C74 W097 U58 K63 60 90 24 -6 Yes 9 12{5}+12{10/3}
Great stellated
truncated
dodecahedron
  2 3 | 5/3  
10/3.10/3.3
Ih C83 W104 U66 K71 60 90 32 2 Yes 13 20{3}+12{10/3}
Truncated
great
icosahedron
  2 5/2 | 3  
6.6.5/2
Ih C71 W095 U55 K60 60 90 32 2 Yes 7 12{5/2}+20{6}
Great
dodecicosahedron
  3 5/3(3/2 5/2) |  
6.10/3.6/5.10/7
Ih C79 W101 U63 K68 60 120 32 -28 No   20{6}+12{10/3}
Great
rhombidodecahedron
  2 5/3 (3/2 5/4) |  
4.10/3.4/3.10/7
Ih C89 W109 U73 K78 60 120 42 -18 No   30{4}+12{10/3}
Icosidodeca-
dodecahedron
  5/3 5 | 3  
6.5/3.6.5
Ih C56 W083 U44 K49 60 120 44 -16 Yes 4 12{5}+12{5/2}+20{6}
Small ditrigonal
dodecicosi-
dodecahedron
  5/3 3 | 5  
10.5/3.10.3
Ih C55 W082 U43 K48 60 120 44 -16 Yes 4 20{3}+12{5/2}+12{10}
Great ditrigonal
dodecicosi-
dodecahedron
  3 5 | 5/3  
10/3.3.10/3.5
Ih C54 W081 U42 K47 60 120 44 -16 Yes 4 20{3}+12{5}+12{10/3}
Great
dodecicosi-
dodecahedron
  5/2 3 | 5/3  
10/3.5/2.10/3.3
Ih C77 W099 U61 K66 60 120 44 -16 Yes 10 20{3}+12{5/2}+12{10/3}
Small icosicosi-
dodecahedron
  5/2 3 | 3  
6.5/2.6.3
Ih C40 W071 U31 K36 60 120 52 -8 Yes 2 20{3}+12{5/2}+20{6}
Rhombidodeca-
dodecahedron
  5/2 5 | 2  
4.5/2.4.5
Ih C48 W076 U38 K43 60 120 54 -6 Yes 3 30{4}+12{5}+12{5/2}
Great
rhombicosi-
dodecahedron
  5/3 3 | 2  
4.5/3.4.3
Ih C84 W105 U67 K72 60 120 62 2 Yes 13 20{3}+30{4}+12{5/2}
Icositruncated
dodeca-
dodecahedron
  5/3 3 5 |  
10/3.6.10
Ih C57 W084 U45 K50 120 180 44 -16 Yes 4 20{6}+12{10}+12{10/3}
Truncated
dodeca-
dodecahedron
  5/3 2 5 |  
10/3.4.10/9
Ih C75 W098 U59 K64 120 180 54 -6 Yes 3 30{4}+12{10}+12{10/3}
Great
truncated
icosidodecahedron
  5/3 2 3 |  
10/3.4.6
Ih C87 W108 U68 K73 120 180 62 2 Yes 13 30{4}+20{6}+12{10/3}
Snub dodeca-
dodecahedron
  | 2 5/2 5  
3.3.5/2.3.5
I C49 W111 U40 K45 60 150 84 -6 Yes 3 60{3}+12{5}+12{5/2}
Inverted
snub dodeca-
dodecahedron
  | 5/3 2 5  
3.5/3.3.3.5
I C76 W114 U60 K65 60 150 84 -6 Yes 9 60{3}+12{5}+12{5/2}
Great
snub
icosidodecahedron
  | 2 5/2 3  
34.5/2
I C73 W116 U57 K62 60 150 92 2 Yes 7 (20+60){3}+12{5/2}
Great
inverted
snub
icosidodecahedron
  | 5/3 2 3  
34.5/3
I C88 W113 U69 K74 60 150 92 2 Yes 13 (20+60){3}+12{5/2}
Great
retrosnub
icosidodecahedron
  | 3/25/3 2  
(34.5/2)/2
I C90 W117 U74 K79 60 150 92 2 Yes 37 (20+60){3}+12{5/2}
Great
snub
dodecicosi-
dodecahedron
  | 5/35/2 3  
33.5/3.3.5/2
I C80 W115 U64 K69 60 180 104 -16 Yes 10 (20+60){3}+(12+12){5/2}
Snub
icosidodeca-
dodecahedron
  | 5/3 3 5  
33.5.5/3
I C58 W112 U46 K51 60 180 104 -16 Yes 4 (20+60){3}+12{5}+12{5/2}
Small snub icos-
icosidodecahedron
  | 5/2 3 3  
35.5/2
Ih C41 W110 U32 K37 60 180 112 -8 Yes 2 (40+60){3}+12{5/2}
Small retrosnub
icosicosi-
dodecahedron
  | 3/23/25/2  
(35.5/3)/2
Ih C91 W118 U72 K77 60 180 112 -8 Yes 38 (40+60){3}+12{5/2}
Great
dirhombicosi-
dodecahedron
  | 3/25/3 3 5/2  
(4.5/3.4.3.
4.5/2.4.3/2)/2
Ih C92 W119 U75 K80 60 240 124 -56 No   40{3}+60{4}+24{5/2}

Name Image Wyth
sym
Vert.
fig
Sym. C# W# U# K# Vert. Edges Faces Chi Orient
able?
Dens. Faces by type
Great disnub
dirhombidodecahedron
*
  | (3/2) 5/3 (3) 5/2  
(5/2.4.3.3.3.4. 5/3.
4.3/2.3/2.3/2.4)/2
Ih -- -- -- -- 60 360 (*) 204 -96 No   120{3}+60{4}+24{5/2}

(*) : The great disnubdirhombidodecahedron has 240 of its 360 edges coinciding in space 120 pairs. Because of this edge-degeneracy, it is not always considered to be a uniform polyhedron.

Column keyEdit

  • 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. This is left blank for non-orientable polyhedra and hemipolyhedra (polyhedra with faces passing through their centers), for which the density is not well-defined.
  • 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.

See alsoEdit

ReferencesEdit

  • Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9.
  • Wenninger, Magnus (1983). Dual Models. Cambridge University Press. ISBN 0-521-54325-8.
  • Coxeter, Harold Scott MacDonald; Longuet-Higgins, M. S.; Miller, J. C. P. (1954). "Uniform polyhedra". Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences. The Royal Society. 246 (916): 401–450. doi:10.1098/rsta.1954.0003. ISSN 0080-4614. JSTOR 91532. MR 0062446.
  • Skilling, J. (1975). "The complete set of uniform polyhedra". Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences. 278 (1278): 111–135. doi:10.1098/rsta.1975.0022. ISSN 0080-4614. JSTOR 74475. MR 0365333.

External linksEdit