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List of uniform polyhedra#Uniform star polyhedra

{{Short description|none}}

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 {{Harvard citation text|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 =

class="wikitable sortable" style="text-align:center;font-size:small;"

! Name

PictureVertex
type		Wythoff
symbol		Sym.C#W#U#K#Vert.EdgesFacesFaces by type
| Tetrahedron60px50px
3.3.3		3 {{pipe}} 2 3TdC15W001U01K064644{3}
| Triangular prism60px50px
3.4.4		2 3 {{pipe}} 2D3hC33a—U76aK01a6952{3}
+3{4}
| Truncated tetrahedron60px50px
3.6.6		2 3 {{pipe}} 3TdC16W006U02K07121884{3}
+4{6}
| Truncated cube60px50px
3.8.8		2 3 {{pipe}} 4OhC21W008U09K142436148{3}
+6{8}
| Truncated dodecahedron60px50px
3.10.10		2 3 {{pipe}} 5IhC29W010U26K3160903220{3}
+12{10}
| Cube60px50px
4.4.4		3 {{pipe}} 2 4OhC18W003U06K1181266{4}
| Pentagonal prism60px50px
4.4.5		2 5 {{pipe}} 2D5hC33b—U76bK01b101575{4}
+2{5}
| Hexagonal prism60px50px
4.4.6		2 6 {{pipe}} 2D6hC33c—U76cK01c121886{4}
+2{6}
| Heptagonal prism60px50px
4.4.7		2 7 {{pipe}} 2D7hC33d—U76dK01d142197{4}
+2{7}
| Octagonal prism60px50px
4.4.8		2 8 {{pipe}} 2D8hC33e—U76eK01e1624108{4}
+2{8}
| Enneagonal prism60px50px
4.4.9		2 9 {{pipe}} 2D9hC33f—U76fK01f1827119{4}
+2{9}
| Decagonal prism60px50px
4.4.10		2 10 {{pipe}} 2D10hC33g—U76gK01g20301210{4}
+2{10}
| Hendecagonal prism60px50px
4.4.11		2 11 {{pipe}} 2D11hC33h—U76hK01h22331311{4}
+2{11}
| Dodecagonal prism60px50px
4.4.12		2 12 {{pipe}} 2D12hC33i—U76iK01i24361412{4}
+2{12}
| Truncated octahedron60px50px
4.6.6		2 4 {{pipe}} 3OhC20W007U08K132436146{4}
+8{6}
| Truncated cuboctahedron60px50px
4.6.8		2 3 4 {{pipe}}OhC23W015U11K1648722612{4}
+8{6}
+6{8}
| Truncated icosidodecahedron60px50px
4.6.10		2 3 5 {{pipe}}IhC31W016U28K331201806230{4}
+20{6}
+12{10}
| Dodecahedron60px50px
5.5.5		3 {{pipe}} 2 5IhC26W005U23K2820301212{5}
| Truncated icosahedron60px50px
5.6.6		2 5 {{pipe}} 3IhC27W009U25K3060903212{5}
+20{6}
| Octahedron60px50px
3.3.3.3		4 {{pipe}} 2 3OhC17W002U05K1061288{3}
| Square antiprism60px50px
3.3.3.4		{{pipe}} 2 2 4D4dC34a—U77aK02a816108{3}
+2{4}
| Pentagonal antiprism60px50px
3.3.3.5		{{pipe}} 2 2 5D5dC34b—U77bK02b10201210{3}
+2{5}
| Hexagonal antiprism60px50px
3.3.3.6		{{pipe}} 2 2 6D6dC34c—U77cK02c12241412{3}
+2{6}
| Heptagonal antiprism60px50px
3.3.3.7		{{pipe}} 2 2 7D7dC34d—U77dK02d14281614{3}
+2{7}
| Octagonal antiprism60px50px
3.3.3.8		{{pipe}} 2 2 8D8dC34e—U77eK02e16321816{3}
+2{8}
| Enneagonal antiprism60px50px
3.3.3.9		{{pipe}} 2 2 9D9dC34f—U77fK02f18362018{3}
+2{9}
| Decagonal antiprism60px50px
3.3.3.10		{{pipe}} 2 2 10D10dC34g—U77gK02g20402220{3}
+2{10}
| Hendecagonal antiprism60px50px
3.3.3.11		{{pipe}} 2 2 11D11dC34h—U77hK02h22442422{3}
+2{11}
| Dodecagonal antiprism60px50px
3.3.3.12		{{pipe}} 2 2 12D12dC34i—U77iK02i24482624{3}
+2{12}
| Cuboctahedron60px50px
3.4.3.4		2 {{pipe}} 3 4| OhC19W011U07K121224148{3}
+6{4}
| Rhombicuboctahedron60px50px
3.4.4.4		3 4 {{pipe}} 2OhC22W013U10K152448268{3}
+(6+12){4}
| Rhombicosidodecahedron60px50px
3.4.5.4		3 5 {{pipe}} 2IhC30W014U27K32601206220{3}
+30{4}
+12{5}
| Icosidodecahedron60px50px
3.5.3.5		2 {{pipe}} 3 5IhC28W012U24K2930603220{3}
+12{5}
| Icosahedron60px50px
3.3.3.3.3		5 {{pipe}} 2 3IhC25W004U22K2712302020{3}
| Snub cube60px50px
3.3.3.3.4		{{pipe}} 2 3 4OC24W017U12K17246038(8+24){3}
+6{4}
| Snub dodecahedron60px50px
3.3.3.3.5		{{pipe}} 2 3 5IC32W018U29K346015092(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 {{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

ImageWyth symVert. figSym.C#W#U#K#Vert.EdgesFacesChiOrient- able?Dens.Faces by type
| Octahemioctahedron60px{{sfrac|3|2}} 3 {{pipe}} 350px 6.{{sfrac|3|2}}.6.3OhC37W068U03K081224120Yes 8{3}+4{6}
| Tetrahemihexahedron60px{{sfrac|3|2}} 3 {{pipe}} 250px 4.{{sfrac|3|2}}.4.3TdC36W067U04K0961271No 4{3}+3{4}
| Cubohemioctahedron60px{{sfrac|4|3}} 4 {{pipe}} 350px 6.{{sfrac|4|3}}.6.4OhC51W078U15K20122410−2No 6{4}+4{6}
| Great dodecahedron60px{{sfrac|5|2}} {{pipe}} 2 550px (5.5.5.5.5)/2IhC44W021U35K40123012−6Yes312{5}
| Great icosahedron60px{{sfrac|5|2}} {{pipe}} 2 350px (3.3.3.3.3)/2IhC69W041U53K581230202Yes720{3}
| Great ditrigonal icosidodecahedron60px{{sfrac|3|2}} {{pipe}} 3 550px (5.3.5.3.5.3)/2IhC61W087U47K52206032−8Yes620{3}+12{5}
| Small rhombihexahedron60px2 4 ({{sfrac|3|2}} {{sfrac|4|2}}) {{pipe}}50px 4.8.{{sfrac|4|3}}.{{sfrac|8|7}}OhC60W086U18K23244818−6No 12{4}+6{8}
| Small cubicuboctahedron60px{{sfrac|3|2}} 4 {{pipe}} 450px 8.{{sfrac|3|2}}.8.4OhC38W069U13K18244820−4Yes28{3}+6{4}+6{8}
| Nonconvex great rhombicuboctahedron60px{{sfrac|3|2}} 4 {{pipe}} 250px 4.{{sfrac|3|2}}.4.4OhC59W085U17K222448262Yes58{3}+(6+12){4}
| Small dodecahemidodecahedron60px{{sfrac|5|4}} 5 {{pipe}} 550px 10.{{sfrac|5|4}}.10.5IhC65W091U51K56306018−12No 12{5}+6{10}
| Great dodecahemicosahedron60px{{sfrac|5|4}} 5 {{pipe}} 350px 6.{{sfrac|5|4}}.6.5IhC81W102U65K70306022−8No 12{5}+10{6}
| Small icosihemidodecahedron60px{{sfrac|3|2}} 3 {{pipe}} 550px 10.{{sfrac|3|2}}.10.3IhC63W089U49K54306026−4No 20{3}+6{10}
| Small dodecicosahedron60px3 5 ({{sfrac|3|2}} {{sfrac|5|4}}) {{pipe}}50px 10.6.{{sfrac|10|9}}.{{sfrac|6|5}}IhC64W090U50K556012032−28No 20{6}+12{10}
| Small rhombidodecahedron60px2 5 ({{sfrac|3|2}} {{sfrac|5|2}}) {{pipe}}50px 10.4.{{sfrac|10|9}}.{{sfrac|4|3}}IhC46W074U39K446012042−18No 30{4}+12{10}
| Small dodecicosidodecahedron60px{{sfrac|3|2}} 5 {{pipe}} 550px 10.{{sfrac|3|2}}.10.5IhC42W072U33K386012044−16Yes220{3}+12{5}+12{10}
| Rhombicosahedron60px2 3 ({{sfrac|5|4}} {{sfrac|5|2}}) {{pipe}}50px 6.4.{{sfrac|6|5}}.{{sfrac|4|3}}IhC72W096U56K616012050−10No 30{4}+20{6}
| Great icosicosidodecahedron60px{{sfrac|3|2}} 5 {{pipe}} 350px 6.{{sfrac|3|2}}.6.5IhC62W088U48K536012052−8Yes620{3}+12{5}+20{6}
| Pentagrammic prism60px2 {{sfrac|5|2}} {{pipe}} 250px {{sfrac|5|2}}.4.4D5hC33b—U78aK03a101572Yes25{4}+2{{mset|{{sfrac|5|2}}}}
| Heptagrammic prism (7/2)60px2 {{sfrac|7|2}} {{pipe}} 250px {{sfrac|7|2}}.4.4D7hC33d—U78bK03b142192Yes27{4}+2{{mset|{{sfrac|7|2}}}}
| Heptagrammic prism (7/3)60px2 {{sfrac|7|3}} {{pipe}} 250px {{sfrac|7|3}}.4.4D7hC33d—U78cK03c142192Yes37{4}+2{{mset|{{sfrac|7|3}}}}
| Octagrammic prism60px2 {{sfrac|8|3}} {{pipe}} 250px {{sfrac|8|3}}.4.4D8hC33e—U78dK03d1624102Yes38{4}+2{{mset|{{sfrac|8|3}}}}
| Pentagrammic antiprism60px{{pipe}} 2 2 {{sfrac|5|2}}50px {{sfrac|5|2}}.3.3.3D5hC34b—U79aK04a1020122Yes210{3}+2{{mset|{{sfrac|5|2}}}}
| Pentagrammic crossed-antiprism60px{{pipe}} 2 2 {{sfrac|5|3}}50px {{sfrac|5|3}}.3.3.3D5dC35a—U80aK05a1020122Yes310{3}+2{{mset|{{sfrac|5|2}}}}
| Heptagrammic antiprism (7/2)60px{{pipe}} 2 2 {{sfrac|7|2}}50px {{sfrac|7|2}}.3.3.3D7hC34d—U79bK04b1428162Yes314{3}+2{{mset|{{sfrac|7|2}}}}
| Heptagrammic antiprism (7/3)60px{{pipe}} 2 2 {{sfrac|7|3}}50px {{sfrac|7|3}}.3.3.3D7dC34d—U79cK04c1428162Yes314{3}+2{{mset|{{sfrac|7|3}}}}
| Heptagrammic crossed-antiprism60px{{pipe}} 2 2 {{sfrac|7|4}}50px {{sfrac|7|4}}.3.3.3D7hC35b—U80bK05b1428162Yes414{3}+2{{mset|{{sfrac|7|3}}}}
| Octagrammic antiprism60px{{pipe}} 2 2 {{sfrac|8|3}}50px {{sfrac|8|3}}.3.3.3D8dC34e—U79dK04d1632182Yes316{3}+2{{mset|{{sfrac|8|3}}}}
| Octagrammic crossed-antiprism60px{{pipe}} 2 2 {{sfrac|8|5}}50px {{sfrac|8|5}}.3.3.3D8dC35c—U80cK05c1632182Yes516{3}+2{{mset|{{sfrac|8|3}}}}
| Small stellated dodecahedron60px5 {{pipe}} 2 {{sfrac|5|2}}50px ({{sfrac|5|2}})5IhC43W020U34K39123012−6Yes312{{mset|{{sfrac|5|2}}}}
| Great stellated dodecahedron60px3 {{pipe}} 2 {{sfrac|5|2}}50px ({{sfrac|5|2}})3IhC68W022U52K572030122Yes712{{mset|{{sfrac|5|2}}}}
| Ditrigonal dodecadodecahedron60px3 {{pipe}} {{sfrac|5|3}} 550px ({{sfrac|5|3}}.5)3IhC53W080U41K46206024−16Yes412{5}+12{{mset|{{sfrac|5|2}}}}
| Small ditrigonal icosidodecahedron60px3 {{pipe}} {{sfrac|5|2}} 350px ({{sfrac|5|2}}.3)3IhC39W070U30K35206032−8Yes220{3}+12{{mset|{{sfrac|5|2}}}}
| Stellated truncated hexahedron60px2 3 {{pipe}} {{sfrac|4|3}}50px {{sfrac|8|3}}.{{sfrac|8|3}}.3OhC66W092U19K242436142Yes78{3}+6{{mset|{{sfrac|8|3}}}}
| Great rhombihexahedron60px2 {{sfrac|4|3}} ({{sfrac|3|2}} {{sfrac|4|2}}) {{pipe}}50px 4.{{sfrac|8|3}}.{{sfrac|4|3}}.{{sfrac|8|5}}OhC82W103U21K26244818−6No 12{4}+6{{mset|{{sfrac|8|3}}}}
| Great cubicuboctahedron60px3 4 {{pipe}} {{sfrac|4|3}}50px {{sfrac|8|3}}.3.{{sfrac|8|3}}.4OhC50W077U14K19244820−4Yes48{3}+6{4}+6{{mset|{{sfrac|8|3}}}}
| Great dodecahemidodecahedron60px{{sfrac|5|3}} {{sfrac|5|2}} {{pipe}} {{sfrac|5|3}}50px {{sfrac|10|3}}.{{sfrac|5|3}}.{{sfrac|10|3}}.{{sfrac|5|2}}IhC86W107U70K75306018−12No 12{{mset|{{sfrac|5|2}}}}+6{{mset|{{sfrac|10|3}}}}
| Small dodecahemicosahedron60px{{sfrac|5|3}} {{sfrac|5|2}} {{pipe}} 350px 6.{{sfrac|5|3}}.6.{{sfrac|5|2}}IhC78W100U62K67306022−8No 12{{mset|{{sfrac|5|2}}}}+10{6}
| Dodecadodecahedron60px2 {{pipe}} 5 {{sfrac|5|2}}50px ({{sfrac|5|2}}.5)2IhC45W073U36K41306024−6Yes312{5}+12{{mset|{{sfrac|5|2}}}}
| Great icosihemidodecahedron60px{{sfrac|3|2}} 3 {{pipe}} {{sfrac|5|3}}50px {{sfrac|10|3}}.{{sfrac|3|2}}.{{sfrac|10|3}}.3IhC85W106U71K76306026−4No 20{3}+6{{mset|{{sfrac|10|3}}}}
| Great icosidodecahedron60px2 {{pipe}} 3 {{sfrac|5|2}}50px ({{sfrac|5|2}}.3)2IhC70W094U54K593060322Yes720{3}+12{{mset|{{sfrac|5|2}}}}
| Cubitruncated cuboctahedron60px{{sfrac|4|3}} 3 4 {{pipe}}50px {{sfrac|8|3}}.6.8OhC52W079U16K21487220−4Yes48{6}+6{8}+6{{mset|{{sfrac|8|3}}}}
| Great truncated cuboctahedron60px{{sfrac|4|3}} 2 3 {{pipe}}50px {{sfrac|8|3}}.4.{{sfrac|6|5}}OhC67W093U20K254872262Yes112{4}+8{6}+6{{mset|{{sfrac|8|3}}}}
| Truncated great dodecahedron60px2 {{sfrac|5|2}} {{pipe}} 550px 10.10.{{sfrac|5|2}}IhC47W075U37K42609024−6Yes312{{mset|{{sfrac|5|2}}}}+12{10}
| Small stellated truncated dodecahedron60px2 5 {{pipe}} {{sfrac|5|3}}50px {{sfrac|10|3}}.{{sfrac|10|3}}.5IhC74W097U58K63609024−6Yes912{5}+12{{mset|{{sfrac|10|3}}}}
| Great stellated truncated dodecahedron60px2 3 {{pipe}} {{sfrac|5|3}}50px {{sfrac|10|3}}.{{sfrac|10|3}}.3IhC83W104U66K716090322Yes1320{3}+12{{mset|{{sfrac|10|3}}}}
| Truncated great icosahedron60px2 {{sfrac|5|2}} {{pipe}} 350px 6.6.{{sfrac|5|2}}IhC71W095U55K606090322Yes712{{mset|{{sfrac|5|2}}}}+20{6}
| Great dodecicosahedron60px3 {{sfrac|5|3}}({{sfrac|3|2}} {{sfrac|5|2}}) {{pipe}}50px 6.{{sfrac|10|3}}.{{sfrac|6|5}}.{{sfrac|10|7}}IhC79W101U63K686012032−28No 20{6}+12{{mset|{{sfrac|10|3}}}}
| Great rhombidodecahedron60px2 {{sfrac|5|3}} ({{sfrac|3|2}} {{sfrac|5|4}}) {{pipe}}50px 4.{{sfrac|10|3}}.{{sfrac|4|3}}.{{sfrac|10|7}}IhC89W109U73K786012042−18No 30{4}+12{{mset|{{sfrac|10|3}}}}
| Icosidodecadodecahedron60px{{sfrac|5|3}} 5 {{pipe}} 350px 6.{{sfrac|5|3}}.6.5IhC56W083U44K496012044−16Yes412{5}+12{{mset|{{sfrac|5|2}}}}+20{6}
| Small ditrigonal dodecicosidodecahedron60px{{sfrac|5|3}} 3 {{pipe}} 550px 10.{{sfrac|5|3}}.10.3IhC55W082U43K486012044−16Yes420{3}+12{{mset|{{sfrac|5|2}}}}+12{10}
| Great ditrigonal dodecicosidodecahedron60px3 5 {{pipe}} {{sfrac|5|3}}50px {{sfrac|10|3}}.3.{{sfrac|10|3}}.5IhC54W081U42K476012044−16Yes420{3}+12{5}+12{{mset|{{sfrac|10|3}}}}
| Great dodecicosidodecahedron60px{{sfrac|5|2}} 3 {{pipe}} {{sfrac|5|3}}50px {{sfrac|10|3}}.{{sfrac|5|2}}.{{sfrac|10|3}}.3IhC77W099U61K666012044−16Yes1020{3}+12{{mset|{{sfrac|5|2}}}}+12{{mset|{{sfrac|10|3}}}}
| Small icosicosidodecahedron60px{{sfrac|5|2}} 3 {{pipe}} 350px 6.{{sfrac|5|2}}.6.3IhC40W071U31K366012052−8Yes220{3}+12{{mset|{{sfrac|5|2}}}}+20{6}
| Rhombidodecadodecahedron60px{{sfrac|5|2}} 5 {{pipe}} 250px 4.{{sfrac|5|2}}.4.5IhC48W076U38K436012054−6Yes330{4}+12{5}+12{{mset|{{sfrac|5|2}}}}
| Nonconvex great rhombicosidodecahedron60px{{sfrac|5|3}} 3 {{pipe}} 250px 4.{{sfrac|5|3}}.4.3IhC84W105U67K7260120622Yes1320{3}+30{4}+12{{mset|{{sfrac|5|2}}}}
| Icositruncated dodecadodecahedron60px3 5 {{sfrac|5|3}} {{pipe}}50px {{sfrac|10|3}}.6.10IhC57W084U45K5012018044−16Yes420{6}+12{10}+12{{mset|{{sfrac|10|3}}}}
| Truncated dodecadodecahedron60px2 5 {{sfrac|5|3}} {{pipe}}50px {{sfrac|10|3}}.4.{{sfrac|10|9}}IhC75W098U59K6412018054−6Yes330{4}+12{10}+12{{mset|{{sfrac|10|3}}}}
| Great truncated icosidodecahedron60px2 3 {{sfrac|5|3}} {{pipe}}50px {{sfrac|10|3}}.4.6IhC87W108U68K73120180622Yes1330{4}+20{6}+12{{mset|{{sfrac|10|3}}}}
| Snub dodecadodecahedron60px{{pipe}} 2 {{sfrac|5|2}} 550px 3.3.{{sfrac|5|2}}.3.5IC49W111U40K456015084−6Yes360{3}+12{5}+12{{mset|{{sfrac|5|2}}}}
| Inverted snub dodecadodecahedron60px{{pipe}} {{sfrac|5|3}} 2 550px 3.{{sfrac|5|3}}.3.3.5IC76W114U60K656015084−6Yes960{3}+12{5}+12{{mset|{{sfrac|5|2}}}}
| Great snub icosidodecahedron60px{{pipe}} 2 {{sfrac|5|2}} 350px 34.{{sfrac|5|2}}IC73W113U57K6260150922Yes7(20+60){3}+12{{mset|{{sfrac|5|2}}}}
| Great inverted snub icosidodecahedron60px{{pipe}} {{sfrac|5|3}} 2 350px 34.{{sfrac|5|3}}IC88W116U69K7460150922Yes13(20+60){3}+12{{mset|{{sfrac|5|2}}}}
| Great retrosnub icosidodecahedron60px{{pipe}} 2 {{sfrac|3|2}} {{sfrac|5|3}}50px (34.{{sfrac|5|2}})/2IC90W117U74K7960150922Yes37(20+60){3}+12{{mset|{{sfrac|5|2}}}}
| Great snub dodecicosidodecahedron60px{{pipe}} {{sfrac|5|3}} {{sfrac|5|2}} 350px 33.{{sfrac|5|3}}.3.{{sfrac|5|2}}IC80W115U64K6960180104−16Yes10(20+60){3}+(12+12){{mset|{{sfrac|5|2}}}}
| Snub icosidodecadodecahedron60px{{pipe}} {{sfrac|5|3}} 3 550px 33.5.3.{{sfrac|5|3}}IC58W112U46K5160180104−16Yes4(20+60){3}+12{5}+12{{mset|{{sfrac|5|2}}}}
| Small snub icosicosidodecahedron60px{{pipe}} {{sfrac|5|2}} 3 350px 35.{{sfrac|5|2}}IhC41W110U32K3760180112−8Yes2(40+60){3}+12{{mset|{{sfrac|5|2}}}}
| Small retrosnub icosicosidodecahedron60px{{pipe}} {{sfrac|3|2}} {{sfrac|3|2}} {{sfrac|5|2}}50px (35.{{sfrac|5|2}})/2IhC91W118U72K7760180112−8Yes38(40+60){3}+12{{mset|{{sfrac|5|2}}}}
| Great dirhombicosidodecahedron60pxnowrap|{{pipe}} {{sfrac|3|2}} {{sfrac|5|3}} 3 {{sfrac|5|2}}50px (4.{{sfrac|5|3}}.4.3.4.{{sfrac|5|2}}.4.{{sfrac|3|2}})/2IhC92W119U75K8060240124−56No 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

ImageWyth
sym		Vert.
fig		Sym.C#W#U#K#Vert.EdgesFacesChiOrient-
able?		Dens.Faces by type
| Great disnub
dirhombidodecahedron		60px{{pipe}} ({{sfrac|3|2}}) {{sfrac|5|3}} (3) {{sfrac|5|2}}50px
({{sfrac|5|2}}.4.3.3.3.4. {{sfrac|5|3}}.
4.{{sfrac|3|2}}.{{sfrac|3|2}}.{{sfrac|3|2}}.4)/2		Ih————60360 (*)204−96No 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 (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, {{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-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

  • {{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

{{Nonconvex polyhedron navigator}}

Uniform polyhedra

ja:一様多面体の一覧