Gourt Home::Gourt :: List of uniform polyhedra
The following list contains all 75 nonprismatic uniform polyhedra, 11 uniform tessellations in the plane, and a samplings of the infinite set of prisms and antiprisms.
Column key
- Solid classes
- R = 5 Platonic solids
- R+= 4 Kepler-Poinsot solids
- 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
- 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)
- T = 11 Planar tessellations
- Bowers style acronym
- Uniform indexing: U01-U80 (Tetrahedron first, Prisms at 76+)
- Kaleido indexing: K01-K80
- 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)
- 67-119 - 53 non-convex uniform
- Chi: the Euler characteristic, χ. Uniform tilings on the plane correspond to a torus topology, with Euler characteristic of zero.
- For the plane tilings, the numbers given of vertices, edges and faces show the ratio of such elements in one period of the pattern, which in each case is a rhombus (sometimes a right-angled rhombus, i.e. a square).
- Note on polyhedron pictures:
- Pictures were generated from VRML model viewing software. Some star polygon faced polyhedra were drawn using a binary filling algorithm, so the center of a pentagram is not filled. Although the definition of "interior" is ambiguous for self-intersecting faces, Polyhedronists usually consider the multileveled interior as solid faced.
- 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.
Table of polyhedra
Convex forms (3 faces/vertex)
| Name | Picture | Solid class | Wythoff symbol | Vertex figure | Bowers' Name | Symmetry group | W# | U# | K# | Vertices | Edges | Faces | Chi | Faces by type |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Tetrahedron | R | 3|2 3 | Tet | Td | W001 | U01 | K06 | 4 | 6 | 4 | 2 | 4{3} |
| Triangular prism | P | 2 3|2 | Trip | D3h | -- | -- | -- | 6 | 9 | 5 | 2 | 2{3}+3{4} |
| Truncated tetrahedron | A | 2 3|3 | Tut | Td | W006 | U02 | K07 | 12 | 18 | 8 | 2 | 4{3}+4{6} |
| Truncated cube | A | 2 3|4 | Tic | Oh | W008 | U09 | K14 | 24 | 36 | 14 | 2 | 8{3}+6{8} |
| Truncated dodecahedron | A | 2 3|5 | Tid | Ih | W010 | U26 | K31 | 60 | 90 | 32 | 2 | 20{3}+12{10} |
| Truncated hexagonal tiling | T | 2 3|6 | Toxat | P6m | -- | -- | -- | 6n | 9n | 3n | 0 | n{12}+2n{3} |
| Cube | R | 3|2 4 | Cube | Oh | W003 | U06 | K11 | 8 | 12 | 6 | 2 | 6{4} |
| Pentagonal prism | P | 2 5|2 | Pip | D5h | -- | U76 | K01 | 10 | 15 | 7 | 2 | 5{4}+2{5} |
| Hexagonal prism | P | 2 6|2 | Hip | D6h | -- | -- | -- | 12 | 18 | 8 | 2 | 6{4}+2{6} |
| Octagonal prism | P | 2 8|2 | Op | D8h | -- | -- | -- | 16 | 24 | 10 | 2 | 8{4}+2{8} |
| Decagonal prism | P | 2 10|2 | Dip | D10h | -- | -- | -- | 20 | 30 | 12 | 2 | 10{4}+2{10} |
| Dodecagonal prism | P | 2 12|2 | Twip | D12h | -- | -- | -- | 24 | 36 | 14 | 2 | 12{4}+2{12} |
| Truncated octahedron | A | 2 4|3 | Toe | Oh | W007 | U08 | K13 | 24 | 36 | 14 | 2 | 6{4}+8{6} |
| Great rhombicuboctahedron | A | 2 3 4| | Girco | Oh | W015 | U11 | K16 | 48 | 72 | 26 | 2 | 12{4}+8{6}+6{8} |
| Great rhombicosidodecahedron | A | 2 3 5| | Grid | Ih | W016 | U28 | K33 | 120 | 180 | 62 | 2 | 30{4}+20{6}+12{10} |
| Great rhombitrihexagonal tiling | T | 2 3 6| | Othat | p6m | -- | -- | -- | 12n | 18n | 6n | 0 | 3n{4}+2n{6}+n{12} |
| Truncated square tiling | T | 2 4|4 | Tosquat | p4m | -- | -- | -- | 4n | 6n | 2n | 0 | n{4}+n{8} |
| Dodecahedron | R | 3|2 5 | Doe | Ih | W005 | U23 | K28 | 20 | 30 | 12 | 2 | 12{5} |
| Truncated icosahedron | A | 2 5|3 | Ti | Ih | W009 | U25 | K30 | 60 | 90 | 32 | 2 | 12{5}+20{6} |
| Hexagonal tiling | T | 3|2 6 | Hexat | p6m | -- | -- | -- | 2n | 3n | n | 0 | n{6} |
Convex forms (4 faces/vertex)
| Name | Picture | Solid class | Wythoff symbol | Vertex figure | Bowers' Name | Symmetry group | W# | U# | K# | Vertices | Edges | Faces | Chi | Faces by type |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Octahedron | R | 4|2 3 | Oct | Oh | W002 | U05 | K10 | 6 | 12 | 8 | 2 | 8{3} |
| Square antiprism | P | |2 2 4 | Squap | D4d | -- | -- | -- | 8 | 16 | 10 | 2 | 8{3}+2{4} |
| Pentagonal antiprism | P | |2 2 5 | Pap | D5d | -- | U77 | K02 | 10 | 20 | 12 | 2 | 10{3}+2{5} |
| Hexagonal antiprism | P | |2 2 6 | Hap | D6d | -- | -- | -- | 12 | 24 | 14 | 2 | 12{3}+2{6} |
| Octagonal antiprism | P | |2 2 8 | Oap | D8d | -- | -- | -- | 16 | 32 | 18 | 2 | 16{3}+2{8} |
| Decagonal antiprism | P | |2 2 10 | Dap | D10d | -- | -- | -- | 20 | 40 | 22 | 2 | 20{3}+2{10} |
| Dodecagonal antiprism | P | |2 2 12 | Twap | D12d | -- | -- | -- | 24 | 48 | 26 | 2 | 24{3}+2{12} |
| Cuboctahedron | A | 2|3 4 | Co | Oh | W011 | U07 | K12 | 12 | 24 | 14 | 2 | 8{3}+6{4} |
| Small rhombicuboctahedron | A | 3 4|2 | Sirco | Oh | W013 | U10 | K15 | 24 | 48 | 26 | 2 | 8{3}+(6+12){4} |
| Small rhombicosidodecahedron | A | 3 5|2 | Srid | Ih | W014 | U27 | K32 | 60 | 120 | 62 | 2 | 20{3}+30{4}+12{5} |
| Small rhombitrihexagonal tiling | T | 3 6|2 | Rothat | p6m | -- | -- | -- | 6n | 12n | 6n | 0 | 2n{3}+3n{4}+n{6} |
| Icosidodecahedron | A | 2|3 5 | Id | Ih | W012 | U24 | K29 | 30 | 60 | 32 | 2 | 20{3}+12{5} |
| Trihexagonal tiling | T | 2|3 6 | That | p6m | -- | -- | -- | 3n | 6n | 3n | 0 | 2n{3}+n{6} |
| Square tiling | T | 4|2 4 | Squat | p4m | -- | -- | -- | n | 2n | n | 0 | n{4} |
Convex forms (5 faces/vertex)
| Name | Picture | Solid class | Wythoff symbol | Vertex figure | Bowers' Name | Symmetry group | W# | U# | K# | Vertices | Edges | Faces | Chi | Faces by type |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Icosahedron | R | 5|2 3 | Ike | Ih | W004 | U22 | K27 | 12 | 30 | 20 | 2 | 20{3} |
| Snub cube | A | |2 3 4 | Snic | O | W017 | U12 | K17 | 24 | 60 | 38 | 2 | (8+24){3}+6{4} |
| Snub dodecahedron | A | |2 3 5 | Snid | I | W018 | U29 | K34 | 60 | 150 | 92 | 2 | (20+60){3}+12{5} |
| Snub hexagonal tiling | T | |2 3 6 | Snathat | p6 | -- | -- | -- | 6n | 15n | 9n | 0 | 8n{3}+n{6} |
| Prismatic trisquare tiling | T | |2 2 (2|2) | Etrat | cmm | -- | -- | -- | 2n | 5n | 3n | 0 | 2n{3}+n{4} |
| Snub square tiling | T | |2 4 4 | Snasquat | p4g | -- | -- | -- | 4n | 10n | 6n | 0 | 4n{3}+2n{4} |
Convex forms (6 faces/vertex)
| Name | Picture | Solid class | Wythoff symbol | Vertex figure | Bowers' Name | Symmetry group | W# | U# | K# | Vertices | Edges | Faces | Chi | Faces by type |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Triangular tiling | T | 6|2 3 | Trat | p6m | -- | -- | -- | n | 3n | 2n | 0 | 2n{3} |
Nonconvex forms with convex faces
| Name | Picture | Solid class | Wythoff symbol | Vertex figure | Bowers' Name | Symmetry group | W# | U# | K# | Vertices | Edges | Faces | Chi | Faces by type |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Tetrahemihexahedron | C+ | 3/2 3|2 | Thah | Td | W067 | U04 | K09 | 6 | 12 | 7 | 1 | 4{3}+3{4} |
| Cubohemioctahedron | C+ | 4/3 4|3 | Cho | Oh | W078 | U15 | K20 | 12 | 24 | 10 | -2 | 6{4}+4{6} |
| Octahemioctahedron | C+ | 3/2 3|3 | Oho | Oh | W068 | U03 | K08 | 12 | 24 | 12 | 0 | 8{3}+4{6} |
| Great dodecahedron | R+ | 5/2|2 5 | Gad | Ih | W021 | U35 | K40 | 12 | 30 | 12 | -6 | 12{5} |
| Great icosahedron | R+ | 5/2|2 3 | Gike | Ih | W041 | U53 | K58 | 12 | 30 | 20 | 2 | 20{3} |
| Great ditrigonal icosidodecahedron | C+ | 3/2|3 5 | Gidtid | Ih | W087 | U47 | K52 | 20 | 60 | 32 | -8 | 20{3}+12{5} |
| Small rhombihexahedron | C+ | 3/2 2 4| | Sroh | Oh | W086 | U18 | K23 | 24 | 48 | 18 | -6 | 12{4}+6{8} |
| Small cubicuboctahedron | C+ | 3/2 4|4 | Socco | Oh | W069 | U13 | K18 | 24 | 48 | 20 | -4 | 8{3}+6{4}+6{8} |
| Uniform great rhombicuboctahedron | C+ | 3/2 4|2 | Querco | Oh | W085 | U17 | K22 | 24 | 48 | 26 | 2 | 8{3}+(6+12){4} |
| Small dodecahemidodecahedron | C+ | 5/4 5|5 | Sidhid | Ih | W091 | U51 | K56 | 30 | 60 | 18 | -12 | 12{5}+6{10} |
| Small icosihemidodecahedron | C+ | 3/2 3|5 | Seihid | Ih | W089 | U49 | K54 | 30 | 60 | 26 | -4 | 20{3}+6{10} |
| Small dodecicosahedron | C+ | 3/2 3 5| | Siddy | Ih | W090 | U50 | K55 | 60 | 120 | 32 | -28 | 20{6}+12{10} |
| Small rhombidodecahedron | C+ | 2 5/2 5| | Sird | Ih | W074 | U39 | K44 | 60 | 120 | 42 | -18 | 30{4}+12{10} |
| Small dodecicosidodecahedron | C+ | 3/2 5|5 | Saddid | Ih | W072 | U33 | K38 | 60 | 120 | 44 | -16 | 20{3}+12{5}+12{10} |
| Rhombicosahedron | C+ | 2 5/2 3| | Ri | Ih | W096 | U56 | K61 | 60 | 120 | 50 | -10 | 30{4}+20{6} |
| Great icosicosidodecahedron | C+ | 3/2 5|3 | Giid | Ih | W088 | U48 | K53 | 60 | 120 | 52 | -8 | 20{3}+12{5}+20{6} |
Nonconvex prismatic forms
| Name | Picture | Solid class | Wythoff symbol | Vertex figure | Bowers' Name | Symmetry group | W# | U# | K# | Vertices | Edges | Faces | Chi | Faces by type |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Pentagrammic prism | P+ | 2 5/2|2 | Stip | D5h | -- | U78 | K03 | 10 | 15 | 7 | 2 | 5{4}+2{5/2} |
| Septagrammic prism (7/3) | P+ | 2 7/3|2 | Giship | D7h | -- | -- | -- | 14 | 21 | 9 | 2 | 7{4}+2{7/3} |
| Septagrammic prism (7/2) | P+ | 2 7/2|2 | Ship | D7h | -- | -- | -- | 14 | 21 | 9 | 2 | 7{4}+2{7/2} |
| Pentagrammic antiprism | P+ | |2 2 5/2 | Stap | D5h | -- | U79 | K04 | 10 | 20 | 12 | 2 | 10{3}+2{5/2} |
| Pentagrammic crossed-antiprism | P+ | |2 2 5/3 | Starp | D5d | -- | U80 | K05 | 10 | 20 | 12 | 2 | 10{3}+2{5/2} |
Other nonconvex forms with nonconvex faces
| Name | Picture | Solid class | Wythoff symbol | Vertex figure | Bowers' Name | Symmetry group | W# | U# | K# | Vertices | Edges | Faces | Chi | Faces by type |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Small stellated dodecahedron | R+ | 5|2 5/2 | Sissid | Ih | W020 | U34 | K39 | 12 | 30 | 12 | -6 | 12{5/2} |
| Great stellated dodecahedron | R+ | 3|2 5/2 | Gissid | Ih | W022 | U52 | K57 | 20 | 30 | 12 | 2 | 12{5/2} |
| Ditrigonal dodecadodecahedron | S+ | 3|5/3 5 | Ditdid | Ih | W080 | U41 | K46 | 20 | 60 | 24 | -16 | 12{5}+12{5/2} |
| Small ditrigonal icosidodecahedron | S+ | 3|5/2 3 | Sidtid | Ih | W070 | U30 | K35 | 20 | 60 | 32 | -8 | 20{3}+12{5/2} |
| Stellated truncated hexahedron | S+ | 2 3|4/3 | Quith | Oh | W092 | U19 | K24 | 24 | 36 | 14 | 2 | 8{3}+6{8/3} |
| Great rhombihexahedron | S+ | 4/33/2 2| | Groh | Oh | W103 | U21 | K26 | 24 | 48 | 18 | -6 | 12{4}+6{8/3} |
| Great cubicuboctahedron | S+ | 3 4|4/3 | Gocco | Oh | W077 | U14 | K19 | 24 | 48 | 20 | -4 | 8{3}+6{4}+6{8/3} |
| Great dodecahemidodecahedron | S+ | 5/35/2|5/3 | Gidhid | Ih | W107 | U70 | K75 | 30 | 60 | 18 | -12 | 12{5/2}+6{10/3} |
| Small dodecahemicosahedron | S+ | 5/35/2|3 | Sidhei | Ih | W100 | U62 | K67 | 30 | 60 | 22 | -8 | 12{5/2}+10{6} |
| Great dodecahemicosahedron | S+ | 5/4 5|3 | Gidhei | Ih | W102 | U65 | K70 | 30 | 60 | 22 | -8 | 12{5}+10{6} |
| Dodecadodecahedron | S+ | 2|5/2 5 | Did | Ih | W073 | U36 | K41 | 30 | 60 | 24 | -6 | 12{5}+12{5/2} |
| Great icosihemidodecahedron | S+ | 3/2 3|5/3 | Geihid | Ih | W106 | U71 | K76 | 30 | 60 | 26 | -4 | 20{3}+6{10/3} |
| Great icosidodecahedron | S+ | 2|5/2 3 | Gid | Ih | W094 | U54 | K59 | 30 | 60 | 32 | 2 | 20{3}+12{5/2} |
| Cubitruncated cuboctahedron | S+ | 4/3 3 4| | Cotco | Oh | W079 | U16 | K21 | 48 | 72 | 20 | -4 | 8{6}+6{8}+6{8/3} |
| Great truncated cuboctahedron | S+ | 4/3 2 3| | Quitco | Oh | W093 | U20 | K25 | 48 | 72 | 26 | 2 | 12{4}+8{6}+6{8/3} |
| Truncated great dodecahedron | S+ | 2 5/2|5 | Tigid | Ih | W075 | U37 | K42 | 60 | 90 | 24 | -6 | 12{5/2}+12{10} |
| Small stellated truncated dodecahedron | S+ | 2 5|5/3 | Quitsissid | Ih | W097 | U58 | K63 | 60 | 90 | 24 | -6 | 12{5}+12{10/3} |
| Great stellated truncated dodecahedron | S+ | 2 3|5/3 | Quitgissid | Ih | W104 | U66 | K71 | 60 | 90 | 32 | 2 | 20{3}+12{10/3} |
| Truncated great icosahedron | S+ | 2 5/2|3 | Tiggy | Ih | W095 | U55 | K60 | 60 | 90 | 32 | 2 | 12{5/2}+20{6} |
| Great dodecicosahedron | S+ | 5/35/2 3| | Giddy | Ih | W101 | U63 | K68 | 60 | 120 | 32 | -28 | 20{6}+12{10/3} |
| Great rhombidodecahedron | S+ | 3/25/3 2| | Gird | Ih | W109 | U73 | K78 | 60 | 120 | 42 | -18 | 30{4}+12{10/3} |
| Icosidodecadodecahedron | S+ | 5/3 5|3 | Ided | Ih | W083 | U44 | K49 | 60 | 120 | 44 | -16 | 12{5}+12{5/2}+20{6} |
| Small ditrigonal dodecicosidodecahedron | S+ | 5/3 3|5 | Sidditdid | Ih | W082 | U43 | K48 | 60 | 120 | 44 | -16 | 20{3}+12{;5/2}+12{10} |
| Great ditrigonal dodecicosidodecahedron | S+ | 3 5|5/3 | Gidditdid | Ih | W081 | U42 | K47 | 60 | 120 | 44 | -16 | 20{3}+12{5}+12{10/3} |
| Great dodecicosidodecahedron | S+ | 5/2 3|5/3 | Gaddid | Ih | W099 | U61 | K66 | 60 | 120 | 44 | -16 | 20{3}+12{5/2}+12{10/3} |
| Small icosicosidodecahedron | S+ | 5/2 3|3 | Siid | Ih | W071 | U31 | K36 | 60 | 120 | 52 | -8 | 20{3}+12{5/2}+20{6} |
| Rhombidodecadodecahedron | S+ | 5/2 5|2 | Raded | Ih | W076 | U38 | K43 | 60 | 120 | 54 | -6 | 30{4}+12{5}+12{5/2} |
| Uniform great rhombicosidodecahedron | S+ | 5/3 3|2 | Qrid | Ih | W105 | U67 | K72 | 60 | 120 | 62 | 2 | 20{3}+30{4}+12{5/2} |
| Snub dodecadodecahedron | S+ | |2 5/2 5 | Siddid | I | W111 | U40 | K45 | 60 | 150 | 84 | -6 | 60{3}+12{5}+12{5/2} |
| Inverted snub dodecadodecahedron | S+ | |5/3 2 5 | Isdid | I | W114 | U60 | K65 | 60 | 150 | 84 | -6 | 60{3}+12{5}+12{5/2} |
| Great snub icosidodecahedron | S+ | |2 5/2 3 | Gosid | I | W116 | U57 | K62 | 60 | 150 | 92 | 2 | (20+60){3}+12{5/2} |
| Great inverted snub icosidodecahedron | S+ | |5/3 2 3 | Gisid | I | W113 | U69 | K74 | 60 | 150 | 92 | 2 | (20+60){3}+12{5/2} |
| Great retrosnub icosidodecahedron | S+ | |3/25/3 2 | Girsid | I | W117 | U74 | K79 | 60 | 150 | 92 | 2 | (20+60){3}+12{5/2} |
| Great snub dodecicosidodecahedron | S+ | |5/35/2 3 | Gisdid | I | W115 | U64 | K69 | 60 | 180 | 104 | -16 | (20+60){3}+(12+12){5/2} |
| Snub icosidodecadodecahedron | S+ | |5/3 3 5 | Sided | I | W112 | U46 | K51 | 60 | 180 | 104 | -16 | (20+60){3}+12{5}+12{5/2} |
| Small snub icosicosidodecahedron | S+ | |5/2 3 3 | Seside | Ih | W110 | U32 | K37 | 60 | 180 | 112 | -8 | (40+60){3}+12{5/2} |
| Small retrosnub icosicosidodecahedron | S+ | |3/23/25/2 | Sirsid | Ih | W118 | U72 | K77 | 60 | 180 | 112 | -8 | (40+60){3}+12{5/2} |
| Great dirhombicosidodecahedron | S+ | |3/25/3 3 | 5/2Gidrid | Ih | W119 | U75 | K80 | 60 | 240 | 124 | -56 | 40{3}+60{4}+24{5/2} |
| Icositruncated dodecadodecahedron | S+ | 5/3 3 5| | Idtid | Ih | W084 | U45 | K50 | 120 | 180 | 44 | -16 | 20{6}+12{10}+12{10/3} |
| Truncated dodecadodecahedron | S+ | 5/3 2 5| | Quitdid | Ih | W098 | U59 | K64 | 120 | 180 | 54 | -6 | 30{4}+12{10}+12{10/3} |
| Great truncated icosidodecahedron | S+ | 5/3 2 3| | Gaquatid | Ih | W108 | U68 | K73 | 120 | 180 | 62 | 2 | 30{4}+20{6}+12{10/3} |
Special case
| Name | Picture | Solid class | Wythoff symbol | Vertex figure | Bowers' Name | Symmetry group | W# | U# | K# | Vertices | Edges | Faces | Chi | Faces by type |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Great disnub dirhombidodecahedron Skilling's figure |
S++ | | (3/2) 5/3 (3) 5/2 | -- | 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.
Reference
External links
- Stella: Polyhedron Navigator - Software for generating and printing nets for all uniform polyhedra
- Paper models
- Uniform indexing: U1-U80, (Tetrahedron first)
- http://mathworld.wolfram.com/UniformPolyhedron.html
- http://www.mathconsult.ch/showroom/unipoly
- http://gratrix.net/polyhedra/uniform/summary
- http://www.it-c.dk/edu/documentation/mathworks/math/math/u/u034.htm
- http://home.earthlink.net/~jbuddenh/uniform/
- Kaleido Indexing: K1-K80 (Triangle prism first)
- http://www.math.technion.ac.il/~rl/kaleido
- http://www.math.technion.ac.il/~rl/docs/uniform.pdf Uniform Solution for Uniform Polyhedra
- http://www.physics.orst.edu/~bulatov/polyhedra/uniform
- http://web.ukonline.co.uk/polyhedra/uniform/uniform.html
- http://www.math.technion.ac.il/~rl/kaleido
- Also
- http://www.polyhedrix.de/e_klintro.htm
"List of uniform polyhedra"