List of uniform polyhedra

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 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:

Indexing

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, 27–80 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 sides

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 polyhedra

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.

There are infinitely many prisms and antiprisms, one for each regular polygon; the ones up to the 12-gonal cases are listed.

Convex uniform polyhedra

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}
Heptagonal prism
4.4.7		 2 7 | 2 D7h C33d — U76d K01d 14 21 9 7{4}
+2{7}
Octagonal prism
4.4.8		 2 8 | 2 D8h C33e — U76e K01e 16 24 10 8{4}
+2{8}
Enneagonal prism
4.4.9		 2 9 | 2 D9h C33f — U76f K01f 18 27 11 9{4}
+2{9}
Decagonal prism
4.4.10		 2 10 | 2 D10h C33g — U76g K01g 20 30 12 10{4}
+2{10}
Hendecagonal prism
4.4.11		 2 11 | 2 D11h C33h — U76h K01h 22 33 13 11{4}
+2{11}
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}
Heptagonal antiprism
3.3.3.7		 | 2 2 7 D7d C34d — U77d K02d 14 28 16 14{3}
+2{7}
Octagonal antiprism
3.3.3.8		 | 2 2 8 D8d C34e — U77e K02e 16 32 18 16{3}
+2{8}
Enneagonal antiprism
3.3.3.9		 | 2 2 9 D9d C34f — U77f K02f 18 36 20 18{3}
+2{9}
Decagonal antiprism
3.3.3.10		 | 2 2 10 D10d C34g — U77g K02g 20 40 22 20{3}
+2{10}
Hendecagonal antiprism
3.3.3.11		 | 2 2 11 D11d C34h — U77h K02h 22 44 24 22{3}
+2{11}
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 polyhedra

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 −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}
Nonconvex 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 dodecahemidodecahedron ⁠5/4⁠ 5 | 5 10.⁠5/4⁠.10.5 Ih C65 W091 U51 K56 30 60 18 −12 No   12{5}+6{10}
Great dodecahemicosahedron ⁠5/4⁠ 5 | 3 6.⁠5/4⁠.6.5 Ih C81 W102 U65 K70 30 60 22 −8 No   12{5}+10{6}
Small icosihemidodecahedron ⁠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 dodecicosidodecahedron ⁠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 icosicosidodecahedron ⁠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 dodecadodecahedron 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 dodecahemidodecahedron ⁠5/3⁠ ⁠5/2⁠ | ⁠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 dodecahemicosahedron ⁠5/3⁠ ⁠5/2⁠ | 3 6.⁠5/3⁠.6.⁠5/2⁠ Ih C78 W100 U62 K67 30 60 22 −8 No   12{⁠5/2⁠}+10{6}
Dodecadodecahedron 2 | 5 ⁠5/2⁠ (⁠5/2⁠.5)2 Ih C45 W073 U36 K41 30 60 24 −6 Yes 3 12{5}+12{⁠5/2⁠}
Great icosihemidodecahedron ⁠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 | 3 ⁠5/2⁠ (⁠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}
Icosidodecadodecahedron ⁠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 dodecicosidodecahedron ⁠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 dodecicosidodecahedron 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 dodecicosidodecahedron ⁠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 icosicosidodecahedron ⁠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}
Rhombidodecadodecahedron ⁠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}
Nonconvex great rhombicosidodecahedron ⁠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 dodecadodecahedron 3 5 ⁠5/3⁠ | ⁠10/3⁠.6.10 Ih C57 W084 U45 K50 120 180 44 −16 Yes 4 20{6}+12{10}+12{⁠10/3⁠}
Truncated dodecadodecahedron 2 5 ⁠5/3⁠ | ⁠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 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 dodecadodecahedron | 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 dodecadodecahedron | ⁠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 W113 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 W116 U69 K74 60 150 92 2 Yes 13 (20+60){3}+12{⁠5/2⁠}
Great retrosnub icosidodecahedron | 2 ⁠3/2⁠ ⁠5/3⁠ (34.⁠5/2⁠)/2 I C90 W117 U74 K79 60 150 92 2 Yes 37 (20+60){3}+12{⁠5/2⁠}
Great snub dodecicosidodecahedron | ⁠5/3⁠ ⁠5/2⁠ 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 icosidodecadodecahedron | ⁠5/3⁠ 3 5 33.5.3.⁠5/3⁠ I C58 W112 U46 K51 60 180 104 −16 Yes 4 (20+60){3}+12{5}+12{⁠5/2⁠}
Small snub icosicosidodecahedron | ⁠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 icosicosidodecahedron | ⁠3/2⁠ ⁠3/2⁠ ⁠5/2⁠ (35.⁠5/2⁠)/2 Ih C91 W118 U72 K77 60 180 112 −8 Yes 38 (40+60){3}+12{⁠5/2⁠}
Great dirhombicosidodecahedron | ⁠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 −56 No   40{3}+60{4}+24{⁠5/2⁠}

Special case

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 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 key

  • 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 also

References

  • 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: 401–450. 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): 111–135. 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): 139–156. 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.

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Charles X dari Prancis

Charles XRaja Prancis dan NavarreBerkuasa16 September 1824 – 2 Agustus 1830Penobatan29 Mei 1825PendahuluLouis XVIIIPenerusLouis Philippe Isebagai Raja PrancisPerdana Menteri See list Comte de VillèleVicomte de MartignacAdipati Polignac Informasi pribadiKelahiran(1757-10-09)9 Oktober 1757Istana Versailles, PrancisKematian6 November 1836(1836-11-06) (umur 79)Görz, Austria (kini Italia)PemakamanBiara Kostanjevica, SloveniaWangsaBourbonNama lengkapCharles Philippe de FranceAyahLouis, Dau...

 

 

JJ Rizal

JJ RizalJJ RizalLahirJJ Rizal12 Februari 1974 (umur 50)Jakarta, IndonesiaPekerjaanSejarawanTahun aktif2000Situs webwww.komunitasbambu.id JJ Rizal, S.S. adalah sejarawan yang terlibat sebagai intelektual publik dengan aktif menyikapi persoalan-persoalan di tengah masyarakat melalui tulisan-tulisan di berbagai media cetak dan online serta nara sumber untuk beberapa stasiun radio dan TV. Ia menyelesaikan kuliah pada 1998 di Jurusan Sejarah, Fakultas Sastra Universitas Indonesia (FSUI)...

 

 

Utah Governor's Mansion

Historic building in Salt Lake City, Utah, U.S. United States historic placeThomas Kearns Mansion and Carriage HouseU.S. National Register of Historic Places Governor's Mansion, March 2010Show map of UtahShow map of the United StatesLocation603 East South Temple StreetSalt Lake City, UtahUnited StatesCoordinates40°46′11″N 111°52′23″W / 40.76972°N 111.87306°W / 40.76972; -111.87306Area9 acres (3.6 ha)Built1900-02ArchitectNeuhausen, Carl M.NRHP refe...

بوابة:علوم عصبية

                                            الثقافة الأعلام والتراجم الجغرافيا التاريخ الرياضيات العلوم المجتمع التقانات الفلسفة الأديان فهرس البوابات  علوم:  تاريخ العلوم    فلسفة العلوم    رياضيات    ف...

 

 

Marco Fossati

Questa voce o sezione sull'argomento calciatori italiani non cita le fonti necessarie o quelle presenti sono insufficienti. Puoi migliorare questa voce aggiungendo citazioni da fonti attendibili secondo le linee guida sull'uso delle fonti. Segui i suggerimenti del progetto di riferimento. Marco Fossati Fossati con la maglia del Monza nel 2019 Nazionalità  Italia Altezza 180 cm Peso 73 kg Calcio Ruolo Centrocampista Squadra  U Cluj Carriera Giovanili 2002-2007 Milan2008-2...

 

 

William M. Daley

William M. DaleyPotret resmi, 2011 Kepala Staf Gedung Putih ke-24Masa jabatan13 Januari 2011 – 27 Januari 2012PresidenBarack ObamaPendahuluRahm EmanuelPenggantiJack LewMenteri Perdagangan Amerika Serikat ke-32Masa jabatan30 Januari 1997 – 19 Juli 2000PresidenBill ClintonPendahuluMickey KantorPenggantiNorman Mineta Informasi pribadiLahirWilliam Michael Daley9 Agustus 1948 (umur 75)Chicago, Illinois, Amerika SerikatPartai politikPartai DemokratSuami/istriBernadette Ke...

Countertop

Horizontal work surface A stainless steel countertop A countertop, also counter top, counter, benchtop, worktop (British English) or kitchen bench (Australian or New Zealand English), bunker (Scottish English) is a raised, firm, flat, and horizontal surface. They are built for work in kitchens or other food preparation areas, bathrooms or lavatories, and workrooms in general. The surface is frequently installed upon and supported by cabinets, positioned at an ergonomic height for the user and...

 

 

Kysuce Protected Landscape Area

Kysuce Protected Landscape AreaCHKO KysuceIUCN category V (protected landscape/seascape)Veľká Rača mountainLocationNorth-western SlovakiaCoordinates49°25′N 18°57′E / 49.417°N 18.950°E / 49.417; 18.950Area654.62 km² (252.7 mi²)Established23 May 1984Governing bodySpráva CHKO Kysuce (Kysuce PLA administration) in Čadca Kysuce Protected Landscape Area (Slovak: Chránená krajinná oblasť Kysuce) is one of the 14 protected landscape areas in Slovakia. I...

 

 

5/42 Evzone Regiment

5/42 Evzone RegimentMen of the 5/42 Evzone Regiment in Odessa c. 1919Active1913–19411948–19541979–presentCountry GreeceBranch Hellenic ArmyTypeInfantrySizeRegiment (currently battalion-size)Part of80th National Guard Higher CommandGarrison/HQLamia (1913–1941)Larissa (1948–1954)Skydra (1980–?)Kalymnos (since 2003)EngagementsWorld War I Macedonian front Vardar Offensive Russian Civil War Allied intervention in the Russian Civil War Southern Front Southern Russia Inter...

Into the Woods (film)

Into the WoodsPoster film Into the WoodsSutradaraRob MarshallProduserRob MarshallJohn DeLucaMarc PlattCallum McDougallDitulis olehJames LapineBerdasarkanInto the Woodsoleh Stephen Sondheim dan James LapinePemeranMeryl StreepEmily BluntJames CordenAnna KendrickChris PineJohnny DeppPenata musikStephen SondheimSinematograferDion BeebePenyuntingWyatt SmithPerusahaanproduksiWalt Disney PicturesLucamar ProductionsDistributorWalt Disney Studios Motion PicturesTanggal rilis 8 Desember 2014 ...

 

 

Историческая наука в СССР

Основная статья: Наука в СССР Директор Института истории АН СССР Б. Д. Греков на художественном маркированном конверте Почты СССР (1982) Истори́ческая нау́ка в СССР (советская историография) — методология и практика исторических исследований в Советском Союзе. Сов...

 

 

Rail shooter

Video game genre Part of a series onAction games Subgenres Action-adventure Metroidvania Battle royale Fighting Beat 'em up Hack and slash Platform fighter Platform Rhythm Action RPG Shooter Artillery Arena First-person Hero Light gun Third-person Tactical Shoot 'em up Bullet hell Twin-stick Sports Racing Stealth Survival Vehicle sim Topics Capture the flag Cover system First-person shooter engine Free look Quick time event WASD keys Lists List of battle royale games List of beat 'em ups List...

Steven Moffat

Steven MoffatSteven MoffatLahirSteven MoffatTahun aktif1988 - SekarangSuami/istriSue Vertue Steven Moffat terlahir pada 18 November 1961 di Paisley, Skotlandia adalah seorang penulis naskah drama atau komedi berkebangsaan Inggris yang telah banyak menuliskan naskah untuk serial televisi sejak akhir tahun 1980-an. Hasil karyanya yang pertama adalah drama serial televisi yang dibuat sebanyak 10 seri, Press Gang. Moffat yang dari dulu merupakan penggemar acara Doctor Who telah menulis bebe...

 

 

Daddy Plays the Horn

1955 studio album by Dexter GordonDaddy Plays the HornStudio album by Dexter GordonReleased1955RecordedSeptember 18, 1955GenreJazzLength41:39LabelBethlehemProducerSteve Backer, Ralph BassDexter Gordon chronology The Duel(1947) Daddy Plays the Horn(1955) Dexter Blows Hot and Cool(1955) Daddy Plays the Horn is a 1955 jazz album by saxophonist Dexter Gordon. Reception Professional ratingsReview scoresSourceRatingAllmusic[1]The Penguin Guide to Jazz[2]Tom HullA−[3 ...

 

 

Teraina

Foto satelit dai Pulau Teraina .(NASA) Pulau Teraina atau juga dikenal sebagai Pulau Washington, merupakan sebuah pulau paling utara dari Kepulauan Line, Kiribati. Pulau ini merupakan sebuah atol koral dengan luas sekitar 14.2 kilometer persegi,[1][2] dan memiliki ketinggian rata-rata 2 meter di atas permukaan laut. Pulau ini juga memiliki sebuah danau air tawar yang terletak di ujung bagian barat. Pulau Teraina pertamakali terlihat oleh dunia barat pada tahun 1798; ditemukan ...

Armoured cavalry

This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources.Find sources: Armoured cavalry – news · newspapers · books · scholar · JSTOR (August 2019)Military with armoured vehicles APC by David E. Graves, Vietnam Combat Artists Program, CAT IX, 1969-70. Courtesy of National Museum of the U.S. Army. Armoured cavalry are mil...

 

 

One more heave

Phrase in British politics One more heave was a slogan used by British Liberal Party leader Jeremy Thorpe during the October 1974 general election and a phrase used (sometimes pejoratively) to describe the political strategy of John Smith, leader of the Labour Party from July 1992 until his death in May 1994. Liberal Party Jeremy Thorpe became leader of the Liberal Party in January 1967. The 1970 general election was disappointing as the Liberals lost six of their twelve seats in the House of...

 

 

Balthus

French artist (1908–2001) For a character of the animated series Adventure Time, see Evergreen (Adventure Time). BalthusBalthasar Klossowski de RolaBalthus by Damian Pettigrew (1996)BornBalthasar Klossowski(1908-02-29)February 29, 1908Paris, FranceDiedFebruary 18, 2001(2001-02-18) (aged 92)Rossinière, SwitzerlandKnown forPainting, drawing, watercolorNotable workThe Street (1933–35)The Mountain (1937)Nude Before a Mirror (1955)Spouse(s)Antoinette de Watteville (Married 1937 - Di...

A Christmas Carol; or, Past, Present, and Future

A play adaption of A Christmas Carol by Charles Dickens Playbill for Stirling's adaptation of A Christmas Carol (1844) A Christmas Carol; or, Past, Present, and Future is a play in three acts (or ‘Staves’) by Edward Stirling at the Adelphi Theatre in London on 5 February 1844.[1][2] Containing songs especially written for the show, the drama was adapted from the novella A Christmas Carol by Charles Dickens which had been published just weeks before in December 1843.[3&...

 

 

Box-office France 1979

Cet article traite du box-office cinéma de 1979 en France. Les films à succès Le dernier film de Louis de Funès numéro un du box-office L'année 1979 est marqué par le retour de la saga du Gendarme avec Le Gendarme et les Extraterrestres avec Louis de Funès. Dès sa sortie le 2 février 1979, le film finit au premier classement du box-office et attire 1 179 731 durant la première semaine d'exploitation et reste sept semaines à la première position du box-office jusqu'au 27 mars 1979...