Regular (2D) polygons  

Convex  Star 
{5} 
{5/2} 
Regular (3D) polyhedra  
Convex  Star 
{5,3} 
{5/2,5} 
Regular 2D tessellations  
Euclidean  Hyperbolic 
{4,4} 
{5,4} 
Regular 4D polytopes  
Convex  Star 
{5,3,3} 
{5/2,5,3} 
Regular 3D tessellations  
Euclidean  Hyperbolic 
{4,3,4} 
{5,3,4} 
This page lists the regular polytopes and regular polytope compounds in Euclidean, spherical and hyperbolic spaces.
The Schläfli symbol describes every regular tessellation of an nsphere, Euclidean and hyperbolic spaces. A Schläfli symbol describing an npolytope equivalently describes a tessellation of an (n  1)sphere. In addition, the symmetry of a regular polytope or tessellation is expressed as a Coxeter group, which Coxeter expressed identically to the Schläfli symbol, except delimiting by square brackets, a notation that is called Coxeter notation. Another related symbol is the CoxeterDynkin diagram which represents a symmetry group with no rings, and the represents regular polytope or tessellation with a ring on the first node. For example, the cube has Schläfli symbol {4,3}, and with its octahedral symmetry, [4,3] or , it is represented by Coxeter diagram .
The regular polytopes are grouped by dimension and subgrouped by convex, nonconvex and infinite forms. Nonconvex forms use the same vertices as the convex forms, but have intersecting facets. Infinite forms tessellate a onelowerdimensional Euclidean space.
Infinite forms can be extended to tessellate a hyperbolic space. Hyperbolic space is like normal space at a small scale, but parallel lines diverge at a distance. This allows vertex figures to have negative angle defects, like making a vertex with seven equilateral triangles and allowing it to lie flat. It cannot be done in a regular plane, but can be at the right scale of a hyperbolic plane.
A more general definition of regular polytopes which do not have simple Schläfli symbols includes regular skew polytopes and regular skew apeirotopes with nonplanar facets or vertex figures.
This table shows a summary of regular polytope counts by dimension.
Dim.  Finite  Euclidean  Hyperbolic  Compounds  

Convex  Star  Skew  Convex  Compact  Star  Paracompact  Convex  Star  
1  1  0  0  1  0  0  0  0  0 
2  ?  ?  ?  1  1  0  0  ?  ? 
3  5  4  ?  3  ?  ?  ?  5  0 
4  6  10  ?  1  4  0  11  26  20 
5  3  0  ?  3  5  4  2  0  0 
6  3  0  ?  1  0  0  5  0  0 
7  3  0  ?  1  0  0  0  3  0 
8  3  0  ?  1  0  0  0  6  0 
9+  3  0  ?  1  0  0  0  ^{[a]}  0 
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. 
A onedimensional polytope or 1polytope is a closed line segment, bounded by its two endpoints. A 1polytope is regular by definition and is represented by Schläfli symbol { },^{[1]}^{[2]} or a Coxeter diagram with a single ringed node, . Norman Johnson calls it a dion^{[3]} and gives it the Schläfli symbol { }.
Although trivial as a polytope, it appears as the edges of polygons and other higher dimensional polytopes.^{[4]} 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.^{[5]}
The twodimensional polytopes are called polygons. Regular polygons are equilateral and cyclic. A pgonal regular polygon is represented by Schläfli symbol {p}.
Usually only convex polygons are considered regular, but star polygons, like the pentagram, can also be considered 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.
Star polygons should be called nonconvex rather than concave because the intersecting edges do not generate new vertices and all the vertices exist on a circle boundary.
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.
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 npointed star regular polygonal stars with Schläfli symbols {n/m} for all m such that m < n/2 (strictly speaking {n/m}={n/(nm)}) and m and n are coprime (as such, all stellations of a polygon with a prime number of sides will be regular stars). Cases where m and n are not coprime are called 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.^{[6]}
In 3dimensional space, a regular skew polygon is called an antiprismatic polygon, with the vertex arrangement of an antiprism, and a subset of edges, zigzagging between top and bottom polygons.
Hexagon  Octagon  Decagons  
D_{3d}, [2^{+},6]  D_{4d}, [2^{+},8]  D_{5d}, [2^{+},10]  

{3}#{ }  {4}#{ }  {5}#{ }  {5/2}#{ }  {5/3}#{ } 
In 4dimensions a regular skew polygon can have vertices on a Clifford torus and related by a Clifford displacement. Unlike antiprismatic skew polygons, skew polygons on double rotations can include an oddnumber of sides.
They can be seen in the Petrie polygons of the convex regular 4polytopes, seen as regular plane polygons in the perimeter of Coxeter plane projection:
In three dimensions, polytopes are called polyhedra:
A regular polyhedron with Schläfli symbol {p,q}, Coxeter diagrams , has a regular face type {p}, and regular vertex figure {q}.
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 (?) 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.^{[7]}
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 KeplerPoinsot 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. 
?  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:
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".^{[8]}
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 (?) 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 SchläfliHess 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+VE=2). Edmund Hess (18431903) 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)[2].
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  ?  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 five dimensions, a regular polytope can be named as 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, zero nonconvex polytopes, 3 4space tessellations, and 5 hyperbolic 4space tessellations. There are no nonconvex regular polytopes in five dimensions or higher.
In dimensions 5 and higher, there are only three kinds of convex regular polytopes.^{[9]}
Name  Schläfli Symbol {p_{1},...,p_{n1}} 
Coxeter  kfaces  Facet type 
Vertex figure 
Dual 

nsimplex  {3^{n1}}  ...  {3^{n2}}  {3^{n2}}  Selfdual  
ncube  {4,3^{n2}}  ...  {4,3^{n3}}  {3^{n2}}  northoplex  
northoplex  {3^{n2},4}  ...  {3^{n2}}  {3^{n3},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} 
Name  Schläfli  Vertices  Edges  Faces  Cells  4faces  5faces  ? 

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 
Name  Schläfli  Vertices  Edges  Faces  Cells  4faces  5faces  6faces  ? 

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 
Name  Schläfli  Vertices  Edges  Faces  Cells  4faces  5faces  6faces  7faces  ? 

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 
Name  Schläfli  Vertices  Edges  Faces  Cells  4faces  5faces  6faces  7faces  8faces  ? 

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 
Name  Schläfli  Vertices  Edges  Faces  Cells  4faces  5faces  6faces  7faces  8faces  9faces  ? 

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 
...
There are no nonconvex regular polytopes in five dimensions or higher.
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.^{[10]}
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 in any dimensions.
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 
In 4dimensions 5 of 6 convex regular 4polytopes generate 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 
There are only 2 convex regular projective hemipolytopes in dimensions 5 or higher.
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 apeirotope, is an infinite polygon, a 3apeirotope, or apeirohedron, is an infinite polyhedron, etc.
There are two main geometric classes of apeirotope:^{[11]}
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 .
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.
{?}  {?i/?} 

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 ?.
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.
There are three regular tessellations of the plane. All three have an Euler characteristic (?) 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.
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  ...  ∞  ...  i?/? 
2  {2,2} 
{2,3} 
{2,4} 
{2,5} 
{2,6} 
{2,7} 
{2,8} 
{2,∞} 
{2,i?/?}  
3  {3,2} 
(tetrahedron) {3,3} 
(octahedron) {3,4} 
(icosahedron) {3,5} 
(deltille) {3,6} 
{3,7} 
{3,8} 
{3,∞} 
{3,i?/?}  
4  {4,2} 
(cube) {4,3} 
(quadrille) {4,4} 
{4,5} 
{4,6} 
{4,7} 
{4,8} 
{4,∞} 
{4,i?/?}  
5  {5,2} 
(dodecahedron) {5,3} 
{5,4} 
{5,5} 
{5,6} 
{5,7} 
{5,8} 
{5,∞} 
{5,i?/?}  
6  {6,2} 
(hextille) {6,3} 
{6,4} 
{6,5} 
{6,6} 
{6,7} 
{6,8} 
{6,∞} 
{6,i?/?}  
7  {7,2} 
{7,3} 
{7,4} 
{7,5} 
{7,6} 
{7,7} 
{7,8} 
{7,∞} 
{7,i?/?}  
8  {8,2} 
{8,3} 
{8,4} 
{8,5} 
{8,6} 
{8,7} 
{8,8} 
{8,∞} 
{8,i?/?}  
...  
∞  {∞,2} 
{∞,3} 
{∞,4} 
{∞,5} 
{∞,6} 
{∞,7} 
{∞,8} 
{∞,∞} 
{∞,i?/?}  
...  
i?/?  {i?/?,2} 
{i?/?,3} 
{i?/?,4} 
{i?/?,5} 
{i?/?,6} 
{i?/?,7} 
{i?/?,8} 
{i?/?,∞} 
{i?/?,i?/?} 
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 KeplerPoinsot 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 regular skew polygon vertex figures.^{[12]}^{[13]}^{[14]} They share the same vertex arrangement and edge arrangement of 3 convex uniform honeycombs.
There are thirty regular apeirohedra in Euclidean 3space.^{[16]} These include those listed above, as well as 8 other "pure" apeirohedra, all related to the cubic honeycomb, {4,3,4}, with others having skew polygon faces: {6,6}_{4}, {4,6}_{4}, {6,4}_{6}, {?,3}^{a}, {?,3}^{b}, {?,4}^{.*3}, {?,4}_{6,4}, {?,6}_{4,4}, and {?,6}_{6,3}.
There are 31 regular skew apeirohedra in hyperbolic 3space:^{[17]}
There is only one nondegenerate regular tessellation of 3space (honeycombs), {4, 3, 4}:^{[18]}
Name  Schläfli {p,q,r} 
Coxeter 
Cell type {p,q} 
Face type {p} 
Edge figure {r} 
Vertex figure {q,r} 
?  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:^{[19]} (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} 
?  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} 
?  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}  {3,4}  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  {4,4,4}  
Order6 dodecahedral honeycomb  {5,3,6}  {5,3}  {5}  {5}  {3,5}  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 some parts inaccessible beyond infinity). 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} 
{∞,3,∞} 






There are no regular hyperbolic starhoneycombs in H^{3}: all forms with a regular star polyhedron as cell, vertex figure or both end up being spherical.
There are three kinds of infinite regular tessellations (honeycombs) that can tessellate Euclidean fourdimensional space:
Name  Schläfli Symbol {p,q,r,s} 
Facet type {p,q,r} 
Cell type {p,q} 
Face type {p} 
Face figure {s} 
Edge figure {r,s} 
Vertex figure {q,r,s} 
Dual 

Tesseractic honeycomb  {4,3,3,4}  {4,3,3}  {4,3}  {4}  {4}  {3,4}  {3,3,4}  Selfdual 
16cell honeycomb  {3,3,4,3}  {3,3,4}  {3,3}  {3}  {3}  {4,3}  {3,4,3}  {3,4,3,3} 
24cell honeycomb  {3,4,3,3}  {3,4,3}  {3,4}  {3}  {3}  {3,3}  {4,3,3}  {3,3,4,3} 
Projected portion of {4,3,3,4} (Tesseractic honeycomb) 
Projected portion of {3,3,4,3} (16cell honeycomb) 
Projected portion of {3,4,3,3} (24cell honeycomb) 
There are also the two improper cases {4,3,4,2} and {2,4,3,4}.
There are three flat regular honeycombs of Euclidean 4space:^{[18]}
There are seven flat regular convex honeycombs of hyperbolic 4space:^{[19]}
There are four flat regular star honeycombs of hyperbolic 4space:^{[19]}
There are seven convex regular honeycombs and four starhoneycombs in H^{4} space.^{[20]} Five convex ones are compact, and two are paracompact.
Five compact regular honeycombs in H^{4}:
Name  Schläfli Symbol {p,q,r,s} 
Facet type {p,q,r} 
Cell type {p,q} 
Face type {p} 
Face figure {s} 
Edge figure {r,s} 
Vertex figure {q,r,s} 
Dual 

Order5 5cell honeycomb  {3,3,3,5}  {3,3,3}  {3,3}  {3}  {5}  {3,5}  {3,3,5}  {5,3,3,3} 
120cell honeycomb  {5,3,3,3}  {5,3,3}  {5,3}  {5}  {3}  {3,3}  {3,3,3}  {3,3,3,5} 
Order5 tesseractic honeycomb  {4,3,3,5}  {4,3,3}  {4,3}  {4}  {5}  {3,5}  {3,3,5}  {5,3,3,4} 
Order4 120cell honeycomb  {5,3,3,4}  {5,3,3}  {5,3}  {5}  {4}  {3,4}  {3,3,4}  {4,3,3,5} 
Order5 120cell honeycomb  {5,3,3,5}  {5,3,3}  {5,3}  {5}  {5}  {3,5}  {3,3,5}  Selfdual 
The two paracompact regular H^{4} honeycombs are: {3,4,3,4}, {4,3,4,3}.
Name  Schläfli Symbol {p,q,r,s} 
Facet type {p,q,r} 
Cell type {p,q} 
Face type {p} 
Face figure {s} 
Edge figure {r,s} 
Vertex figure {q,r,s} 
Dual 

Order4 24cell honeycomb  {3,4,3,4}  {3,4,3}  {3,4}  {3}  {4}  {3,4}  {4,3,4}  {4,3,4,3} 
Cubic honeycomb honeycomb  {4,3,4,3}  {4,3,4}  {4,3}  {4}  {3}  {4,3}  {3,4,3}  {3,4,3,4} 
Noncompact solutions exist as Lorentzian Coxeter groups, and can be visualized with open domains in hyperbolic space (the fundamental 5cell having some parts inaccessible beyond infinity). All honeycombs which are not shown in the set of tables below and do not have 2 in their Schläfli symbol are noncompact.
Spherical/Euclidean/hyperbolic(compact/paracompact/noncompact) honeycombs {p,q,r,s}  




There are four regular starhoneycombs in H^{4} space:
Name  Schläfli Symbol {p,q,r,s} 
Facet type {p,q,r} 
Cell type {p,q} 
Face type {p} 
Face figure {s} 
Edge figure {r,s} 
Vertex figure {q,r,s} 
Dual  Density 

Small stellated 120cell honeycomb  {5/2,5,3,3}  {5/2,5,3}  {5/2,5}  {5/2}  {3}  {3,3}  {5,3,3}  {3,3,5,5/2}  5 
Pentagrammicorder 600cell honeycomb  {3,3,5,5/2}  {3,3,5}  {3,3}  {3}  {5/2}  {5,5/2}  {3,5,5/2}  {5/2,5,3,3}  5 
Order5 icosahedral 120cell honeycomb  {3,5,5/2,5}  {3,5,5/2}  {3,5}  {3}  {5}  {5/2,5}  {5,5/2,5}  {5,5/2,5,3}  10 
Great 120cell honeycomb  {5,5/2,5,3}  {5,5/2,5}  {5,5/2}  {5}  {3}  {5,3}  {5/2,5,3}  {3,5,5/2,5}  10 
There is only one flat regular honeycomb of Euclidean 5space: (previously listed above as tessellations)^{[18]}
There are five flat regular regular honeycombs of hyperbolic 5space, all paracompact: (previously listed above as tessellations)^{[19]}
The hypercubic honeycomb is the only family of regular honeycombs that can tessellate each dimension, five or higher, formed by hypercube facets, four around every ridge.
Name  Schläfli {p_{1}, p_{2}, ..., p_{n−1}} 
Facet type 
Vertex figure 
Dual 

Square tiling  {4,4}  {4}  {4}  Selfdual 
Cubic honeycomb  {4,3,4}  {4,3}  {3,4}  Selfdual 
Tesseractic honeycomb  {4,3^{2},4}  {4,3^{2}}  {3^{2},4}  Selfdual 
5cube honeycomb  {4,3^{3},4}  {4,3^{3}}  {3^{3},4}  Selfdual 
6cube honeycomb  {4,3^{4},4}  {4,3^{4}}  {3^{4},4}  Selfdual 
7cube honeycomb  {4,3^{5},4}  {4,3^{5}}  {3^{5},4}  Selfdual 
8cube honeycomb  {4,3^{6},4}  {4,3^{6}}  {3^{6},4}  Selfdual 
nhypercubic honeycomb  {4,3^{n2},4}  {4,3^{n2}}  {3^{n2},4}  Selfdual 
In E^{5}, there are also the improper cases {4,3,3,4,2}, {2,4,3,3,4}, {3,3,4,3,2}, {2,3,3,4,3}, {3,4,3,3,2}, and {2,3,4,3,3}. In E^{n}, {4,3^{n3},4,2} and {2,4,3^{n3},4} are always improper Euclidean tessellations.
There are 5 regular honeycombs in H^{5}, all paracompact, which include infinite (Euclidean) facets or vertex figures: {3,4,3,3,3}, {3,3,4,3,3}, {3,3,3,4,3}, {3,4,3,3,4}, and {4,3,3,4,3}.
There are no compact regular tessellations of hyperbolic space of dimension 5 or higher and no paracompact regular tessellations in hyperbolic space of dimension 6 or higher.
Name  Schläfli Symbol {p,q,r,s,t} 
Facet type {p,q,r,s} 
4face type {p,q,r} 
Cell type {p,q} 
Face type {p} 
Cell figure {t} 
Face figure {s,t} 
Edge figure {r,s,t} 
Vertex figure {q,r,s,t} 
Dual 

5orthoplex honeycomb  {3,3,3,4,3}  {3,3,3,4}  {3,3,3}  {3,3}  {3}  {3}  {4,3}  {3,4,3}  {3,3,4,3}  {3,4,3,3,3} 
24cell honeycomb honeycomb  {3,4,3,3,3}  {3,4,3,3}  {3,4,3}  {3,4}  {3}  {3}  {3,3}  {3,3,3}  {4,3,3,3}  {3,3,3,4,3} 
16cell honeycomb honeycomb  {3,3,4,3,3}  {3,3,4,3}  {3,3,4}  {3,3}  {3}  {3}  {3,3}  {4,3,3}  {3,4,3,3}  selfdual 
Order4 24cell honeycomb honeycomb  {3,4,3,3,4}  {3,4,3,3}  {3,4,3}  {3,4}  {3}  {4}  {3,4}  {3,3,4}  {4,3,3,4}  {4,3,3,4,3} 
Tesseractic honeycomb honeycomb  {4,3,3,4,3}  {4,3,3,4}  {4,3,3}  {4,3}  {4}  {3}  {4,3}  {3,4,3}  {3,3,4,3}  {3,4,3,3,4} 
Since there are no regular star npolytopes for n >= 5, that could be potential cells or vertex figures, there are no more hyperbolic star honeycombs in H^{n} for n >= 5.
There are no regular compact or paracompact tessellations of hyperbolic space of dimension 6 or higher. However, any Schläfli symbol of the form {p,q,r,s,...} not covered above (p,q,r,s,... natural numbers above 2, or infinity) will form a noncompact tessellation of hyperbolic nspace.
For any natural number n, there are npointed star regular polygonal stars with Schläfli symbols {n/m} for all m such that m < n/2 (strictly speaking {n/m}={n/(nm)}) and m and n are coprime. When m and n are not coprime, the star polygon obtained will be a regular polygon with n/m sides. A new figure is obtained by rotating these regular n/mgons one vertex to the left on the original polygon until the number of vertices rotated equals n/m minus one, and combining these figures. An extreme case of this is where n/m is 2, producing a figure consisting of n/2 straight line segments; this is called a degenerate star polygon.
In other cases where n and m have a common factor, a star polygon for a lower n is obtained, and rotated versions can be combined. These figures are called star figures, improper star polygons or compound polygons. The same notation {n/m} is often used for them, although authorities such as Grünbaum (1994) regard (with some justification) the form k{n} as being more correct, where usually k = m.
A further complication comes when we compound two or more star polygons, as for example two pentagrams, differing by a rotation of 36°, inscribed in a decagon. This is correctly written in the form k{n/m}, as 2{5/2}, rather than the commonly used {10/4}.
Coxeter's extended notation for compounds is of the form c{m,n,...}[d{p,q,...}]e{s,t,...}, indicating that d distinct {p,q,...}'s together cover the vertices of {m,n,...} c times and the facets of {s,t,...} e times. If no regular {m,n,...} exists, the first part of the notation is removed, leaving [d{p,q,...}]e{s,t,...}; the opposite holds if no regular {s,t,...} exists. The dual of c{m,n,...}[d{p,q,...}]e{s,t,...} is e{t,s,...}[d{q,p,...}]c{n,m,...}. If c or e are 1, they may be omitted. For compound polygons, this notation reduces to {nk}[k{n/m}]{nk}: for example, the hexagram may be written thus as {6}[2{3}]{6}.
2{2} 
3{2} 
4{2} 
5{2} 
6{2} 
7{2} 
8{2} 
9{2} 
10{2} 
11{2} 
12{2} 
13{2} 
14{2} 
15{2}  
2{3} 
3{3} 
4{3} 
5{3} 
6{3} 
7{3} 
8{3} 
9{3} 
10{3} 
2{4} 
3{4} 
4{4} 
5{4} 
6{4} 
7{4} 
2{5} 
3{5} 
4{5} 
5{5} 
6{5} 
2{5/2} 
3{5/2} 
4{5/2} 
5{5/2} 
6{5/2} 
2{6} 
3{6} 
4{6} 
5{6}  
2{7} 
3{7} 
4{7} 
2{7/2} 
3{7/2} 
4{7/2} 
2{7/3} 
3{7/3} 
4{7/3} 
2{8} 
3{8} 
2{8/3} 
3{8/3}  
2{9} 
3{9} 
2{9/2} 
3{9/2} 
2{9/4} 
3{9/4} 
2{10} 
3{10} 
2{10/3} 
3{10/3}  
2{11} 
2{11/2} 
2{11/3} 
2{11/4} 
2{11/5} 
2{12} 
2{12/5} 
2{13} 
2{13/2} 
2{13/3} 
2{13/4} 
2{13/5} 
2{13/6}  
2{14} 
2{14/3} 
2{14/5} 
2{15} 
2{15/2} 
2{15/4} 
2{15/7} 
Regular skew polygons also create compounds, seen in the edges of prismatic compound of antiprisms, for instance:
Compound skew squares 
Compound skew hexagons 
Compound skew decagons  
Two {2}#{ }  Three {2}#{ }  Two {3}#{ }  Two {5/3}#{ } 
A regular polyhedron compound can be defined as a compound which, like a regular polyhedron, is vertextransitive, edgetransitive, and facetransitive. With this definition there are 5 regular compounds.
Symmetry  [4,3], O_{h}  [5,3]^{+}, I  [5,3], I_{h}  

Duality  Selfdual  Dual pairs  
Image  
Spherical  
Polyhedra  2 {3,3}  5 {3,3}  10 {3,3}  5 {4,3}  5 {3,4} 
Coxeter  {4,3}[2{3,3}]{3,4}  {5,3}[5{3,3}]{3,5}  2{5,3}[10{3,3}]2{3,5}  2{5,3}[5{4,3}]  [5{3,4}]2{3,5} 
Coxeter's notation for regular compounds is given in the table above, incorporating Schläfli symbols. The material inside the square brackets, [d{p,q}], denotes the components of the compound: d separate {p,q}'s. The material before the square brackets denotes the vertex arrangement of the compound: c{m,n}[d{p,q}] is a compound of d {p,q}'s sharing the vertices of an {m,n} counted c times. The material after the square brackets denotes the facet arrangement of the compound: [d{p,q}]e{s,t} is a compound of d {p,q}'s sharing the faces of {s,t} counted e times. These may be combined: thus c{m,n}[d{p,q}]e{s,t} is a compound of d {p,q}'s sharing the vertices of {m,n} counted c times and the faces of {s,t} counted e times. This notation can be generalised to compounds in any number of dimensions.^{[21]}
There are eighteen twoparameter families of regular compound tessellations of the Euclidean plane. In the hyperbolic plane, five oneparameter families and seventeen isolated cases are known, but the completeness of this listing has not yet been proven.
The Euclidean and hyperbolic compound families 2 {p,p} (4 p p an integer) are analogous to the spherical stella octangula, 2 {3,3}.
Selfdual  Duals  Selfdual  

2 {4,4}  2 {6,3}  2 {3,6}  2 {∞,∞} 
{{4,4}} or a{4,4} or {4,4}[2{4,4}]{4,4} + or 
[2{6,3}]{3,6}  a{6,3} or {6,3}[2{3,6}] + or 
{{∞,∞}} or a{∞,∞} or {4,∞}[2{∞,∞}]{∞,4} + or 
3 {6,3}  3 {3,6}  3 {∞,∞}  
2{3,6}[3{6,3}]{6,3}  {3,6}[3{3,6}]2{6,3} + + 
+ + 
Coxeter lists 32 regular compounds of regular 4polytopes in his book Regular Polytopes.^{[22]}McMullen adds six in his paper New Regular Compounds of 4Polytopes.^{[23]} In the following tables, the superscript (var) indicates that the labeled compounds are distinct from the other compounds with the same symbols.
Compound  Constituent  Symmetry  Vertex arrangement  Cell arrangement 

120 {3,3,3}  5cell  [5,3,3], order 14400^{[22]}  {5,3,3}  {3,3,5} 
120 {3,3,3}^{(var)}  5cell  order 1200^{[23]}  {5,3,3}  {3,3,5} 
720 {3,3,3}  5cell  [5,3,3], order 14400^{[23]}  6{5,3,3}  6{3,3,5} 
5 {3,4,3}  24cell  [5,3,3], order 14400^{[22]}  {3,3,5}  {5,3,3} 
Compound 1  Compound 2  Symmetry  Vertex arrangement (1)  Cell arrangement (1)  Vertex arrangement (2)  Cell arrangement (2) 

3 {3,3,4}^{[24]}  3 {4,3,3}  [3,4,3], order 1152^{[22]}  {3,4,3}  2{3,4,3}  2{3,4,3}  {3,4,3} 
15 {3,3,4}  15 {4,3,3}  [5,3,3], order 14400^{[22]}  {3,3,5}  2{5,3,3}  2{3,3,5}  {5,3,3} 
75 {3,3,4}  75 {4,3,3}  [5,3,3], order 14400^{[22]}  5{3,3,5}  10{5,3,3}  10{3,3,5}  5{5,3,3} 
75 {3,3,4}  75 {4,3,3}  [5,3,3], order 14400^{[22]}  {5,3,3}  2{3,3,5}  2{5,3,3}  {3,3,5} 
75 {3,3,4}  75 {4,3,3}  order 600^{[23]}  {5,3,3}  2{3,3,5}  2{5,3,3}  {3,3,5} 
300 {3,3,4}  300 {4,3,3}  [5,3,3]^{+}, order 7200^{[22]}  4{5,3,3}  8{3,3,5}  8{5,3,3}  4{3,3,5} 
600 {3,3,4}  600 {4,3,3}  [5,3,3], order 14400^{[22]}  8{5,3,3}  16{3,3,5}  16{5,3,3}  8{3,3,5} 
25 {3,4,3}  25 {3,4,3}  [5,3,3], order 14400^{[22]}  {5,3,3}  5{5,3,3}  5{3,3,5}  {3,3,5} 
There are two different compounds of 75 tesseracts: one shares the vertices of a 120cell, while the other shares the vertices of a 600cell. It immediately follows therefore that the corresponding dual compounds of 75 16cells are also different.
Compound  Symmetry  Vertex arrangement  Cell arrangement 

5 {5,5/2,5}  [5,3,3]^{+}, order 7200^{[22]}  {5,3,3}  {3,3,5} 
10 {5,5/2,5}  [5,3,3], order 14400^{[22]}  2{5,3,3}  2{3,3,5} 
5 {5/2,5,5/2}  [5,3,3]^{+}, order 7200^{[22]}  {5,3,3}  {3,3,5} 
10 {5/2,5,5/2}  [5,3,3], order 14400^{[22]}  2{5,3,3}  2{3,3,5} 
Compound 1  Compound 2  Symmetry  Vertex arrangement (1)  Cell arrangement (1)  Vertex arrangement (2)  Cell arrangement (2) 

5 {3,5,5/2}  5 {5/2,5,3}  [5,3,3]^{+}, order 7200^{[22]}  {5,3,3}  {3,3,5}  {5,3,3}  {3,3,5} 
10 {3,5,5/2}  10 {5/2,5,3}  [5,3,3], order 14400^{[22]}  2{5,3,3}  2{3,3,5}  2{5,3,3}  2{3,3,5} 
5 {5,5/2,3}  5 {3,5/2,5}  [5,3,3]^{+}, order 7200^{[22]}  {5,3,3}  {3,3,5}  {5,3,3}  {3,3,5} 
10 {5,5/2,3}  10 {3,5/2,5}  [5,3,3], order 14400^{[22]}  2{5,3,3}  2{3,3,5}  2{5,3,3}  2{3,3,5} 
5 {5/2,3,5}  5 {5,3,5/2}  [5,3,3]^{+}, order 7200^{[22]}  {5,3,3}  {3,3,5}  {5,3,3}  {3,3,5} 
10 {5/2,3,5}  10 {5,3,5/2}  [5,3,3], order 14400^{[22]}  2{5,3,3}  2{3,3,5}  2{5,3,3}  2{3,3,5} 
There are also fourteen partially regular compounds, that are either vertextransitive or celltransitive but not both. The seven vertextransitive partially regular compounds are the duals of the seven celltransitive partially regular compounds.
Compound 1 Vertextransitive 
Compound 2 Celltransitive 
Symmetry 

2 16cells^{[25]}  2 tesseracts  [4,3,3], order 384^{[22]} 
25 24cell^{(var)}  25 24cell^{(var)}  order 600^{[23]} 
100 24cell  100 24cell  [5,3,3]^{+}, order 7200^{[22]} 
200 24cell  200 24cell  [5,3,3], order 14400^{[22]} 
5 600cell  5 120cell  [5,3,3]^{+}, order 7200^{[22]} 
10 600cell  10 120cell  [5,3,3], order 14400^{[22]} 
Compound 1 Vertextransitive 
Compound 2 Celltransitive 
Symmetry 

5 {3,3,5/2}  5 {5/2,3,3}  [5,3,3]^{+}, order 7200^{[22]} 
10 {3,3,5/2}  10 {5/2,3,3}  [5,3,3], order 14400^{[22]} 
Although the 5cell and 24cell are both selfdual, their dual compounds (the compound of two 5cells and compound of two 24cells) are not considered to be regular, unlike the compound of two tetrahedra and the various dual polygon compounds, because they are neither vertexregular nor cellregular: they are not facetings or stellations of any regular 4polytope.
The only regular Euclidean compound honeycombs are an infinite family of compounds of cubic honeycombs, all sharing vertices and faces with another cubic honeycomb. This compound can have any number of cubic honeycombs. The Coxeter notation is {4,3,4}[d{4,3,4}]{4,3,4}.
There are no regular compounds in five or six dimensions. There are three known sevendimensional compounds (16, 240, or 480 7simplices), and six known eightdimensional ones (16, 240, or 480 8cubes or 8orthoplexes). There is also one compound of nsimplices in ndimensional space provided that n is one less than a power of two, and also two compounds (one of ncubes and a dual one of northoplexes) in ndimensional space if n is a power of two.
The Coxeter notation for these compounds are (using ?^{n} = {3^{n1}}, ?^{n} = {3^{n2},4}, ?_{n} = {4,3^{n2}}:
The general cases (where n = 2^{k} and d = 2^{2k  k  1}, k = 2, 3, 4, ...):
A known family of regular Euclidean compound honeycombs in five or more dimensions is an infinite family of compounds of hypercubic honeycombs, all sharing vertices and faces with another hypercubic honeycomb. This compound can have any number of hypercubic honeycombs. The Coxeter notation is ?_{n}[d?_{n}]?_{n} where ?_{n} = {?} when n = 2 and {4,3^{n3},4} when n >= 3.
The abstract polytopes arose out of an attempt to study polytopes apart from the geometrical space they are embedded in. They include the tessellations of spherical, Euclidean and hyperbolic space, tessellations of other manifolds, and many other objects that do not have a welldefined topology, but instead may be characterised by their "local" topology. There are infinitely many in every dimension. See this atlas for a sample. Some notable examples of abstract regular polytopes that do not appear elsewhere in this list are the 11cell, {3,5,3}, and the 57cell, {5,3,5}, which have regular projective polyhedra as cells and vertex figures.
The elements of an abstract polyhedron are its body (the maximal element), its faces, edges, vertices and the null polytope or empty set. These abstract elements can be mapped into ordinary space or realised as geometrical figures. Some abstract polyhedra have wellformed or faithful realisations, others do not. A flag is a connected set of elements of each dimension  for a polyhedron that is the body, a face, an edge of the face, a vertex of the edge, and the null polytope. An abstract polytope is said to be regular if its combinatorial symmetries are transitive on its flags  that is to say, that any flag can be mapped onto any other under a symmetry of the polyhedron. Abstract regular polytopes remain an active area of research.
Five such regular abstract polyhedra, which can not be realised faithfully, were identified by H. S. M. Coxeter in his book Regular Polytopes (1977) and again by J. M. Wills in his paper "The combinatorially regular polyhedra of index 2" (1987).^{[26]} They are all topologically equivalent to toroids. Their construction, by arranging n faces around each vertex, can be repeated indefinitely as tilings of the hyperbolic plane. In the diagrams below, the hyperbolic tiling images have colors corresponding to those of the polyhedra images.
Polyhedron  Medial rhombic triacontahedron 
Dodecadodecahedron 
Medial triambic icosahedron 
Ditrigonal dodecadodecahedron 
Excavated dodecahedron 

Vertex figure  {5}, {5/2} 
(5.5/2)^{2} 
{5}, {5/2} 
(5.5/3)^{3} 

Faces  30 rhombi 
12 pentagons 12 pentagrams 
20 hexagons 
12 pentagons 12 pentagrams 
20 hexagrams 
Tiling  {4, 5} 
{5, 4} 
{6, 5} 
{5, 6} 
{6, 6} 
?  6  6  16  16  20 
These occur as dual pairs as follows:
Fundamental convex regular and uniform honeycombs in dimensions 29
 

Space  Family  / /  
E^{2}  Uniform tiling  {3^{[3]}}  δ_{3}  hδ_{3}  qδ_{3}  Hexagonal 
E^{3}  Uniform convex honeycomb  {3^{[4]}}  δ_{4}  hδ_{4}  qδ_{4}  
E^{4}  Uniform 4honeycomb  {3^{[5]}}  δ_{5}  hδ_{5}  qδ_{5}  24cell honeycomb 
E^{5}  Uniform 5honeycomb  {3^{[6]}}  δ_{6}  hδ_{6}  qδ_{6}  
E^{6}  Uniform 6honeycomb  {3^{[7]}}  δ_{7}  hδ_{7}  qδ_{7}  2_{22} 
E^{7}  Uniform 7honeycomb  {3^{[8]}}  δ_{8}  hδ_{8}  qδ_{8}  1_{33} o 3_{31} 
E^{8}  Uniform 8honeycomb  {3^{[9]}}  δ_{9}  hδ_{9}  qδ_{9}  1_{52} o 2_{51} o 5_{21} 
E^{9}  Uniform 9honeycomb  {3^{[10]}}  δ_{10}  hδ_{10}  qδ_{10}  
E^{n1}  Uniform (n1)honeycomb  {3^{[n]}}  δ_{n}  hδ_{n}  qδ_{n}  1_{k2} o 2_{k1} o k_{21} 