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/vertex figures or both.
This list includes these:
- all 75 nonprismatic uniform polyhedra;
- a few representatives of theββinfinite sets of prisms and antiprisms;
- one degenerate polyhedron, Skilling's figure with overlapping edges.
It was proven in Sopov (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:
- The uniform polyhedron compounds.
- 40 potential uniform polyhedra with degenerate vertex figures which have overlapping edges (not counted by Coxeter);
- The uniform tilings (infinite polyhedra)
- 11 Euclidean convex uniform tilings;
- 28 Euclidean nonconvex or apeirogonal uniform tilings;
- Infinite number of uniform tilings in hyperbolic plane.
- Any polygons or 4-polytopes
Indexingβ»
Four numbering schemes for the uniform polyhedra are in common use, distinguished by letters:
- β» Coxeter et al., 1954, showed the convex forms as figures 15 through 32; three prismatic forms, figures 33β35; and the nonconvex forms, figures 36β92.
- β» Wenninger, 1974, has 119 figures: 1β5 for the Platonic solids, 6β18 for the Archimedean solids, 19β66 for stellated forms including the 4 regular nonconvex polyhedra. And ended with 67β119 for the nonconvex uniform polyhedra.
- β» Kaleido, 1993: The 80 figures were grouped by symmetry: 1β5 as representatives of the infinite families of prismatic forms with dihedral symmetry, 6β9 with tetrahedral symmetry, 10β26 with octahedral symmetry, 27β80 with icosahedral symmetry.
- β» Mathematica, 1993, follows the Kaleido series with the 5 prismatic forms movedββto last, so that the nonprismatic forms become 1β75.
Names of polyhedra by number of sidesβ»
There are generic geometric names for the most common polyhedra. The 5 Platonic solids are called a tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron with 4, 6, 8, 12, and 20 sides respectively. The regular hexahedron is a cube.
Table of polyhedraβ»
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 similaritiesββto be shown.
There are infinitely many prisms. And antiprisms, one for each regular polygon; the ones up to the 12-gonal cases are listed.
Convex uniform polyhedraβ»
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 | β | 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 | β | U76b | K01b | 10 | 15 | 7 | 5{4} +2{5} | |
Hexagonal prism | 4.4.6 |
2 6 | 2 | D6h | C33c | β | U76c | K01c | 12 | 18 | 8 | 6{4} +2{6} | |
Heptagonal prism | 4.4.7 |
2 7 | 2 | D7h | C33d | β | U76d | K01d | 14 | 21 | 9 | 7{4} +2{7} | |
Octagonal prism | 4.4.8 |
2 8 | 2 | D8h | C33e | β | U76e | K01e | 16 | 24 | 10 | 8{4} +2{8} | |
Enneagonal prism | 4.4.9 |
2 9 | 2 | D9h | C33f | β | U76f | K01f | 18 | 27 | 11 | 9{4} +2{9} | |
Decagonal prism | 4.4.10 |
2 10 | 2 | D10h | C33g | β | U76g | K01g | 20 | 30 | 12 | 10{4} +2{10} | |
Hendecagonal prism | 4.4.11 |
2 11 | 2 | D11h | C33h | β | U76h | K01h | 22 | 33 | 13 | 11{4} +2{11} | |
Dodecagonal prism | 4.4.12 |
2 12 | 2 | D12h | C33i | β | 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 | β | U77a | K02a | 8 | 16 | 10 | 8{3} +2{4} | |
Pentagonal antiprism | 3.3.3.5 |
| 2 2 5 | D5d | C34b | β | U77b | K02b | 10 | 20 | 12 | 10{3} +2{5} | |
Hexagonal antiprism | 3.3.3.6 |
| 2 2 6 | D6d | C34c | β | U77c | K02c | 12 | 24 | 14 | 12{3} +2{6} | |
Heptagonal antiprism | 3.3.3.7 |
| 2 2 7 | D7d | C34d | β | U77d | K02d | 14 | 28 | 16 | 14{3} +2{7} | |
Octagonal antiprism | 3.3.3.8 |
| 2 2 8 | D8d | C34e | β | U77e | K02e | 16 | 32 | 18 | 16{3} +2{8} | |
Enneagonal antiprism | 3.3.3.9 |
| 2 2 9 | D9d | C34f | β | U77f | K02f | 18 | 36 | 20 | 18{3} +2{9} | |
Decagonal antiprism | 3.3.3.10 |
| 2 2 10 | D10d | C34g | β | U77g | K02g | 20 | 40 | 22 | 20{3} +2{10} | |
Hendecagonal antiprism | 3.3.3.11 |
| 2 2 11 | D11d | C34h | β | U77h | K02h | 22 | 44 | 24 | 22{3} +2{11} | |
Dodecagonal antiprism | 3.3.3.12 |
| 2 2 12 | D12d | C34i | β | 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 polyhedraβ»
The forms containing only convex faces are listed first, followed by the forms with star faces. Again infinitely many prisms and antiprisms exist; they are listed here up to the 8-sided ones.
The uniform polyhedra | 5/2 3 3, | 5/2 3/2 3/2, | 5/3 5/2 3, | 3/2 5/3 3 5/2, and | (3/2) 5/3 (3) 5/2 have some faces occurring as coplanar pairs. (Coxeter et al. 1954, pp. 423, 425, 426; Skilling 1975, p. 123)
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 | β2 | 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 | β6 | 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 | β8 | 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 | β6 | No | 12{4}+6{8} | ||
Small cubicuboctahedron |
3/2 4 | 4 | 8.3/2.8.4 |
Oh | C38 | W069 | U13 | K18 | 24 | 48 | 20 | β4 | 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 | β12 | 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 | β8 | 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 | β4 | 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 | β28 | 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 | β18 | 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 | β16 | 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 | β10 | 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 | β8 | Yes | 6 | 20{3}+12{5}+20{6} | |
Pentagrammic prism |
2 5/2 | 2 | 5/2.4.4 |
D5h | C33b | β | 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 | β | 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 | β | 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 | β | 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 | β | 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 | β | 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 | β | 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 | β | 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 | β | 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 | β | 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 | β | U80c | K05c | 16 | 32 | 18 | 2 | Yes | 5 | 16{3}+2{8/3} | |
Small stellated dodecahedron |
5 | 2 5/2 | (5/2) |
Ih | C43 | W020 | U34 | K39 | 12 | 30 | 12 | β6 | Yes | 3 | 12{5/2} | |
Great stellated dodecahedron |
3 | 2 5/2 | (5/2) |
Ih | C68 | W022 | U52 | K57 | 20 | 30 | 12 | 2 | Yes | 7 | 12{5/2} | |
Ditrigonal dodeca- dodecahedron |
3 | 5/3 5 | (5/3.5) |
Ih | C53 | W080 | U41 | K46 | 20 | 60 | 24 | β16 | Yes | 4 | 12{5}+12{5/2} | |
Small ditrigonal icosidodecahedron |
3 | 5/2 3 | (5/2.3) |
Ih | C39 | W070 | U30 | K35 | 20 | 60 | 32 | β8 | 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 | β6 | 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 | β4 | Yes | 4 | 8{3}+6{4}+6{8/3} | |
Great dodecahemi- dodecahedron |
5/3 5/2 | 5/3 | 10/3.5/3.10/3.5/2 |
Ih | C86 | W107 | U70 | K75 | 30 | 60 | 18 | β12 | No | 12{5/2}+6{10/3} | ||
Small dodecahemi- cosahedron |
5/3 5/2 | 3 | 6.5/3.6.5/2 |
Ih | C78 | W100 | U62 | K67 | 30 | 60 | 22 | β8 | No | 12{5/2}+10{6} | ||
Dodeca- dodecahedron |
2 | 5 5/2 | (5/2.5) |
Ih | C45 | W073 | U36 | K41 | 30 | 60 | 24 | β6 | 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 | β4 | No | 20{3}+6{10/3} | ||
Great icosidodecahedron |
2 | 3 5/2 | (5/2.3) |
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 | β4 | 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 | β6 | 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 | β6 | 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 | β28 | 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 | β18 | 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 | β16 | 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 | β16 | 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 | β16 | 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 | β16 | 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 | β8 | 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 | β6 | 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 |
3 5 5/3 | | 10/3.6.10 |
Ih | C57 | W084 | U45 | K50 | 120 | 180 | 44 | β16 | Yes | 4 | 20{6}+12{10}+12{10/3} | |
Truncated dodeca- dodecahedron |
2 5 5/3 | | 10/3.4.10/9 |
Ih | C75 | W098 | U59 | K64 | 120 | 180 | 54 | β6 | Yes | 3 | 30{4}+12{10}+12{10/3} | |
Great truncated icosidodecahedron |
2 3 5/3 | | 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 | β6 | 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 | β6 | Yes | 9 | 60{3}+12{5}+12{5/2} | |
Great snub icosidodecahedron |
| 2 5/2 3 | 3.5/2 |
I | C73 | W113 | U57 | K62 | 60 | 150 | 92 | 2 | Yes | 7 | (20+60){3}+12{5/2} | |
Great inverted snub icosidodecahedron |
| 5/3 2 3 | 3.5/3 |
I | C88 | W116 | U69 | K74 | 60 | 150 | 92 | 2 | Yes | 13 | (20+60){3}+12{5/2} | |
Great retrosnub icosidodecahedron |
| 2 3/2 5/3 | (3.5/2)/2 |
I | C90 | W117 | U74 | K79 | 60 | 150 | 92 | 2 | Yes | 37 | (20+60){3}+12{5/2} | |
Great snub dodecicosi- dodecahedron |
| 5/3 5/2 3 | 3.5/3.3.5/2 |
I | C80 | W115 | U64 | K69 | 60 | 180 | 104 | β16 | Yes | 10 | (20+60){3}+(12+12){5/2} | |
Snub icosidodeca- dodecahedron |
| 5/3 3 5 | 3.5.3.5/3 |
I | C58 | W112 | U46 | K51 | 60 | 180 | 104 | β16 | Yes | 4 | (20+60){3}+12{5}+12{5/2} | |
Small snub icos- icosidodecahedron |
| 5/2 3 3 | 3.5/2 |
Ih | C41 | W110 | U32 | K37 | 60 | 180 | 112 | β8 | Yes | 2 | (40+60){3}+12{5/2} | |
Small retrosnub icosicosi- dodecahedron |
| 3/2 3/2 5/2 | (3.5/2)/2 |
Ih | C91 | W118 | U72 | K77 | 60 | 180 | 112 | β8 | Yes | 38 | (40+60){3}+12{5/2} | |
Great dirhombicosi- dodecahedron |
| 3/2 5/3 3 5/2 | (4.5/3.4.3. 4.5/2.4.3/2)/2 |
Ih | C92 | W119 | U75 | K80 | 60 | 240 | 124 | β56 | No | 40{3}+60{4}+24{5/2} |
Special caseβ»
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 | β | β | β | β | 60 | 360 (*) | 204 | β96 | No | 120{3}+60{4}+24{5/2} |
The great disnub dirhombidodecahedron has 240 of its 360 edges coinciding in space in 120 pairs. Because of this edge-degeneracy, it is not always considered to be a uniform polyhedron.
Column keyβ»
- Uniform indexing: U01βU80 (Tetrahedron first, Prisms at 76+)
- Kaleido software indexing: K01βK80 (Kn = Unβ5 for n = 6 to 80) (prisms 1β5, Tetrahedron etc. 6+)
- Magnus Wenninger Polyhedron Models: W001-W119
- 1β18: 5 convex regular and 13 convex semiregular
- 20β22, 41: 4 non-convex regular
- 19β66: Special 48 stellations/compounds (Nonregulars not given on this list)
- 67β109: 43 non-convex non-snub uniform
- 110β119: 10 non-convex snub uniform
- Chi: the Euler characteristic, Ο. 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 alsoβ»
- List of uniform polyhedra by vertex figure
- List of uniform polyhedra by Wythoff symbol
- List of uniform polyhedra by Schwarz triangle
Referencesβ»
- 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. 246 (916). The Royal Society: 401β450. Bibcode:1954RSPTA.246..401C. doi:10.1098/rsta.1954.0003. ISSN 0080-4614. JSTOR 91532. MR 0062446. S2CID 202575183.
- 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): 111β135. Bibcode:1975RSPTA.278..111S. doi:10.1098/rsta.1975.0022. ISSN 0080-4614. JSTOR 74475. MR 0365333. S2CID 122634260.
- Sopov, S. P. (1970). "A proof of the completeness on the list of elementary homogeneous polyhedra". Ukrainskiui Geometricheskiui Sbornik (8): 139β156. MR 0326550.
- 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.
External linksβ»
- Stella: Polyhedron Navigator β Software able to generate and print nets for all uniform polyhedra. Used to create most images on this page.
- Paper models
- Uniform indexing: U1-U80, (Tetrahedron first)
- Uniform Polyhedra (80), Paul Bourke
- Weisstein, Eric W. "Uniform Polyhedron". MathWorld.
- http://www.mathconsult.ch/showroom/unipoly
- https://web.archive.org/web/20171110075259/http://gratrix.net/polyhedra/uniform/summary/
- http://www.it-c.dk/edu/documentation/mathworks/math/math/u/u034.htm
- http://www.buddenbooks.com/jb/uniform/
- Kaleido Indexing: K1-K80 (Pentagonal prism first)
- Also