View source for List of uniform polyhedra

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

Uniform polyhedra can be divided between [[Convex polytope|convex]] forms with convex [[regular polygon]] faces and star forms. Star forms have either regular [[star polygon]] [[face (geometry)|faces]] or [[vertex figure]]s or both.

This list includes these:
* all 75 nonprismatic [[uniform polyhedron|uniform]] [[polyhedron|polyhedra]];
* a few representatives of the infinite sets of [[prism (geometry)|prism]]s and [[antiprism]]s;
* one [[#Special case|degenerate]] polyhedron, [[Skilling's figure]] with overlapping edges.

It was proven in {{Harvard citation text|Sopov|1970}} that there are only 75 [[uniform polyhedron|uniform polyhedra]] other than the infinite families of [[prism (geometry)|prisms]] and [[antiprism]]s.  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 compound]]s.
* 40 potential [[List of uniform polyhedra by Schwarz triangle|uniform polyhedra with degenerate]] [[vertex figure]]s which have overlapping edges (not counted by [[Coxeter]]);
* The uniform tilings (infinite polyhedra)
** 11 Euclidean [[Euclidean tilings of regular polygons#Archimedean.2C uniform or semiregular tilings|convex uniform tilings]];
** 28 Euclidean [[Uniform tiling#Expanded lists of uniform tilings|nonconvex or apeirogonal uniform tilings]];
** Infinite number of [[uniform tilings in hyperbolic plane]].
* Any [[polygons]] or [[4-polytopes]]

==Indexing==

Four numbering schemes for the uniform polyhedra are in common use, distinguished by letters:

* ['''C'''] Coxeter et al., 1954, showed the [[convex polyhedron|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 in three dimensions|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 [[Geometry|geometric]] names for the most common [[Polyhedron|polyhedra]]. The 5 [[Platonic solid]]s 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 configuration]]s 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 ===
{| class="wikitable sortable" style="text-align:center;font-size:small;"
! Name|| Picture|| [[Vertex configuration|Vertex<br>type]]||Wythoff<br>symbol|| Sym.|| C#|| W#|| U#|| K#|| Vert.|| Edges|| Faces||  Faces by type
|-
|| [[Tetrahedron]]|| [[Image:Tetrahedron.png|60px]]|| [[Image:Tetrahedron vertfig.svg|50px]]<br>3.3.3|| 3 {{pipe}} 2 3|| T<sub>d</sub>|| C15|| W001|| U01|| K06|| 4|| 6|| 4|| 4{3}
|-
|| [[Triangular prism]]|| [[Image:Triangular prism.png|60px]]|| [[Image:Triangular prism vertfig.png|50px]]<br>3.4.4|| 2 3 {{pipe}} 2|| D<sub>3h</sub>|| C33a|| —|| U76a|| K01a|| 6|| 9|| 5|| 2{3}<br>+3{4}
|-
|| [[Truncated tetrahedron]]|| [[Image:Truncated tetrahedron.png|60px]]|| [[Image:Truncated tetrahedron vertfig.png|50px]]<br>3.6.6|| 2 3 {{pipe}} 3|| T<sub>d</sub>|| C16|| W006|| U02|| K07|| 12|| 18|| 8|| 4{3}<br>+4{6}
|-
|| [[Truncated cube]]|| [[Image:Truncated hexahedron.png|60px]]|| [[Image:Truncated cube vertfig.svg|50px]]<br>3.8.8|| 2 3 {{pipe}} 4|| O<sub>h</sub>|| C21|| W008|| U09|| K14|| 24|| 36|| 14||8{3}<br>+6{8}
|-
|| [[Truncated dodecahedron]]|| [[Image:Truncated dodecahedron.png|60px]]|| [[Image:Truncated dodecahedron vertfig.png|50px]]<br>3.10.10|| 2 3 {{pipe}} 5|| I<sub>h</sub>|| C29|| W010|| U26|| K31|| 60|| 90|| 32
|| 20{3}<br>+12{10}
|-
|| [[Cube]]|| [[Image:hexahedron.png|60px]]|| [[Image:Cube vertfig.png|50px]]<br>4.4.4|| 3 {{pipe}} 2 4|| O<sub>h</sub>|| C18|| W003|| U06|| K11|| 8|| 12|| 6||6{4}
|-
|| [[Pentagonal prism]]|| [[Image:Pentagonal prism.png|60px]]|| [[Image:Pentagonal prism vertfig.png|50px]]<br>4.4.5|| 2 5 {{pipe}} 2|| D<sub>5h</sub>|| C33b|| —|| U76b|| K01b|| 10|| 15|| 7|| 5{4}<br>+2{5}
|-
|| [[Hexagonal prism]]|| [[Image:Hexagonal prism.png|60px]]|| [[Image:Hexagonal prism vertfig.png|50px]]<br>4.4.6|| 2 6 {{pipe}} 2|| D<sub>6h</sub>|| C33c|| —|| U76c||K01c||  12|| 18|| 8|| 6{4}<br>+2{6}
|-
|| [[Heptagonal prism]]|| [[Image:Prism 7.png|60px]]|| [[Image:Heptagonal prism vertfig.png|50px]]<br>4.4.7|| 2 7 {{pipe}} 2|| D<sub>7h</sub>|| C33d|| —|| U76d||K01d||  14|| 21|| 9|| 7{4}<br>+2{7}
|-
|| [[Octagonal prism]]|| [[Image:Octagonal prism.png|60px]]|| [[Image:Octagonal prism vertfig.png|50px]]<br>4.4.8|| 2 8 {{pipe}} 2|| D<sub>8h</sub>|| C33e|| —|| U76e|| K01e|| 16|| 24|| 10|| 8{4}<br>+2{8}
|-
|| [[Enneagonal prism]]|| [[Image:Prism 9.png|60px]]|| [[Image:Enneagonal prism vertfig.png|50px]]<br>4.4.9|| 2 9 {{pipe}} 2|| D<sub>9h</sub>|| C33f|| —|| U76f||K01f||  18|| 27|| 11|| 9{4}<br>+2{9}
|-
|| [[Decagonal prism]]|| [[Image:Decagonal prism.png|60px]]|| [[Image:Decagonal prism vf.png|50px]]<br>4.4.10|| 2 10 {{pipe}} 2|| D<sub>10h</sub>|| C33g|| —|| U76g|| K01g|| 20|| 30|| 12|| 10{4}<br>+2{10}
|-
|| [[Hendecagonal prism]]|| [[Image:Hendecagonal prism.png|60px]]|| [[Image:Hendecagonal prism vf.png|50px]]<br>4.4.11|| 2 11 {{pipe}} 2|| D<sub>11h</sub>|| C33h|| —|| U76h|| K01h|| 22|| 33|| 13|| 11{4}<br>+2{11}
|-
|| [[Dodecagonal prism]]|| [[Image:Dodecagonal prism.png|60px]]|| [[Image:Dodecagonal prism vf.png|50px]]<br>4.4.12|| 2 12 {{pipe}} 2|| D<sub>12h</sub>|| C33i|| —|| U76i|| K01i|| 24|| 36|| 14|| 12{4}<br>+2{12}
|-
|| [[Truncated octahedron]]|| [[Image:Truncated octahedron.png|60px]]|| [[Image:Truncated octahedron vertfig.png|50px]]<br>4.6.6|| 2 4 {{pipe}} 3|| O<sub>h</sub>|| C20|| W007|| U08|| K13|| 24|| 36|| 14|| 6{4}<br>+8{6}
|-
|| [[Truncated cuboctahedron]]|| [[Image:Great rhombicuboctahedron.png|60px]]|| [[Image:Great rhombicuboctahedron vertfig.png|50px]]<br>4.6.8|| 2 3 4 {{pipe}} || O<sub>h</sub>|| C23|| W015|| U11|| K16|| 48|| 72|| 26|| 12{4}<br>+8{6}<br>+6{8}
|-
|| [[Truncated icosidodecahedron]]|| [[Image:Great rhombicosidodecahedron.png|60px]]|| [[Image:Great rhombicosidodecahedron vertfig.png|50px]]<br>4.6.10|| 2 3 5 {{pipe}} || I<sub>h</sub>|| C31|| W016|| U28|| K33|| 120|| 180|| 62|| 30{4}<br>+20{6}<br>+12{10}
|-
|| [[Dodecahedron]]|| [[Image:Dodecahedron.png|60px]]|| [[Image:Dodecahedron vertfig.png|50px]]<br>5.5.5|| 3 {{pipe}} 2 5|| I<sub>h</sub>|| C26|| W005|| U23|| K28|| 20|| 30|| 12|| 12{5}
|-
|| [[Truncated icosahedron]]|| [[Image:Truncated icosahedron.png|60px]]|| [[Image:Truncated icosahedron vertfig.png|50px]]<br>5.6.6|| 2 5 {{pipe}} 3|| I<sub>h</sub>|| C27|| W009|| U25|| K30|| 60|| 90|| 32|| 12{5}<br>+20{6}
|-
|| [[Octahedron]]|| [[Image:Octahedron.png|60px]]|| [[Image:Octahedron vertfig.svg|50px]]<br>3.3.3.3|| 4 {{pipe}} 2 3|| O<sub>h</sub>|| C17|| W002|| U05|| K10|| 6|| 12|| 8|| 8{3}
|-
|| [[Square antiprism]]|| [[Image:Square antiprism.png|60px]]|| [[Image:Square antiprism vertfig.png|50px]]<br>3.3.3.4||  {{pipe}} 2 2 4|| D<sub>4d</sub>|| C34a|| —|| U77a|| K02a|| 8|| 16|| 10|| 8{3}<br>+2{4}
|-
|| [[Pentagonal antiprism]]|| [[Image:Pentagonal antiprism.png|60px]]|| [[Image:Pentagonal antiprism vertfig.png|50px]]<br>3.3.3.5||  {{pipe}} 2 2 5|| D<sub>5d</sub>|| C34b|| —|| U77b|| K02b|| 10|| 20|| 12|| 10{3}<br>+2{5}
|-
|| [[Hexagonal antiprism]]|| [[Image:Hexagonal antiprism.png|60px]]|| [[Image:Hexagonal antiprism vertfig.png|50px]]<br>3.3.3.6||  {{pipe}} 2 2 6|| D<sub>6d</sub>|| C34c|| —|| U77c|| K02c|| 12|| 24|| 14|| 12{3}<br>+2{6}
|-
|| [[Heptagonal antiprism]]|| [[Image:Antiprism 7.png|60px]]|| [[Image:Heptagonal antiprism vertfig.png|50px]]<br>3.3.3.7||  {{pipe}} 2 2 7|| D<sub>7d</sub>|| C34d|| —|| U77d|| K02d|| 14|| 28|| 16|| 14{3}<br>+2{7}
|-
|| [[Octagonal antiprism]]|| [[Image:Octagonal antiprism.png|60px]]|| [[Image:Octagonal antiprism vertfig.png|50px]]<br>3.3.3.8||  {{pipe}} 2 2 8|| D<sub>8d</sub>|| C34e|| —|| U77e|| K02e|| 16|| 32|| 18|| 16{3}<br>+2{8}
|-
|| [[Enneagonal antiprism]]|| [[Image:Enneagonal antiprism.png|60px]]|| [[Image:Enneagonal antiprism vertfig.png|50px]]<br>3.3.3.9||  {{pipe}} 2 2 9|| D<sub>9d</sub>|| C34f|| —|| U77f|| K02f|| 18|| 36|| 20|| 18{3}<br>+2{9}
|-
|| [[Decagonal antiprism]]|| [[Image:Decagonal antiprism.png|60px]]|| [[Image:Decagonal antiprism vf.png|50px]]<br>3.3.3.10||  {{pipe}} 2 2 10|| D<sub>10d</sub>|| C34g|| —|| U77g||K02g|| 20|| 40|| 22|| 20{3}<br>+2{10}
|-
|| Hendecagonal antiprism|| [[Image:Hendecagonal antiprism.png|60px]]|| [[Image:Hendecagonal antiprism vf.png|50px]]<br>3.3.3.11||  {{pipe}} 2 2 11|| D<sub>11d</sub>|| C34h|| —|| U77h||K02h|| 22|| 44|| 24|| 22{3}<br>+2{11}
|-
|| Dodecagonal antiprism|| [[Image:Dodecagonal antiprism.png|60px]]|| [[Image:Dodecagonal antiprism vf.png|50px]]<br>3.3.3.12||  {{pipe}} 2 2 12|| D<sub>12d</sub>|| C34i|| —|| U77i||K02i||  24|| 48|| 26|| 24{3}<br>+2{12}
|-
|| [[Cuboctahedron]]|| [[Image:Cuboctahedron.png|60px]]|| [[Image:Cuboctahedron vertfig.png|50px]]<br>3.4.3.4|| 2 {{pipe}} 3 4||| O<sub>h</sub>|| C19|| W011|| U07|| K12|| 12|| 24|| 14|| 8{3}<br>+6{4}
|-
|| [[Rhombicuboctahedron]]|| [[Image:Small rhombicuboctahedron.png|60px]]|| [[Image:Small rhombicuboctahedron vertfig.png|50px]]<br>3.4.4.4|| 3 4 {{pipe}} 2|| O<sub>h</sub>|| C22|| W013|| U10|| K15|| 24|| 48|| 26|| 8{3}<br>+(6+12){4}
|-
|| [[Rhombicosidodecahedron]]|| [[Image:Small rhombicosidodecahedron.png|60px]]|| [[Image:Small rhombicosidodecahedron vertfig.png|50px]]<br>3.4.5.4|| 3 5 {{pipe}} 2|| I<sub>h</sub>|| C30|| W014|| U27|| K32|| 60|| 120|| 62|| 20{3}<br>+30{4}<br>+12{5}
|-
|| [[Icosidodecahedron]]|| [[Image:Icosidodecahedron.png|60px]]|| [[Image:Icosidodecahedron vertfig.png|50px]]<br>3.5.3.5|| 2 {{pipe}} 3 5|| I<sub>h</sub>|| C28|| W012|| U24|| K29|| 30|| 60|| 32|| 20{3}<br>+12{5}
|-
|| [[Icosahedron]]|| [[Image:Icosahedron.png|60px]]|| [[Image:Icosahedron vertfig.png|50px]]<br>3.3.3.3.3|| 5 {{pipe}} 2 3|| I<sub>h</sub>|| C25|| W004|| U22|| K27|| 12|| 30|| 20|| 20{3}
|-
|| [[Snub cube]]|| [[Image:Snub hexahedron.png|60px]]|| [[Image:Snub cube vertfig.png|50px]]<br>3.3.3.3.4||  {{pipe}} 2 3 4|| O|| C24|| W017|| U12|| K17|| 24|| 60|| 38|| (8+24){3}<br>+6{4}
|-
|| [[Snub dodecahedron]]|| [[Image:Snub dodecahedron ccw.png|60px]]|| [[Image:Snub dodecahedron vertfig.png|50px]]<br>3.3.3.3.5||  {{pipe}} 2 3 5|| I|| C32|| W018|| U29|| K34|| 60|| 150|| 92|| (20+60){3}<br>+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 {{pipe}} {{sfrac|5|2}} 3 3, {{pipe}} {{sfrac|5|2}} {{sfrac|3|2}} {{sfrac|3|2}}, {{pipe}} {{sfrac|5|3}} {{sfrac|5|2}} 3, {{pipe}} {{sfrac|3|2}} {{sfrac|5|3}} 3 {{sfrac|5|2}}, and {{pipe}} ({{sfrac|3|2}}) {{sfrac|5|3}} (3) {{sfrac|5|2}} have some faces occurring as coplanar pairs. (Coxeter et al. 1954, pp. 423, 425, 426; Skilling 1975, p. 123)

{| class="wikitable sortable" style="text-align:center;font-size:small;"
!Name|| Image|| [[Wythoff symbol|Wyth sym]]|| [[Vertex configuration|Vert. fig]]|| Sym.|| C#|| W#|| U#|| K#|| Vert.|| Edges|| Faces|| Chi|| [[Orientability|Orient- able?]]|| Dens.|| Faces by type
|-
|| [[Octahemioctahedron]]|| [[Image:Octahemioctahedron.png|60px]]|| {{sfrac|3|2}} 3 {{pipe}} 3|| [[Image:Octahemioctahedron vertfig.png|50px]] 6.{{sfrac|3|2}}.6.3|| O<sub>h</sub>|| C37||W068|| U03|| K08|| 12|| 24|| 12|| 0|| Yes|| &nbsp;|| 8{3}+4{6}
|-
|| [[Tetrahemihexahedron]]|| [[Image:Tetrahemihexahedron.png|60px]]|| {{sfrac|3|2}} 3 {{pipe}} 2|| [[Image:Tetrahemihexahedron vertfig.svg|50px]] 4.{{sfrac|3|2}}.4.3|| T<sub>d</sub>|| C36||W067|| U04|| K09|| 6|| 12|| 7|| 1|| No|| &nbsp;|| 4{3}+3{4}
|-
|| [[Cubohemioctahedron]]|| [[Image:Cubohemioctahedron.png|60px]]|| {{sfrac|4|3}} 4 {{pipe}} 3|| [[Image:Cubohemioctahedron vertfig.png|50px]] 6.{{sfrac|4|3}}.6.4|| O<sub>h</sub>|| C51||W078|| U15|| K20|| 12|| 24|| 10|| −2|| No|| &nbsp;|| 6{4}+4{6}
|-
|| [[Great dodecahedron]]|| [[Image:Great dodecahedron.png|60px]]|| {{sfrac|5|2}} {{pipe}} 2 5|| [[Image:Great dodecahedron vertfig.png|50px]] (5.5.5.5.5)/2|| I<sub>h</sub>|| C44||W021|| U35|| K40|| 12|| 30|| 12|| −6|| Yes|| 3|| 12{5}
|-
|| [[Great icosahedron]]|| [[Image:Great icosahedron.png|60px]]|| {{sfrac|5|2}} {{pipe}} 2 3|| [[Image:Great icosahedron vertfig.svg|50px]] (3.3.3.3.3)/2|| I<sub>h</sub>|| C69||W041|| U53|| K58|| 12|| 30|| 20|| 2|| Yes|| 7|| 20{3}
|-
|| [[Great ditrigonal icosidodecahedron]]|| [[Image:Great ditrigonal icosidodecahedron.png|60px]]|| {{sfrac|3|2}} {{pipe}} 3 5|| [[Image:Great ditrigonal icosidodecahedron vertfig.png|50px]] (5.3.5.3.5.3)/2|| I<sub>h</sub>|| C61||W087|| U47|| K52|| 20|| 60|| 32|| −8|| Yes|| 6|| 20{3}+12{5}
|-
|| [[Small rhombihexahedron]]|| [[Image:Small rhombihexahedron.png|60px]]|| 2 4 ({{sfrac|3|2}} {{sfrac|4|2}}) {{pipe}} || [[Image:Small rhombihexahedron vertfig.png|50px]] 4.8.{{sfrac|4|3}}.{{sfrac|8|7}}|| O<sub>h</sub>|| C60||W086|| U18|| K23|| 24|| 48|| 18|| −6|| No|| &nbsp;|| 12{4}+6{8}
|-
|| [[Small cubicuboctahedron]]|| [[Image:Small cubicuboctahedron.png|60px]]|| {{sfrac|3|2}} 4 {{pipe}} 4|| [[Image:Small cubicuboctahedron vertfig.png|50px]] 8.{{sfrac|3|2}}.8.4|| O<sub>h</sub>|| C38||W069|| U13|| K18|| 24|| 48|| 20|| −4|| Yes|| 2|| 8{3}+6{4}+6{8}
|-
|| [[Nonconvex great rhombicuboctahedron]]|| [[Image:Uniform great rhombicuboctahedron.png|60px]]|| {{sfrac|3|2}} 4 {{pipe}} 2|| [[Image:Uniform great rhombicuboctahedron vertfig.png|50px]] 4.{{sfrac|3|2}}.4.4|| O<sub>h</sub>|| C59||W085|| U17|| K22|| 24|| 48|| 26|| 2|| Yes|| 5|| 8{3}+(6+12){4}
|-
|| [[Small dodecahemidodecahedron]]|| [[Image:Small dodecahemidodecahedron.png|60px]]|| {{sfrac|5|4}} 5 {{pipe}} 5|| [[Image:Small dodecahemidodecahedron vertfig.png|50px]] 10.{{sfrac|5|4}}.10.5|| I<sub>h</sub>|| C65||W091|| U51|| K56|| 30|| 60|| 18|| −12|| No|| &nbsp;|| 12{5}+6{10}
|-
|| [[Great dodecahemicosahedron]]|| [[Image:Great dodecahemicosahedron.png|60px]]|| {{sfrac|5|4}} 5 {{pipe}} 3|| [[Image:Great dodecahemicosahedron vertfig.png|50px]] 6.{{sfrac|5|4}}.6.5|| I<sub>h</sub>||C81|| W102|| U65|| K70|| 30|| 60|| 22|| −8|| No|| &nbsp;|| 12{5}+10{6}
|-
|| [[Small icosihemidodecahedron]]|| [[Image:Small icosihemidodecahedron.png|60px]]|| {{sfrac|3|2}} 3 {{pipe}} 5|| [[Image:Small icosihemidodecahedron vertfig.svg|50px]] 10.{{sfrac|3|2}}.10.3|| I<sub>h</sub>|| C63||W089|| U49|| K54|| 30|| 60|| 26|| −4|| No|| &nbsp;|| 20{3}+6{10}
|-
|| [[Small dodecicosahedron]]|| [[Image:Small dodecicosahedron.png|60px]]|| 3 5 ({{sfrac|3|2}} {{sfrac|5|4}}) {{pipe}} || [[Image:Small dodecicosahedron vertfig.png|50px]] 10.6.{{sfrac|10|9}}.{{sfrac|6|5}}|| I<sub>h</sub>|| C64||W090|| U50|| K55|| 60|| 120|| 32|| −28|| No|| &nbsp;|| 20{6}+12{10}
|-
|| [[Small rhombidodecahedron]]|| [[Image:Small rhombidodecahedron.png|60px]]|| 2 5 ({{sfrac|3|2}} {{sfrac|5|2}}) {{pipe}} || [[Image:Small rhombidodecahedron vertfig.png|50px]] 10.4.{{sfrac|10|9}}.{{sfrac|4|3}}|| I<sub>h</sub>|| C46||W074|| U39|| K44|| 60|| 120|| 42|| −18|| No|| &nbsp;|| 30{4}+12{10}
|-
|| [[Small dodecicosidodecahedron]]|| [[Image:Small dodecicosidodecahedron.png|60px]]|| {{sfrac|3|2}} 5 {{pipe}} 5|| [[Image:Small dodecicosidodecahedron vertfig.png|50px]] 10.{{sfrac|3|2}}.10.5|| I<sub>h</sub>||C42|| W072|| U33|| K38|| 60|| 120|| 44|| −16|| Yes|| 2|| 20{3}+12{5}+12{10}
|-
|| [[Rhombicosahedron]]|| [[Image:Rhombicosahedron.png|60px]]|| 2 3 ({{sfrac|5|4}} {{sfrac|5|2}}) {{pipe}} || [[Image:Rhombicosahedron vertfig.png|50px]] 6.4.{{sfrac|6|5}}.{{sfrac|4|3}}|| I<sub>h</sub>||C72|| W096|| U56|| K61|| 60|| 120|| 50|| −10|| No|| &nbsp;|| 30{4}+20{6}
|-
|| [[Great icosicosidodecahedron]]|| [[Image:Great icosicosidodecahedron.png|60px]]|| {{sfrac|3|2}} 5 {{pipe}} 3|| [[Image:Great icosicosidodecahedron vertfig.png|50px]] 6.{{sfrac|3|2}}.6.5|| I<sub>h</sub>|| C62||W088|| U48|| K53|| 60|| 120|| 52|| −8|| Yes|| 6|| 20{3}+12{5}+20{6}
|-
|| [[Pentagrammic prism]]|| [[Image:Pentagrammic prism.png|60px]]|| 2 {{sfrac|5|2}} {{pipe}} 2|| [[Image:Pentagrammic prism vertfig.png|50px]] {{sfrac|5|2}}.4.4|| D<sub>5h</sub>|| C33b|| —|| U78a|| K03a|| 10|| 15|| 7|| 2|| Yes|| 2|| 5{4}+2{{mset|{{sfrac|5|2}}}}
|-
|| Heptagrammic prism (7/2)|| [[Image:Heptagrammic prism 7-2.png|60px]]|| 2 {{sfrac|7|2}} {{pipe}} 2|| [[Image:Septagrammic prism vertfig.png|50px]] {{sfrac|7|2}}.4.4|| D<sub>7h</sub>|| C33d|| —|| U78b|| K03b|| 14|| 21|| 9|| 2|| Yes|| 2|| 7{4}+2{{mset|{{sfrac|7|2}}}}
|-
|| Heptagrammic prism (7/3)|| [[Image:Heptagrammic prism 7-3.png|60px]]|| 2 {{sfrac|7|3}} {{pipe}} 2|| [[Image:Septagrammic prism-3-7 vertfig.png|50px]] {{sfrac|7|3}}.4.4|| D<sub>7h</sub>|| C33d|| —|| U78c|| K03c|| 14|| 21|| 9|| 2|| Yes|| 3|| 7{4}+2{{mset|{{sfrac|7|3}}}}
|-
|| [[Octagrammic prism]]|| [[File:Prism 8-3.png|60px]]|| 2 {{sfrac|8|3}} {{pipe}} 2|| [[File:Octagrammic prism vertfig.png|50px]] {{sfrac|8|3}}.4.4|| D<sub>8h</sub>|| C33e|| —|| U78d|| K03d|| 16|| 24 || 10|| 2|| Yes|| 3|| 8{4}+2{{mset|{{sfrac|8|3}}}}
|-
|| [[Pentagrammic antiprism]]|| [[Image:Pentagrammic antiprism.png|60px]]|| {{pipe}} 2 2 {{sfrac|5|2}}|| [[Image:Pentagrammic antiprism vertfig.png|50px]] {{sfrac|5|2}}.3.3.3|| D<sub>5h</sub>|| C34b|| —|| U79a|| K04a|| 10|| 20|| 12|| 2|| Yes|| 2|| 10{3}+2{{mset|{{sfrac|5|2}}}}
|-
|| [[Pentagrammic crossed-antiprism]]|| [[Image:Pentagrammic crossed antiprism.png|60px]]|| {{pipe}} 2 2 {{sfrac|5|3}}|| [[Image:Pentagrammic crossed-antiprism vertfig.png|50px]] {{sfrac|5|3}}.3.3.3|| D<sub>5d</sub>|| C35a|| —|| U80a|| K05a|| 10|| 20|| 12|| 2|| Yes|| 3|| 10{3}+2{{mset|{{sfrac|5|2}}}}
|-
|| Heptagrammic antiprism (7/2)|| [[File:Antiprism 7-2.png|60px]]|| {{pipe}} 2 2 {{sfrac|7|2}} || [[Image:Heptagrammic antiprism-2-7 vertfig.png|50px]] {{sfrac|7|2}}.3.3.3|| D<sub>7h</sub>|| C34d|| —|| U79b|| K04b|| 14|| 28|| 16|| 2|| Yes|| 3|| 14{3}+2{{mset|{{sfrac|7|2}}}}
|-
|| Heptagrammic antiprism (7/3)|| [[File:Antiprism 7-3.png|60px]]||{{pipe}} 2 2 {{sfrac|7|3}} || [[Image:Heptagrammic antiprism-3-7 vertfig.png|50px]] {{sfrac|7|3}}.3.3.3|| D<sub>7d</sub>|| C34d|| —|| U79c|| K04c|| 14|| 28|| 16|| 2|| Yes|| 3|| 14{3}+2{{mset|{{sfrac|7|3}}}}
|-
|| Heptagrammic crossed-antiprism|| [[File:Antiprism 7-4.png|60px]]||{{pipe}} 2 2 {{sfrac|7|4}} || [[Image:Heptagrammic antiprism-4-7 vertfig.png|50px]] {{sfrac|7|4}}.3.3.3|| D<sub>7h</sub>|| C35b|| —|| U80b|| K05b|| 14|| 28|| 16|| 2|| Yes|| 4|| 14{3}+2{{mset|{{sfrac|7|3}}}}
|-
|| [[Octagrammic antiprism]]|| [[File:Antiprism 8-3.png|60px]]||{{pipe}} 2 2 {{sfrac|8|3}} || [[Image:Octagrammic antiprism-3-8 vertfig.png|50px]] {{sfrac|8|3}}.3.3.3|| D<sub>8d</sub>|| C34e|| —|| U79d|| K04d|| 16|| 32|| 18|| 2|| Yes|| 3|| 16{3}+2{{mset|{{sfrac|8|3}}}}
|-
|| [[Octagrammic crossed-antiprism]]|| [[File:Antiprism 8-5.png|60px]]||{{pipe}} 2 2 {{sfrac|8|5}} || [[Image:Octagrammic antiprism-5-8 vertfig.png|50px]] {{sfrac|8|5}}.3.3.3|| D<sub>8d</sub>|| C35c|| —|| U80c|| K05c|| 16|| 32|| 18|| 2|| Yes|| 5|| 16{3}+2{{mset|{{sfrac|8|3}}}}
|-
|| [[Small stellated dodecahedron]]|| [[Image:Small stellated dodecahedron.png|60px]]|| 5 {{pipe}} 2 {{sfrac|5|2}}|| [[Image:Small stellated dodecahedron vertfig.png|50px]] ({{sfrac|5|2}})<sup>5</sup>|| I<sub>h</sub>|| C43|| W020|| U34|| K39|| 12|| 30|| 12|| −6|| Yes|| 3|| 12{{mset|{{sfrac|5|2}}}}
|-
|| [[Great stellated dodecahedron]]|| [[Image:Great stellated dodecahedron.png|60px]]|| 3 {{pipe}} 2 {{sfrac|5|2}}|| [[Image:Great stellated dodecahedron vertfig.png|50px]] ({{sfrac|5|2}})<sup>3</sup>|| I<sub>h</sub>|| C68|| W022|| U52|| K57|| 20|| 30|| 12|| 2|| Yes|| 7|| 12{{mset|{{sfrac|5|2}}}}
|-
|| [[Ditrigonal dodecadodecahedron]]|| [[Image:Ditrigonal dodecadodecahedron.png|60px]]|| 3 {{pipe}} {{sfrac|5|3}} 5|| [[Image:Ditrigonal dodecadodecahedron vertfig.png|50px]] ({{sfrac|5|3}}.5)<sup>3</sup>|| I<sub>h</sub>|| C53|| W080|| U41|| K46|| 20|| 60|| 24|| −16|| Yes|| 4|| 12{5}+12{{mset|{{sfrac|5|2}}}}
|-
|| [[Small ditrigonal icosidodecahedron]]|| [[Image:Small ditrigonal icosidodecahedron.png|60px]]|| 3 {{pipe}} {{sfrac|5|2}} 3|| [[Image:Small ditrigonal icosidodecahedron vertfig.png|50px]] ({{sfrac|5|2}}.3)<sup>3</sup>|| I<sub>h</sub>|| C39|| W070|| U30|| K35|| 20|| 60|| 32|| −8|| Yes|| 2|| 20{3}+12{{mset|{{sfrac|5|2}}}}
|-
|| [[Stellated truncated hexahedron]]|| [[Image:Stellated truncated hexahedron.png|60px]]|| 2 3 {{pipe}} {{sfrac|4|3}}|| [[Image:Stellated truncated hexahedron vertfig.png|50px]] {{sfrac|8|3}}.{{sfrac|8|3}}.3|| O<sub>h</sub>|| C66|| W092|| U19|| K24|| 24|| 36|| 14|| 2|| Yes|| 7|| 8{3}+6{{mset|{{sfrac|8|3}}}}
|-
|| [[Great rhombihexahedron]]|| [[Image:Great rhombihexahedron.png|60px]]|| 2 {{sfrac|4|3}} ({{sfrac|3|2}} {{sfrac|4|2}}) {{pipe}} || [[Image:Great rhombihexahedron vertfig.png|50px]] 4.{{sfrac|8|3}}.{{sfrac|4|3}}.{{sfrac|8|5}}|| O<sub>h</sub>|| C82|| W103|| U21|| K26|| 24|| 48|| 18|| −6|| No|| &nbsp;|| 12{4}+6{{mset|{{sfrac|8|3}}}}
|-
|| [[Great cubicuboctahedron]]|| [[Image:Great cubicuboctahedron.png|60px]]|| 3 4 {{pipe}} {{sfrac|4|3}}|| [[Image:Great cubicuboctahedron vertfig.png|50px]] {{sfrac|8|3}}.3.{{sfrac|8|3}}.4|| O<sub>h</sub>|| C50|| W077|| U14|| K19|| 24|| 48|| 20|| −4|| Yes|| 4|| 8{3}+6{4}+6{{mset|{{sfrac|8|3}}}}
|-
|| [[Great dodecahemidodecahedron]]|| [[Image:Great dodecahemidodecahedron.png|60px]]|| {{sfrac|5|3}} {{sfrac|5|2}} {{pipe}} {{sfrac|5|3}}|| [[Image:Great dodecahemidodecahedron vertfig.png|50px]] {{sfrac|10|3}}.{{sfrac|5|3}}.{{sfrac|10|3}}.{{sfrac|5|2}}|| I<sub>h</sub>|| C86|| W107|| U70|| K75|| 30|| 60|| 18|| −12|| No|| &nbsp;|| 12{{mset|{{sfrac|5|2}}}}+6{{mset|{{sfrac|10|3}}}}
|-
|| [[Small dodecahemicosahedron]]|| [[Image:Small dodecahemicosahedron.png|60px]]|| {{sfrac|5|3}} {{sfrac|5|2}} {{pipe}} 3|| [[Image:Small dodecahemicosahedron vertfig.png|50px]] 6.{{sfrac|5|3}}.6.{{sfrac|5|2}}|| I<sub>h</sub>|| C78|| W100|| U62|| K67|| 30|| 60|| 22|| −8|| No|| &nbsp;|| 12{{mset|{{sfrac|5|2}}}}+10{6}
|-
|| [[Dodecadodecahedron]]|| [[Image:Dodecadodecahedron.png|60px]]|| 2 {{pipe}} 5 {{sfrac|5|2}}|| [[Image:Dodecadodecahedron vertfig.png|50px]] ({{sfrac|5|2}}.5)<sup>2</sup>|| I<sub>h</sub>|| C45|| W073|| U36|| K41|| 30|| 60|| 24|| −6|| Yes|| 3|| 12{5}+12{{mset|{{sfrac|5|2}}}}
|-
|| [[Great icosihemidodecahedron]]|| [[Image:Great icosihemidodecahedron.png|60px]]|| {{sfrac|3|2}} 3 {{pipe}} {{sfrac|5|3}}|| [[Image:Great icosihemidodecahedron vertfig.png|50px]] {{sfrac|10|3}}.{{sfrac|3|2}}.{{sfrac|10|3}}.3|| I<sub>h</sub>|| C85|| W106|| U71|| K76|| 30|| 60|| 26|| −4|| No|| &nbsp;|| 20{3}+6{{mset|{{sfrac|10|3}}}}
|-
|| [[Great icosidodecahedron]]|| [[Image:Great icosidodecahedron.png|60px]]|| 2 {{pipe}} 3 {{sfrac|5|2}}|| [[Image:Great icosidodecahedron vertfig.png|50px]] ({{sfrac|5|2}}.3)<sup>2</sup>|| I<sub>h</sub>|| C70|| W094|| U54|| K59|| 30|| 60|| 32|| 2|| Yes|| 7|| 20{3}+12{{mset|{{sfrac|5|2}}}}
|-
|| [[Cubitruncated cuboctahedron]]|| [[Image:Cubitruncated cuboctahedron.png|60px]]|| {{sfrac|4|3}} 3 4 {{pipe}} || [[Image:Cubitruncated cuboctahedron vertfig.png|50px]] {{sfrac|8|3}}.6.8|| O<sub>h</sub>|| C52|| W079|| U16|| K21|| 48|| 72|| 20|| −4|| Yes|| 4|| 8{6}+6{8}+6{{mset|{{sfrac|8|3}}}}
|-
|| [[Great truncated cuboctahedron]]|| [[Image:Great truncated cuboctahedron.png|60px]]|| {{sfrac|4|3}} 2 3 {{pipe}} || [[Image:Great truncated cuboctahedron vertfig.png|50px]] {{sfrac|8|3}}.4.{{sfrac|6|5}}|| O<sub>h</sub>|| C67|| W093|| U20|| K25|| 48|| 72|| 26|| 2|| Yes|| 1|| 12{4}+8{6}+6{{mset|{{sfrac|8|3}}}}
|-
|| [[Truncated great dodecahedron]]|| [[Image:Great truncated dodecahedron.png|60px]]|| 2 {{sfrac|5|2}} {{pipe}} 5|| [[Image:Truncated great dodecahedron vertfig.png|50px]] 10.10.{{sfrac|5|2}}|| I<sub>h</sub>|| C47|| W075|| U37|| K42|| 60|| 90|| 24|| −6|| Yes|| 3|| 12{{mset|{{sfrac|5|2}}}}+12{10}
|-
|| [[Small stellated truncated dodecahedron]]|| [[Image:Small stellated truncated dodecahedron.png|60px]]|| 2 5 {{pipe}} {{sfrac|5|3}}|| [[Image:Small stellated truncated dodecahedron vertfig.png|50px]] {{sfrac|10|3}}.{{sfrac|10|3}}.5|| I<sub>h</sub>|| C74|| W097|| U58|| K63|| 60|| 90|| 24|| −6|| Yes|| 9|| 12{5}+12{{mset|{{sfrac|10|3}}}}
|-
|| [[Great stellated truncated dodecahedron]]|| [[Image:Great stellated truncated dodecahedron.png|60px]]|| 2 3 {{pipe}} {{sfrac|5|3}}|| [[Image:Great stellated truncated dodecahedron vertfig.png|50px]] {{sfrac|10|3}}.{{sfrac|10|3}}.3|| I<sub>h</sub>|| C83|| W104|| U66|| K71|| 60|| 90|| 32|| 2|| Yes|| 13|| 20{3}+12{{mset|{{sfrac|10|3}}}}
|-
|| [[Truncated great icosahedron]]|| [[Image:Great truncated icosahedron.png|60px]]|| 2 {{sfrac|5|2}} {{pipe}} 3|| [[Image:Great truncated icosahedron vertfig.png|50px]] 6.6.{{sfrac|5|2}}|| I<sub>h</sub>|| C71|| W095|| U55|| K60|| 60|| 90|| 32|| 2|| Yes|| 7|| 12{{mset|{{sfrac|5|2}}}}+20{6}
|-
|| [[Great dodecicosahedron]]|| [[Image:Great dodecicosahedron.png|60px]]|| 3 {{sfrac|5|3}}({{sfrac|3|2}} {{sfrac|5|2}}) {{pipe}} || [[Image:Great dodecicosahedron vertfig.png|50px]] 6.{{sfrac|10|3}}.{{sfrac|6|5}}.{{sfrac|10|7}}|| I<sub>h</sub>|| C79|| W101|| U63|| K68|| 60|| 120|| 32|| −28|| No|| &nbsp;|| 20{6}+12{{mset|{{sfrac|10|3}}}}
|-
|| [[Great rhombidodecahedron]]|| [[Image:Great rhombidodecahedron.png|60px]]|| 2 {{sfrac|5|3}} ({{sfrac|3|2}} {{sfrac|5|4}}) {{pipe}} || [[Image:Great rhombidodecahedron vertfig.png|50px]] 4.{{sfrac|10|3}}.{{sfrac|4|3}}.{{sfrac|10|7}}|| I<sub>h</sub>|| C89|| W109|| U73|| K78|| 60|| 120|| 42|| −18|| No|| &nbsp;|| 30{4}+12{{mset|{{sfrac|10|3}}}}
|-
|| [[Icosidodecadodecahedron]]|| [[Image:Icosidodecadodecahedron.png|60px]]|| {{sfrac|5|3}} 5 {{pipe}} 3|| [[Image:Icosidodecadodecahedron vertfig.png|50px]] 6.{{sfrac|5|3}}.6.5||I<sub>h</sub>|| C56|| W083|| U44|| K49|| 60|| 120|| 44|| −16|| Yes|| 4|| 12{5}+12{{mset|{{sfrac|5|2}}}}+20{6}
|-
|| [[Small ditrigonal dodecicosidodecahedron]]|| [[Image:Small ditrigonal dodecicosidodecahedron.png|60px]]|| {{sfrac|5|3}} 3 {{pipe}} 5|| [[Image:Small ditrigonal dodecicosidodecahedron vertfig.png|50px]] 10.{{sfrac|5|3}}.10.3|| I<sub>h</sub>|| C55|| W082|| U43|| K48|| 60|| 120|| 44|| −16|| Yes|| 4|| 20{3}+12{{mset|{{sfrac|5|2}}}}+12{10}
|-
|| [[Great ditrigonal dodecicosidodecahedron]]|| [[Image:Great ditrigonal dodecicosidodecahedron.png|60px]]|| 3 5 {{pipe}} {{sfrac|5|3}}|| [[Image:Great ditrigonal dodecicosidodecahedron vertfig.png|50px]] {{sfrac|10|3}}.3.{{sfrac|10|3}}.5|| I<sub>h</sub>|| C54|| W081|| U42|| K47|| 60|| 120|| 44|| −16|| Yes|| 4|| 20{3}+12{5}+12{{mset|{{sfrac|10|3}}}}
|-
|| [[Great dodecicosidodecahedron]]|| [[Image:Great dodecicosidodecahedron.png|60px]]|| {{sfrac|5|2}} 3 {{pipe}} {{sfrac|5|3}}|| [[Image:Great dodecicosidodecahedron vertfig.png|50px]] {{sfrac|10|3}}.{{sfrac|5|2}}.{{sfrac|10|3}}.3|| I<sub>h</sub>|| C77|| W099|| U61|| K66|| 60|| 120|| 44|| −16|| Yes|| 10|| 20{3}+12{{mset|{{sfrac|5|2}}}}+12{{mset|{{sfrac|10|3}}}}
|-
|| [[Small icosicosidodecahedron]]|| [[Image:Small icosicosidodecahedron.png|60px]]|| {{sfrac|5|2}} 3 {{pipe}} 3|| [[Image:Small icosicosidodecahedron vertfig.png|50px]] 6.{{sfrac|5|2}}.6.3|| I<sub>h</sub>|| C40|| W071|| U31|| K36|| 60|| 120|| 52|| −8|| Yes|| 2|| 20{3}+12{{mset|{{sfrac|5|2}}}}+20{6}
|-
|| [[Rhombidodecadodecahedron]]|| [[Image:Rhombidodecadodecahedron.png|60px]]|| {{sfrac|5|2}} 5 {{pipe}} 2|| [[Image:Rhombidodecadodecahedron vertfig.png|50px]] 4.{{sfrac|5|2}}.4.5|| I<sub>h</sub>|| C48|| W076|| U38|| K43|| 60|| 120|| 54|| −6|| Yes|| 3|| 30{4}+12{5}+12{{mset|{{sfrac|5|2}}}}
|-
|| [[Nonconvex great rhombicosidodecahedron]]|| [[Image:Uniform great rhombicosidodecahedron.png|60px]]|| {{sfrac|5|3}} 3 {{pipe}} 2|| [[Image:Uniform great rhombicosidodecahedron vertfig.png|50px]] 4.{{sfrac|5|3}}.4.3|| I<sub>h</sub>|| C84|| W105|| U67|| K72|| 60|| 120|| 62|| 2|| Yes|| 13|| 20{3}+30{4}+12{{mset|{{sfrac|5|2}}}}
|-
|| [[Icositruncated dodecadodecahedron]]|| [[Image:Icositruncated dodecadodecahedron.png|60px]]|| 3 5 {{sfrac|5|3}} {{pipe}} || [[Image:Icositruncated dodecadodecahedron vertfig.png|50px]] {{sfrac|10|3}}.6.10|| I<sub>h</sub>|| C57|| W084|| U45|| K50|| 120|| 180|| 44|| −16|| Yes|| 4|| 20{6}+12{10}+12{{mset|{{sfrac|10|3}}}}
|-
|| [[Truncated dodecadodecahedron]]|| [[Image:Truncated dodecadodecahedron.png|60px]]|| 2 5 {{sfrac|5|3}} {{pipe}} || [[Image:Truncated dodecadodecahedron vertfig.png|50px]] {{sfrac|10|3}}.4.{{sfrac|10|9}}|| I<sub>h</sub>|| C75|| W098|| U59|| K64|| 120|| 180|| 54|| −6|| Yes|| 3|| 30{4}+12{10}+12{{mset|{{sfrac|10|3}}}}
|-
|| [[Great truncated icosidodecahedron]]|| [[Image:Great truncated icosidodecahedron.png|60px]]|| 2 3 {{sfrac|5|3}} {{pipe}} || [[Image:Great truncated icosidodecahedron vertfig.png|50px]] {{sfrac|10|3}}.4.6|| I<sub>h</sub>|| C87|| W108|| U68|| K73|| 120|| 180|| 62|| 2|| Yes|| 13|| 30{4}+20{6}+12{{mset|{{sfrac|10|3}}}}
|-
|| [[Snub dodecadodecahedron]]|| [[Image:Snub dodecadodecahedron.png|60px]]|| {{pipe}} 2 {{sfrac|5|2}} 5|| [[Image:Snub dodecadodecahedron vertfig.png|50px]] 3.3.{{sfrac|5|2}}.3.5|| I|| C49|| W111|| U40|| K45|| 60|| 150|| 84|| −6|| Yes|| 3|| 60{3}+12{5}+12{{mset|{{sfrac|5|2}}}}
|-
|| [[Inverted snub dodecadodecahedron]]|| [[Image:Inverted snub dodecadodecahedron.png|60px]]|| {{pipe}} {{sfrac|5|3}} 2 5|| [[Image:Inverted snub dodecadodecahedron vertfig.png|50px]] 3.{{sfrac|5|3}}.3.3.5|| I|| C76|| W114|| U60|| K65|| 60|| 150|| 84|| −6|| Yes|| 9|| 60{3}+12{5}+12{{mset|{{sfrac|5|2}}}}
|-
|| [[Great snub icosidodecahedron]]|| [[Image:Great snub icosidodecahedron.png|60px]]|| {{pipe}} 2 {{sfrac|5|2}} 3|| [[Image:Great snub icosidodecahedron vertfig.png|50px]] 3<sup>4</sup>.{{sfrac|5|2}}|| I|| C73|| W113|| U57|| K62|| 60|| 150|| 92|| 2|| Yes|| 7|| (20+60){3}+12{{mset|{{sfrac|5|2}}}}
|-
|| [[Great inverted snub icosidodecahedron]]|| [[Image:Great inverted snub icosidodecahedron.png|60px]]|| {{pipe}} {{sfrac|5|3}} 2 3|| [[Image:Great inverted snub icosidodecahedron vertfig.png|50px]] 3<sup>4</sup>.{{sfrac|5|3}}|| I|| C88|| W116|| U69|| K74|| 60|| 150|| 92|| 2|| Yes|| 13|| (20+60){3}+12{{mset|{{sfrac|5|2}}}}
|-
|| [[Great retrosnub icosidodecahedron]]|| [[Image:Great retrosnub icosidodecahedron.png|60px]]|| {{pipe}} 2 {{sfrac|3|2}} {{sfrac|5|3}}|| [[Image:Great retrosnub icosidodecahedron vertfig.png|50px]] (3<sup>4</sup>.{{sfrac|5|2}})/2|| I|| C90|| W117|| U74|| K79|| 60|| 150|| 92|| 2|| Yes|| 37|| (20+60){3}+12{{mset|{{sfrac|5|2}}}}
|-
|| [[Great snub dodecicosidodecahedron]]|| [[Image:Great snub dodecicosidodecahedron.png|60px]]|| {{pipe}} {{sfrac|5|3}} {{sfrac|5|2}} 3|| [[Image:Great snub dodecicosidodecahedron vertfig.png|50px]] 3<sup>3</sup>.{{sfrac|5|3}}.3.{{sfrac|5|2}}|| I|| C80|| W115|| U64|| K69|| 60|| 180|| 104|| −16|| Yes|| 10|| (20+60){3}+(12+12){{mset|{{sfrac|5|2}}}}
|-
|| [[Snub icosidodecadodecahedron]]|| [[Image:Snub icosidodecadodecahedron.png|60px]]|| {{pipe}} {{sfrac|5|3}} 3 5|| [[Image:Snub icosidodecadodecahedron vertfig.png|50px]] 3<sup>3</sup>.5.3.{{sfrac|5|3}}|| I|| C58|| W112|| U46|| K51|| 60|| 180|| 104|| −16|| Yes|| 4|| (20+60){3}+12{5}+12{{mset|{{sfrac|5|2}}}}
|-
|| [[Small snub icosicosidodecahedron]]|| [[Image:Small snub icosicosidodecahedron.png|60px]]|| {{pipe}} {{sfrac|5|2}} 3 3|| [[Image:Small snub icosicosidodecahedron vertfig.png|50px]] 3<sup>5</sup>.{{sfrac|5|2}}|| I<sub>h</sub>|| C41|| W110|| U32|| K37|| 60|| 180|| 112|| −8|| Yes|| 2|| (40+60){3}+12{{mset|{{sfrac|5|2}}}}
|-
|| [[Small retrosnub icosicosidodecahedron]]|| [[Image:Small retrosnub icosicosidodecahedron.png|60px]]|| {{pipe}} {{sfrac|3|2}} {{sfrac|3|2}} {{sfrac|5|2}}|| [[Image:Small retrosnub icosicosidodecahedron vertfig.png|50px]] (3<sup>5</sup>.{{sfrac|5|2}})/2|| I<sub>h</sub>|| C91|| W118|| U72|| K77|| 60|| 180|| 112|| −8|| Yes|| 38|| (40+60){3}+12{{mset|{{sfrac|5|2}}}}
|-
|| [[Great dirhombicosidodecahedron]]|| [[Image:Great dirhombicosidodecahedron.png|60px]]|| nowrap|{{pipe}} {{sfrac|3|2}} {{sfrac|5|3}} 3 {{sfrac|5|2}}
|| [[Image:Great dirhombicosidodecahedron vertfig.png|50px]] (4.{{sfrac|5|3}}.4.3.4.{{sfrac|5|2}}.4.{{sfrac|3|2}})/2|| I<sub>h</sub>|| C92|| W119|| U75|| K80|| 60|| 240|| 124|| −56|| No|| &nbsp;|| 40{3}+60{4}+24{{mset|{{sfrac|5|2}}}}
|}

==== {{anchor|Special case}}Special case ====
{| class="wikitable sortable" style="text-align:center;font-size:small;"
!Name|| Image|| [[Wythoff symbol|Wyth<br>sym]]|| [[Vertex configuration|Vert.<br>fig]]|| Sym.|| C#|| W#|| U#|| K#|| Vert.|| Edges|| Faces|| Chi|| [[Orientability|Orient-<br>able?]]|| Dens.|| Faces by type
|-
|| [[Great disnub dirhombidodecahedron|Great disnub<br>dirhombidodecahedron]]|| [[Image:Great disnub dirhombidodecahedron.png|60px]]||  {{pipe}} ({{sfrac|3|2}}) {{sfrac|5|3}} (3) {{sfrac|5|2}}|| [[Image:Great disnub dirhombidodecahedron vertfig.png|50px]]<br>({{sfrac|5|2}}.4.3.3.3.4. {{sfrac|5|3}}.<br>4.{{sfrac|3|2}}.{{sfrac|3|2}}.{{sfrac|3|2}}.4)/2|| I<sub>h</sub>|| —||—|| —|| —|| 60|| 360 (*)|| 204|| −96|| No|| &nbsp;|| 120{3}+60{4}+24{{mset|{{sfrac|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 (K<sub>''n''</sub>&nbsp;=&nbsp;U<sub>''n''–5</sub> for ''n''&nbsp;=&nbsp;6&nbsp;to&nbsp;80) (prisms 1–5, Tetrahedron etc. 6+)
* [[List of Wenninger polyhedron models|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]], {{mvar|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-[[orientability|orientable]] polyhedra and [[hemipolyhedron|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==
* [[List of uniform polyhedra by vertex figure]]
* [[List of uniform polyhedra by Wythoff symbol]]
* [[List of uniform polyhedra by Schwarz triangle]]

==References==
*{{Cite journal|last1=Coxeter|first1=Harold Scott MacDonald|author1-link=Harold Scott MacDonald Coxeter|last2=Longuet-Higgins|first2=M. S.|last3=Miller|first3=J. C. P.|title=Uniform polyhedra|jstor=91532|mr=0062446|year=1954|journal=Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences|issn=0080-4614|volume=246|issue=916|pages=401–450|publisher=The Royal Society|doi=10.1098/rsta.1954.0003|bibcode=1954RSPTA.246..401C|s2cid=202575183}}
*{{Cite journal |  last1=Skilling | first1=J. | title=The complete set of uniform polyhedra | jstor=74475 | mr=0365333  | year=1975 | journal=Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences | issn=0080-4614 | volume=278 | pages=111–135 | doi=10.1098/rsta.1975.0022 | issue=1278| bibcode=1975RSPTA.278..111S | s2cid=122634260 }}
*{{Cite journal|last1=Sopov|first1=S. P.|year=1970|title=A proof of the completeness on the list of elementary homogeneous polyhedra|journal=Ukrainskiui Geometricheskiui Sbornik|issue=8|pages=139–156|mr=0326550}}
*{{cite book | first=Magnus | last=Wenninger | authorlink=Magnus Wenninger  | title=Polyhedron Models | publisher=Cambridge University Press | year=1974 | isbn=0-521-09859-9 }}
* {{cite book | first=Magnus | last=Wenninger | authorlink=Magnus Wenninger  | title=Dual Models | publisher=Cambridge University Press | year=1983 | isbn=0-521-54325-8 }}

== External links ==
* [http://www.software3d.com/Stella.php Stella: Polyhedron Navigator] – Software able to generate and print nets for all uniform polyhedra.  Used to create most images on this page.
* [http://www.software3d.com/Uniform.php Paper models]
* Uniform indexing: U1-U80, (Tetrahedron first)
** [https://web.archive.org/web/20060911064351/http://local.wasp.uwa.edu.au/~pbourke/geometry/polyhedra/ Uniform Polyhedra (80), Paul Bourke]
** {{MathWorld|UniformPolyhedron|Uniform Polyhedron}}
** http://www.mathconsult.ch/showroom/unipoly
***[http://www.mathconsult.ch/showroom/unipoly/list.html All uniform polyhedra by rotation group]
** 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)
** https://www.math.technion.ac.il/~rl/kaleido
*** https://web.archive.org/web/20110927223146/http://www.math.technion.ac.il/~rl/docs/uniform.pdf Uniform Solution for Uniform Polyhedra
** http://bulatov.org/polyhedra/uniform
** http://www.orchidpalms.com/polyhedra/uniform/uniform.html
* Also
** http://www.polyedergarten.de/polyhedrix/e_klintro.htm
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[[Category:Uniform polyhedra| ]]
[[Category:Mathematics-related lists|Uniform polyhedra]]

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