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: all 75 nonprismatic uniform polyhedra; a few representatives of the infinite sets of prisms and antiprisms; one degenerate polyhedron, Skilling's figure with overlapping edges.
It was proven in that there are only 75 uniform polyhedra other than the infinite families of prisms and antiprisms. John Skilling discovered an overlooked degenerate example, by relaxing the condition that only two faces may meet at an edge. This is a degenerate uniform polyhedron rather than a uniform polyhedron, because some pairs of edges coincide.
Not included are: The uniform polyhedron compounds. 40 potential uniform polyhedra with degenerate vertex figures which have overlapping edges (not counted by Coxeter); The uniform tilings (infinite polyhedra) 11 Euclidean convex uniform tilings; 28 Euclidean nonconvex or apeirogonal uniform tilings; Infinite number of uniform tilings in hyperbolic plane. Any polygons or 4-polytopes
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 VertextypeWythoffsymbol 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 36 148{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 66{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 — U76cK01c 12 18 8 6{4}+2{6} Heptagonal prism 4.4.7 2 7 2 D7h C33d — U76dK01d 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 — U76fK01f 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 — U77gK02g 20 40 22 20{3}+2{10} Hendecagonal antiprism 3.3.3.11 2 2 11 D11d C34h — U77hK02h 22 44 24 22{3}+2{11} Dodecagonal antiprism 3.3.3.12 2 2 12 D12d C34i — U77iK02i 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 history of publication of the uniform star polyhedra is given in Coxeter et al. 1954.
The uniform polyhedra 3 3, , 3, 3 , and () (3) have some faces occurring as coplanar pairs. (Coxeter et al. 1954, pp. 423, 425, 426; Skilling 1975, p. 123)