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# List of regular polytopes

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.

## Regular polytope summary count 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

## One-dimensional regular polytopes

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.

## Three-dimensional regular polytopes

In three dimensions, the regular polytopes are called polyhedra:

A regular polyhedron with Schläfli symbol $\left\{p,q\right\}$ has a regular face type $\left\{p\right\}$, and regular vertex figure $\left\{q\right\}$.

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 $\left\{p,q\right\}$ 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 $\left\{p\right\}$ and $\left\{q\right\}$ limited to: $\left\{3\right\}, \left\{4\right\}, \left\{5\right\}$, {5/2}, and $\left\{6\right\}$.

Beyond Euclidean space, there is an infinite set of regular hyperbolic tilings.

## Four-dimensional regular polytopes

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

• 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 $\left\{p,q,r\right\}$ is constrained by the existence of the regular polyhedra $\left\{p,q\right\}, \left\{q,r\right\}$.

Each will exist in a space dependent upon this expression:

$sin left \left(frac\left\{pi\right\}\left\{p\right\} right \right) sin left\left(frac\left\{pi\right\}\left\{r\right\}right\right) - cosleft\left(frac\left\{pi\right\}\left\{q\right\}right\right)$
$> 0$ : 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 $chi$ for polychora is $chi = V+F-E-C$ and is zero for all forms.

## Five-dimensional regular polytopes

In five dimensions, a regular polytope can be named as $\left\{p,q,r,s\right\}$ where $\left\{p,q,r\right\}$ is the hypercell (or teron) type, $\left\{p,q\right\}$ is the cell type, $\left\{p\right\}$ is the face type, and $\left\{s\right\}$ is the face figure, $\left\{r,s\right\}$ is the edge figure, and $\left\{q,r,s\right\}$ 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 $\left\{p,q,r,s\right\}$ exists only if $\left\{p,q,r\right\}$ and $\left\{q,r,s\right\}$ are regular polychora.

The space it fits in is based on the expression:

$frac\left\{cos^2left\left(frac\left\{pi\right\}\left\{q\right\}right\right)\right\}\left\{sin^2left\left(frac\left\{pi\right\}\left\{p\right\}right\right)\right\} + frac\left\{cos^2left\left(frac\left\{pi\right\}\left\{r\right\}right\right)\right\}\left\{sin^2left\left(frac\left\{pi\right\}\left\{s\right\}right\right)\right\}$
$< 1$ : 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.

## Classical convex polytopes

### Two dimensions

The Schläfli symbol $\left\{p\right\}$ represents a regular p-gon:

The infinite set of convex regular polygons are:

Name Schläfli
Symbol
{p}
Henagon {1}
digon {2}
equilateral triangle
(2-simplex)
{3}
square
(2-cube)
(2-orthoplex)
{4}
pentagon {5}
hexagon {6}
heptagon {7}
octagon {8}
nonagon {9}
decagon {10}
hendecagon {11}
dodecagon {12}
tridecagon {13}
dodecagon {16}
icosagon {20}
...n-gon {n}
apeirogon {}

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

A henagon {1}, and digon {2}, can be considered a degenerate regular polygon.

### Three dimensions

The convex regular polyhedra are called the 5 Platonic solids. (The vertex figure is given with each vertex count.)

Name Schläfli
{p,q}
Faces
{p}
Edges Vertices
{q}
χ Symmetry Dual
Tetrahedron
(3-simplex)
{3,3} 4
{3}
6 4
{3}
2 Td Self-dual
Cube (hexahedron)
(3-cube)
{4,3} 6
{4}
12 8
{3}
2 Oh Octahedron
Octahedron
(3-orthoplex)
{3,4} 8
{3}
12 6
{4}
2 Oh Cube
Dodecahedron {5,3} 12
{5}
30 20
{3}
2 Ih Icosahedron
Icosahedron {3,5} 20
{3}
30 12
{5}
2 Ih 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).

### Four dimensions

The 6 convex polychora are as follows:

Name
Schläfli
{p,q,r}
Cells
{p,q}
Faces
{p}
Edges
{r}
Vertices
{q,r}
χ Dual
{r,q,p}
5-cell
(pentachoron)
(4-simplex)
{3,3,3} 5
{3,3}
10
{3}
10
{3}
5
{3,3}
0 Self-dual
8-cell
(Tesseract)
(4-cube)
{4,3,3} 8
{4,3}
24
{4}
32
{3}
16
{3,3}
0 16-cell
16-cell
(4-orthoplex)
{3,3,4} 16
{3,3}
32
{3}
24
{4}
8
{3,4}
0 Tesseract
24-cell {3,4,3} 24
{3,4}
96
{3}
96
{3}
24
{4,3}
0 Self-dual
120-cell {5,3,3} 120
{5,3}
720
{5}
1200
{3}
600
{3,3}
0 600-cell
600-cell {3,3,5} 600
{3,3}
1200
{3}
720
{5}
120
{3,5}
0 120-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
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)

### Five dimensions

There are three kinds of convex regular polytopes in five dimensions:

Name projective
graph
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-orthoplex
(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

### Higher dimensions

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

## Finite non-convex polytopes - star-polytopes

### Two dimensions

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} {10/3}

### Three dimensions

The regular star polyhedra are called 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
{p,q}
Faces
{p}
Edges Vertices
{q}
χ Symmetry Dual
Small stellated dodecahedron {5/2,5} 12
{5/2}
30 12
{5}
-6 Ih Great dodecahedron
Great dodecahedron {5,5/2} 12
{5}
30 12
{5/2}
-6 Ih Small stellated dodecahedron
Great stellated dodecahedron {5/2,3} 12
{5/2}
30 20
{3}
2 Ih Great icosahedron
Great icosahedron {3,5/2} 20
{3}
30 12
{5/2}
2 Ih Great stellated dodecahedron

### Four dimensions

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

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
{p,q,r}
Cells
{p,q}
Faces
{p}
Edges
{r}
Vertices and
Vertex figure
{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}

### Higher dimensions

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

## 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.

### Two dimensions

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 {∞}.

......

### Three dimensions

#### Euclidean (plane) tilings

There are three regular tessellations of the plane.

Name Schläfli
Symbol
{p,q}
Face
type
{p}
Vertex
figure

{q}
χ Symmetry Dual
Square tiling {4,4} {4} {4} 0 p4m Self-dual
Triangular tiling {3,6} {3} {6} 0 p6m Hexagonal tiling
Hexagonal tiling {6,3} {6} {3} 0 p6m Triangular tiling

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

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 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}
χ Symmetry Dual
Order-5 square tiling {4,5} {4} {5} 0 *542 {5,4}
Order-4 pentagonal tiling {5,4} {5} {4} 0 *542 {4,5}
Order-7 triangular tiling {3,7} {3} {7} 0 *732 {7,3}
Order-3 heptagonal tiling {7,3} {7} {3} 0 *732 {3,7}
Order-6 square tiling {4,6} {4} {6} 0 *642 {6,4}
Order-4 hexagonal tiling {6,4} {6} {4} 0 *642 {4,6}
Order-5 pentagonal tiling {5,5} {5} {5} 0 *552 Self-dual
Order-8 triangular tiling {3,8} {3} {8} 0 *832 {8,3}
Order-3 octagonal tiling {8,3} {8} {3} 0 *832 {3,8}
Order-7 square tiling {4,7} {4} {7} 0 *742 {7,4}
Order-4 heptagonal tiling {7,4} {7} {4} 0 *742 {4,7}
Order-6 pentagonal tiling {5,6} {5} {6} 0 *652 {6,5}
Order-5 hexagonal tiling {6,5} {6} {5} 0 *652 {5,6}
Order-9 triangular tiling {3,9} {3} {9} 0 *932 {9,3}
Order-3 enneagonal tiling {9,3} {9} {3} 0 *932 {3,9}
Order-8 square tiling {4,8} {4} {8} 0 *842 {8,4}
Order-4 octagonal tiling {8,4} {8} {4} 0 *842 {4,8}
Order-7 pentagonal tiling {5,7} {5} {7} 0 *752 {7,5}
Order-5 heptagonal tiling {7,5} {7} {5} 0 *752 {5,7}
Order-6 hexagonal tiling {6,6} {6} {6} 0 *662 Self-dual

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.

 {4,5} {5,4} {3,7} {7,3}

### Four dimensions

#### 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 dimensions

#### 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)

#### 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

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}

### Higher dimensions

#### Tessellations of Euclidean Space

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 space

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: $\left\{3,4,3,3,3\right\}, \left\{3,3,4,3,3\right\}, \left\{3,3,3,4,3\right\}, \left\{3,4,3,3,4\right\}, \left\{4,3,3,4,3\right\}$.

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 just goes on for ever.

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 $\left\{4,4\right\}, \left\{6,3\right\}$ and $\left\{3,6\right\}$ above, as well as (in the plane) polytopes of type: $\left\{infty,3\right\}$, $\left\{infty,4\right\}$ and $\left\{infty,6\right\}$, 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.

## 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