Regular (2D) polygons  

Convex  Star 
{5} 
{5/2} 
Regular (3D) polyhedra  
Convex  Star 
{5,3} 
{5/2,5} 
Regular 4D polytopes  
Convex  Star 
{5,3,3} 
{5/2,5,3} 
Regular 2D tessellations  
Euclidean  Hyperbolic 
{4,4} 
{5,4} 
Regular 3D tessellations  
Euclidean  Hyperbolic 
{4,3,4} 
{5,3,4} 
This article lists the regular polytopes and regular polytope compounds in Euclidean, spherical and hyperbolic spaces.
This table shows a summary of regular polytope counts by rank.
Rank  Finite  Euclidean  Hyperbolic  Abstract  Compounds  

Compact  Paracompact  
Convex  Star  Skew^{[a]}^{[1]}  Convex  Skew^{[a]}^{[1]}  Convex  Star  Convex  Convex  Star  
1  1  none  none  none  none  none  none  none  1  none  none 
2  none  1  none  1  none  none  
3  5  4  9  3  3  5  none  
4  6  10  18  1  7  4  none  11  26  20  
5  3  none  3  3  15  5  4  2  none  none  
6  3  none  3  1  7  none  none  5  none  none  
7  3  none  3  1  7  none  none  none  3  none  
8  3  none  3  1  7  none  none  none  6  none  
9+  3  none  3  1  7  none  none  none  ^{[b]}  none 
There are no Euclidean regular star tessellations in any number of dimensions.
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 p and its mirror image point p', and the line segment between them. 
There is only one polytope of rank 1 (1polytope), the closed line segment bounded by its two endpoints. Every realization of this 1polytope is regular. It has the Schläfli symbol { },^{[2]}^{[3]} or a Coxeter diagram with a single ringed node, . Norman Johnson calls it a dion^{[4]} and gives it the Schläfli symbol { }.
Although trivial as a polytope, it appears as the edges of polygons and other higher dimensional polytopes.^{[5]} 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.^{[6]}
The polytopes of rank 2 (2polytopes) are called polygons. Regular polygons are equilateral and cyclic. A pgonal regular polygon is represented by Schläfli symbol {p}.
Many sources only cosider 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.
The Schläfli symbol {p} represents a regular pgon.
Name  Triangle (2simplex) 
Square (2orthoplex) (2cube) 
Pentagon (2pentagonal polytope) 
Hexagon  Heptagon  Octagon  

Schläfli  {3}  {4}  {5}  {6}  {7}  {8}  
Symmetry  D_{3}, [3]  D_{4}, [4]  D_{5}, [5]  D_{6}, [6]  D_{7}, [7]  D_{8}, [8]  
Coxeter  
Image  
Name  Nonagon (Enneagon) 
Decagon  Hendecagon  Dodecagon  Tridecagon  Tetradecagon  
Schläfli  {9}  {10}  {11}  {12}  {13}  {14}  
Symmetry  D_{9}, [9]  D_{10}, [10]  D_{11}, [11]  D_{12}, [12]  D_{13}, [13]  D_{14}, [14]  
Dynkin  
Image  
Name  Pentadecagon  Hexadecagon  Heptadecagon  Octadecagon  Enneadecagon  Icosagon  ...pgon 
Schläfli  {15}  {16}  {17}  {18}  {19}  {20}  {p} 
Symmetry  D_{15}, [15]  D_{16}, [16]  D_{17}, [17]  D_{18}, [18]  D_{19}, [19]  D_{20}, [20]  D_{p}, [p] 
Dynkin  
Image 
The regular digon {2} can be considered to be a degenerate regular polygon. It can be realized nondegenerately in some nonEuclidean spaces, such as on the surface of a sphere or torus. For example, digon can be realised nondegenerately as a spherical lune. A monogon {1} could also be realised on the sphere as a single point with a great circle through it.^{[7]} However, a monogon is not a valid abstract polytope because its single edge is incident to only one vertex rather than two.
Name  Monogon  Digon 

Schläfli symbol  {1}  {2} 
Symmetry  D_{1}, [ ]  D_{2}, [2] 
Coxeter diagram  or  
Image 
There exist infinitely many regular star polytopes in two dimensions, whose Schläfli symbols consist of rational numbers {n/m}. They are called star polygons and share the same vertex arrangements of the convex regular polygons.
In general, for any natural number n, there are regular npointed stars with Schläfli symbols {n/m} for all m such that m < n/2 (strictly speaking (({1))}) and m and n are coprime (as such, all stellations of a polygon with a prime number of sides will be regular stars). Symbols where m and n are not coprime may be used to represent compound polygons.
Name  Pentagram  Heptagrams  Octagram  Enneagrams  Decagram  ...ngrams  

Schläfli  {5/2}  {7/2}  {7/3}  {8/3}  {9/2}  {9/4}  {10/3}  {p/q} 
Symmetry  D_{5}, [5]  D_{7}, [7]  D_{8}, [8]  D_{9}, [9],  D_{10}, [10]  D_{p}, [p]  
Coxeter  
Image 
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 do not appear to have 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.^{[8]}
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 P and Q, written P#Q, can be constructed as follows:
Alternatively, the blend is the polygon ⟨ρ_{0}σ_{0}, ρ_{1}σ_{1}⟩ where ρ and σ are the generating mirrors of P and Q placed in orthogonal subspaces.^{[9]} The blending operation is commutative, associative and idempotent.
Every regular skew polygon can be expressed as the blend of a unique^{[a]} set of planar polygons.^{[9]} If P and Q share no factors then Dim(P#Q) = Dim(P) + Dim(Q).
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 ({n/m}#{} where n is odd) or an antiprism ({n/m}#{} where n 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.
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 3dimensional polygons, skew polygons on double rotations can include an oddnumber of sides.
Polytopes of rank 3 are called polyhedra:
A regular polyhedron with Schläfli symbol {p,q}, Coxeter diagrams , has a regular face type {p}, and regular vertex figure {1}.
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 {p,q} 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 {p} and {q} limited to: {3}, {4}, {5}, {5/2}, and {6}.
Beyond Euclidean space, there is an infinite set of regular hyperbolic tilings.
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 (χ) of 2.
Name  Schläfli {p,q} 
Coxeter 
Image (solid) 
Image (sphere) 
Faces {p} 
Edges  Vertices {q} 
Symmetry  Dual 

Tetrahedron (3simplex) 
{3,3}  4 {3} 
6  4 {3} 
T_{d} [3,3] (*332) 
(self)  
Hexahedron Cube (3cube) 
{4,3}  6 {4} 
12  8 {3} 
O_{h} [4,3] (*432) 
Octahedron  
Octahedron (3orthoplex) 
{3,4}  8 {3} 
12  6 {4} 
O_{h} [4,3] (*432) 
Cube  
Dodecahedron  {5,3}  12 {5} 
30  20 {3} 
I_{h} [5,3] (*532) 
Icosahedron  
Icosahedron  {3,5}  20 {3} 
30  12 {5} 
I_{h} [5,3] (*532) 
Dodecahedron 
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.^{[10]}
The first few cases (n from 2 to 6) are listed below.
Name  Schläfli {2,p} 
Coxeter diagram 
Image (sphere) 
Faces {2}_{π/p} 
Edges  Vertices {p} 
Symmetry  Dual 

Digonal hosohedron  {2,2}  2 {2}_{π/2} 
2  2 {2}_{π/2} 
D_{2h} [2,2] (*222) 
Self  
Trigonal hosohedron  {2,3}  3 {2}_{π/3} 
3  2 {3} 
D_{3h} [2,3] (*322) 
Trigonal dihedron  
Square hosohedron  {2,4}  4 {2}_{π/4} 
4  2 {4} 
D_{4h} [2,4] (*422) 
Square dihedron  
Pentagonal hosohedron  {2,5}  5 {2}_{π/5} 
5  2 {5} 
D_{5h} [2,5] (*522) 
Pentagonal dihedron  
Hexagonal hosohedron  {2,6}  6 {2}_{π/6} 
6  2 {6} 
D_{6h} [2,6] (*622) 
Hexagonal dihedron 
Name  Schläfli {p,2} 
Coxeter diagram 
Image (sphere) 
Faces {p} 
Edges  Vertices {2} 
Symmetry  Dual 

Digonal dihedron  {2,2}  2 {2}_{π/2} 
2  2 {2}_{π/2} 
D_{2h} [2,2] (*222) 
Self  
Trigonal dihedron  {3,2}  2 {3} 
3  3 {2}_{π/3} 
D_{3h} [3,2] (*322) 
Trigonal hosohedron  
Square dihedron  {4,2}  2 {4} 
4  4 {2}_{π/4} 
D_{4h} [4,2] (*422) 
Square hosohedron  
Pentagonal dihedron  {5,2}  2 {5} 
5  5 {2}_{π/5} 
D_{5h} [5,2] (*522) 
Pentagonal hosohedron  
Hexagonal dihedron  {6,2}  2 {6} 
6  6 {2}_{π/6} 
D_{6h} [6,2] (*622) 
Hexagonal hosohedron 
Stardihedra and hosohedra {p/q,2} and {2,p/q} also exist for any star polygon {p/q}.
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.
Name  Image (skeletonic) 
Image (solid) 
Image (sphere) 
Stellation diagram 
Schläfli {p,q} and Coxeter 
Faces {p} 
Edges  Vertices {q} verf. 
χ  Density  Symmetry  Dual 

Small stellated dodecahedron  {5/2,5} 
12 {5/2} 
30  12 {5} 
−6  3  I_{h} [5,3] (*532) 
Great dodecahedron  
Great dodecahedron  {5,5/2} 
12 {5} 
30  12 {5/2} 
−6  3  I_{h} [5,3] (*532) 
Small stellated dodecahedron  
Great stellated dodecahedron  {5/2,3} 
12 {5/2} 
30  20 {3} 
2  7  I_{h} [5,3] (*532) 
Great icosahedron  
Great icosahedron  {3,5/2} 
20 {3} 
30  12 {5/2} 
2  7  I_{h} [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}.
Regular skew polyhedra are generalizations to the set of regular polyhedron which include the possibility of nonplanar vertex figures.
For 4dimensional skew polyhedra, Coxeter offered a modified Schläfli symbol {l,mn} for these figures, with {l,m} implying the vertex figure, m lgons around a vertex, and ngonal holes. Their vertex figures are skew polygons, zigzagging between two planes.
The regular skew polyhedra, represented by {l,mn}, follow this equation:
Four of them can be seen in 4dimensions as a subset of faces of four regular 4polytopes, sharing the same vertex arrangement and edge arrangement:
{4, 6  3}  {6, 4  3}  {4, 8  3}  {8, 4  3} 

Regular 4polytopes with Schläfli symbol have cells of type , faces of type , edge figures , and vertex figures .
The existence of a regular 4polytope is constrained by the existence of the regular polyhedra . A suggested name for 4polytopes is "polychoron".^{[11]}
Each will exist in a space dependent upon this expression:
These constraints allow for 21 forms: 6 are convex, 10 are nonconvex, one is a Euclidean 3space honeycomb, and 4 are hyperbolic honeycombs.
The Euler characteristic for convex 4polytopes is zero:
The 6 convex regular 4polytopes are shown in the table below. All these 4polytopes have an Euler characteristic (χ) of 0.
Name 
Schläfli {p,q,r} 
Coxeter 
Cells {p,q} 
Faces {p} 
Edges {r} 
Vertices {q,r} 
Dual {r,q,p} 

5cell (4simplex) 
{3,3,3}  5 {3,3} 
10 {3} 
10 {3} 
5 {3,3} 
(self)  
8cell (4cube) (Tesseract) 
{4,3,3}  8 {4,3} 
24 {4} 
32 {3} 
16 {3,3} 
16cell  
16cell (4orthoplex) 
{3,3,4}  16 {3,3} 
32 {3} 
24 {4} 
8 {3,4} 
Tesseract  
24cell  {3,4,3}  24 {3,4} 
96 {3} 
96 {3} 
24 {4,3} 
(self)  
120cell  {5,3,3}  120 {5,3} 
720 {5} 
1200 {3} 
600 {3,3} 
600cell  
600cell  {3,3,5}  600 {3,3} 
1200 {3} 
720 {5} 
120 {3,5} 
120cell 
5cell  8cell  16cell  24cell  120cell  600cell 

{3,3,3}  {4,3,3}  {3,3,4}  {3,4,3}  {5,3,3}  {3,3,5} 
Wireframe (Petrie polygon) skew orthographic projections  
Solid orthographic projections  
tetrahedral envelope (cell/ vertexcentered) 
cubic envelope (cellcentered) 
cubic envelope (cellcentered) 
cuboctahedral envelope (cellcentered) 
truncated rhombic triacontahedron envelope (cellcentered) 
Pentakis icosidodecahedral envelope (vertexcentered) 
Wireframe Schlegel diagrams (Perspective projection)  
(cellcentered) 
(cellcentered) 
(cellcentered) 
(cellcentered) 
(cellcentered) 
(vertexcentered) 
Wireframe stereographic projections (Hyperspherical)  
Di4topes and hoso4topes exist as regular tessellations of the 3sphere.
Regular di4topes (2 facets) include: {3,3,2}, {3,4,2}, {4,3,2}, {5,3,2}, {3,5,2}, {p,2,2}, and their hoso4tope duals (2 vertices): {2,3,3}, {2,4,3}, {2,3,4}, {2,3,5}, {2,5,3}, {2,2,p}. 4polytopes of the form {2,p,2} are the same as {2,2,p}. There are also the cases {p,2,q} which have dihedral cells and hosohedral vertex figures.
Schläfli {2,p,q} 
Coxeter 
Cells {2,p}_{π/q} 
Faces {2}_{π/p,π/q} 
Edges  Vertices  Vertex figure {p,q} 
Symmetry  Dual 

{2,3,3}  4 {2,3}_{π/3} 
6 {2}_{π/3,π/3} 
4  2  {3,3} 
[2,3,3]  {3,3,2}  
{2,4,3}  6 {2,4}_{π/3} 
12 {2}_{π/4,π/3} 
8  2  {4,3} 
[2,4,3]  {3,4,2}  
{2,3,4}  8 {2,3}_{π/4} 
12 {2}_{π/3,π/4} 
6  2  {3,4} 
[2,4,3]  {4,3,2}  
{2,5,3}  12 {2,5}_{π/3} 
30 {2}_{π/5,π/3} 
20  2  {5,3} 
[2,5,3]  {3,5,2}  
{2,3,5}  20 {2,3}_{π/5} 
30 {2}_{π/3,π/5} 
12  2  {3,5} 
[2,5,3]  {5,3,2} 
There are ten regular star 4polytopes, which are called the Schläfli–Hess 4polytopes. Their vertices are based on the convex 120cell {5,3,3} and 600cell {3,3,5}.
Ludwig Schläfli found four of them and skipped the last six because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zerohole tori: F+V−E=2). Edmund Hess (1843–1903) completed the full list of ten in his German book Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder (1883)[1].
There are 4 unique edge arrangements and 7 unique face arrangements from these 10 regular star 4polytopes, shown as orthogonal projections:
Name 
Wireframe  Solid  Schläfli {p, q, r} Coxeter 
Cells {p, q} 
Faces {p} 
Edges {r} 
Vertices {q, r} 
Density  χ  Symmetry group  Dual {r, q,p} 

Icosahedral 120cell (faceted 600cell) 
{3,5,5/2} 
120 {3,5} 
1200 {3} 
720 {5/2} 
120 {5,5/2} 
4  480  H_{4} [5,3,3] 
Small stellated 120cell  
Small stellated 120cell  {5/2,5,3} 
120 {5/2,5} 
720 {5/2} 
1200 {3} 
120 {5,3} 
4  −480  H_{4} [5,3,3] 
Icosahedral 120cell  
Great 120cell  {5,5/2,5} 
120 {5,5/2} 
720 {5} 
720 {5} 
120 {5/2,5} 
6  0  H_{4} [5,3,3] 
Selfdual  
Grand 120cell  {5,3,5/2} 
120 {5,3} 
720 {5} 
720 {5/2} 
120 {3,5/2} 
20  0  H_{4} [5,3,3] 
Great stellated 120cell  
Great stellated 120cell  {5/2,3,5} 
120 {5/2,3} 
720 {5/2} 
720 {5} 
120 {3,5} 
20  0  H_{4} [5,3,3] 
Grand 120cell  
Grand stellated 120cell  {5/2,5,5/2} 
120 {5/2,5} 
720 {5/2} 
720 {5/2} 
120 {5,5/2} 
66  0  H_{4} [5,3,3] 
Selfdual  
Great grand 120cell  {5,5/2,3} 
120 {5,5/2} 
720 {5} 
1200 {3} 
120 {5/2,3} 
76  −480  H_{4} [5,3,3] 
Great icosahedral 120cell  
Great icosahedral 120cell (great faceted 600cell) 
{3,5/2,5} 
120 {3,5/2} 
1200 {3} 
720 {5} 
120 {5/2,5} 
76  480  H_{4} [5,3,3] 
Great grand 120cell  
Grand 600cell  {3,3,5/2} 
600 {3,3} 
1200 {3} 
720 {5/2} 
120 {3,5/2} 
191  0  H_{4} [5,3,3] 
Great grand stellated 120cell  
Great grand stellated 120cell  {5/2,3,3} 
120 {5/2,3} 
720 {5/2} 
1200 {3} 
600 {3,3} 
191  0  H_{4} [5,3,3] 
Grand 600cell 
There are 4 failed potential regular star 4polytopes permutations: {3,5/2,3}, {4,3,5/2}, {5/2,3,4}, {5/2,3,5/2}. Their cells and vertex figures exist, but they do not cover a hypersphere with a finite number of repetitions.
In addition to the 16 planar 4polytopes above there are 18 finite skew polytopes.^{[12]} One of these is obtained as the Petrial of the tesseract, and the other 17 can be formed by applying the kappa operation to the planar polytopes and the Petrial of the tesseract.
5polytopes can be given the symbol where is the 4face type, is the cell type, is the face type, and is the face figure, is the edge figure, and is the vertex figure.
A regular 5polytope exists only if and are regular 4polytopes.
The space it fits in is based on the expression:
Enumeration of these constraints produce 3 convex polytopes, no star polytopes, 3 tessellations of Euclidean 4space, and 5 tessellations of paracompact hyperbolic 4space. The only no nonconvex regular polytopes for ranks 5 and higher are skews.
In dimensions 5 and higher, there are only three kinds of convex regular polytopes.^{[13]}
Name  Schläfli Symbol {p_{1},...,p_{n−1}} 
Coxeter  kfaces  Facet type 
Vertex figure 
Dual 

nsimplex  {3^{n−1}}  ...  {3^{n−2}}  {3^{n−2}}  Selfdual  
ncube  {4,3^{n−2}}  ...  {4,3^{n−3}}  {3^{n−2}}  northoplex  
northoplex  {3^{n−2},4}  ...  {3^{n−2}}  {3^{n−3},4}  ncube 
There are also improper cases where some numbers in the Schläfli symbol are 2. For example, {p,q,r,...2} is an improper regular spherical polytope whenever {p,q,r...} is a regular spherical polytope, and {2,...p,q,r} is an improper regular spherical polytope whenever {...p,q,r} is a regular spherical polytope. Such polytopes may also be used as facets, yielding forms such as {p,q,...2...y,z}.
Name  Schläfli Symbol {p,q,r,s} Coxeter 
Facets {p,q,r} 
Cells {p,q} 
Faces {p} 
Edges  Vertices  Face figure {s} 
Edge figure {r,s} 
Vertex figure {q,r,s} 

5simplex  {3,3,3,3} 
6 {3,3,3} 
15 {3,3} 
20 {3} 
15  6  {3}  {3,3}  {3,3,3} 
5cube  {4,3,3,3} 
10 {4,3,3} 
40 {4,3} 
80 {4} 
80  32  {3}  {3,3}  {3,3,3} 
5orthoplex  {3,3,3,4} 
32 {3,3,3} 
80 {3,3} 
80 {3} 
40  10  {4}  {3,4}  {3,3,4} 
5simplex 
5cube 
5orthoplex 
Name  Schläfli  Vertices  Edges  Faces  Cells  4faces  5faces  χ 

6simplex  {3,3,3,3,3}  7  21  35  35  21  7  0 
6cube  {4,3,3,3,3}  64  192  240  160  60  12  0 
6orthoplex  {3,3,3,3,4}  12  60  160  240  192  64  0 
6simplex 
6cube 
6orthoplex 
Name  Schläfli  Vertices  Edges  Faces  Cells  4faces  5faces  6faces  χ 

7simplex  {3,3,3,3,3,3}  8  28  56  70  56  28  8  2 
7cube  {4,3,3,3,3,3}  128  448  672  560  280  84  14  2 
7orthoplex  {3,3,3,3,3,4}  14  84  280  560  672  448  128  2 
7simplex 
7cube 
7orthoplex 
Name  Schläfli  Vertices  Edges  Faces  Cells  4faces  5faces  6faces  7faces  χ 

8simplex  {3,3,3,3,3,3,3}  9  36  84  126  126  84  36  9  0 
8cube  {4,3,3,3,3,3,3}  256  1024  1792  1792  1120  448  112  16  0 
8orthoplex  {3,3,3,3,3,3,4}  16  112  448  1120  1792  1792  1024  256  0 
8simplex 
8cube 
8orthoplex 
Name  Schläfli  Vertices  Edges  Faces  Cells  4faces  5faces  6faces  7faces  8faces  χ 

9simplex  {3^{8}}  10  45  120  210  252  210  120  45  10  2 
9cube  {4,3^{7}}  512  2304  4608  5376  4032  2016  672  144  18  2 
9orthoplex  {3^{7},4}  18  144  672  2016  4032  5376  4608  2304  512  2 
9simplex 
9cube 
9orthoplex 
Name  Schläfli  Vertices  Edges  Faces  Cells  4faces  5faces  6faces  7faces  8faces  9faces  χ 

10simplex  {3^{9}}  11  55  165  330  462  462  330  165  55  11  0 
10cube  {4,3^{8}}  1024  5120  11520  15360  13440  8064  3360  960  180  20  0 
10orthoplex  {3^{8},4}  20  180  960  3360  8064  13440  15360  11520  5120  1024  0 
10simplex 
10cube 
10orthoplex 
There are no regular star polytopes of rank 5 or higher, with the exception of degenerate polytopes created by the star product of lower rank star polytopes. e.g. hosotopes and ditopes.
A projective regular (n+1)polytope exists when an original regular nspherical tessellation, {p,q,...}, is centrally symmetric. Such a polytope is named hemi{p,q,...}, and contain half as many elements. Coxeter gives a symbol {p,q,...}/2, while McMullen writes {p,q,...}_{h/2} with h as the coxeter number.^{[14]}
Evensided regular polygons have hemi2ngon projective polygons, {2p}/2.
There are 4 regular projective polyhedra related to 4 of 5 Platonic solids.
The hemicube and hemioctahedron generalize as hemincubes and heminorthoplexes to any rank.
Name  Coxeter McMullen 
Image  Faces  Edges  Vertices  χ 

Hemicube  {4,3}/2 {4,3}_{3} 
3  6  4  1  
Hemioctahedron  {3,4}/2 {3,4}_{3} 
4  6  3  1  
Hemidodecahedron  {5,3}/2 {5,3}_{5} 
6  15  10  1  
Hemiicosahedron  {3,5}/2 {3,5}_{5} 
10  15  6  1 
5 of 6 convex regular 4polytopes are centrally symmetric generating projective 4polytopes. The 3 special cases are hemi24cell, hemi600cell, and hemi120cell.
Name  Coxeter symbol 
McMullen Symbol 
Cells  Faces  Edges  Vertices  χ 

Hemitesseract  {4,3,3}/2  {4,3,3}_{4}  4  12  16  8  0 
Hemi16cell  {3,3,4}/2  {3,3,4}_{4}  8  16  12  4  0 
Hemi24cell  {3,4,3}/2  {3,4,3}_{6}  12  48  48  12  0 
Hemi120cell  {5,3,3}/2  {5,3,3}_{15}  60  360  600  300  0 
Hemi600cell  {3,3,5}/2  {3,3,5}_{15}  300  600  360  60  0 
Only 2 of 3 regular spereical polytopes are centrally symmetric for ranks 5 or higher: they are the hemi versions of the regular hypercube and orthoplex. They are tabulated below for rank 5, for example:
Name  Schläfli  4faces  Cells  Faces  Edges  Vertices  χ 

hemipenteract  {4,3,3,3}/2  5  20  40  40  16  1 
hemipentacross  {3,3,3,4}/2  16  40  40  20  5  1 
An apeirotope or infinite polytope is a polytope which has infinitely many facets. An napeirotope is an infinite npolytope: a 2apeirotope or apeirogon is an infinite polygon, a 3apeirotope or apeirohedron is an infinite polyhedron, etc.
There are two main geometric classes of apeirotope:^{[15]}
The straight apeirogon is a regular tessellation of the line, subdividing it into infinitely many equal segments. It has infinitely many vertices and edges. Its Schläfli symbol is {∞}, and Coxeter diagram .
It exists as the limit of the pgon as p tends to infinity, as follows:
Name  Monogon  Digon  Triangle  Square  Pentagon  Hexagon  Heptagon  pgon  Apeirogon 

Schläfli  {1}  {2}  {3}  {4}  {5}  {6}  {7}  {p}  {∞} 
Symmetry  D_{1}, [ ]  D_{2}, [2]  D_{3}, [3]  D_{4}, [4]  D_{5}, [5]  D_{6}, [6]  D_{7}, [7]  [p]  
Coxeter  or  
Image 
Apeirogons in the hyperbolic plane, most notably the regular apeirogon, {∞}, can have a curvature just like finite polygons of the Euclidean plane, with the vertices circumscribed by horocycles or hypercycles rather than circles.
Regular apeirogons that are scaled to converge at infinity have the symbol {∞} and exist on horocycles, while more generally they can exist on hypercycles.
{∞}  {πi/λ} 

Apeirogon on horocycle 
Apeirogon on hypercycle 
Above are two regular hyperbolic apeirogons in the Poincaré disk model, the right one shows perpendicular reflection lines of divergent fundamental domains, separated by length λ.
A skew apeirogon in two dimensions forms a zigzag line in the plane. If the zigzag is even and symmetrical, then the apeirogon is regular.
Skew apeirogons can be constructed in any number of dimensions. In three dimensions, a regular skew apeirogon traces out a helical spiral and may be either left or righthanded.
2dimensions  3dimensions 

Zigzag apeirogon 
Helix apeirogon 
There are three regular tessellations of the plane. All three have an Euler characteristic (χ) of 0.
Name  Square tiling (quadrille) 
Triangular tiling (deltille) 
Hexagonal tiling (hextille) 

Symmetry  p4m, [4,4], (*442)  p6m, [6,3], (*632)  
Schläfli {p,q}  {4,4}  {3,6}  {6,3} 
Coxeter diagram  
Image 
There are two improper regular tilings: {∞,2}, an apeirogonal dihedron, made from two apeirogons, each filling half the plane; and secondly, its dual, {2,∞}, an apeirogonal hosohedron, seen as an infinite set of parallel lines.
{∞,2}, 
{2,∞}, 
There are no regular plane tilings of star polygons. There are many enumerations that fit in the plane (1/p + 1/q = 1/2), like {8/3,8}, {10/3,5}, {5/2,10}, {12/5,12}, etc., but none repeat periodically.
Tessellations of hyperbolic 2space are hyperbolic tilings. There are infinitely many regular tilings in H^{2}. As stated above, every positive integer pair {p,q} such that 1/p + 1/q < 1/2 gives a hyperbolic tiling. In fact, for the general Schwarz triangle (p, q, r) the same holds true for 1/p + 1/q + 1/r < 1.
There are a number of different ways to display the hyperbolic plane, including the Poincaré disc model which maps the plane into a circle, as shown below. It should be recognized that all of the polygon faces in the tilings below are equalsized and only appear to get smaller near the edges due to the projection applied, very similar to the effect of a camera fisheye lens.
There are infinitely many flat regular 3apeirotopes (apeirohedra) as regular tilings of the hyperbolic plane, of the form {p,q}, with p+q<pq/2. (previously listed above as tessellations)
A sampling:
Regular hyperbolic tiling table  

Spherical (improper/Platonic)/Euclidean/hyperbolic (Poincaré disc: compact/paracompact/noncompact) tessellations with their Schläfli symbol  
p \ q  2  3  4  5  6  7  8  ...  ∞  ...  iπ/λ 
2  {2,2} 
{2,3} 
{2,4} 
{2,5} 
{2,6} 
{2,7} 
{2,8} 
{2,∞} 
{2,iπ/λ}  
3  {3,2} 
(tetrahedron) {3,3} 
(octahedron) {3,4} 
(icosahedron) {3,5} 
(deltille) {3,6} 
{3,7} 
{3,8} 
{3,∞} 
{3,iπ/λ}  
4  {4,2} 
(cube) {4,3} 
(quadrille) {4,4} 
{4,5} 
{4,6} 
{4,7} 
{4,8} 
{4,∞} 
{4,iπ/λ}  
5  {5,2} 
(dodecahedron) {5,3} 
{5,4} 
{5,5} 
{5,6} 
{5,7} 
{5,8} 
{5,∞} 
{5,iπ/λ}  
6  {6,2} 
(hextille) {6,3} 
{6,4} 
{6,5} 
{6,6} 
{6,7} 
{6,8} 
{6,∞} 
{6,iπ/λ}  
7  {7,2} 
{7,3} 
{7,4} 
{7,5} 
{7,6} 
{7,7} 
{7,8} 
{7,∞} 
{7,iπ/λ}  
8  {8,2} 
{8,3} 
{8,4} 
{8,5} 
{8,6} 
{8,7} 
{8,8} 
{8,∞} 
{8,iπ/λ}  
...  
∞  {∞,2} 
{∞,3} 
{∞,4} 
{∞,5} 
{∞,6} 
{∞,7} 
{∞,8} 
{∞,∞} 
{∞,iπ/λ}  
...  
iπ/λ  {iπ/λ,2} 
{iπ/λ,3} 
{iπ/λ,4} 
{iπ/λ,5} 
{iπ/λ,6} 
{iπ/λ,7} 
{iπ/λ,8} 
{iπ/λ,∞} 
{iπ/λ, iπ/λ} 
The tilings {p, ∞} have ideal vertices, on the edge of the Poincaré disc model. Their duals {∞, p} have ideal apeirogonal faces, meaning that they are inscribed in horocycles. One could go further (as is done in the table above) and find tilings with ultraideal vertices, outside the Poincaré disc, which are dual to tiles inscribed in hypercycles; in what is symbolised {p, iπ/λ} above, infinitely many tiles still fit around each ultraideal vertex.^{[16]} (Parallel lines in extended hyperbolic space meet at an ideal point; ultraparallel lines meet at an ultraideal point.)^{[17]}
There are 2 infinite forms of hyperbolic tilings whose faces or vertex figures are star polygons: {m/2, m} and their duals {m, m/2} with m = 7, 9, 11, .... The {m/2, m} tilings are stellations of the {m, 3} tilings while the {m, m/2} dual tilings are facetings of the {3, m} tilings and greatenings of the {m, 3} tilings.
The patterns {m/2, m} and {m, m/2} continue for odd m < 7 as polyhedra: when m = 5, we obtain the small stellated dodecahedron and great dodecahedron, and when m = 3, the case degenerates to a tetrahedron. The other two Kepler–Poinsot polyhedra (the great stellated dodecahedron and great icosahedron) do not have regular hyperbolic tiling analogues. If m is even, depending on how we choose to define {m/2}, we can either obtain degenerate double covers of other tilings or compound tilings.
Name  Schläfli  Coxeter diagram  Image  Face type {p} 
Vertex figure {q} 
Density  Symmetry  Dual 

Order7 heptagrammic tiling  {7/2,7}  {7/2} 
{7} 
3  *732 [7,3] 
Heptagrammicorder heptagonal tiling  
Heptagrammicorder heptagonal tiling  {7,7/2}  {7} 
{7/2} 
3  *732 [7,3] 
Order7 heptagrammic tiling  
Order9 enneagrammic tiling  {9/2,9}  {9/2} 
{9} 
3  *932 [9,3] 
Enneagrammicorder enneagonal tiling  
Enneagrammicorder enneagonal tiling  {9,9/2}  {9} 
{9/2} 
3  *932 [9,3] 
Order9 enneagrammic tiling  
Order11 hendecagrammic tiling  {11/2,11}  {11/2} 
{11} 
3  *11.3.2 [11,3] 
Hendecagrammicorder hendecagonal tiling  
Hendecagrammicorder hendecagonal tiling  {11,11/2}  {11} 
{11/2} 
3  *11.3.2 [11,3] 
Order11 hendecagrammic tiling  
Orderp pgrammic tiling  {p/2,p}  {p/2}  {p}  3  *p32 [p,3] 
pgrammicorder pgonal tiling  
pgrammicorder pgonal tiling  {p,p/2}  {p}  {p/2}  3  *p32 [p,3] 
Orderp pgrammic tiling 
There are three regular skew apeirohedra in Euclidean 3space, with planar faces.^{[18]}^{[19]}^{[20]} They share the same vertex arrangement and edge arrangement of 3 convex uniform honeycombs.
Regular skew polyhedra with planar faces  

{4,64} 
{6,44} 
{6,63} 
Allowing for skew faces, there are 24 regular apeirohedra in Euclidean 3space.^{[22]} These include 12 apeirhedra created by blends with the Euclidean apeirohedra, and 12 pure apeirohedra, including the 3 above, which cannot be expressed as a nontrivial blend.
Those pure apeirohedra are:
There are 31 regular skew apeirohedra with convex faces in hyperbolic 3space with compact or paracompact symmetry:^{[23]}
There is only one nondegenerate regular tessellation of 3space (honeycombs), {4, 3, 4}:^{[24]}
Name  Schläfli {p,q,r} 
Coxeter 
Cell type {p,q} 
Face type {p} 
Edge figure {r} 
Vertex figure {q,r} 
χ  Dual 

Cubic honeycomb  {4,3,4}  {4,3}  {4}  {4}  {3,4}  0  Selfdual 
There are six improper regular tessellations, pairs based on the three regular Euclidean tilings. Their cells and vertex figures are all regular hosohedra {2,n}, dihedra, {n,2}, and Euclidean tilings. These improper regular tilings are constructionally related to prismatic uniform honeycombs by truncation operations. They are higherdimensional analogues of the order2 apeirogonal tiling and apeirogonal hosohedron.
Schläfli {p,q,r} 
Coxeter diagram 
Cell type {p,q} 
Face type {p} 
Edge figure {r} 
Vertex figure {q,r} 

{2,4,4}  {2,4}  {2}  {4}  {4,4}  
{2,3,6}  {2,3}  {2}  {6}  {3,6}  
{2,6,3}  {2,6}  {2}  {3}  {6,3}  
{4,4,2}  {4,4}  {4}  {2}  {4,2}  
{3,6,2}  {3,6}  {3}  {2}  {6,2}  
{6,3,2}  {6,3}  {6}  {2}  {3,2} 
There are ten flat regular honeycombs of hyperbolic 3space:^{[25]} (previously listed above as tessellations)
 

Tessellations of hyperbolic 3space can be called hyperbolic honeycombs. There are 15 hyperbolic honeycombs in H^{3}, 4 compact and 11 paracompact.
Name  Schläfli Symbol {p,q,r} 
Coxeter 
Cell type {p,q} 
Face type {p} 
Edge figure {r} 
Vertex figure {q,r} 
χ  Dual 

Icosahedral honeycomb  {3,5,3}  {3,5}  {3}  {3}  {5,3}  0  Selfdual  
Order5 cubic honeycomb  {4,3,5}  {4,3}  {4}  {5}  {3,5}  0  {5,3,4}  
Order4 dodecahedral honeycomb  {5,3,4}  {5,3}  {5}  {4}  {3,4}  0  {4,3,5}  
Order5 dodecahedral honeycomb  {5,3,5}  {5,3}  {5}  {5}  {3,5}  0  Selfdual 
There are also 11 paracompact H^{3} honeycombs (those with infinite (Euclidean) cells and/or vertex figures): {3,3,6}, {6,3,3}, {3,4,4}, {4,4,3}, {3,6,3}, {4,3,6}, {6,3,4}, {4,4,4}, {5,3,6}, {6,3,5}, and {6,3,6}.
Name  Schläfli Symbol {p,q,r} 
Coxeter 
Cell type {p,q} 
Face type {p} 
Edge figure {r} 
Vertex figure {q,r} 
χ  Dual 

Order6 tetrahedral honeycomb  {3,3,6}  {3,3}  {3}  {6}  {3,6}  0  {6,3,3}  
Hexagonal tiling honeycomb  {6,3,3}  {6,3}  {6}  {3}  {3,3}  0  {3,3,6}  
Order4 octahedral honeycomb  {3,4,4}  {3,4}  {3}  {4}  {4,4}  0  {4,4,3}  
Square tiling honeycomb  {4,4,3}  {4,4}  {4}  {3}  {4,3}  0  {3,3,4}  
Triangular tiling honeycomb  {3,6,3}  {3,6}  {3}  {3}  {6,3}  0  Selfdual  
Order6 cubic honeycomb  {4,3,6}  {4,3}  {4}  {4}  {3,6}  0  {6,3,4}  
Order4 hexagonal tiling honeycomb  {6,3,4}  {6,3}  {6}  {4}  {3,4}  0  {4,3,6}  
Order4 square tiling honeycomb  {4,4,4}  {4,4}  {4}  {4}  {4,4}  0  Selfdual  
Order6 dodecahedral honeycomb  {5,3,6}  {5,3}  {5}  {5}  {3,6}  0  {6,3,5}  
Order5 hexagonal tiling honeycomb  {6,3,5}  {6,3}  {6}  {5}  {3,5}  0  {5,3,6}  
Order6 hexagonal tiling honeycomb  {6,3,6}  {6,3}  {6}  {6}  {3,6}  0  Selfdual 
Noncompact solutions exist as Lorentzian Coxeter groups, and can be visualized with open domains in hyperbolic space (the fundamental tetrahedron having ultraideal vertices). All honeycombs with hyperbolic cells or vertex figures and do not have 2 in their Schläfli symbol are noncompact.
{p,3} \ r  2  3  4  5  6  7  8  ... ∞ 

{2,3} 
{2,3,2} 
{2,3,3}  {2,3,4}  {2,3,5}  {2,3,6}  {2,3,7}  {2,3,8}  {2,3,∞} 
{3,3} 
{3,3,2} 
{3,3,3} 
{3,3,4} 
{3,3,5} 
{3,3,6} 
{3,3,7} 
{3,3,8} 
{3,3,∞} 
{4,3} 
{4,3,2} 
{4,3,3} 
{4,3,4} 
{4,3,5} 
{4,3,6} 
{4,3,7} 
{4,3,8} 
{4,3,∞} 
{5,3} 
{5,3,2} 
{5,3,3} 
{5,3,4} 
{5,3,5} 
{5,3,6} 
{5,3,7} 
{5,3,8} 
{5,3,∞} 
{6,3} 
{6,3,2} 
{6,3,3} 
{6,3,4} 
{6,3,5} 
{6,3,6} 
{6,3,7} 
{6,3,8} 
{6,3,∞} 
{7,3} 
{7,3,2}  {7,3,3} 
{7,3,4} 
{7,3,5} 
{7,3,6} 
{7,3,7} 
{7,3,8} 
{7,3,∞} 
{8,3} 
{8,3,2}  {8,3,3} 
{8,3,4} 
{8,3,5} 
{8,3,6} 
{8,3,7} 
{8,3,8} 
{8,3,∞} 
... {∞,3} 
{∞,3,2}  {∞,3,3} 
{∞,3,4} 
{∞,3,5} 
{∞,3,6} 
{∞,3,7} 
{∞,3,8} 