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 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, 46-80 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

The generic geometric names for the most common polyhedra. The 5 regular polyhedra are called a tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron with 4, 6, 8, 12, and 20 sides respectively.

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.

Convex uniform polyhedra

NamePictureVertex
type
Wythoff
symbol
Sym.C#W#U#K#Vert.EdgesFacesFaces by type
Tetrahedron
3.3.3
3 | 2 3TdC15W001U01K064644{3}
Triangular prism
3.4.4
2 3 | 2D3hC33a--U76aK01a6952{3}
+3{4}
Truncated tetrahedron
3.6.6
2 3 | 3TdC16W006U02K07121884{3}
+4{6}
Truncated cube
3.8.8
2 3 | 4OhC21W008U09K142436148{3}
+6{8}
Truncated dodecahedron
3.10.10
2 3 | 5IhC29W010U26K31609032 20{3}
+12{10}
Cube
4.4.4
3 | 2 4OhC18W003U06K1181266{4}
Pentagonal prism
4.4.5
2 5 | 2D5hC33b--U76bK01b101575{4}
+2{5}
Hexagonal prism
4.4.6
2 6 | 2D6hC33c--U76cK01c121886{4}
+2{6}
Octagonal prism
4.4.8
2 8 | 2D8hC33e--U76eK01e1624108{4}
+2{8}
Decagonal prism
4.4.10
2 10 | 2D10hC33g--U76gK01g20301210{4}
+2{10}
Dodecagonal prism
4.4.12
2 12 | 2D12hC33i--U76iK01i24361412{4}
+2{12}
Truncated octahedron
4.6.6
2 4 | 3OhC20W007U08K132436146{4}
+8{6}
Truncated cuboctahedron
4.6.8
2 3 4 |OhC23W015U11K1648722612{4}
+8{6}
+6{8}
Truncated icosidodecahedron
4.6.10
2 3 5 |IhC31W016U28K331201806230{4}
+20{6}
+12{10}
Dodecahedron
5.5.5
3 | 2 5IhC26W005U23K2820301212{5}
Truncated icosahedron
5.6.6
2 5 | 3IhC27W009U25K3060903212{5}
+20{6}
Octahedron
3.3.3.3
4 | 2 3OhC17W002U05K1061288{3}
Square antiprism
3.3.3.4
| 2 2 4D4dC34a--U77aK02a816108{3}
+2{4}
Pentagonal antiprism
3.3.3.5
| 2 2 5D5dC34b--U77bK02b10201210{3}
+2{5}
Hexagonal antiprism
3.3.3.6
| 2 2 6D6dC34c--U77cK02c12241412{3}
+2{6}
Octagonal antiprism
3.3.3.8
| 2 2 8D8dC34e--U77eK02e16321816{3}
+2{8}
Decagonal antiprism
3.3.3.10
| 2 2 10D10dC34g--U77gK02g20402220{3}
+2{10}
Dodecagonal antiprism
3.3.3.12
| 2 2 12D12dC34i--U77iK02i24482624{3}
+2{12}
Cuboctahedron
3.4.3.4
2 | 3 4OhC19W011U07K121224148{3}
+6{4}
Rhombicuboctahedron
3.4.4.4
3 4 | 2OhC22W013U10K152448268{3}
+(6+12){4}
Rhombicosidodecahedron
3.4.5.4
3 5 | 2IhC30W014U27K32601206220{3}
+30{4}
+12{5}
Icosidodecahedron
3.5.3.5
2 | 3 5IhC28W012U24K2930603220{3}
+12{5}
Icosahedron
3.3.3.3.3
5 | 2 3IhC25W004U22K2712302020{3}
Snub cube
3.3.3.3.4
| 2 3 4OC24W017U12K17246038(8+24){3}
+6{4}
Snub dodecahedron
3.3.3.3.5
| 2 3 5IC32W018U29K346015092(20+60){3}
+12{5}

Uniform star polyhedra

NameImageWyth
sym
Vert.
fig
Sym.C#W#U#K#Vert.EdgesFacesChiOrient
able?
Dens.Faces by type
Octahemioctahedron3/2 3 | 3
6.3/2.6.3
OhC37W068U03K081224120Yes 8{3}+4{6}
Tetrahemihexahedron3/2 3 | 2
4.3/2.4.3
TdC36W067U04K0961271No 4{3}+3{4}
Cubohemioctahedron4/3 4 | 3
6.4/3.6.4
OhC51W078U15K20122410-2No 6{4}+4{6}
Great
dodecahedron
5/2 | 2 5
(5.5.5.5.5)/2
IhC44W021U35K40123012-6Yes312{5}
Great
icosahedron
5/2 | 2 3
(3.3.3.3.3)/2
IhC69W041U53K581230202Yes720{3}
Great
ditrigonal
icosidodecahedron
3/2 | 3 5
(5.3.5.3.5.3)/2
IhC61W087U47K52206032-8Yes620{3}+12{5}
Small
rhombihexahedron
2 4 (3/2 4/2) |
4.8.4/3.8/7
OhC60W086U18K23244818-6No 12{4}+6{8}
Small
cubicuboctahedron
3/2 4 | 4
8.3/2.8.4
OhC38W069U13K18244820-4Yes28{3}+6{4}+6{8}
Great
rhombicuboctahedron
3/2 4 | 2
4.3/2.4.4
OhC59W085U17K222448262Yes58{3}+(6+12){4}
Small dodecahemi-
dodecahedron
5/4 5 | 5
10.5/4.10.5
IhC65W091U51K56306018-12No 12{5}+6{10}
Great dodecahem-
icosahedron
5/4 5 | 3
6.5/4.6.5
IhC81W102U65K70306022-8No 12{5}+10{6}
Small icosihemi-
dodecahedron
3/2 3 | 5
10.3/2.10.3
IhC63W089U49K54306026-4No 20{3}+6{10}
Small
dodecicosahedron
3 5 (3/2 5/4) |
10.6.10/9.6/5
IhC64W090U50K556012032-28No 20{6}+12{10}
Small
rhombidodecahedron
2 5 (3/2 5/2) |
10.4.10/9.4/3
IhC46W074U39K446012042-18No 30{4}+12{10}
Small dodecicosi-
dodecahedron
3/2 5 | 5
10.3/2.10.5
IhC42W072U33K386012044-16Yes220{3}+12{5}+12{10}
Rhombicosahedron2 3 (5/4 5/2) |
6.4.6/5.4/3
IhC72W096U56K616012050-10No 30{4}+20{6}
Great
icosicosi-
dodecahedron
3/2 5 | 3
6.3/2.6.5
IhC62W088U48K536012052-8Yes620{3}+12{5}+20{6}
Pentagrammic
prism
2 5/2 | 2
5/2.4.4
D5hC33b--U78aK03a101572Yes25{4}+2{5/2}
Heptagrammic
prism (7/2)
2 7/2 | 2
7/2.4.4
D7hC33d--U78bK03b142192Yes27{4}+2{7/2}
Heptagrammic
prism (7/3)
2 7/3 | 2
7/3.4.4
D7hC33d--U78cK03c142192Yes37{4}+2{7/3}
Octagrammic
prism
2 8/3 | 2
8/3.4.4
D8hC33e--U78dK03d1624102Yes38{4}+2{8/3}
Pentagrammic antiprism| 2 2 5/2
5/2.3.3.3
D5hC34b--U79aK04a1020122Yes210{3}+2{5/2}
Pentagrammic
crossed-antiprism
| 2 2 5/3
5/3.3.3.3
D5dC35a--U80aK05a1020122Yes310{3}+2{5/2}
Heptagrammic
antiprism (7/2)
| 2 2 7/2
7/2.3.3.3
D7hC34d--U79bK04b1428162Yes314{3}+2{7/2}
Heptagrammic
antiprism (7/3)
| 2 2 7/3
7/3.3.3.3
D7dC34d--U79cK04c1428162Yes314{3}+2{7/3}
Heptagrammic
crossed-antiprism
| 2 2 7/4
7/4.3.3.3
D7hC35b--U80bK05b1428162Yes414{3}+2{7/3}
Octagrammic
antiprism
| 2 2 8/3
8/3.3.3.3
D8dC34e--U79dK04d1632182Yes316{3}+2{8/3}
Octagrammic
crossed-antiprism
| 2 2 8/5
8/5.3.3.3
D8dC35c--U80cK05c1632182Yes516{3}+2{8/3}
Small
stellated
dodecahedron
5 | 2 5/2
(5/2)5
IhC43W020U34K39123012-6Yes312{5/2}
Great
stellated
dodecahedron
3 | 2 5/2
(5/2)3
IhC68W022U52K572030122Yes712{5/2}
Ditrigonal
dodeca-
dodecahedron
3 | 5/3 5
(5/3.5)3
IhC53W080U41K46206024-16Yes412{5}+12{5/2}
Small
ditrigonal
icosidodecahedron
3 | 5/2 3
(5/2.3)3
IhC39W070U30K35206032-8Yes220{3}+12{5/2}
Stellated
truncated
hexahedron
2 3 | 4/3
8/3.8/3.3
OhC66W092U19K242436142Yes78{3}+6{8/3}
Great
rhombihexahedron
2 4/3 (3/2 4/2) |
4.8/3.4/3.8/5
OhC82W103U21K26244818-6No 12{4}+6{8/3}
Great
cubicuboctahedron
3 4 | 4/3
8/3.3.8/3.4
OhC50W077U14K19244820-4Yes48{3}+6{4}+6{8/3}
Great dodecahemi-
dodecahedron
5/35/2 | 5/3
10/3.5/3.10/3.5/2
IhC86W107U70K75306018-12No 12{5/2}+6{10/3}
Small dodecahemi-
cosahedron
5/35/2 | 3
6.5/3.6.5/2
IhC78W100U62K67306022-8No 12{5/2}+10{6}
Dodeca-
dodecahedron
2 | 5/2 5
(5/2.5)2
IhC45W073U36K41306024-6Yes312{5}+12{5/2}
Great icosihemi-
dodecahedron
3/2 3 | 5/3
10/3.3/2.10/3.3
IhC85W106U71K76306026-4No 20{3}+6{10/3}
Great
icosidodecahedron
2 | 5/2 3
(5/2.3)2
IhC70W094U54K593060322Yes720{3}+12{5/2}
Cubitruncated
cuboctahedron
4/3 3 4 |
8/3.6.8
OhC52W079U16K21487220-4Yes48{6}+6{8}+6{8/3}
Great
truncated
cuboctahedron
4/3 2 3 |
8/3.4.6/5
OhC67W093U20K254872262Yes112{4}+8{6}+6{8/3}
Truncated
great
dodecahedron
2 5/2 | 5
10.10.5/2
IhC47W075U37K42609024-6Yes312{5/2}+12{10}
Small stellated
truncated
dodecahedron
2 5 | 5/3
10/3.10/3.5
IhC74W097U58K63609024-6Yes912{5}+12{10/3}
Great stellated
truncated
dodecahedron
2 3 | 5/3
10/3.10/3.3
IhC83W104U66K716090322Yes1320{3}+12{10/3}
Truncated
great
icosahedron
2 5/2 | 3
6.6.5/2
IhC71W095U55K606090322Yes712{5/2}+20{6}
Great
dodecicosahedron
3 5/3(3/2 5/2) |
6.10/3.6/5.10/7
IhC79W101U63K686012032-28No 20{6}+12{10/3}
Great
rhombidodecahedron
2 5/3 (3/2 5/4) |
4.10/3.4/3.10/7
IhC89W109U73K786012042-18No 30{4}+12{10/3}
Icosidodeca-
dodecahedron
5/3 5 | 3
6.5/3.6.5
IhC56W083U44K496012044-16Yes412{5}+12{5/2}+20{6}
Small ditrigonal
dodecicosi-
dodecahedron
5/3 3 | 5
10.5/3.10.3
IhC55W082U43K486012044-16Yes420{3}+12{5/2}+12{10}
Great ditrigonal
dodecicosi-
dodecahedron
3 5 | 5/3
10/3.3.10/3.5
IhC54W081U42K476012044-16Yes420{3}+12{5}+12{10/3}
Great
dodecicosi-
dodecahedron
5/2 3 | 5/3
10/3.5/2.10/3.3
IhC77W099U61K666012044-16Yes1020{3}+12{5/2}+12{10/3}
Small icosicosi-
dodecahedron
5/2 3 | 3
6.5/2.6.3
IhC40W071U31K366012052-8Yes220{3}+12{5/2}+20{6}
Rhombidodeca-
dodecahedron
5/2 5 | 2
4.5/2.4.5
IhC48W076U38K436012054-6Yes330{4}+12{5}+12{5/2}
Great
rhombicosi-
dodecahedron
5/3 3 | 2
4.5/3.4.3
IhC84W105U67K7260120622Yes1320{3}+30{4}+12{5/2}
Icositruncated
dodeca-
dodecahedron
5/3 3 5 |
10/3.6.10
IhC57W084U45K5012018044-16Yes420{6}+12{10}+12{10/3}
Truncated
dodeca-
dodecahedron
5/3 2 5 |
10/3.4.10/9
IhC75W098U59K6412018054-6Yes330{4}+12{10}+12{10/3}
Great
truncated
icosidodecahedron
5/3 2 3 |
10/3.4.6
IhC87W108U68K73120180622Yes1330{4}+20{6}+12{10/3}
Snub dodeca-
dodecahedron
| 2 5/2 5
3.3.5/2.3.5
IC49W111U40K456015084-6Yes360{3}+12{5}+12{5/2}
Inverted
snub dodeca-
dodecahedron
| 5/3 2 5
3.5/3.3.3.5
IC76W114U60K656015084-6Yes960{3}+12{5}+12{5/2}
Great
snub
icosidodecahedron
| 2 5/2 3
34.5/2
IC73W116U57K6260150922Yes7(20+60){3}+12{5/2}
Great
inverted
snub
icosidodecahedron
| 5/3 2 3
34.5/3
IC88W113U69K7460150922Yes13(20+60){3}+12{5/2}
Great
retrosnub
icosidodecahedron
| 3/25/3 2
(34.5/2)/2
IC90W117U74K7960150922Yes37(20+60){3}+12{5/2}
Great
snub
dodecicosi-
dodecahedron
| 5/35/2 3
33.5/3.3.5/2
IC80W115U64K6960180104-16Yes10(20+60){3}+(12+12){5/2}
Snub
icosidodeca-
dodecahedron
| 5/3 3 5
33.5.5/3
IC58W112U46K5160180104-16Yes4(20+60){3}+12{5}+12{5/2}
Small snub icos-
icosidodecahedron
| 5/2 3 3
35.5/2
IhC41W110U32K3760180112-8Yes2(40+60){3}+12{5/2}
Small retrosnub
icosicosi-
dodecahedron
| 3/23/25/2
(35.5/3)/2
IhC91W118U72K7760180112-8Yes38(40+60){3}+12{5/2}
Great
dirhombicosi-
dodecahedron
| 3/25/3 3 5/2
(4.5/3.4.3.
4.5/2.4.3/2)/2
IhC92W119U75K8060240124-56No 40{3}+60{4}+24{5/2}

NameImageWyth
sym
Vert.
fig
Sym.C#W#U#K#Vert.EdgesFacesChiOrient
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--------60360 (*)204-96No 120{3}+60{4}+24{5/2}

(*) : The great disnubdirhombidodecahedron has 240 of its 360 edges coinciding in space 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

  • 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.
  • 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. The Royal Society. 246 (916): 401–450. doi:10.1098/rsta.1954.0003. ISSN 0080-4614. JSTOR 91532. MR 0062446.
  • 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. doi:10.1098/rsta.1975.0022. ISSN 0080-4614. JSTOR 74475. MR 0365333.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.