This page lists the regular polytopes in Euclidean, spherical and hyperbolic spaces.
The Schläfli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each.
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 one lower dimensional 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 7 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.
Contents |
Overview
This table shows a summary of regular polytope counts by dimension
| Dimension | Convex | Nonconvex | Convex Euclidean tessellations |
Convex hyperbolic tessellations |
Nonconvex hyperbolic tessellations |
Abstract Polytopes |
|---|---|---|---|---|---|---|
| 1 | 1 line segment | 0 | 0 | 0 | 0 | 1 |
| 2 | ∞ polygons | ∞ star polygons | 1 | 1 | 0 | ∞ |
| 3 | 5 Platonic solids | 4 Kepler-Poinsot solids | 3 tilings | ∞ | ∞ | ∞ |
| 4 | 6 convex polychora | 10 Schläfli-Hess polychora | 1 honeycomb | 4 | 0 | ∞ |
| 5 | 3 convex 5-polytopes | 0 nonconvex 5-polytopes | 3 tessellations | 5 | 4 | ∞ |
| 6+ | 3 | 0 | 1 | 0 | 0 | ∞ |
Tessellations
The classical convex polytopes may be considered tessellations, or tilings, of spherical space. Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.
One-dimensional regular polytope
There is only one polytope in 1 dimensions, whose boundaries are the two endpoints of a line segment, represented by the empty Schläfli symbol {}.
Two-dimensional regular polytopes
The two dimensional polytopes are called polygons. Regular polygons are equilateral and cyclic. A p-gonal 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 complete.
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.
Convex
The Schläfli symbol {p} represents a regular p-gon.
| Name | Triangle (2-simplex) |
Square (2-orthoplex) (2-cube) |
Pentagon | Hexagon | Heptagon | Octagon | |
|---|---|---|---|---|---|---|---|
| Schläfli | {3} | {4} | {5} | {6} | {7} | {8} | |
| Image |  |
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| Name | Enneagon | Decagon | Hendecagon | Dodecagon | Triskaidecagon | Tetradecagon | |
| Schläfli | {9} | {10} | {11} | {12} | {13} | {14} | |
| Image |  |
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| Name | Pentadecagon | Hexadecagon | Heptadecagon | Octadecagon | Enneadecagon | Icosagon | ...n-gon |
| Schläfli | {15} | {16} | {17} | {18} | {19} | {20} | {n} |
| Image |  |
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Degenerate (Spherical)
The regular henagon {1} and regular digon {2} can be considered degenerate regular polygons. They can exist nondegenerately in non-Euclidean spaces like on the surface of a sphere or torus.
| Name | Henagon | Digon |
|---|---|---|
| Schläfli | {1} | {2} |
| Image |  |
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Non-convex
There exist infinitely many non-convex regular 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 n-pointed non-convex regular polygonal stars with Schläfli symbols {n/m} for all m such that m < n/2 (strictly speaking {n/m}={n/(n-m)}) and m and n are coprime.
| Name | Pentagram | Heptagrams | Octagram | Enneagrams | Decagram | ...n-agrams | ||
|---|---|---|---|---|---|---|---|---|
| Schläfli | {5/2} | {7/2} | {7/3} | {8/3} | {9/2} | {9/4} | {10/3} | {n/m} |
| Image |  |
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Tessellation
There is one tessellation of the line, giving one polytope, the (two-dimensional) apeirogon. This has infinitely many vertices and edges. Its Schläfli symbol is {∞}.
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Three-dimensional regular polytopes
In three dimensions, the regular polytopes are called polyhedra:
A regular polyhedron with Schläfli symbol {p,q} 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:
- 1 / p + 1 / q > 1 / 2 : Polyhedron (existing in Euclidean 3-space)
- 1 / p + 1 / q = 1 / 2 : Euclidean plane tiling
- 1 / p + 1 / q < 1 / 2 : Hyperbolic plane tiling
By enumerating the permutations, we find 5 convex forms, 4 nonconvex forms and 3 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.
Convex
The convex regular polyhedra are called the 5 Platonic solids. The vertex figure is given with each vertex count. All these polyhedra have a Euler characteristic (χ) of 2.
| Name | Schläfli {p,q} |
Image (transparent) |
Image (solid) |
Image (sphere) |
Faces {p} |
Edges | Vertices {q} |
Symmetry | Dual |
|---|---|---|---|---|---|---|---|---|---|
| Tetrahedron (3-simplex) (Pyramid) |
{3,3} |  |
 |
 |
4 {3} |
6 | 4 {3} |
Td | (self) |
| Cube (3-cube) (Hexahedron) |
{4,3} |  |
 |
 |
6 {4} |
12 | 8 {3} |
Oh | Octahedron |
| Octahedron (3-orthoplex) |
{3,4} |  |
 |
 |
8 {3} |
12 | 6 {4} |
Oh | Cube |
| Dodecahedron | {5,3} |  |
 |
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12 {5} |
30 | 20 {3}2 |
Ih | Icosahedron |
| Icosahedron | {3,5} |  |
 |
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20 {3} |
30 | 12 {5} |
Ih | Dodecahedron |
Degenerate (spherical)
In spherical geometry, the hosohedron {2,n} and dihedron {n,2} can be considered regular polyhedra (tilings of the sphere).
Some include:
| Name | Schläfli {p,q} |
Image (sphere) |
Faces {p} |
Edges | Vertices {q} |
Symmetry | Dual |
|---|---|---|---|---|---|---|---|
| Hengonal dihedron | {1,2} |  |
2 {1} |
1 | 1 {2} |
C1v (*22) |
Hengonal hosohedron |
| Hengonal hosohedron | {2,1} |  |
1 {2} |
1 | 2 {1} |
C1v (*22) |
Hengonal dihedron |
| Digonal dihedron Digonal hosohedron |
{2,2} |  |
2 {2} |
2 | 2 {2} |
D2h (*222) |
Self |
| Trigonal hosohedron | {2,3} |  |
3 {2} |
3 | 2 {3} |
D3h (*322) |
Trigonal dihedron |
| Trigonal dihedron | {3,2} |  |
2 {3} |
3 | 3 {2} |
D3h (*322) |
Trigonal hosohedron |
| Hexagonal hosohedron | {2,6} |  |
6 {2} |
6 | 2 {6} |
D3h (*622) |
Hexagonal dihedron |
Non-convex
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 nonconvex 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 | Schläfli {p,q} |
Image (transparent) |
Image (solid) |
Image (sphere) |
Faces {p} |
Edges | Vertices {q} |
χ | Density | Symmetry | Dual |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Small stellated dodecahedron | {5/2,5} |  |
 |
 |
12 {5/2} |
30 | 12 {5} |
-6 | 3 | Ih | Great dodecahedron |
| Great dodecahedron | {5,5/2} |  |
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12 {5} |
30 | 12 {5/2} |
-6 | 3 | Ih | Small stellated dodecahedron |
| Great stellated dodecahedron | {5/2,3} |  |
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12 {5/2} |
30 | 20 {3} |
2 | 7 | Ih | Great icosahedron |
| Great icosahedron | {3,5/2} |  |
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20 {3} |
30 | 12 {5/2} |
2 | 7 | Ih | Great stellated dodecahedron |
Tessellations
Euclidean tilings
There are three regular tessellations of the plane. All three have a Euler characteristic (χ) of 0.
| Name | Schläfli {p,q} | Image | Face type {p} |
Vertex figure {q} |
Symmetry | Dual |
|---|---|---|---|---|---|---|
| Square tiling (Quadrille) |
{4,4} |  |
{4} | {4} | *442 (p4m) |
(self) |
| Triangular tiling (Deltille) |
{3,6} |  |
{3} | {6} | *632 (p6m) |
Hexagonal tiling |
| Hexagonal tiling (Hextille) |
{6,3} |  |
{6} | {3} | *632 (p6m) |
Triangular tiling |
There is one degenerate regular tiling, {∞,2}, made from two apeirogons, each filling half the plane. This tiling is related to a 2-faced dihedron, {p,2}, on the sphere.
Euclidean star-tilings
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.
Hyperbolic tilings
Tessellations of hyperbolic 2-space can be called a hyperbolic tiling. There are infinitely many regular tilings in H2. As stated above, every positive integer pairs {p,q} such that 1/p + 1/q < 1/2 is a hyperbolic tiling.
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,...
There are a number of different ways to display the hyperbolic plane, including the Poincaré disc model below 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 equal-sized and only appear to get smaller near the edges due to the projection applied, very similar to the effect of a camera fisheye lens.
A sampling:
| p \ q | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|
| 3 | (tetrahedron) {3,3} |
(octahedron) {3,4} |
(icosahedron) {3,5} |
(deltille) |  {3,7} |
 {3,8} |
 {3,9} |
| 4 | (cube) {4,3} |
(quadrille) {4,4} |
 {4,5} |
 {4,6} |
 {4,7} |
 {4,8} |
{4,9} |
| 5 | (dodecahedron) {5,3} |
 {5,4} |
 {5,5} |
 {5,6} |
 {5,7} |
{5,8} | {5,9} |
| 6 | (hextille) {6,3} |
 {6,4} |
 {6,5} |
 {6,6} |
{6,7} | {6,8} | {6,9} |
| 7 |  {7,3} |
 {7,4} |
 {7,5} |
{7,6} | {7,7} | {7,8} | {7,9} |
| 8 |  {8,3} |
 {8,4} |
{8,5} | {8,6} | {8,7} | {8,8} | {8,9} |
| 9 |  {9,3} |
{9,4} | {9,5} | {9,6} | {9,7} | {9,8} | {9,9} |
Four-dimensional regular polytopes
Regular polychora with Schläfli symbol {p,q,r} have cells of type {p,q}, faces of type {p}, edge figures {r}, and vertex figures {q,r}.
- A vertex figure (of a polychoron) is a polyhedron, seen by the arrangement of neighboring vertices around a given vertex. For regular polychora, this vertex figure is a regular polyhedron.
- An edge figure is a polygon, seen by the arrangement of faces around an edge. For regular polychora, this edge figure will always be a regular polygon.
The existence of a regular polychoron {p,q,r} is constrained by the existence of the regular polyhedra {p,q},{q,r}.
Each will exist in a space dependent upon this expression:
- > 0 : Hyperspherical 3-space honeycomb or 4-space polychoron
- = 0 : Euclidean 3-space honeycomb
- < 0 : 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.
The Euler characteristic χ for polychora is χ = V + F − E − C and is zero for all forms.
Convex
The 6 convex polychora are shown in the table below. All these polychora have a Euler characteristic (χ) of 0.
| Name |
Schläfli {p,q,r} |
Cells {p,q} |
Faces {p} |
Edges {r} |
Vertices {q,r} |
Dual {r,q,p} |
|---|---|---|---|---|---|---|
| 5-cell (4-simplex) (Pentachoron) |
{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 |
| 5-cell | 8-cell | 16-cell | 24-cell | 120-cell | 600-cell |
|---|---|---|---|---|---|
| {3,3,3} | {4,3,3} | {3,3,4} | {3,4,3} | {5,3,3} | {3,3,5} |
| Wireframe orthographic projections | |||||
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| Solid orthographic projections (cell-centered) | |||||
tetrahedral envelope |
cubic envelope |
octahedral envelope |
 cuboctahedral envelope |
 truncated rhombic triacontahedron envelope |
 pentakis dodecahedral envelope |
| Wireframe Schlegel diagrams (Perspective projection) | |||||
 (Cell-centered) |
 (Cell-centered) |
 (Cell-centered) |
 (Cell-centered) |
 (Cell-centered) |
 (Vertex-centered) |
| Wireframe stereographic projections (Hyperspherical) | |||||
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Degenerate (Spherical)
Ditopes and hosotopes exist as regular tessellations of the 3-sphere.
Regular ditopes (2 facets) include: {3,3,2}, {3,4,2}, {4,3,2}, {5,3,2}, {3,5,2}, {p,2,2}, and their hosotope duals (2 vertices): {2,3,3}, {2,4,3}, {2,3,4}, {2,3,5}, {2,5,3}.
Non-convex
There are ten regular star polychora, which can be called Schläfli-Hess polychora and their vertices are based on the convex 120-cell {5,3,3} and 600-cell {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 zero-hole toruses: F+V-E=2). Edmund Hess (1843-1903) completed the full list of ten in his 1883 German book Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder[1].
There are 4 unique edge arrangements and 7 unique face arrangements from these 10 nonconvex polychora, shown as orthogonal projections:
| Name |
Wireframe | Solid | Schläfli {p, q,r} Coxeter-Dynkin |
Cells {p, q} |
Faces {p} |
Edges {r} |
Vertices {q, r} |
Density | χ | Symmetry group | Dual {r, q,p} |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Icosahedral 120-cell |  |
 |
{3,5,5/2}     |
120 {3,5} |
1200 {3}  |
720 {5/2}  |
120 {5,5/2} |
4 | 480 | H4 | Small stellated 120-cell |
| Small stellated 120-cell |  |
 |
{5/2,5,3} |
120 {5/2,5} |
720 {5/2} |
1200 {3} |
120 {5,3} |
4 | -480 | H4 | Icosahedral 120-cell |
| Great 120-cell | |
 |
{5,5/2,5} |
120 {5,5/2} |
720 {5}  |
720 {5} |
120 {5/2,5} |
6 | 0 | H4 | Self-dual |
| Grand 120-cell | |
 |
{5,3,5/2} |
120 {5,3} |
720 {5} |
720 {5/2} |
120 {3,5/2} |
20 | 0 | H4 | Great stellated 120-cell |
| Great stellated 120-cell |  |
 |
{5/2,3,5} |
120 {5/2,3} |
720 {5/2} |
720 {5} |
120 {3,5} |
20 | 0 | H4 | Grand 120-cell |
| Grand stellated 120-cell | |
 |
{5/2,5,5/2} |
120 {5/2,5} |
720 {5/2} |
720 {5/2} |
120 {5,5/2} |
66 | 0 | H4 | Self-dual |
| Great grand 120-cell | |
 |
{5,5/2,3} |
120 {5,5/2} |
720 {5} |
1200 {3} |
120 {5/2,3} |
76 | -480 | H4 | Great icosahedral 120-cell |
| Great icosahedral 120-cell | |
 |
{3,5/2,5} |
120 {3,5/2} |
1200 {3} |
720 {5} |
120 {5/2,5} |
76 | 480 | H4 | Great grand 120-cell |
| Grand 600-cell | |
 |
{3,3,5/2} |
600 {3,3} |
1200 {3} |
720 {5/2} |
120 {3,5/2} |
191 | 0 | H4 | Great grand stellated 120-cell |
| Great grand stellated 120-cell |  |
 |
{5/2,3,3} |
120 {5/2,3} |
720 {5/2} |
1200 {3} |
600 {3,3} |
191 | 0 | H4 | Grand 600-cell |
There are 4 failed potential nonconvex regular polychora 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.
Tessellations of Euclidean 3-space
There is only one regular tessellation of 3-space (honeycombs):
| Name | Schläfli symbol {p,q,r} |
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 | Self-dual |
Tessellations of hyperbolic 3-space
Tessellations of hyperbolic 3-space can be called hyperbolic honeycombs. There are 4 regular honeycombs in H3:
| Name | Schläfli Symbol {p,q,r} |
Cell type {p,q} |
Face type {p} |
Edge figure {r} |
Vertex figure {q,r} |
χ | Dual |
|---|---|---|---|---|---|---|---|
| Order-3 icosahedral honeycomb | {3,5,3} | {3,5} | {3} | {3} | {5,3} | 0 | Self-dual |
| Order-5 cubic honeycomb | {4,3,5} | {4,3} | {4} | {5} | {3,5} | 0 | {5,3,4} |
| Order-4 dodecahedral honeycomb | {5,3,4} | {5,3} | {5} | {4} | {3,4} | 0 | {4,3,5} |
| Order-5 dodecahedral honeycomb | {5,3,5} | {5,3} | {5} | {5} | {3,5} | 0 | Self-dual |
Here are some projected images: The first shows the perspective from the center of the disc in a Beltrami-Klein model, and the second and third from the outside with a Poincaré disk model.
 {5,3,4} (8 dodecahedra at a vertex) |
 {4,3,5} (20 cubes at a vertex) |
 {3,5,3} (12 icosahedra at a vertex) |
There are also 11 H3 honeycombs which have 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}, {6,3,6}.
Five-dimensional regular polytopes and higher
In five dimensions, a regular polytope can be named as {p,q,r,s} where {p,q,r} is the hypercell (or teron) type, {p,q} is the cell type, {p} is the face type, and {s} is the face figure, {r,s} is the edge figure, and {q,r,s} is the vertex figure.
A 5-polytope has been called a polyteron, and if infinite (i.e. a honeycomb).
- A vertex figure (of a 5-polytope) is a polychoron, seen by the arrangement of neighboring vertices to each vertex.
- An edge figure (of a 5-polytope) is a polyhedron, seen by the arrangement of faces around each edge.
- A face figure (of a 5-polytope) is a polygon, seen by the arrangement of cells around each face.
A regular polytope {p,q,r,s} exists only if {p,q,r} and {q,r,s} are regular polychora.
The space it fits in is based on the expression:
- < 1 : Spherical 4-space tessellation or 5-space polytope
- = 1 : Euclidean 4-space tessellation
- > 1 : hyperbolic 4-space tessellation
Enumeration of these constraints produce 3 convex polytopes, zero nonconvex polytopes, 3 4-space tessellations, and 5 hyperbolic 4-space tessellations. There are no non-convex regular polytopes in five dimensions or higher.
Convex
In dimensions 5 and higher, there are only three kinds of convex regular polytopes. [Coxeter, Regular Polytopes, Table I: Regular polytopes, (iii) The three regular polytopes in n dimensions (n>=5), pp. 294-295]
| Name | Schläfli Symbol {p1,p2,...,pn-1} |
Facet type |
Vertex figure |
Dual |
|---|---|---|---|---|
| n-simplex | {3,3,3,...,3} | {3,3,...,3} | {3,3,...,3} | Self-dual |
| n-cube | {4,3,3,...,3} | {4,3,...,3} | {3,3,...,3} | n-orthoplex |
| n-orthoplex | {3,...,3,3,4} | {3,...,3,3} | {3,...,3,4} | n-cube |
5-dimensions
| Name | Schläfli Symbol {p,q,r,s} |
Facets {p,q,r} |
Cells {p,q} |
Faces {p} |
Edges | Vertices | Face figure {s} |
Edge figure {r,s} |
Vertex figure {q,r,s} |
|---|---|---|---|---|---|---|---|---|---|
| 5-simplex | {3,3,3,3} | 6 {3,3,3} |
15 {3,3} |
20 {3} |
15 | 6 | {3} | {3,3} | {3,3,3} |
| 5-cube | {4,3,3,3} | 10 {4,3,3} |
40 {4,3} |
80 {4} |
80 | 32 | {3} | {3,3} | {3,3,3} |
| 5-orthoplex | {3,3,3,4} | 32 {3,3,3} |
80 {3,3} |
80 {3} |
40 | 10 | {4} | {3,4} | {3,3,4} |
 5-simplex |
 5-cube |
 5-orthoplex |
6-dimensions
| Name | Schläfli symbol |
Vertices | Edges | Faces | Cells | 4-faces | 5-faces | χ |
|---|---|---|---|---|---|---|---|---|
| 6-simplex | {3,3,3,3,3} | 7 | 21 | 35 | 35 | 21 | 7 | 0 |
| 6-cube | {4,3,3,3,3} | 64 | 192 | 240 | 160 | 60 | 12 | 0 |
| 6-orthoplex | {3,3,3,3,4} | 12 | 60 | 160 | 240 | 192 | 64 | 0 |
 6-simplex |
 6-cube |
 6-orthoplex |
7-dimensions
| Name | Schläfli symbol |
Vertices | Edges | Faces | Cells | 4-faces | 5-faces | 6-faces | χ |
|---|---|---|---|---|---|---|---|---|---|
| 7-simplex | {3,3,3,3,3,3} | 8 | 28 | 56 | 70 | 56 | 28 | 8 | 2 |
| 7-cube | {4,3,3,3,3,3} | 128 | 448 | 672 | 560 | 280 | 84 | 14 | 2 |
| 7-orthoplex | {3,3,3,3,3,4} | 14 | 84 | 280 | 560 | 672 | 448 | 128 | 2 |
 7-simplex |
 7-cube |
 7-orthoplex |
8-dimensions
| Name | Schläfli symbol |
Vertices | Edges | Faces | Cells | 4-faces | 5-faces | 6-faces | 7-faces | χ |
|---|---|---|---|---|---|---|---|---|---|---|
| 8-simplex | {3,3,3,3,3,3,3} | 9 | 36 | 84 | 126 | 126 | 84 | 36 | 9 | 0 |
| 8-cube | {4,3,3,3,3,3,3} | 256 | 1024 | 1792 | 1792 | 1120 | 448 | 112 | 16 | 0 |
| 8-orthoplex | {3,3,3,3,3,3,4} | 16 | 112 | 448 | 1120 | 1792 | 1792 | 1024 | 256 | 0 |
 8-simplex |
 8-cube |
 8-orthoplex |
9-dimensions
| Name | Schläfli symbol |
Vertices | Edges | Faces | Cells | 4-faces | 5-faces | 6-faces | 7-faces | 8-faces | χ |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 9-simplex | {38} | 10 | 45 | 120 | 210 | 252 | 210 | 120 | 45 | 10 | 2 |
| 9-cube | {4,37} | 512 | 2304 | 4608 | 5376 | 4032 | 2016 | 672 | 144 | 18 | 2 |
| 9-orthoplex | {37,4} | 18 | 144 | 672 | 2016 | 4032 | 5376 | 4608 | 2304 | 512 | 2 |
9-simplex |
 9-cube |
9-orthoplex |
10-dimensions
| Name | Schläfli symbol |
Vertices | Edges | Faces | Cells | 4-faces | 5-faces | 6-faces | 7-faces | 8-faces | 9-faces | χ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 10-simplex | {38} | 11 | 55 | 165 | 330 | 462 | 462 | 330 | 165 | 55 | 11 | 0 |
| 10-cube | {4,38} | 1024 | 5120 | 11520 | 15360 | 13440 | 8064 | 3360 | 960 | 180 | 20 | 0 |
| 10-orthoplex | {38,4} | 20 | 180 | 960 | 3360 | 8064 | 13440 | 15360 | 11520 | 5120 | 1024 | 0 |
 10-simplex |
 10-cube |
 10-orthoplex |
...
Nonconvex
There are no non-convex regular polytopes in five dimensions or higher.
Tessellations of Euclidean 4-space
There are three kinds of infinite regular tessellations (honeycombs) that can tessellate four dimensional 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} | Self-dual |
| Hexadecachoric honeycomb | {3,3,4,3} | {3,3,4} | {3,3} | {3} | {3} | {4,3} | {3,4,3} | {3,4,3,3} |
| Icositetrachoric 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} (Hexadecachoronic honeycomb) |
 Projected portion of {3,4,3,3} (Icositetrachoronic honeycomb) |
The hypercube 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 {p1, p2, ..., pn−1} |
Facet type |
Vertex figure |
Dual |
|---|---|---|---|---|
| Square tiling | {4,4} | {4} | {4} | Self-dual |
| Cubic honeycomb | {4,3,4} | {4,3} | {3,4} | Self-dual |
| Tesseractic honeycomb | {4,32,4} | {4,32} | {32,4} | Self-dual |
| Penteractic honeycomb | {4,33,4} | {4,33} | {33,4} | Self-dual |
| Hexeractic honeycomb | {4,34,4} | {4,34} | {34,4} | Self-dual |
| Hepteractic honeycomb | {4,35,4} | {4,35} | {35,4} | Self-dual |
| Octeractic honeycomb | {4,36,4} | {4,36} | {36,4} | Self-dual |
| n-hypercube honeycomb | {4,3n-2,4} | {4,3n-2} | {3n-2,4} | Self-dual |
Tessellations of hyperbolic 4-space
There are five kinds of convex regular honeycombs and four kinds of star-honeycombs in H4 space. [Coxeter, The Beauty of Geometry: Twelve Essays, Chapter 10: Regular honeycombs in hyperbolic space, Summary tables IV p213]
Five convex regular honeycombs in H4:
| 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 |
|---|---|---|---|---|---|---|---|---|
| Order-5 pentachoric honeycomb | {3,3,3,5} | {3,3,3} | {3,3} | {3} | {5} | {3,5} | {3,3,5} | {5,3,3,3} |
| Order-3 hecatonicosachoric honeycomb | {5,3,3,3} | {5,3,3} | {5,3} | {5} | {3} | {3,3} | {3,3,3} | {3,3,3,5} |
| Order-5 tesseractic honeycomb | {4,3,3,5} | {4,3,3} | {4,3} | {4} | {5} | {3,5} | {3,3,5} | {5,3,3,4} |
| Order-4 hecatonicosachoric honeycomb | {5,3,3,4} | {5,3,3} | {5,3} | {5} | {4} | {3,4} | {3,3,4} | {4,3,3,5} |
| Order-5 hecatonicosachoric honeycomb | {5,3,3,5} | {5,3,3} | {5,3} | {5} | {5} | {3,5} | {3,3,5} | Self-dual |
There are four regular star-honeycombs in H4 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 |
|---|---|---|---|---|---|---|---|---|
| Order-3 small stellated hecatonicosachoric honeycomb | {5/2,5,3,3} | {5/2,5,3} | {5/2,5} | {5} | {5} | {3,3} | {5,3,3} | {3,3,5,5/2} |
| Pentagrammic-order hexacosichoric 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} |
| Order-5 icosahedral hecatonicosachoric 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} |
| Order-3 great hecatonicosachoric 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} |
There are also 2 H4 honeycombs with infinite (Euclidean) facets or vertex figures: {3,4,3,4}, {4,3,4,3}
There are no finite-faceted regular tessellations of hyperbolic space of dimension 5 or higher.
There are 5 regular honeycombs in H5 with 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},{4,3,3,4,3}.
Even allowing for infinite (Euclidean) facets and/or vertex figures, there are no regular tessellations of hyperbolic space of dimension 6 or higher.
Apeirotopes
An apeirotope is, like any other polytope, an unbounded hyper-surface. The difference is that whereas a polytope's hyper-surface curls back on itself to close round a finite volume of hyperspace, an apeirotope does not curl back.
Some people regard apeirotopes as just a special kind of polytope, while others regard them as rather different things.
Two dimensions
A regular apeirogon is a regular division of an infinitely long line into equal segments, joined by vertices. It has regular embeddings in the plane, and in higher-dimensional spaces. In two dimensions it can form a straight line or a zig-zag. In three dimensions, it traces out a helical spiral. The zig-zag and spiral forms are said to be skew.
Three dimensions
An apeirohedron is an infinite polyhedral surface. Like an apeirogon, it can be flat or skew. A flat apeirohedron is just a tiling of the plane. A skew apeirohedron is an intricate honeycomb-like structure which divides space into two regions.
There are thirty regular apeirohedra in Euclidean space. See section 7E of Abstract Regular Polytopes, by McMullen and Schulte. These include the tessellations of type {4,4},{6,3} and {3,6} above, as well as (in the plane) polytopes of type: 
, 
and 
, and in 3-dimensional space, blends of these with either an apeirogon or a line segment, and the "pure" 3-dimensional apeirohedra (12 in number)
Four and higher dimensions
The apeirochora have not been completely classified as of 2006.
Abstract polytopes
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 well-defined 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 polytopes that do not appear elsewhere in this list are the 11-cell and the 57-cell.
See also
- Polygon
- Polyhedron
- Regular polyhedron (5 regular Platonic solids and 4 Kepler-Poinsot solids)
- Polychoron
- Convex regular 4-polytope (6 regular polychora)
- Schläfli-Hess polychoron (10 regular star polychora)
- Tessellation
- Regular polytope
References
- Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294-296)
- Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213)
- D. M. Y. Sommerville, An Introduction to the Geometry of n Dimensions. New York, E. P. Dutton, 1930. 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes
External links
- The Platonic Solids
- Kepler-Poinsot Polyhedra
- Regular 4d Polytope Foldouts
- Multidimensional Glossary (Look up Hexacosichoron and Hecatonicosachoron)
- Polytope Viewer
- Polytopes and optimal packing of p points in n dimensional spheres
- An atlas of small regular polytopes
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