+ Selected regular polytopesRegular (2D) polygonsConvexStar{5}{5/2}Regular (3D) polyhedraConvexStar{5,3}{5/2,5}Regular 4D polytopesConvexStar{5,3,3}{5/2,5,3}Regular 2D tessellationsEuclideanHyperbolic{4,4}{5,4}Regular 3D tessellationsEuclideanHyperbolic{4,3,4}{5,3,4} This article lists the regular polytopes in Euclidean, spherical and hyperbolic spaces.
Overview This table shows a summary of regular polytope counts by rank.
Rank Finite Infinite Euclidean / Spherical Abstract Euclidean Hyperbolic Abstract Compact Paracompact Convex Star Skew Convex Skew Convex Star Convex = = +1 = +2 = -1 = = +111000010000000020101100135493324461018?17?40115303?315?5426303?17?0057+303?17?000
1-polytopes A Coxeter diagram represent mirror "planes" as nodes, and puts a ring around a node if a point is not on the plane. A dion { }, , is a point and its mirror image point , and the line segment between them. There is only one polytope of rank 1 (1-polytope), the closed line segment bounded by its two endpoints. Every realization of this 1-polytope is regular. It has the Schläfli symbol { }, or a Coxeter diagram with a single ringed node, . Norman Johnson calls it a dion and gives it the Schläfli symbol { }.
Although trivial as a polytope, it appears as the edges of polygons and other higher dimensional polytopes. It is used in the definition of uniform prisms like Schläfli symbol { }×{p}, or Coxeter diagram as a Cartesian product of a line segment and a regular polygon.
2-polytopes (polygons) The polytopes of rank 2 (2-polytopes) are called polygons. Regular polygons are equilateral and cyclic. A -gonal regular polygon is represented by Schläfli symbol {p}.
Many sources only consider convex polygons, but star polygons, like the pentagram, when considered, can also be regular. They use the same vertices as the convex forms, but connect in an alternate connectivity which passes around the circle more than once to be completed.
Convex The Schläfli symbol {p} represents a regular -gon.
Name()()()() {3}{4}{5}{6}{7}{8}SymmetryD3, D4, D5, D6, D7, D8, CoxeterImageNameNonagon(Enneagon)DecagonHendecagonDodecagonTridecagonTetradecagonSchläfli{9}{10}{11}{12}{13}{14}SymmetryD9, D10, D11, D12, D13, D14, CoxeterImageNamePentadecagonHexadecagonHeptadecagonOctadecagonEnneadecagonIcosagon p-gon Schläfli{15}{16}{17}{18}{19}{20}{p}SymmetryD15, D16, D17, D18, D19, D20, Dp, [p]CoxeterImage
Spherical The regular digon {2} can be considered to be a degenerate regular polygon. It can be realized non-degenerately in some non-Euclidean spaces, such as on the surface of a sphere or torus. For example, digon can be realised non-degenerately as a spherical lune. A monogon {1} could also be realised on the sphere as a single point with a great circle through it.Coxeter, Regular Complex Polytopes, p. 9 However, a monogon is not a valid abstract polytope because its single edge is incident to only one vertex rather than two.
NameMonogonDigon{1}{2}SymmetryD1, [ ]D2, Coxeter diagram or Image
Stars There exist infinitely many regular star polytopes in two dimensions, whose Schläfli symbols consist of rational numbers . They are called star polygons and share the same vertex arrangements of the convex regular polygons.
In general, for any natural number , there are regular -pointed stars with Schläfli symbols for all such that (strictly speaking ) and and are coprime (as such, all stellations of a polygon with a prime number of sides will be regular stars). Symbols where and are not coprime may be used to represent compound polygons.
NamePentagram HeptagramsOctagram EnneagramsDecagram...n-grams{5/2}{7/2}{7/3}{8/3}{9/2}{9/4}{10/3}{p/q}SymmetryD5, D7, D8, D9, ,D10, Dp, [p]CoxeterImage
+ Regular star polygons up to 20 sides{11/2}{11/3}{11/4}{11/5}{12/5}{13/2}{13/3}{13/4}{13/5}{13/6}{14/3}{14/5}{15/2}{15/4}{15/7}{16/3}{16/5}{16/7}{17/2}{17/3}{17/4}{17/5}{17/6}{17/7}{17/8}{18/5}{18/7}{19/2}{19/3}{19/4}{19/5}{19/6}{19/7}{19/8}{19/9}{20/3}{20/7}{20/9}
Star polygons that can only exist as spherical tilings, similarly to the monogon and digon, may exist (for example: {3/2}, {5/3}, {5/4}, {7/4}, {9/5}), however these have not been studied in detail.
There also exist failed star polygons, such as the piangle, which do not cover the surface of a circle finitely many times.
Skew polygons In addition to the planar regular polygons there are infinitely many regular skew polygons. Skew polygons can be created via the blending operation.
The blend of two polygons and , written , can be constructed as follows: take the cartesian product of their vertices . add edges where is an edge of and is an edge of . select an arbitrary connected component of the result. Alternatively, the blend is the polygon where and are the generating mirrors of and placed in orthogonal subspaces. The blending operation is commutative, associative and idempotent.
Every regular skew polygon can be expressed as the blend of a unique set of planar polygons. If and share no factors then .
In 3 space The regular finite polygons in 3 dimensions are exactly the blends of the planar polygons (dimension 2) with the digon (dimension 1). They have vertices corresponding to a prism ( where is odd) or an antiprism ( where is even). All polygons in 3 space have an even number of vertices and edges.
Several of these appear as the Petrie polygons of regular polyhedra.
In 4 space The regular finite polygons in 4 dimensions are exactly the polygons formed as a blend of two distinct planar polygons. They have vertices lying on a Clifford torus and related by a Clifford displacement. Unlike 3-dimensional polygons, skew polygons on double rotations can include an odd-number of sides.
3-polytopes (polyhedra) Polytopes of dimension 3 are called polyhedra:
A regular polyhedron with Schläfli symbol , Coxeter diagrams , has a regular face type , and regular vertex figure .
A vertex figure (of a polyhedron) is a polygon, seen by connecting those vertices which are one edge away from a given vertex. For regular polyhedra, this vertex figure is always a regular (and planar) polygon.
Existence of a regular polyhedron is constrained by an inequality, related to the vertex figure's angle defect:
By enumerating the permutations, we find five convex forms, four star forms and three plane tilings, all with polygons and limited to: {3}, {4}, {5}, {5/2}, and {6}.
Beyond Euclidean space, there is an infinite set of regular hyperbolic tilings.
Convex The five convex regular polyhedra are called the Platonic solids. The vertex figure is given with each vertex count. All these polyhedra have an Euler characteristic () of 2.
NameSchläfliCoxeterImage(solid)Image(sphere)FacesEdgesVerticesSymmetryDualTetrahedron(3-simplex){3,3}4{3}64{3}Td[3,3](*332)(self)HexahedronCube (3-cube){4,3}6{4}128{3}Oh[4,3](*432)OctahedronOctahedron(3-orthoplex){3,4}8{3}126{4}Oh[4,3](*432)CubeDodecahedron{5,3}12{5}3020{3}Ih[5,3](*532)IcosahedronIcosahedron{3,5}20{3}3012{5}Ih[5,3](*532)Dodecahedron
Spherical In spherical geometry, regular spherical polyhedra (tilings of the sphere) exist that would otherwise be degenerate as polytopes. These are the hosohedra {2,n} and their dual dihedra {n,2}. Coxeter calls these cases "improper" tessellations.
The first few cases (n from 2 to 6) are listed below. + HosohedraNameSchläfli{2,p}CoxeterdiagramImage(sphere)Faces{2}π/pEdgesVertices{p}SymmetryDualDigonal hosohedron{2,2}2{2}π/222{2}π/2D2h[2,2](*222)SelfTrigonal hosohedron{2,3}3{2}π/332{3}D3h[2,3](*322)Trigonal dihedronSquare hosohedron{2,4}4{2}π/442{4}D4h[2,4](*422)Square dihedronPentagonal hosohedron{2,5}5{2}π/552{5}D5h[2,5](*522)Pentagonal dihedronHexagonal hosohedron{2,6}6{2}π/662{6}D6h[2,6](*622)Hexagonal dihedron
+ DihedraNameSchläfli{p,2}CoxeterdiagramImage(sphere)Faces{p}EdgesVertices{2}SymmetryDualDigonal dihedron{2,2}2{2}π/222{2}π/2D2h[2,2](*222)SelfTrigonal dihedron{3,2}2{3}33{2}π/3D3h[3,2](*322)Trigonal hosohedronSquare dihedron{4,2}2{4}44{2}π/4D4h[4,2](*422)Square hosohedronPentagonal dihedron{5,2}2{5}55{2}π/5D5h[5,2](*522)Pentagonal hosohedronHexagonal dihedron{6,2}2{6}66{2}π/6D6h[6,2](*622)Hexagonal hosohedron Star-dihedra and hosohedra and also exist for any star polygon .
Stars The regular star polyhedra are called the Kepler–Poinsot polyhedra and there are four of them, based on the vertex arrangements of the dodecahedron {5,3} and icosahedron {3,5}:
As spherical tilings, these star forms overlap the sphere multiple times, called its density, being 3 or 7 for these forms. The tiling images show a single spherical polygon face in yellow. NameImage(skeletonic)Image(solid)Image(sphere)StellationdiagramSchläfli andCoxeterFacesEdgesVerticesverf.χDensitySymmetryDualSmall stellated dodecahedron{5/2,5}12{5/2}3012{5} −63Ih[5,3](*532)Great dodecahedronGreat dodecahedron{5,5/2}12{5}3012{5/2} −63Ih[5,3](*532)Small stellated dodecahedronGreat stellated dodecahedron{5/2,3}12{5/2}3020{3}27Ih[5,3](*532)Great icosahedronGreat icosahedron{3,5/2}20{3}3012{5/2}27Ih[5,3](*532)Great stellated dodecahedron
There are infinitely many failed star polyhedra. These are also spherical tilings with star polygons in their Schläfli symbols, but they do not cover a sphere finitely many times. Some examples are {5/2,4}, {5/2,9}, {7/2,3}, {5/2,5/2}, {7/2,7/3}, {4,5/2}, and {3,7/3}.
Skew polyhedra
Regular skew polyhedra are generalizations to the set of regular polyhedron which include the possibility of nonplanar vertex figures.
For 4-dimensional skew polyhedra, Coxeter offered a modified Schläfli symbol {l,m|n} for these figures, with {l,m} implying the vertex figure, m l-gons around a vertex, and -gonal holes. Their vertex figures are skew polygons, zig-zagging between two planes.
The regular skew polyhedra, represented by {l,m|n}, follow this equation:
Four of them can be seen in 4-dimensions as a subset of faces of four regular 4-polytopes, sharing the same vertex arrangement and edge arrangement:
{4, 6 3}{6, 4 3}{4, 8 3}{8, 4 3}
4-polytopes (polychora) Regular 4-polytopes with Schläfli symbol have cells of type , faces of type , edge figures , and vertex figures . A vertex figure (of a 4-polytope) is a polyhedron, seen by the arrangement of neighboring vertices around a given vertex. For regular 4-polytopes, this vertex figure is a regular polyhedron. An edge figure is a polygon, seen by the arrangement of faces around an edge. For regular 4-polytopes, this edge figure will always be a regular polygon.
The existence of a regular 4-polytope is constrained by the existence of the regular polyhedra . A suggested name for 4-polytopes is "polychoron".
Each will exist in a space dependent upon this expression:
: Hyperspherical 3-space honeycomb or 4-polytope : Euclidean 3-space honeycomb : Hyperbolic 3-space honeycomb
These constraints allow for 21 forms: 6 are convex, 10 are nonconvex, one is a Euclidean 3-space honeycomb, and 4 are hyperbolic honeycombs.
Convex The 6 convex regular 4-polytopes are shown in the table below. All these 4-polytopes have an Euler characteristic () of 0.
Name Schläfli{p,q,r} Coxeter Cells {p,q} Faces {p} Edges {r} Vertices {q,r} Dual {r,q,p} 5-cell(4-simplex) {3,3,3} 5{3,3} 10{3} 10 {3} 5 {3,3} (self) 8-cell(4-cube)(Tesseract) {4,3,3} 8 {4,3} 24 {4} 32 {3} 16 {3,3} 16-cell 16-cell(4-orthoplex) {3,3,4} 16 {3,3} 32 {3} 24 {4} 8 {3,4} Tesseract 24-cell {3,4,3} 24 {3,4} 96 {3} 96 {3} 24 {4,3} (self) 120-cell {5,3,3} 120 {5,3} 720 {5} 1200 {3} 600 {3,3} 600-cell 600-cell {3,3,5} 600 {3,3} 1200 {3} 720 {5} 120 {3,5} 120-cell