List of regular polytopes

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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 can't be done in a regular plane, but can be at the right scale of a hyperbolic plane.

1 Regular polytope summary count by dimension
2 Two-dimensional regular polytopes
3 Three-dimensional regular polytopes
4 Four-dimensional regular polytopes
5 Five-dimensional regular polytopes
6 Classical Convex Polytopes
7 Finite Non-Convex Polytopes - star-polytopes
8 Tessellations
9 Apeirotopes
10 Abstract Polytopes
11 See also
12 References
13 External links

Dimension	Convex	Nonconvex	Convex Euclidean tessellations	Convex hyperbolic tessellations	Nonconvex hyperbolic tessellations
2	∞ polygons	∞ star polygons	1	1
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

The two dimensional polytopes are called polygons. Regular polygons are equilateral and cyclic.

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.

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 polyhedral vertex figure is an imaginary polygon which can be 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's an infinite set of regular hyperbolic tilings.

Regular polychora with Schläfli symbol symbol ${p, q, r}$ have cells of type ${p, q}$ , faces of type ${p}$ , edge figures ${r}$ , and vertex figures ${q, r}$ .

A polychoral vertex figure is an imaginary polyhedron that can be seen by the arrangement of neighboring vertices around a given vertex. For regular polychora, this vertex figure is a regular polyhedron.

A polychoral edge figure is an imaginary polygon that can be 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 surface polychoron (in 4-space)
- $= 0$ : 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.

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 5-polytope can be called a tetracomb.

A polyteron vertex figure is an imaginary polychoron that can be seen by the arrangement of neighboring vertices to each vertex.

A polyteron edge figure is an imaginary polyhedron that can be seen by the arrangement of faces around each edge.

A polyteron face figure is an imaginary polygon that can be 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 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.

The Schläfli symbol $p$ represents a regular p-gon:

The infinite set of convex regular polygons are:

Name	Schläfli Symbol {p}
digon	{2}
equilateral triangle	{3}
square	{4}
pentagon	{5}
hexagon	{6}
heptagon	{7}
octagon	{8}
enneagon	{9}
decagon	{10}
Hendecagon	{11}
Dodecagon	{12}
...n-gon	{n}
apeirogon	{∞}

{2}	{3}	{4}	{5}	{6}	{7}	{8}	{9}	{10}	{11}	{12}

A digon, {2}, can be considered a degenerate regular polygon.

The convex regular polyhedra are called the 5 Platonic solids:

Name	Schläfli Symbol {p,q}	Faces {p}	Edges	Vertices {q}	χ	Symmetry	dual
Tetrahedron	{3,3}	4 {3}	6	4 {3}	2	T_d	Self-dual
Cube (hexahedron)	{4,3}	6 {4}	12	8 {3}	2	O_h	Octahedron
Octahedron	{3,4}	8 {3}	12	6 {4}	2	O_h	Cube
Dodecahedron	{5,3}	12 {5}	30	20 {3}	2	I_h	Icosahedron
Icosahedron	{3,5}	20 {3}	30	12 {5}	2	I_h	Dodecahedron

{3,3}	{4,3}	{3,4}	{5,3}	{3,5}

In spherical geometry, hosohedron, {2,n} and dihedron {n,2} can be considered regular polyhedra (tilings of the sphere).

The 6 convex polychora are as follows:

Name	Schläfli Symbol {p,q,r}	Cells {p,q}	Faces {p}	Edges {r}	Vertices {q,r}	Dual {r,q,p}
5-cell (pentachoron)	{3,3,3}	5 {3,3}	10 {3}	10 {3}	5 {3,3}	Self-dual
8-cell (Tesseract)	{4,3,3}	8 {4,3}	24 {4}	32 {3}	16 {3,3}	16-cell
16-cell	{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-dual
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

Stereoscopic projections (Schlegel diagrams)

{3,3,3}	{4,3,3}	{3,3,4}	{3,4,3}	{5,3,3}	{3,3,5}

There are three kinds of convex regular polytopes in five 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}	Dual
5-simplex (or hexateron)	{3,3,3,3}	6 {3,3,3}	15 {3,3}	20 {3}	15	6	{3}	{3,3}	{3,3,3}	Self-dual
5-hypercube (or decateron or penteract)	{4,3,3,3}	10 {4,3,3}	40 {4,3}	80 {4}	80	32	{3}	{3,3}	{3,3,3}	pentacross
5-cross-polytope (or triacontakaiditeron or pentacross)	{3,3,3,4}	32 {3,3,3}	80 {3,3}	80 {3}	40	10	{4}	{3,4}	{3,3,4}	penteract

In dimensions 5 and higher , there are only three kinds of convex regular polytopes. [Coxeter, Regular Polytopes, Tables I: Regular polytopes, (iii) The three regular polytopes in n-dimensions (n>=5), pp. 294-295]

Name	Schläfli Symbol {p₁,p₂,...,p_n-1}	Facet type	Vertex figure	Dual
n-simplex	{3,3,3,...,3}	{3,3,...,3}	{3,3,...,3}	Self-dual
n-hypercube	{4,3,3,...,3}	{4,3,...,3}	{3,3,...,3}	cross-n-polytope
n-cross-polytope	{3,...,3,3,4}	{3,...,3,3}	{3,...,3,4}	measure n-polytope

There exist infinitely many non-convex regular polytopes in two dimensions, whose Schläfli symbols consist of rational numbers {m/n}. They are called star 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	Schläfli Symbol {n/m}
pentagram	{5/2}
heptagrams	{7/2}, {7/3}
octagram	{8/3}
enneagrams	{9/2}, {9/4}
decagram	{10/3}
hendecagrams	{11/2} {11/3}, {11/4}, {11/5}
dodecagram	{12/5}
...n-agrams	{n/m}

{5/2}	{7/2}	{7/3}	{8/3}	{9/2}	{9/4}

The regular star polyhedra are call the Kepler-Poinsot solids and there are four of them, based on the vertices of the dodecahedron {5,3} and icosahedron {3,5}:

Name	Schläfli Symbol {p,q}	Faces {p}	Edges	Vertices {q}	χ	Symmetry	Dual
Small stellated dodecahedron	{5/2,5}	12 {5/2}	30	12 {5}	-6	I_h	Great dodecahedron
Great dodecahedron	{5,5/2}	12 {5}	30	12 {5/2}	-6	I_h	Small stellated dodecahedron
Great stellated dodecahedron	{5/2,3}	12 {5/2}	30	20 {3}	2	I_h	Great icosahedron
Great icosahedron	{3,5/2}	20 {3}	30	12 {5/2}	2	I_h	Great stellated dodecahedron

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 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.

Name	Schläfli Symbol {p,q,r}	Cells {p,q}	Faces {p}	Edges {r}	Vertices {q,r}	χ	Dual {r,q,p}
Great grand stellated 120-cell	{5/2,3,3}	120 {5/2,3}	720 {5/2}	1200 {3}	600 {3,3}	0	Grand 600-cell
Grand 600-cell	{3,3,5/2}	600 {3,3}	1200 {3}	720 {5/2}	120 {3,5/2}	0	Great grand stellated 120-cell
Great stellated 120-cell	{5/2,3,5}	120 {5/2,3}	720 {5/2}	720 {5}	120 {3,5}	0	Grand 120-cell
Grand 120-cell	{5,3,5/2}	120 {5,3}	720 {5}	720 {5/2}	120 {3,5/2}	0	Great stellated 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}	0	Self-dual
Small stellated 120-cell	{5/2,5,3}	120 {5/2,5}	720 {5/2}	1200 {3}	120 {5,3}	-480	Icosahedral 120-cell
Icosahedral 120-cell	{3,5,5/2}	120 {3,5}	1200 {3}	720 {5/2}	120 {5,5/2}	480	Small stellated 120-cell
Great icosahedral 120-cell	{3,5/2,5}	120 {3,5/2}	1200 {3}	720 {5}	120 {5/2,5}	480	Great grand 120-cell
Great grand 120-cell	{5,5/2,3}	120 {5,5/2}	720 {5}	1200 {3}	120 {5/2,3}	-480	Great icosahedral 120-cell
Great 120-cell	{5,5/2,5}	120 {5,5/2}	720 {5}	720 {5}	120 {5/2,5}	0	Self-dual

There are 7 unique face arrangements from these 10 nonconvex polychora, shown as orthogonal projections:

{3,5,5/2}	{5,5/2,5} and {5,3,5/2}	{5/2,5,3}	{5,5/2,3}
{5/2,3,5} and {5/2,5,5/2}	{3,5/2,5} and {3,3,5/2}	{5/2,3,3}

There are no non-convex regular polytopes in five dimensions or higher.

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.

There is one tessellation of the line, giving one polytope, the (two-dimensional) apeirogon. This has infinitely many vertices and edges. Its Schläfli is {∞}.

There are three regular tessellations of the plane.

Name	Schläfli Symbol {p,q}	Face type {p}	Vertex figure {q}	Symmetry	Dual
Square tiling	{4,4}	{4}	{4}	p4m	Self-dual
Triangular tiling	{3,6}	{3}	{6}	p6m	Hexagonal tiling
Hexagonal tiling	{6,3}	{6}	{3}	p6m	Triangular tiling

{4,4}	{3,6}	{6,3}

There's 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.

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 2-space can be called hyperbolic tilings.

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.

A sampling:

Name	Schläfli Symbol {p,q}	Face type {p}	Vertex figure {q}	Symmetry	Dual
Order-5 square tiling	{4,5}	{4}	{5}	*542	{5,4}
Order-4 pentagonal tiling	{5,4}	{5}	{4}	*542	{4,5}
Order-7 triangular tiling	{3,7}	{3}	{7}	*732	{7,3}
Order-3 heptagonal tiling	{7,3}	{7}	{3}	*732	{3,7}
Order-6 square tiling	{4,6}	{4}	{6}	*642	{6,4}
Order-4 hexagonal tiling	{6,4}	{6}	{4}	*642	{4,6}
Order-5 pentagonal tiling	{5,5}	{5}	{5}	*552	Self-dual
Order-8 triangular tiling	{3,8}	{3}	{8}	*832	{8,3}
Order-3 octagonal tiling	{8,3}	{8}	{3}	*832	{3,8}
Order-7 square tiling	{4,7}	{4}	{7}	*742	{7,4}
Order-4 heptagonal tiling	{7,4}	{7}	{4}	*742	{4,7}
Order-6 pentagonal tiling	{5,6}	{5}	{6}	*652	{6,5}
Order-5 hexagonal tiling	{6,5}	{6}	{5}	*652	{5,6}
Order-9 triangular tiling	{3,9}	{3}	{9}	*932	{9,3}
Order-3 enneagonal tiling	{9,3}	{9}	{3}	*932	{3,9}
Order-8 square tiling	{4,8}	{4}	{8}	*842	{8,4}
Order-4 octagonal tiling	{8,4}	{8}	{4}	*842	{4,8}
Order-7 pentagonal tiling	{5,7}	{5}	{7}	*752	{7,5}
Order-5 heptagonal tiling	{7,5}	{7}	{5}	*752	{5,7}
Order-6 hexagonal tiling	{6,6}	{6}	{6}	*662	Self-dual

There are 2 infinite forms of hyperbolic tilings of 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.

{4,5}

{5,4}

{3,7}

{7,3}

{4,3,4}

Tessellations of 3-space are called honeycombs. There is only one regular honeycomb:

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 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}	Dual
Order-3 icosahedral honeycomb	{3,5,3}	{3,5}	{3}	{3}	{5,3}	Self-dual
Order-5 cubic honeycomb	{4,3,5}	{4,3}	{4}	{5}	{3,5}	{5,3,4}
Order-4 dodecahedral honeycomb	{5,3,4}	{5,3}	{5}	{4}	{3,4}	{4,3,5}
Order-5 dodecahedral honeycomb	{5,3,5}	{5,3}	{5}	{5}	{3,5}	Self-dual

Some projected images, Poincaré disk models. The first shows the perspective from the center of the disc, and the second and third from the outside.

{5,3,4}
(8 dodecahedra at a vertex)

{4,3,5}
(20 cubes at a vertex)

{5,3,3}
(4 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}.

There are three kinds of infinite regular tessellations (tetracombs) 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 tetracomb	{4,3,3,4}	{4,3,3}	{4,3}	{4}	{4}	{3,4}	{3,3,4}	Self-dual
Hexadecachoronic tetracomb	{3,3,4,3}	{3,3,4}	{3,3}	{3}	{3}	{4,3}	{3,4,3}	{3,4,3,3}
Icositetrachoronic tetracomb	{3,4,3,3}	{3,4,3}	{3,4}	{3}	{3}	{3,3}	{4,3,3}	{3,3,4,3}

There are five kinds of convex regular tetracombs 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]

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 pentachoronic tetracomb	{3,3,3,5}	{3,3,3}	{3,3}	{3}	{5}	{3,5}	{3,3,5}	{5,3,3,3}
Order-3 hecatonicosachoronic tetracomb	{5,3,3,3}	{5,3,3}	{5,3}	{5}	{3}	{3,3}	{3,3,3}	{3,3,3,5}
Order-5 tesseractic tetracomb	{4,3,3,5}	{4,3,3}	{4,3}	{4}	{5}	{3,5}	{3,3,5}	{5,3,3,4}
Order-4 hecatonicosachoronic tetracomb	{5,3,3,4}	{5,3,3}	{5,3}	{5}	{4}	{3,4}	{3,3,4}	{4,3,3,5}
Order-5 hecatonicosachoronic tetracomb	{5,3,3,5}	{5,3,3}	{5,3}	{5}	{5}	{3,5}	{3,3,5}	Self-dual
Order-3 small stellated hecatonicosachoronic tetracomb	{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 hexacosichoron tetracomb	{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 hecatonicosachoronic tetracomb	{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 hecatonicosachoronic tetracomb	{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 is only one infinite regular polytope that can tessellate each dimension, five or higher, formed by hypercube facets, four around every ridge.

Name	Schläfli Symbol {p₁, p₂, ..., p_n−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 tetracomb	{4,3,3,4}	{4,3,3}	{3,3,4}	Self-dual
Penteractic pentacomb	{4,3,3,3,4}	{4,3,3,3}	{3,3,3,4}	Self-dual
Hexeractic hexacomb	{4,3,3,3,3,4}	{4,3,3,3,3}	{3,3,3,3,4}	Self-dual
Order-4 n-hypercubic tessellation	{4,3,...,3,4}	{4,3,...,3}	{3,...,3,4}	Self-dual

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.

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 just goes on for ever.

Some people regard apeirotopes as just a special kind of polytope, while others regard them as rather different beasts.

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.

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: $\{\infty,3\}$ , $\{\infty,4\}$ and $\{\infty,6\}$ , 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)

The apeirochora have not been completely classified as of 2006.

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.

Polygon
- Regular polygon
- Star polygon
Polyhedron
- Regular polyhedron (5 regular Platonic solids and 4 Kepler-Poinsot solids)
  - Uniform polyhedron
Polychoron
- Convex regular polychoron (6 regular polychora)
  - Uniform polychoron
- Schläfli-Hess polychoron (10 regular star polychora)
Tessellation
- Tilings of regular polygons
- Convex uniform honeycomb
Regular polytope
- Uniform polytope

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)

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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Categories: Mathematics-related lists | Polytopes

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