List of regular polytopes and compounds

Last updated

Example regular polytopes
Regular (2D) polygons
ConvexStar
Regular pentagon.svg
{5}
Star polygon 5-2.svg
{5/2}
Regular (3D) polyhedra
ConvexStar
Dodecahedron.png
{5,3}
Small stellated dodecahedron.png
{5/2,5}
Regular 4D polytopes
ConvexStar
Schlegel wireframe 120-cell.png
{5,3,3}
Ortho solid 010-uniform polychoron p53-t0.png
{5/2,5,3}
Regular 2D tessellations
EuclideanHyperbolic
Uniform tiling 44-t0.svg
{4,4}
H2-5-4-dual.svg
{5,4}
Regular 3D tessellations
EuclideanHyperbolic
Cubic honeycomb.png
{4,3,4}
Hyperbolic orthogonal dodecahedral honeycomb.png
{5,3,4}

This article lists the regular polytopes and regular polytope compounds in Euclidean, spherical and hyperbolic spaces.

Contents

The Schläfli symbol describes every regular tessellation of an n-sphere, Euclidean and hyperbolic spaces. A Schläfli symbol describing an n-polytope 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 Coxeter-Dynkin 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 CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png, it is represented by Coxeter diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png.

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 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.

[1] Overview

This table shows a summary of regular polytope counts by dimension.

Note that the Euclidean and hyperbolic tilings are given one dimension more than what would be expected. This is because of an analogy with finite polytopes: a convex regular n-polytope can be seen as a tessellation of (n−1)-dimensional spherical space. Thus the three regular tilings of the Euclidean plane (by triangles, squares, and hexagons) are listed under dimension three rather than two.

Dim.FiniteEuclideanHyperbolicCompounds
CompactParacompact
ConvexStarSkewConvexConvexStarConvexConvexStar
11nonenone1nonenonenonenonenone
211nonenone
354?35none
4610?14none112620
53none?3542nonenone
63none?1nonenone5nonenone
73none?1nonenonenone3none
83none?1nonenonenone6none
9+3none?1nonenonenone [lower-alpha 1] none

There are no Euclidean regular star tessellations in any number of dimensions.

One dimension

Coxeter node markup1.png 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 { }, CDel node 1.png, is a point p and its mirror image point p', and the line segment between them.

A one-dimensional polytope or 1-polytope is a closed line segment, bounded by its two endpoints. A 1-polytope is regular by definition and is represented by Schläfli symbol { }, [2] [3] or a Coxeter diagram with a single ringed node, CDel node 1.png. Norman Johnson calls it a dion [4] and gives it the Schläfli symbol { }.

Although trivial as a polytope, it appears as the edges of polygons and other higher dimensional polytopes. [5] It is used in the definition of uniform prisms like Schläfli symbol { }×{p}, or Coxeter diagram CDel node 1.pngCDel 2.pngCDel node 1.pngCDel p.pngCDel node.png as a Cartesian product of a line segment and a regular polygon. [6]

Two dimensions (polygons)

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 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.

Convex

The Schläfli symbol {p} represents a regular p-gon.

Name Triangle
(2-simplex)
Square
(2-orthoplex)
(2-cube)
Pentagon
(2-pentagonal
polytope
)
Hexagon Heptagon Octagon
Schläfli {3}{4}{5}{6}{7}{8}
SymmetryD3, [3]D4, [4]D5, [5]D6, [6]D7, [7]D8, [8]
Coxeter CDel node 1.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel node 1.pngCDel 7.pngCDel node.pngCDel node 1.pngCDel 8.pngCDel node.png
Image Regular triangle.svg Regular quadrilateral.svg Regular pentagon.svg Regular hexagon.svg Regular heptagon.svg Regular octagon.svg
Name Nonagon
(Enneagon)
Decagon Hendecagon Dodecagon Tridecagon Tetradecagon
Schläfli{9}{10}{11}{12}{13}{14}
SymmetryD9, [9]D10, [10]D11, [11]D12, [12]D13, [13]D14, [14]
DynkinCDel node 1.pngCDel 9.pngCDel node.pngCDel node 1.pngCDel 10.pngCDel node.pngCDel node 1.pngCDel 11.pngCDel node.pngCDel node 1.pngCDel 12.pngCDel node.pngCDel node 1.pngCDel 13.pngCDel node.pngCDel node 1.pngCDel 14.pngCDel node.png
Image Regular nonagon.svg Regular decagon.svg Regular hendecagon.svg Regular dodecagon.svg Regular tridecagon.svg Regular tetradecagon.svg
Name Pentadecagon Hexadecagon Heptadecagon Octadecagon Enneadecagon Icosagon ...p-gon
Schläfli{15}{16}{17}{18}{19}{20}{p}
SymmetryD15, [15]D16, [16]D17, [17]D18, [18]D19, [19]D20, [20]Dp, [p]
DynkinCDel node 1.pngCDel 15.pngCDel node.pngCDel node 1.pngCDel 16.pngCDel node.pngCDel node 1.pngCDel 17.pngCDel node.pngCDel node 1.pngCDel 18.pngCDel node.pngCDel node 1.pngCDel 19.pngCDel node.pngCDel node 1.pngCDel 20.pngCDel node.pngCDel node 1.pngCDel p.pngCDel node.png
Image Regular pentadecagon.svg Regular hexadecagon.svg Regular heptadecagon.svg Regular octadecagon.svg Regular enneadecagon.svg Regular icosagon.svg Disk 1.svg

Spherical

The regular digon {2} can be considered to be a degenerate regular polygon. It can be realized non-degenerately in some non-Euclidean spaces, such as on the surface of a sphere or torus. For example, digon can be realised non-degenerately as a spherical lune. A monogon {1} could also be realised on the sphere as a single point with a great circle through it. [7] However, a monogon is not a valid abstract polytope because its single edge is incident to only one vertex rather than two.

Name Monogon Digon
Schläfli symbol {1}{2}
SymmetryD1, [ ]D2, [2]
Coxeter diagram CDel node.png or CDel node h.pngCDel 2x.pngCDel node.pngCDel node 1.pngCDel 2x.pngCDel node.png
Image Monogon.svg Digon.svg

Stars

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 n-pointed 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 ...n-grams
Schläfli {5/2}{7/2}{7/3}{8/3}{9/2}{9/4}{10/3}{p/q}
SymmetryD5, [5]D7, [7]D8, [8]D9, [9],D10, [10]Dp, [p]
Coxeter CDel node 1.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.pngCDel node 1.pngCDel 7.pngCDel rat.pngCDel d2.pngCDel node.pngCDel node 1.pngCDel 7.pngCDel rat.pngCDel d3.pngCDel node.pngCDel node 1.pngCDel 8.pngCDel rat.pngCDel d3.pngCDel node.pngCDel node 1.pngCDel 9.pngCDel rat.pngCDel d2.pngCDel node.pngCDel node 1.pngCDel 9.pngCDel rat.pngCDel d4.pngCDel node.pngCDel node 1.pngCDel 10.pngCDel rat.pngCDel d3.pngCDel node.pngCDel node 1.pngCDel p.pngCDel rat.pngCDel dq.pngCDel node.png
Image Star polygon 5-2.svg Star polygon 7-2.svg Star polygon 7-3.svg Star polygon 8-3.svg Star polygon 9-2.svg Star polygon 9-4.svg Star polygon 10-3.svg  
Regular star polygons up to 20 sides
Regular star polygon 11-2.svg
{11/2}
Regular star polygon 11-3.svg
{11/3}
Regular star polygon 11-4.svg
{11/4}
Regular star polygon 11-5.svg
{11/5}
Regular star polygon 12-5.svg
{12/5}
Regular star polygon 13-2.svg
{13/2}
Regular star polygon 13-3.svg
{13/3}
Regular star polygon 13-4.svg
{13/4}
Regular star polygon 13-5.svg
{13/5}
Regular star polygon 13-6.svg
{13/6}
Regular star polygon 14-3.svg
{14/3}
Regular star polygon 14-5.svg
{14/5}
Regular star polygon 15-2.svg
{15/2}
Regular star polygon 15-4.svg
{15/4}
Regular star polygon 15-7.svg
{15/7}
Regular star polygon 16-3.svg
{16/3}
Regular star polygon 16-5.svg
{16/5}
Regular star polygon 16-7.svg
{16/7}
Regular star polygon 17-2.svg
{17/2}
Regular star polygon 17-3.svg
{17/3}
Regular star polygon 17-4.svg
{17/4}
Regular star polygon 17-5.svg
{17/5}
Regular star polygon 17-6.svg
{17/6}
Regular star polygon 17-7.svg
{17/7}
Regular star polygon 17-8.svg
{17/8}
Regular star polygon 18-5.svg
{18/5}
Regular star polygon 18-7.svg
{18/7}
Regular star polygon 19-2.svg
{19/2}
Regular star polygon 19-3.svg
{19/3}
Regular star polygon 19-4.svg
{19/4}
Regular star polygon 19-5.svg
{19/5}
Regular star polygon 19-6.svg
{19/6}
Regular star polygon 19-7.svg
{19/7}
Regular star polygon 19-8.svg
{19/8}
Regular star polygon 19-9.svg
{19/9}
Regular star polygon 20-3.svg
{20/3}
Regular star polygon 20-7.svg
{20/7}
Regular star polygon 20-9.svg
{20/9}

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. [1]

Skew polygons

In 3-dimensional space, a regular skew polygon is called an antiprismatic polygon, with the vertex arrangement of an antiprism, and a subset of edges, zig-zagging between top and bottom polygons.

Example regular skew zig-zag polygons
HexagonOctagonDecagons
D3d, [2+,6]D4d, [2+,8]D5d, [2+,10]
{3}#{ }{4}#{ }{5}#{ }{5/2}#{ }{5/3}#{ }
Skew polygon in triangular antiprism.png Skew polygon in square antiprism.png Regular skew polygon in pentagonal antiprism.png Regular skew polygon in pentagrammic antiprism.png Regular skew polygon in pentagrammic crossed-antiprism.png

In 4-dimensions 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 odd-number of sides.

They can be seen in the Petrie polygons of the convex regular 4-polytopes, seen as regular plane polygons in the perimeter of Coxeter plane projection:

PentagonOctagonDodecagonTriacontagon
4-simplex t0.svg
5-cell
4-orthoplex.svg
16-cell
24-cell t0 F4.svg
24-cell
600-cell graph H4.svg
600-cell

Three dimensions (polyhedra)

In three dimensions, polytopes are called polyhedra:

A regular polyhedron with Schläfli symbol {p,q}, Coxeter diagrams CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png, 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.

Convex

The five convex regular polyhedra are called the Platonic solids. The vertex figure is given with each vertex count. All these polyhedra have an Euler characteristic (χ) of 2.

Name Schläfli
{p,q}
Coxeter
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png
Image
(solid)
Image
(sphere)
Faces
{p}
Edges Vertices
{q}
Symmetry Dual
Tetrahedron
(3-simplex)
{3,3}CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png Polyhedron 4b.png Uniform tiling 332-t2.png 4
{3}
64
{3}
Td
[3,3]
(*332)
(self)
Hexahedron
Cube
(3-cube)
{4,3}CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png Polyhedron 6.png Uniform tiling 432-t0.png 6
{4}
128
{3}
Oh
[4,3]
(*432)
Octahedron
Octahedron
(3-orthoplex)
{3,4}CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png Polyhedron 8.png Uniform tiling 432-t2.png 8
{3}
126
{4}
Oh
[4,3]
(*432)
Cube
Dodecahedron {5,3}CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png Polyhedron 12.png Uniform tiling 532-t0.png 12
{5}
3020
{3}
Ih
[5,3]
(*532)
Icosahedron
Icosahedron {3,5}CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png Polyhedron 20.png Uniform tiling 532-t2.png 20
{3}
3012
{5}
Ih
[5,3]
(*532)
Dodecahedron

Spherical

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. [8]

The first few cases (n from 2 to 6) are listed below.

Hosohedra
Name Schläfli
{2,p}
Coxeter
diagram
Image
(sphere)
Faces
{2}π/p
Edges Vertices
{p}
Symmetry Dual
Digonal hosohedron{2,2}CDel node 1.pngCDel 2x.pngCDel node.pngCDel 2x.pngCDel node.png Spherical digonal hosohedron.png 2
{2}π/2
22
{2}π/2
D2h
[2,2]
(*222)
Self
Trigonal hosohedron{2,3}CDel node 1.pngCDel 2x.pngCDel node.pngCDel 3.pngCDel node.png Spherical trigonal hosohedron.png 3
{2}π/3
32
{3}
D3h
[2,3]
(*322)
Trigonal dihedron
Square hosohedron{2,4}CDel node 1.pngCDel 2x.pngCDel node.pngCDel 4.pngCDel node.png Spherical square hosohedron.png 4
{2}π/4
42
{4}
D4h
[2,4]
(*422)
Square dihedron
Pentagonal hosohedron{2,5}CDel node 1.pngCDel 2x.pngCDel node.pngCDel 5.pngCDel node.png Spherical pentagonal hosohedron.png 5
{2}π/5
52
{5}
D5h
[2,5]
(*522)
Pentagonal dihedron
Hexagonal hosohedron{2,6}CDel node 1.pngCDel 2x.pngCDel node.pngCDel 6.pngCDel node.png Spherical hexagonal hosohedron.png 6
{2}π/6
62
{6}
D6h
[2,6]
(*622)
Hexagonal dihedron
Dihedra
Name Schläfli
{p,2}
Coxeter
diagram
Image
(sphere)
Faces
{p}
Edges Vertices
{2}
Symmetry Dual
Digonal dihedron{2,2}CDel node 1.pngCDel 2x.pngCDel node.pngCDel 2x.pngCDel node.png Digonal dihedron.png 2
{2}π/2
22
{2}π/2
D2h
[2,2]
(*222)
Self
Trigonal dihedron{3,2}CDel node 1.pngCDel 3.pngCDel node.pngCDel 2x.pngCDel node.png Trigonal dihedron.png 2
{3}
33
{2}π/3
D3h
[3,2]
(*322)
Trigonal hosohedron
Square dihedron{4,2}CDel node 1.pngCDel 4.pngCDel node.pngCDel 2x.pngCDel node.png Tetragonal dihedron.png 2
{4}
44
{2}π/4
D4h
[4,2]
(*422)
Square hosohedron
Pentagonal dihedron{5,2}CDel node 1.pngCDel 5.pngCDel node.pngCDel 2x.pngCDel node.png Pentagonal dihedron.png 2
{5}
55
{2}π/5
D5h
[5,2]
(*522)
Pentagonal hosohedron
Hexagonal dihedron{6,2}CDel node 1.pngCDel 6.pngCDel node.pngCDel 2x.pngCDel node.png Hexagonal dihedron.png 2
{6}
66
{2}π/6
D6h
[6,2]
(*622)
Hexagonal hosohedron

Star-dihedra and hosohedra {p/q,2} and {2,p/q} also exist for any star polygon {p/q}.

Stars

The regular star polyhedra are called the Kepler–Poinsot polyhedra and there are four of them, based on the vertex arrangements of the dodecahedron {5,3} and icosahedron {3,5}:

As spherical tilings, these star forms overlap the sphere multiple times, called its density, being 3 or 7 for these forms. The tiling images show a single spherical polygon face in yellow.

NameImage
(skeletonic)
Image
(solid)
Image
(sphere)
Stellation
diagram
Schläfli
{p,q} and
Coxeter
Faces
{p}
EdgesVertices
{q}
verf.
χ Density Symmetry Dual
Small stellated dodecahedron Skeleton St12, size m.png Small stellated dodecahedron (gray with yellow face).svg Small stellated dodecahedron tiling.png First stellation of dodecahedron facets.svg {5/2,5}
CDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node 1.png
12
{5/2}
Star polygon 5-2.svg
3012
{5}
Regular pentagon.svg
−63Ih
[5,3]
(*532)
Great dodecahedron
Great dodecahedron Skeleton Gr12, size m.png Great dodecahedron (gray with yellow face).svg Great dodecahedron tiling.png Second stellation of dodecahedron facets.svg {5,5/2}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png
12
{5}
Regular pentagon.svg
3012
{5/2}
Star polygon 5-2.svg
−63Ih
[5,3]
(*532)
Small stellated dodecahedron
Great stellated dodecahedron Skeleton GrSt12, size s.png Great stellated dodecahedron (gray with yellow face).svg Great stellated dodecahedron tiling.png Third stellation of dodecahedron facets.svg {5/2,3}
CDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node 1.png
12
{5/2}
Star polygon 5-2.svg
3020
{3}
Regular triangle.svg
27Ih
[5,3]
(*532)
Great icosahedron
Great icosahedron Skeleton Gr20, size m.png Great icosahedron (gray with yellow face).svg Great icosahedron tiling.png Great icosahedron stellation facets.svg {3,5/2}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png
20
{3}
Regular triangle.svg
3012
{5/2}
Star polygon 5-2.svg
27Ih
[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}.

Skew polyhedra

Regular skew polyhedra are generalizations to the set of regular polyhedron which include the possibility of nonplanar vertex figures.

For 4-dimensional skew polyhedra, Coxeter offered a modified Schläfli symbol {l,m|n} for these figures, with {l,m} implying the vertex figure, m l-gons around a vertex, and n-gonal holes. Their vertex figures are skew polygons, zig-zagging between two planes.

The regular skew polyhedra, represented by {l,m|n}, follow this equation:

2 sin(π/l) sin(π/m) = cos(π/n)

Four of them can be seen in 4-dimensions as a subset of faces of four regular 4-polytopes, sharing the same vertex arrangement and edge arrangement:

4-simplex t03.svg 4-simplex t12.svg 24-cell t03 F4.svg 24-cell t12 F4.svg
{4, 6 | 3}{6, 4 | 3}{4, 8 | 3}{8, 4 | 3}

Four dimensions

Regular 4-polytopes with Schläfli symbol have cells of type , faces of type , edge figures , and vertex figures .

The existence of a regular 4-polytope is constrained by the existence of the regular polyhedra . A suggested name for 4-polytopes is "polychoron". [9]

Each will exist in a space dependent upon this expression:

 : Hyperspherical 3-space honeycomb or 4-polytope
 : Euclidean 3-space honeycomb
 : 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 for convex 4-polytopes is zero:

Convex

The 6 convex regular 4-polytopes are shown in the table below. All these 4-polytopes have an Euler characteristic (χ) of 0.

Name
Schläfli
{p,q,r}
Coxeter
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png
Cells
{p,q}
Faces
{p}
Edges
{r}
Vertices
{q,r}
Dual
{r,q,p}
5-cell
(4-simplex)
{3,3,3}CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png5
{3,3}
10
{3}
10
{3}
5
{3,3}
(self)
8-cell
(4-cube)
(Tesseract)
{4,3,3}CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png8
{4,3}
24
{4}
32
{3}
16
{3,3}
16-cell
16-cell
(4-orthoplex)
{3,3,4}CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png16
{3,3}
32
{3}
24
{4}
8
{3,4}
Tesseract
24-cell {3,4,3}CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png24
{3,4}
96
{3}
96
{3}
24
{4,3}
(self)
120-cell {5,3,3}CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png120
{5,3}
720
{5}
1200
{3}
600
{3,3}
600-cell
600-cell {3,3,5}CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png600
{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 (Petrie polygon) skew orthographic projections
Complete graph K5.svg 4-cube graph.svg 4-orthoplex.svg 24-cell graph F4.svg Cell120Petrie.svg Cell600Petrie.svg
Solid orthographic projections
Tetrahedron.png
tetrahedral
envelope
(cell/
vertex-centered)
Hexahedron.png
cubic envelope
(cell-centered)
16-cell ortho cell-centered.png
cubic envelope
(cell-centered)
Ortho solid 24-cell.png
cuboctahedral
envelope

(cell-centered)
Ortho solid 120-cell.png
truncated rhombic
triacontahedron
envelope

(cell-centered)
Ortho solid 600-cell.png
Pentakis
icosidodecahedral

envelope
(vertex-centered)
Wireframe Schlegel diagrams (Perspective projection)
Schlegel wireframe 5-cell.png
(cell-centered)
Schlegel wireframe 8-cell.png
(cell-centered)
Schlegel wireframe 16-cell.png
(cell-centered)
Schlegel wireframe 24-cell.png
(cell-centered)
Schlegel wireframe 120-cell.png
(cell-centered)
Schlegel wireframe 600-cell vertex-centered.png
(vertex-centered)
Wireframe stereographic projections (Hyperspherical)
Stereographic polytope 5cell.png Stereographic polytope 8cell.png Stereographic polytope 16cell.png Stereographic polytope 24cell.png Stereographic polytope 120cell.png Stereographic polytope 600cell.png

Spherical

Di-4-topes and hoso-4-topes exist as regular tessellations of the 3-sphere.

Regular di-4-topes (2 facets) include: {3,3,2}, {3,4,2}, {4,3,2}, {5,3,2}, {3,5,2}, {p,2,2}, and their hoso-4-tope duals (2 vertices): {2,3,3}, {2,4,3}, {2,3,4}, {2,3,5}, {2,5,3}, {2,2,p}. 4-polytopes 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.

Regular hoso-4-topes as 3-sphere honeycombs
Schläfli
{2,p,q}
Coxeter
CDel node 1.pngCDel 2x.pngCDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png
Cells
{2,p}π/q
Faces
{2}π/p,π/q
Edges Vertices Vertex figure
{p,q}
Symmetry Dual
{2,3,3}CDel node 1.pngCDel 2x.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png4
{2,3}π/3
Spherical trigonal hosohedron.png
6
{2}π/3,π/3
42{3,3}
Uniform tiling 332-t0-1-.png
[2,3,3]{3,3,2}
{2,4,3}CDel node 1.pngCDel 2x.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png6
{2,4}π/3
Spherical square hosohedron.png
12
{2}π/4,π/3
82{4,3}
Uniform tiling 432-t0.png
[2,4,3]{3,4,2}
{2,3,4}CDel node 1.pngCDel 2x.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png8
{2,3}π/4
Spherical trigonal hosohedron.png
12
{2}π/3,π/4
62{3,4}
Uniform tiling 432-t2.png
[2,4,3]{4,3,2}
{2,5,3}CDel node 1.pngCDel 2x.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png12
{2,5}π/3
Spherical trigonal hosohedron.png
30
{2}π/5,π/3
202{5,3}
Uniform tiling 532-t0.png
[2,5,3]{3,5,2}
{2,3,5}CDel node 1.pngCDel 2x.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png20
{2,3}π/5
Spherical pentagonal hosohedron.png
30
{2}π/3,π/5
122{3,5}
Uniform tiling 532-t2.png
[2,5,3]{5,3,2}

Stars

There are ten regular star 4-polytopes, which are called the Schläfli–Hess 4-polytopes. 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 tori: F+V−E=2). Edmund Hess (1843–1903) completed the full list of ten in his German book Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder (1883).

There are 4 unique edge arrangements and 7 unique face arrangements from these 10 regular star 4-polytopes, shown as orthogonal projections:

Name
WireframeSolid Schläfli
{p, q, r}
Coxeter
Cells
{p, q}
Faces
{p}
Edges
{r}
Vertices
{q, r}
Density χ Symmetry group Dual
{r, q,p}
Icosahedral 120-cell
(faceted 600-cell)
Schlafli-Hess polychoron-wireframe-3.png Ortho solid 007-uniform polychoron 35p-t0.png {3,5,5/2}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png
120
{3,5}
Icosahedron.png
1200
{3}
Regular triangle.svg
720
{5/2}
Star polygon 5-2.svg
120
{5,5/2}
Great dodecahedron.png
4480H4
[5,3,3]
Small stellated 120-cell
Small stellated 120-cell Schlafli-Hess polychoron-wireframe-2.png Ortho solid 010-uniform polychoron p53-t0.png {5/2,5,3}
CDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node 1.png
120
{5/2,5}
Small stellated dodecahedron.png
720
{5/2}
Star polygon 5-2.svg
1200
{3}
Regular triangle.svg
120
{5,3}
Dodecahedron.png
4−480H4
[5,3,3]
Icosahedral 120-cell
Great 120-cell Schlafli-Hess polychoron-wireframe-3.png Ortho solid 008-uniform polychoron 5p5-t0.png {5,5/2,5}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.pngCDel 5.pngCDel node.png
120
{5,5/2}
Great dodecahedron.png
720
{5}
Regular pentagon.svg
720
{5}
Regular pentagon.svg
120
{5/2,5}
Small stellated dodecahedron.png
60H4
[5,3,3]
Self-dual
Grand 120-cell Schlafli-Hess polychoron-wireframe-3.png Ortho solid 009-uniform polychoron 53p-t0.png {5,3,5/2}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png
120
{5,3}
Dodecahedron.png
720
{5}
Regular pentagon.svg
720
{5/2}
Star polygon 5-2.svg
120
{3,5/2}
Great icosahedron.png
200H4
[5,3,3]
Great stellated 120-cell
Great stellated 120-cell Schlafli-Hess polychoron-wireframe-4.png Ortho solid 012-uniform polychoron p35-t0.png {5/2,3,5}
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node 1.png
120
{5/2,3}
Great stellated dodecahedron.png
720
{5/2}
Star polygon 5-2.svg
720
{5}
Regular pentagon.svg
120
{3,5}
Icosahedron.png
200H4
[5,3,3]
Grand 120-cell
Grand stellated 120-cell Schlafli-Hess polychoron-wireframe-4.png Ortho solid 013-uniform polychoron p5p-t0.png {5/2,5,5/2}
CDel node 1.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png
120
{5/2,5}
Small stellated dodecahedron.png
720
{5/2}
Star polygon 5-2.svg
720
{5/2}
Star polygon 5-2.svg
120
{5,5/2}
Great dodecahedron.png
660H4
[5,3,3]
Self-dual
Great grand 120-cell Schlafli-Hess polychoron-wireframe-2.png Ortho solid 011-uniform polychoron 53p-t0.png {5,5/2,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.pngCDel 3.pngCDel node.png
120
{5,5/2}
Great dodecahedron.png
720
{5}
Regular pentagon.svg
1200
{3}
Regular triangle.svg
120
{5/2,3}
Great stellated dodecahedron.png
76−480H4
[5,3,3]
Great icosahedral 120-cell
Great icosahedral 120-cell
(great faceted 600-cell)
Schlafli-Hess polychoron-wireframe-4.png Ortho solid 014-uniform polychoron 3p5-t0.png {3,5/2,5}
CDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.pngCDel 3.pngCDel node 1.png
120
{3,5/2}
Great icosahedron.png
1200
{3}
Regular triangle.svg
720
{5}
Regular pentagon.svg
120
{5/2,5}
Small stellated dodecahedron.png
76480H4
[5,3,3]
Great grand 120-cell
Grand 600-cell Schlafli-Hess polychoron-wireframe-4.png Ortho solid 015-uniform polychoron 33p-t0.png {3,3,5/2}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png
600
{3,3}
Tetrahedron.png
1200
{3}
Regular triangle.svg
720
{5/2}
Star polygon 5-2.svg
120
{3,5/2}
Great icosahedron.png
1910H4
[5,3,3]
Great grand stellated 120-cell
Great grand stellated 120-cell Schlafli-Hess polychoron-wireframe-1.png Ortho solid 016-uniform polychoron p33-t0.png {5/2,3,3}
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node 1.png
120
{5/2,3}
Great stellated dodecahedron.png
720
{5/2}
Star polygon 5-2.svg
1200
{3}
Regular triangle.svg
600
{3,3}
Tetrahedron.png
1910H4
[5,3,3]
Grand 600-cell

There are 4 failed potential regular star 4-polytopes 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.

Five and more dimensions

In five dimensions, a regular polytope can be named as where is the 4-face type, is the cell type, is the face type, and is the face figure, is the edge figure, and is the vertex figure.

A vertex figure (of a 5-polytope) is a 4-polytope, 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 5-polytope exists only if and are regular 4-polytopes.

The space it fits in is based on the expression:

 : Spherical 4-space tessellation or 5-space polytope
 : Euclidean 4-space tessellation
 : 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. [10]

Name Schläfli
Symbol
{p1,...,pn−1}
Coxeter k-facesFacet
type
Vertex
figure
Dual
n-simplex {3n−1}CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel 3.pngCDel node.png {3n−2}{3n−2}Self-dual
n-cube {4,3n−2}CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel 3.pngCDel node.png{4,3n−3}{3n−2}n-orthoplex
n-orthoplex {3n−2,4}CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel 4.pngCDel node.png{3n−2}{3n−3,4}n-cube

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}.

5 dimensions

Name Schläfli
Symbol
{p,q,r,s}
Coxeter
Facets
{p,q,r}
Cells
{p,q}
Faces
{p}
EdgesVerticesFace
figure
{s}
Edge
figure
{r,s}
Vertex
figure

{q,r,s}
5-simplex {3,3,3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6
{3,3,3}
15
{3,3}
20
{3}
156{3}{3,3}{3,3,3}
5-cube {4,3,3,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
10
{4,3,3}
40
{4,3}
80
{4}
8032{3}{3,3}{3,3,3}
5-orthoplex {3,3,3,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
32
{3,3,3}
80
{3,3}
80
{3}
4010{4}{3,4}{3,3,4}
5-simplex t0.svg
5-simplex
5-cube graph.svg
5-cube
5-orthoplex.svg
5-orthoplex

6 dimensions

Name Schläfli VerticesEdgesFacesCells4-faces5-facesχ
6-simplex {3,3,3,3,3}72135352170
6-cube {4,3,3,3,3}6419224016060120
6-orthoplex {3,3,3,3,4}1260160240192640
6-simplex t0.svg
6-simplex
6-cube graph.svg
6-cube
6-orthoplex.svg
6-orthoplex

7 dimensions

Name Schläfli VerticesEdgesFacesCells4-faces5-faces6-facesχ
7-simplex {3,3,3,3,3,3}8285670562882
7-cube {4,3,3,3,3,3}12844867256028084142
7-orthoplex {3,3,3,3,3,4}14842805606724481282
7-simplex t0.svg
7-simplex
7-cube graph.svg
7-cube
7-orthoplex.svg
7-orthoplex

8 dimensions

Name Schläfli VerticesEdgesFacesCells4-faces5-faces6-faces7-facesχ
8-simplex {3,3,3,3,3,3,3}93684126126843690
8-cube {4,3,3,3,3,3,3}2561024179217921120448112160
8-orthoplex {3,3,3,3,3,3,4}1611244811201792179210242560
8-simplex t0.svg
8-simplex
8-cube.svg
8-cube
8-orthoplex.svg
8-orthoplex

9 dimensions

Name Schläfli VerticesEdgesFacesCells4-faces5-faces6-faces7-faces8-facesχ
9-simplex {38}104512021025221012045102
9-cube {4,37}51223044608537640322016672144182
9-orthoplex {37,4}18144672201640325376460823045122
9-simplex t0.svg
9-simplex
9-cube.svg
9-cube
9-orthoplex.svg
9-orthoplex

10 dimensions

Name Schläfli VerticesEdgesFacesCells4-faces5-faces6-faces7-faces8-faces9-facesχ
10-simplex {39}115516533046246233016555110
10-cube {4,38}1024512011520153601344080643360960180200
10-orthoplex {38,4}2018096033608064134401536011520512010240
10-simplex t0.svg
10-simplex
10-cube.svg
10-cube
10-orthoplex.svg
10-orthoplex

...

Non-convex

There are no non-convex regular polytopes in five dimensions or higher, excluding hosotopes formed from lower-dimensional non-convex regular polytopes.

Regular projective polytopes

A projective regular (n+1)-polytope exists when an original regular n-spherical 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. [11]

Even-sided regular polygons have hemi-2n-gon projective polygons, {2p}/2.

There are 4 regular projective polyhedra related to 4 of 5 Platonic solids.

The hemi-cube and hemi-octahedron generalize as hemi-n-cubes and hemi-n-orthoplexes in any dimensions.

Regular projective polyhedra

3-dimensional regular hemi-polytopes
NameCoxeter
McMullen
ImageFacesEdgesVertices χ
Hemi-cube {4,3}/2
{4,3}3
Hemicube.svg 3641
Hemi-octahedron {3,4}/2
{3,4}3
Hemi-octahedron2.png 4631
Hemi-dodecahedron {5,3}/2
{5,3}5
Hemi-dodecahedron.png 615101
Hemi-icosahedron {3,5}/2
{3,5}5
Hemi-icosahedron2.png 101561

Regular projective 4-polytopes

In 4-dimensions 5 of 6 convex regular 4-polytopes generate projective 4-polytopes. The 3 special cases are hemi-24-cell, hemi-600-cell, and hemi-120-cell.

4-dimensional regular hemi-polytopes
NameCoxeter
symbol
McMullen
Symbol
CellsFacesEdgesVertices χ
Hemi-tesseract {4,3,3}/2{4,3,3}44121680
Hemi-16-cell {3,3,4}/2{3,3,4}48161240
Hemi-24-cell {3,4,3}/2{3,4,3}6124848120
Hemi-120-cell {5,3,3}/2{5,3,3}15603606003000
Hemi-600-cell {3,3,5}/2{3,3,5}15300600360600

Regular projective 5-polytopes

There are only 2 convex regular projective hemi-polytopes in dimensions 5 or higher: they are the hemi versions of the regular hypercube and orthoplex. They are tabulated below in dimension 5, for example:

Name Schläfli 4-facesCellsFacesEdgesVertices χ
hemi-penteract {4,3,3,3}/25204040161
hemi-pentacross {3,3,3,4}/21640402051

Apeirotopes

An apeirotope or infinite polytope is a polytope which has infinitely many facets. An n-apeirotope is an infinite n-polytope: a 2-apeirotope or apeirogon is an infinite polygon, a 3-apeirotope or apeirohedron is an infinite polyhedron, etc.

There are two main geometric classes of apeirotope: [12]

One dimension (apeirogons)

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 CDel node 1.pngCDel infin.pngCDel node.png.

... Regular apeirogon.svg ...

It exists as the limit of the p-gon as p tends to infinity, as follows:

Name Monogon Digon Triangle Square Pentagon Hexagon Heptagon p-gon Apeirogon
Schläfli {1}{2}{3}{4}{5}{6}{7}{p}{∞}
SymmetryD1, [ ]D2, [2]D3, [3]D4, [4]D5, [5]D6, [6]D7, [7][p]
Coxeter CDel node.png or CDel node h.pngCDel 2x.pngCDel node.pngCDel node 1.pngCDel 2x.pngCDel node.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel node 1.pngCDel 7.pngCDel node.pngCDel node 1.pngCDel p.pngCDel node.pngCDel node 1.pngCDel infin.pngCDel node.png
Image Monogon.svg Digon.svg Regular triangle.svg Regular quadrilateral.svg Regular pentagon.svg Regular hexagon.svg Regular heptagon.svg Regular apeirogon.svg

Apeirogons in the hyperbolic plane, most notably the regular apeirogon, {∞}, can have a curvature just like finite polygons of the Euclidean plane, with the vertices circumscribed by horocycles or hypercycles rather than circles.

Regular apeirogons that are scaled to converge at infinity have the symbol {∞} and exist on horocycles, while more generally they can exist on hypercycles.

{∞}{πi/λ}
Hyperbolic apeirogon example.png
Apeirogon on horocycle
Pseudogon example.png
Apeirogon on hypercycle

Above are two regular hyperbolic apeirogons in the Poincaré disk model, the right one shows perpendicular reflection lines of divergent fundamental domains, separated by length λ.

Skew apeirogons

A skew apeirogon in two dimensions forms a zig-zag line in the plane. If the zig-zag 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 right-handed.

2-dimensions3-dimensions
Regular zig-zag.svg
Zig-zag apeirogon
Triangular helix.png
Helix apeirogon

Two dimensions (apeirohedra)

Euclidean tilings

There are three regular tessellations of the plane. All three have an Euler characteristic (χ) of 0.

Name Square tiling
(quadrille)
Triangular tiling
(deltille)
Hexagonal tiling
(hextille)
Symmetry p4m, [4,4], (*442)p6m, [6,3], (*632)
Schläfli {p,q}{4,4}{3,6}{6,3}
Coxeter diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
Image Uniform tiling 44-t0.png Uniform tiling 63-t2.png Uniform tiling 63-t0.png

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.

Apeirogonal tiling.png
{∞,2}, CDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.png
Apeirogonal hosohedron.png
{2,∞}, CDel node 1.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png

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 are hyperbolic tilings . There are infinitely many regular tilings in H2. 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 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.

There are infinitely many flat regular 3-apeirotopes (apeirohedra) as regular tilings of the hyperbolic plane, of the form {p,q}, with p+q<pq/2. (previously listed above as tessellations)

  • {3,7}, {3,8}, {3,9} ... {3,∞}
  • {4,5}, {4,6}, {4,7} ... {4,∞}
  • {5,4}, {5,5}, {5,6} ... {5,∞}
  • {6,4}, {6,5}, {6,6} ... {6,∞}
  • {7,3}, {7,4}, {7,5} ... {7,∞}
  • {8,3}, {8,4}, {8,5} ... {8,∞}
  • {9,3}, {9,4}, {9,5} ... {9,∞}
  • ...
  • {∞,3}, {∞,4}, {∞,5} ... {∞,∞}

A sampling:

Regular hyperbolic tiling table
Spherical (improper/Platonic)/Euclidean/hyperbolic (Poincaré disc: compact/paracompact/noncompact) tessellations with their Schläfli symbol
p \ q2345678......iπ/λ
2 Spherical digonal hosohedron.png
{2,2}
CDel node 1.pngCDel 2x.pngCDel node.pngCDel 2x.pngCDel node.png
Spherical trigonal hosohedron.png
{2,3}
CDel node 1.pngCDel 2x.pngCDel node.pngCDel 3.pngCDel node.png
Spherical square hosohedron.png
{2,4}
CDel node 1.pngCDel 2x.pngCDel node.pngCDel 4.pngCDel node.png
Spherical pentagonal hosohedron.png
{2,5}
CDel node 1.pngCDel 2x.pngCDel node.pngCDel 5.pngCDel node.png
Spherical hexagonal hosohedron.png
{2,6}
CDel node 1.pngCDel 2x.pngCDel node.pngCDel 6.pngCDel node.png
Spherical heptagonal hosohedron.png
{2,7}
CDel node 1.pngCDel 2x.pngCDel node.pngCDel 7.pngCDel node.png
Spherical octagonal hosohedron.png
{2,8}
CDel node 1.pngCDel 2x.pngCDel node.pngCDel 8.pngCDel node.png
E2 tiling 22i-4.png
{2,}
CDel node 1.pngCDel 2x.pngCDel node.pngCDel infin.pngCDel node.png
H2 tiling 22i-4.png
{2,iπ/λ}
CDel node 1.pngCDel 2x.pngCDel node.pngCDel ultra.pngCDel node.png
3 Trigonal dihedron.png

{3,2}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 2x.pngCDel node.png
Uniform tiling 332-t0-1-.png
(tetrahedron)
{3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 432-t2.png
(octahedron)
{3,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 532-t2.png
(icosahedron)
{3,5}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
Uniform tiling 63-t2.png
(deltille)
{3,6}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
Uniform tiling 37-t0.png

{3,7}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
Uniform tiling 38-t0.png

{3,8}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 23i-4.png

{3,}
CDel node 1.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
H2 tiling 2312j-4.png

{3,iπ/λ}
CDel node 1.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png
4 Tetragonal dihedron.png

{4,2}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 2x.pngCDel node.png
Uniform tiling 432-t0.png
(cube)
{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 44-t0.svg
(quadrille)
{4,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 45-t0.png

{4,5}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.png
Uniform tiling 46-t0.png

{4,6}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node.png
Uniform tiling 47-t0.png

{4,7}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 7.pngCDel node.png
Uniform tiling 48-t0.png

{4,8}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 24i-4.png

{4,}
CDel node 1.pngCDel 4.pngCDel node.pngCDel infin.pngCDel node.png
H2 tiling 2412j-4.png
{4,iπ/λ}
CDel node 1.pngCDel 4.pngCDel node.pngCDel ultra.pngCDel node.png
5 Pentagonal dihedron.png

{5,2}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 2x.pngCDel node.png
Uniform tiling 532-t0.png
(dodecahedron)
{5,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
H2-5-4-dual.svg

{5,4}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 55-t0.png

{5,5}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.png
Uniform tiling 56-t0.png

{5,6}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 6.pngCDel node.png
Uniform tiling 57-t0.png

{5,7}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 7.pngCDel node.png
Uniform tiling 58-t0.png

{5,8}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 25i-4.png

{5,}
CDel node 1.pngCDel 5.pngCDel node.pngCDel infin.pngCDel node.png
H2 tiling 2512j-4.png
{5,iπ/λ}
CDel node 1.pngCDel 5.pngCDel node.pngCDel ultra.pngCDel node.png
6 Hexagonal dihedron.png

{6,2}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 2x.pngCDel node.png
Uniform tiling 63-t0.svg
(hextille)
{6,3}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 64-t0.png

{6,4}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 65-t0.png

{6,5}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.png
Uniform tiling 66-t2.png

{6,6}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.png
Uniform tiling 67-t0.png

{6,7}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 7.pngCDel node.png
Uniform tiling 68-t0.png

{6,8}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 26i-4.png

{6,}
CDel node 1.pngCDel 6.pngCDel node.pngCDel infin.pngCDel node.png
H2 tiling 2612j-4.png
{6,iπ/λ}
CDel node 1.pngCDel 6.pngCDel node.pngCDel ultra.pngCDel node.png
7 {7,2}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 2x.pngCDel node.png
Heptagonal tiling.svg
{7,3}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 74-t0.png
{7,4}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 75-t0.png
{7,5}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 5.pngCDel node.png
Uniform tiling 76-t0.png
{7,6}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 6.pngCDel node.png
Uniform tiling 77-t2.png
{7,7}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 7.pngCDel node.png
Uniform tiling 78-t0.png
{7,8}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 27i-4.png
{7,}
CDel node 1.pngCDel 7.pngCDel node.pngCDel infin.pngCDel node.png
{7,iπ/λ}
CDel node 1.pngCDel 7.pngCDel node.pngCDel ultra.pngCDel node.png
8 {8,2}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 2x.pngCDel node.png
H2-8-3-dual.svg
{8,3}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 84-t0.png
{8,4}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 85-t0.png
{8,5}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 5.pngCDel node.png
Uniform tiling 86-t0.png
{8,6}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node.png
Uniform tiling 87-t0.png
{8,7}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 7.pngCDel node.png
Uniform tiling 88-t2.png
{8,8}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 28i-4.png
{8,}
CDel node 1.pngCDel 8.pngCDel node.pngCDel infin.pngCDel node.png
{8,iπ/λ}
CDel node 1.pngCDel 8.pngCDel node.pngCDel ultra.pngCDel node.png
...
E2 tiling 22i-1.png
{,2}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 2x.pngCDel node.png
H2-I-3-dual.svg
{,3}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png
H2 tiling 24i-1.png
{,4}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node.png
H2 tiling 25i-1.png
{,5}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 5.pngCDel node.png
H2 tiling 26i-1.png
{,6}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 6.pngCDel node.png
H2 tiling 27i-1.png
{,7}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 7.pngCDel node.png
H2 tiling 28i-1.png
{,8}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 2ii-1.png
{,}
CDel node 1.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.png
H2 tiling 2i12j-4.png
{,iπ/λ}
CDel node 1.pngCDel infin.pngCDel node.pngCDel ultra.pngCDel node.png
...
iπ/λ H2 tiling 22i-1.png
{iπ/λ,2}
CDel node 1.pngCDel ultra.pngCDel node.pngCDel 2x.pngCDel node.png
H2 tiling 2312j-1.png
{iπ/λ,3}
CDel node 1.pngCDel ultra.pngCDel node.pngCDel 3.pngCDel node.png
H2 tiling 2412j-1.png
{iπ/λ,4}
CDel node 1.pngCDel ultra.pngCDel node.pngCDel 4.pngCDel node.png
H2 tiling 2512j-1.png
{iπ/λ,5}
CDel node 1.pngCDel ultra.pngCDel node.pngCDel 5.pngCDel node.png
H2 tiling 2612j-1.png
{iπ/λ,6}
CDel node 1.pngCDel ultra.pngCDel node.pngCDel 6.pngCDel node.png
{iπ/λ,7}
CDel node 1.pngCDel ultra.pngCDel node.pngCDel 7.pngCDel node.png
{iπ/λ,8}
CDel node 1.pngCDel ultra.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 2i12j-1.png
{iπ/λ,}
CDel node 1.pngCDel ultra.pngCDel node.pngCDel infin.pngCDel node.png
H2 tiling 212j12j-1.png

{iπ/λ, iπ/λ}
CDel node 1.pngCDel ultra.pngCDel node.pngCDel ultra.pngCDel node.png

The tilings {p, ∞} have ideal vertices, on the edge of the Poincaré disc model. Their duals {∞, p} have ideal apeirogonal faces, meaning that they are inscribed in horocycles. One could go further (as is done in the table above) and find tilings with ultra-ideal vertices, outside the Poincaré disc, which are dual to tiles inscribed in hypercycles; in what is symbolised {p, iπ/λ} above, infinitely many tiles still fit around each ultra-ideal vertex. [13] (Parallel lines in extended hyperbolic space meet at an ideal point; ultraparallel lines meet at an ultra-ideal point.) [14]

Hyperbolic star-tilings

There are 2 infinite forms of hyperbolic tilings whose faces or vertex figures are star polygons: {m/2, m} and their duals {m, m/2} with m = 7, 9, 11, .... The {m/2, m} tilings are stellations of the {m, 3} tilings while the {m, m/2} dual tilings are facetings of the {3, m} tilings and greatenings of the {m, 3} tilings.

The patterns {m/2, m} and {m, m/2} continue for odd m< 7 as polyhedra: when m = 5, we obtain the small stellated dodecahedron and great dodecahedron, and when m = 3, the case degenerates to a tetrahedron. The other two Kepler–Poinsot polyhedra (the great stellated dodecahedron and great icosahedron) do not have regular hyperbolic tiling analogues. If m is even, depending on how we choose to define {m/2}, we can either obtain degenerate double covers of other tilings or compound tilings.

Name Schläfli Coxeter diagram ImageFace type
{p}
Vertex figure
{q}
Density Symmetry Dual
Order-7 heptagrammic tiling {7/2,7}CDel node 1.pngCDel 7.pngCDel rat.pngCDel d2.pngCDel node.pngCDel 7.pngCDel node.png Hyperbolic tiling 7-2 7.png {7/2}
Star polygon 7-2.svg
{7}
Regular heptagon.svg
3*732
[7,3]
Heptagrammic-order heptagonal tiling
Heptagrammic-order heptagonal tiling {7,7/2}CDel node 1.pngCDel 7.pngCDel node.pngCDel 7.pngCDel rat.pngCDel d2.pngCDel node.png Hyperbolic tiling 7 7-2.png {7}
Regular heptagon.svg
{7/2}
Star polygon 7-2.svg
3*732
[7,3]
Order-7 heptagrammic tiling
Order-9 enneagrammic tiling {9/2,9}CDel node 1.pngCDel 9.pngCDel rat.pngCDel d2.pngCDel node.pngCDel 9.pngCDel node.png Hyperbolic tiling 9-2 9.png {9/2}
Star polygon 9-2.svg
{9}
Regular nonagon.svg
3*932
[9,3]
Enneagrammic-order enneagonal tiling
Enneagrammic-order enneagonal tiling {9,9/2}CDel node 1.pngCDel 9.pngCDel node.pngCDel 9.pngCDel rat.pngCDel d2.pngCDel node.png Hyperbolic tiling 9 9-2.png {9}
Regular nonagon.svg
{9/2}
Star polygon 9-2.svg
3*932
[9,3]
Order-9 enneagrammic tiling
Order-11 hendecagrammic tiling {11/2,11}CDel node 1.pngCDel 11.pngCDel rat.pngCDel d2.pngCDel node.pngCDel 11.pngCDel node.png Order-11 hendecagrammic tiling.png {11/2}
Star polygon 11-2.svg
{11}
Regular hendecagon.svg
3*11.3.2
[11,3]
Hendecagrammic-order hendecagonal tiling
Hendecagrammic-order hendecagonal tiling {11,11/2}CDel node 1.pngCDel 11.pngCDel node.pngCDel 11.pngCDel rat.pngCDel d2.pngCDel node.png Hendecagrammic-order hendecagonal tiling.png {11}
Regular hendecagon.svg
{11/2}
Star polygon 11-2.svg
3*11.3.2
[11,3]
Order-11 hendecagrammic tiling
Order-pp-grammic tiling{p/2,p}CDel node 1.pngCDel p.pngCDel rat.pngCDel d2.pngCDel node.pngCDel p.pngCDel node.png {p/2}{p}3*p32
[p,3]
p-grammic-order p-gonal tiling
p-grammic-order p-gonal tiling{p,p/2}CDel node 1.pngCDel p.pngCDel node.pngCDel p.pngCDel rat.pngCDel d2.pngCDel node.png {p}{p/2}3*p32
[p,3]
Order-pp-grammic tiling

Skew apeirohedra in Euclidean 3-space

There are three regular skew apeirohedra in Euclidean 3-space, with regular skew polygon vertex figures. [15] [16] [17] They share the same vertex arrangement and edge arrangement of 3 convex uniform honeycombs.

  • 6 squares around each vertex: {4,6|4}
  • 4 hexagons around each vertex: {6,4|4}
  • 6 hexagons around each vertex: {6,6|3}
12 "pure" apeirohedra in Euclidean 3-space based on the structure of the cubic honeycomb, {4,3,4}. A p petrie dual operator replaces faces with petrie polygons; d is a dual operator reverses vertices and faces; phk is a kth facetting operator; e is a halving operator, and s skewing halving operator. Pure 3-dimensional apeirohedra chart.png
12 "pure" apeirohedra in Euclidean 3-space based on the structure of the cubic honeycomb, {4,3,4}. A π petrie dual operator replaces faces with petrie polygons; δ is a dual operator reverses vertices and faces; φk is a kth facetting operator; η is a halving operator, and σ skewing halving operator.
Regular skew polyhedra
Mucube.png
{4,6|4}
Muoctahedron.png
{6,4|4}
Mutetrahedron.png
{6,6|3}

There are thirty regular apeirohedra in Euclidean 3-space. [19] 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.

Skew apeirohedra in hyperbolic 3-space

There are 31 regular skew apeirohedra in hyperbolic 3-space: [20]

  • 14 are compact: {8,10|3}, {10,8|3}, {10,4|3}, {4,10|3}, {6,4|5}, {4,6|5}, {10,6|3}, {6,10|3}, {8,8|3}, {6,6|4}, {10,10|3},{6,6|5}, {8,6|3}, and {6,8|3}.
  • 17 are paracompact: {12,10|3}, {10,12|3}, {12,4|3}, {4,12|3}, {6,4|6}, {4,6|6}, {8,4|4}, {4,8|4}, {12,6|3}, {6,12|3}, {12,12|3}, {6,6|6}, {8,6|4}, {6,8|4}, {12,8|3}, {8,12|3}, and {8,8|4}.

Three dimensions (4-apeirotopes)

Tessellations of Euclidean 3-space

Edge framework of cubic honeycomb, {4,3,4} Cubic honeycomb.png
Edge framework of cubic honeycomb, {4,3,4}

There is only one non-degenerate regular tessellation of 3-space ( honeycombs ), {4, 3, 4}: [21]

Name Schläfli
{p,q,r}
Coxeter
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png
Cell
type
{p,q}
Face
type
{p}
Edge
figure
{r}
Vertex
figure

{q,r}
χ Dual
Cubic honeycomb {4,3,4}CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png{4,3}{4}{4}{3,4}0Self-dual

Improper tessellations of Euclidean 3-space

Regular {2,4,4} honeycomb, seen projected into a sphere. Order-4 square hosohedral honeycomb-sphere.png
Regular {2,4,4} honeycomb, seen projected into a sphere.

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 higher-dimensional analogues of the order-2 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} CDel node 1.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png{2,4}{2}{4}{4,4}
{2,3,6} CDel node 1.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png{2,3}{2}{6}{3,6}
{2,6,3} CDel node 1.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png{2,6}{2}{3}{6,3}
{4,4,2} CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.png{4,4}{4}{2}{4,2}
{3,6,2} CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.png{3,6}{3}{2}{6,2}
{6,3,2} CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png{6,3}{6}{2}{3,2}

Tessellations of hyperbolic 3-space

There are ten flat regular honeycombs of hyperbolic 3-space: [22] (previously listed above as tessellations)

  • 4 are compact: {3,5,3}, {4,3,5}, {5,3,4}, and {5,3,5}
  • while 6 are paracompact: {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}.
4 compact regular honeycombs
H3 534 CC center.png
{5,3,4}
H3 535 CC center.png
{5,3,5}
H3 435 CC center.png
{4,3,5}
H3 353 CC center.png
{3,5,3}
4 of 11 paracompact regular honeycombs
H3 344 CC center.png
{3,4,4}
H3 363 FC boundary.png
{3,6,3}
H3 443 FC boundary.png
{4,4,3}
H3 444 FC boundary.png
{4,4,4}

Tessellations of hyperbolic 3-space can be called hyperbolic honeycombs . There are 15 hyperbolic honeycombs in H3, 4 compact and 11 paracompact.

4 compact regular honeycombs
Name Schläfli
Symbol
{p,q,r}
Coxeter
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png
Cell
type
{p,q}
Face
type
{p}
Edge
figure
{r}
Vertex
figure

{q,r}
χ Dual
Icosahedral honeycomb {3,5,3}CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png {3,5} {3}{3} {5,3} 0Self-dual
Order-5 cubic honeycomb {4,3,5}CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png {4,3} {4}{5} {3,5} 0{5,3,4}
Order-4 dodecahedral honeycomb {5,3,4}CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png {5,3} {5}{4} {3,4} 0{4,3,5}
Order-5 dodecahedral honeycomb {5,3,5}CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png {5,3} {5}{5} {3,5} 0Self-dual

There are also 11 paracompact H3 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}.

11 paracompact regular honeycombs
Name Schläfli
Symbol
{p,q,r}
Coxeter
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png
Cell
type
{p,q}
Face
type
{p}
Edge
figure
{r}
Vertex
figure

{q,r}
χ Dual
Order-6 tetrahedral honeycomb {3,3,6}CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png {3,3} {3}{6} {3,6} 0{6,3,3}
Hexagonal tiling honeycomb {6,3,3}CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png {6,3} {6}{3} {3,3} 0{3,3,6}
Order-4 octahedral honeycomb {3,4,4}CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png {3,4} {3}{4} {4,4} 0{4,4,3}
Square tiling honeycomb {4,4,3}CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png {4,4} {4}{3} {4,3} 0{3,3,4}
Triangular tiling honeycomb {3,6,3}CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png {3,6} {3}{3} {6,3} 0Self-dual
Order-6 cubic honeycomb {4,3,6}CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png {4,3} {4}{4} {3,6} 0{6,3,4}
Order-4 hexagonal tiling honeycomb {6,3,4}CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png {6,3} {6}{4} {3,4} 0{4,3,6}
Order-4 square tiling honeycomb {4,4,4}CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png {4,4} {4}{4} {4,4} 0Self-dual
Order-6 dodecahedral honeycomb {5,3,6}CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png {5,3} {5}{5} {3,6} 0{6,3,5}
Order-5 hexagonal tiling honeycomb {6,3,5}CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png {6,3} {6}{5} {3,5} 0{5,3,6}
Order-6 hexagonal tiling honeycomb {6,3,6}CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png {6,3} {6}{6} {3,6} 0Self-dual

Noncompact solutions exist as Lorentzian Coxeter groups, and can be visualized with open domains in hyperbolic space (the fundamental tetrahedron having ultra-ideal vertices). All honeycombs with hyperbolic cells or vertex figures and do not have 2 in their Schläfli symbol are noncompact.

Spherical (improper/Platonic)/Euclidean/hyperbolic(compact/paracompact/noncompact) honeycombs {p,3,r}
{p,3} \ r2345678...
{2,3}
Spherical trigonal hosohedron.png
Spherical trigonal hosohedron.png
{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}
Uniform polyhedron-33-t0.png
Tetrahedron.png
{3,3,2}
Schlegel wireframe 5-cell.png
{3,3,3}
Schlegel wireframe 16-cell.png
{3,3,4}
Schlegel wireframe 600-cell vertex-centered.png
{3,3,5}
H3 336 CC center.png
{3,3,6}
Hyperbolic honeycomb 3-3-7 poincare cc.png
{3,3,7}
Hyperbolic honeycomb 3-3-8 poincare cc.png
{3,3,8}
Hyperbolic honeycomb 3-3-i poincare cc.png
{3,3,}
{4,3}
Uniform polyhedron-43-t0.svg
Hexahedron.png
{4,3,2}
Schlegel wireframe 8-cell.png
{4,3,3}
Cubic honeycomb.png
{4,3,4}
H3 435 CC center.png
{4,3,5}
H3 436 CC center.png
{4,3,6}
Hyperbolic honeycomb 4-3-7 poincare cc.png
{4,3,7}
Hyperbolic honeycomb 4-3-8 poincare cc.png
{4,3,8}
Hyperbolic honeycomb 4-3-i poincare cc.png
{4,3,}
{5,3}
Uniform polyhedron-53-t0.svg
Dodecahedron.png
{5,3,2}
Schlegel wireframe 120-cell.png
{5,3,3}
H3 534 CC center.png
{5,3,4}
H3 535 CC center.png
{5,3,5}
H3 536 CC center.png
{5,3,6}
Hyperbolic honeycomb 5-3-7 poincare cc.png
{5,3,7}
Hyperbolic honeycomb 5-3-8 poincare cc.png
{5,3,8}
Hyperbolic honeycomb 5-3-i poincare cc.png
{5,3,}
{6,3}
Uniform tiling 63-t0.svg
Uniform tiling 63-t0.png
{6,3,2}
H3 633 FC boundary.png
{6,3,3}
H3 634 FC boundary.png
{6,3,4}
H3 635 FC boundary.png
{6,3,5}
H3 636 FC boundary.png
{6,3,6}
Hyperbolic honeycomb 6-3-7 poincare.png
{6,3,7}
Hyperbolic honeycomb 6-3-8 poincare.png
{6,3,8}
Hyperbolic honeycomb 6-3-i poincare.png
{6,3,}
{7,3}
Heptagonal tiling.svg
{7,3,2} Hyperbolic honeycomb 7-3-3 poincare vc.png
{7,3,3}
Hyperbolic honeycomb 7-3-4 poincare vc.png
{7,3,4}
Hyperbolic honeycomb 7-3-5 poincare vc.png
{7,3,5}
Hyperbolic honeycomb 7-3-6 poincare.png
{7,3,6}
Hyperbolic honeycomb 7-3-7 poincare.png
{7,3,7}
Hyperbolic honeycomb 7-3-8 poincare.png
{7,3,8}
Hyperbolic honeycomb 7-3-i poincare.png
{7,3,}
{8,3}
H2-8-3-dual.svg
{8,3,2} Hyperbolic honeycomb 8-3-3 poincare vc.png
{8,3,3}
Hyperbolic honeycomb 8-3-4 poincare vc.png
{8,3,4}
Hyperbolic honeycomb 8-3-5 poincare vc.png
{8,3,5}
Hyperbolic honeycomb 8-3-6 poincare.png
{8,3,6}
Hyperbolic honeycomb 8-3-7 poincare.png
{8,3,7}
Hyperbolic honeycomb 8-3-8 poincare.png
{8,3,8}
Hyperbolic honeycomb 8-3-i poincare.png
{8,3,}
... {,3}
H2-I-3-dual.svg
{,3,2} Hyperbolic honeycomb i-3-3 poincare vc.png
{,3,3}
Hyperbolic honeycomb i-3-4 poincare vc.png
{,3,4}
Hyperbolic honeycomb i-3-5 poincare vc.png
{,3,5}
Hyperbolic honeycomb i-3-6 poincare.png
{,3,6}
Hyperbolic honeycomb i-3-7 poincare.png
{,3,7}
Hyperbolic honeycomb i-3-8 poincare.png
{,3,8}
Hyperbolic honeycomb i-3-i poincare.png
{,3,}
{p,4,r}
{p,4} \ r23456
{2,4}
Spherical square hosohedron.png
Spherical square hosohedron.png
{2,4,2}
{2,4,3} Order-4 square hosohedral honeycomb-sphere.png
{2,4,4}
{2,4,5} {2,4,6} {2,4,}
{3,4}
Uniform polyhedron-43-t2.svg
Octahedron.png
{3,4,2}
Schlegel wireframe 24-cell.png
{3,4,3}
H3 344 CC center.png
{3,4,4}
Hyperbolic honeycomb 3-4-5 poincare cc.png
{3,4,5}
Hyperbolic honeycomb 3-4-6 poincare cc.png
{3,4,6}
Hyperbolic honeycomb 3-4-i poincare cc.png
{3,4,}
{4,4}
Uniform tiling 44-t0.svg
Uniform tiling 44-t0.png
{4,4,2}
H3 443 FC boundary.png
{4,4,3}
H3 444 FC boundary.png
{4,4,4}
Hyperbolic honeycomb 4-4-5 poincare.png
{4,4,5}
Hyperbolic honeycomb 4-4-6 poincare.png
{4,4,6}
Hyperbolic honeycomb 4-4-i poincare.png
{4,4,}
{5,4}
H2-5-4-dual.svg
{5,4,2} Hyperbolic honeycomb 5-4-3 poincare vc.png
{5,4,3}
Hyperbolic honeycomb 5-4-4 poincare.png
{5,4,4}
Hyperbolic honeycomb 5-4-5 poincare.png
{5,4,5}
Hyperbolic honeycomb 5-4-6 poincare.png
{5,4,6}
Hyperbolic honeycomb 5-4-i poincare.png
{5,4,}
{6,4}
Uniform tiling 55-t0.png
{6,4,2} Hyperbolic honeycomb 6-4-3 poincare vc.png
{6,4,3}
Hyperbolic honeycomb 6-4-4 poincare.png
{6,4,4}
Hyperbolic honeycomb 6-4-5 poincare.png
{6,4,5}
Hyperbolic honeycomb 6-4-6 poincare.png
{6,4,6}
Hyperbolic honeycomb 6-4-i poincare.png
{6,4,}
{,4}
H2 tiling 24i-1.png
{,4,2} Hyperbolic honeycomb i-4-3 poincare vc.png
{,4,3}
Hyperbolic honeycomb i-4-4 poincare.png
{,4,4}
Hyperbolic honeycomb i-4-5 poincare.png
{,4,5}
Hyperbolic honeycomb i-4-6 poincare.png
{,4,6}
Hyperbolic honeycomb i-4-i poincare.png
{,4,}
{p,5,r}
{p,5} \ r23456
{2,5}
Spherical pentagonal hosohedron.png
Spherical pentagonal hosohedron.png
{2,5,2}
{2,5,3} {2,5,4} {2,5,5} {2,5,6} {2,5,}
{3,5}
Uniform polyhedron-53-t2.svg
Icosahedron.png
{3,5,2}
H3 353 CC center.png
{3,5,3}
Hyperbolic honeycomb 3-5-4 poincare cc.png
{3,5,4}
Hyperbolic honeycomb 3-5-5 poincare cc.png
{3,5,5}
Hyperbolic honeycomb 3-5-6 poincare cc.png
{3,5,6}
Hyperbolic honeycomb 3-5-i poincare cc.png
{3,5,}
{4,5}
Uniform tiling 45-t0.png
{4,5,2} Hyperbolic honeycomb 4-5-3 poincare vc.png
{4,5,3}
Hyperbolic honeycomb 4-5-4 poincare.png
{4,5,4}
Hyperbolic honeycomb 4-5-5 poincare.png
{4,5,5}
Hyperbolic honeycomb 4-5-6 poincare.png
{4,5,6}
Hyperbolic honeycomb 4-5-i poincare.png
{4,5,}
{5,5}
Uniform tiling 55-t0.png
{5,5,2} Hyperbolic honeycomb 5-5-3 poincare vc.png
{5,5,3}
Hyperbolic honeycomb 5-5-4 poincare.png
{5,5,4}
Hyperbolic honeycomb 5-5-5 poincare.png
{5,5,5}
Hyperbolic honeycomb 5-5-6 poincare.png
{5,5,6}
Hyperbolic honeycomb 5-5-i poincare.png
{5,5,}
{6,5}
Uniform tiling 65-t0.png
{6,5,2} Hyperbolic honeycomb 6-5-3 poincare vc.png
{6,5,3}
Hyperbolic honeycomb 6-5-4 poincare.png
{6,5,4}
Hyperbolic honeycomb 6-5-5 poincare.png
{6,5,5}
Hyperbolic honeycomb 6-5-6 poincare.png
{6,5,6}
Hyperbolic honeycomb 6-5-i poincare.png
{6,5,}
{,5}
H2 tiling 25i-1.png
{,5,2} Hyperbolic honeycomb i-5-3 poincare vc.png
{,5,3}
Hyperbolic honeycomb i-5-4 poincare.png
{,5,4}
Hyperbolic honeycomb i-5-5 poincare.png
{,5,5}
Hyperbolic honeycomb i-5-6 poincare.png
{,5,6}