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

Overview[edit]

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

Dimension Convex Nonconvex Convex
Euclidean
tessellations
Convex
hyperbolic
tessellations
Nonconvex
hyperbolic
tessellations
Hyperbolic Tessellations
with infinite cells
and/or vertex figures
Abstract
Polytopes
1 1 line segment 0 1 0 0 0 1
2 polygons star polygons 1 1 0 0
3 5 Platonic solids 4 Kepler–Poinsot solids 3 tilings
4 6 convex polychora 10 Schläfli–Hess polychora 1 honeycomb 4 0 11
5 3 convex 5-polytopes 0 3 tetracombs 5 4 2
6 3 convex 6-polytopes 0 1 pentacombs 0 0 5
7+ 3 0 1 0 0 0

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

Tessellations[edit]

The classical convex polytopes may be considered tessellations, or tilings, of spherical space. Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three-dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.

One-dimensional regular polytope[edit]

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[edit]

The two-dimensional polytopes are called polygons. Regular polygons are equilateral and cyclic. A p-gonal regular polygon is represented by Schläfli symbol {p}.

Usually only convex polygons are considered regular, but star polygons, like the pentagram, can also be considered regular. They use the same vertices as the convex forms, but connect in an alternate connectivity which passes around the circle more than once to complete.

Star polygons should be called nonconvex rather than concave because the intersecting edges do not generate new vertices and all the vertices exist on a circle boundary.

Convex[edit]

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

Name Triangle
(2-simplex)
(trig)
Square
(2-orthoplex)
(2-cube)
(square)
Pentagon
(peg)
Hexagon
(hig)
Heptagon
(heg)
Octagon
(oc)
Schläfli {3} {4} {5} {6} {7} {8}
Coxeter CDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node.png CDel node 1.pngCDel 6.pngCDel node.png CDel node 1.pngCDel 7.pngCDel node.png CDel 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 Enneagon
(en)
Decagon
(dec)
Hendecagon
(heng)
Dodecagon
(dog)
Tridecagon
(tad)
Tetradecagon
(ted)
Schläfli {9} {10} {11} {12} {13} {14}
Dynkin CDel node 1.pngCDel 9.pngCDel node.png CDel node 1.pngCDel 10.pngCDel node.png CDel node 1.pngCDel 11.pngCDel node.png CDel node 1.pngCDel 12.pngCDel node.png CDel node 1.pngCDel 13.pngCDel node.png CDel 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
(ped)
Hexadecagon
(hed)
Heptadecagon Octadecagon Enneadecagon Icosagon ...p-gon
Schläfli {15} {16} {17} {18} {19} {20} {p}
Dynkin CDel node 1.pngCDel 15.pngCDel node.png CDel node 1.pngCDel 16.pngCDel node.png CDel node 1.pngCDel 17.pngCDel node.png CDel node 1.pngCDel 18.pngCDel node.png CDel node 1.pngCDel 19.pngCDel node.png CDel node 1.pngCDel 20.pngCDel node.png CDel 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

Degenerate (spherical)[edit]

The regular henagon {1} and regular digon {2} can be considered degenerate regular polygons. They can exist nondegenerately in non-Euclidean spaces like on the surface of a sphere or torus.

Name Henagon Digon
Schläfli symbol {1} {2}
Coxeter diagram CDel node.png CDel node 1.png
Image Henagon.svg Digon.svg

Non-convex[edit]

There exist infinitely many non-convex regular polytopes in two dimensions, whose Schläfli symbols consist of rational numbers {n/m}. They are called star polygons and share the same vertex arrangements of the convex regular polygons.

In general, for any natural number n, there are n-pointed non-convex regular polygonal stars with Schläfli symbols {n/m} for all m such that m < n/2 (strictly speaking {n/m}={n/(n−m)}) and m and n are coprime.

Name Pentagram Heptagrams Octagram Enneagrams Decagram ...n-agrams
Schläfli {5/2}
(star)
{7/2}
(hag)
{7/3}
(gahg)
{8/3}
(og)
{9/2}
(eng)
{9/4}
(geng)
{10/3}
(dag)
{p/q}
Coxeter CDel node 1.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png CDel node 1.pngCDel 7.pngCDel rat.pngCDel d2.pngCDel node.png CDel node 1.pngCDel 7.pngCDel rat.pngCDel d3.pngCDel node.png CDel node 1.pngCDel 8.pngCDel rat.pngCDel d3.pngCDel node.png CDel node 1.pngCDel 9.pngCDel rat.pngCDel d2.pngCDel node.png CDel node 1.pngCDel 9.pngCDel rat.pngCDel d4.pngCDel node.png CDel node 1.pngCDel 10.pngCDel rat.pngCDel d3.pngCDel node.png CDel 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  

Tessellation[edit]

There is one tessellation of the line, giving one polytope, the (two-dimensional) apeirogon (aze). This has infinitely many vertices and edges. Its Schläfli symbol is {∞}, and Coxeter diagram CDel node 1.pngCDel infin.pngCDel node.png.

...Regular apeirogon.png...

Three-dimensional regular polytopes[edit]

In three dimensions, polytopes are called polyhedra:

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

A vertex figure (of a polyhedron) is a polygon, seen by connecting those vertices which are one edge away from a given vertex. For regular polyhedra, this vertex figure is always a regular (and planar) polygon.

Existence of a regular polyhedron \{p,q\} is constrained by an inequality, related to the vertex figure's angle defect:

1/p + 1/q > 1/2 : Polyhedron (existing in Euclidean 3-space)
1/p + 1/q = 1/2 : Euclidean plane tiling
1/p + 1/q < 1/2 : Hyperbolic plane tiling

By enumerating the permutations, we find 5 convex forms, 4 nonconvex forms and 3 plane tilings, all with polygons \{p\} and \{q\} limited to: \{3\}, \{4\}, \{5\}, \{\frac{5}{2}\}, and \{6\}.

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

Convex[edit]

The convex regular polyhedra are called the 5 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
(transparent)
Image
(solid)
Image
(sphere)
Faces
{p}
Edges Vertices
{q}
Symmetry Dual
Tetrahedron
(3-simplex)
(Triangular pyramid)
(tet)
{3,3} CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png Tetrahedron.svg Tetrahedron.png Uniform tiling 332-t0-1-.png 4
{3}
6 4
{3}
Td (self)
Cube
(3-cube)
(Hexahedron)
(cube)
{4,3} CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png Hexahedron.svg Hexahedron.png Uniform tiling 432-t0.png 6
{4}
12 8
{3}
Oh Octahedron
Octahedron
(3-orthoplex)
(oct)
{3,4} CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png Octahedron.svg Octahedron.png Uniform tiling 432-t2.png 8
{3}
12 6
{4}
Oh Cube
Dodecahedron
(doe)
{5,3} CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png POV-Ray-Dodecahedron.svg Dodecahedron.png Uniform tiling 532-t0.png 12
{5}
30 20
{3}
Ih Icosahedron
Icosahedron
(ike)
{3,5} CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png Icosahedron.svg Icosahedron.png Uniform tiling 532-t2.png 20
{3}
30 12
{5}
Ih Dodecahedron

Degenerate (spherical)[edit]

In spherical geometry, the hosohedra {2,n}, dihedra {n,2} and henagonal henahedron {1,1} can be considered regular polyhedra (tilings of the sphere).

Some include:

Name Schläfli
{p,q}
Coxeter
diagram
Image
(sphere)
Faces
{p}
Edges Vertices
{q}
Symmetry Dual
Henagonal henahedron {1,1} CDel node.png Spherical henagonal henahedron.png 1
{1}
0 1
{1}
C1
(*1)
Self
Henagonal dihedron {1,2} CDel node 1.pngCDel 2.pngCDel node.png Hengonal dihedron.png 2
{1}
1 1
{2}
C1v
(*22)
Henagonal hosohedron
Henagonal hosohedron {2,1} CDel node.pngCDel 2.pngCDel node.png Henagonal hosohedron.png 1
{2}
1 2
{1}
C1v
(*22)
Henagonal dihedron
Digonal dihedron
Digonal hosohedron
{2,2} CDel node 1.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png Digonal dihedron.png 2
{2}
2 2
{2}
D2h
(*222)
Self
Trigonal hosohedron {2,3} CDel node 1.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png Trigonal hosohedron.png 3
{2}
3 2
{3}
D3h
(*322)
Trigonal dihedron
Trigonal dihedron {3,2} CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png Trigonal dihedron.png 2
{3}
3 3
{2}
D3h
(*322)
Trigonal hosohedron
Hexagonal hosohedron {2,6} CDel node 1.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.png Hexagonal hosohedron.png 6
{2}
6 2
{6}
D6h
(*622)
Hexagonal dihedron
Hexagonal dihedron {6,2} CDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.png Hexagonal dihedron.png 2
{6}
6 6
{2}
D6h
(*622)
Hexagonal hosohedron

Non-convex[edit]

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

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

Name Image
(transparent)
Image
(solid)
Image
(sphere)
Stellation
diagram
Schläfli
{p,q} and
Coxeter-Dynkin
Faces
{p}
Edges Vertices
{q}
verf.
χ Density Symmetry Dual
Small stellated dodecahedron
(sissid)
SmallStellatedDodecahedron.jpg Small stellated dodecahedron.png 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}
Pentagram.svg
30 12
{5}
Pentagon.svg
−6 3 Ih Great dodecahedron
Great dodecahedron
(gad)
GreatDodecahedron.jpg Great dodecahedron.png 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}
Pentagon.svg
30 12
{5/2}
Pentagram.svg
−6 3 Ih Small stellated dodecahedron
Great stellated dodecahedron
(gissid)
GreatStellatedDodecahedron.jpg Great stellated dodecahedron.png 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}
Pentagram.svg
30 20
{3}
Triangle.Equilateral.svg
2 7 Ih Great icosahedron
Great icosahedron
(gike)
GreatIcosahedron.jpg Great icosahedron.png Great icosahedron tiling.png Sixteenth stellation of icosahedron facets.png {3,5/2}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png
20
{3}
Triangle.Equilateral.svg
30 12
{5/2}
Pentagram.svg
2 7 Ih Great stellated dodecahedron

Tessellations[edit]

Euclidean tilings[edit]

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

Name Square tiling
(Quadrille)
(squat)
Triangular tiling
(Deltille)
(trat)
Hexagonal tiling
(Hextille)
(hexat)
Schläfli {p,q} {4,4} {3,6} {6,3}
Coxeter diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png CDel 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
Symmetry *442
(p4m)
*632
(p6m)

There is one degenerate regular tiling, {∞,2} (azat), 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[edit]

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[edit]

Tessellations of hyperbolic 2-space can be called 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 (pqr) 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.

A sampling:

Spherical (Platonic)/Euclidean/hyperbolic (Poincaré disc) tessellations with their Schläfli symbol
p \ q 3 4 5 6 7 8 ...
3 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
4 Uniform tiling 432-t0.png
(cube)
{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 44-t0.png
(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
5 Uniform tiling 532-t0.png
(dodecahedron)
{5,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 54-t0.png

{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
6 Uniform tiling 63-t0.png
(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
7 Uniform tiling 73-t0.png
{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
8 Uniform tiling 83-t0.png
{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
...
H2 tiling 23i-1.png
{∞,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
Hyperbolic star-tilings[edit]

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, we obtain the tetrahedron. If m is even, depending on how we choose to define {m/2}, we can either obtain degenerate double covers of other tilings or compound tilings.

Name Schläfli Coxeter diagram Image Face type
{p}
Vertex figure
{q}
Density Symmetry Dual
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}
Heptagon.svg
3 *732 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}
Heptagon.svg
{7/2}
Star polygon 7-2.svg
3 *732 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}
Nonagon.svg
3 *932 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}
Nonagon.svg
{9/2}
Star polygon 9-2.svg
3 *932 Order-9 enneagrammic tiling
Order-p p-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-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 Order-p p-grammic tiling

Four-dimensional regular polytopes[edit]

Regular 4-polytopes (called polychora) with Schläfli symbol \{p,q,r\} have cells of type \{p,q\}, faces of type \{p\}, edge figures \{r\}, and vertex figures \{q,r\}.

  • A vertex figure (of a polychoron) is a polyhedron, seen by the arrangement of neighboring vertices around a given vertex. For regular polychora, this vertex figure is a regular polyhedron.
  • An edge figure is a polygon, seen by the arrangement of faces around an edge. For regular polychora, this edge figure will always be a regular polygon.

The existence of a regular polychoron \{p,q,r\} is constrained by the existence of the regular polyhedra \{p,q\}, \{q,r\}.

Each will exist in a space dependent upon this expression:

\sin \left ( \frac{\pi}{p} \right ) \sin \left(\frac{\pi}{r}\right) - \cos\left(\frac{\pi}{q}\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.

Convex[edit]

The 6 convex regular polychora are shown in the table below. All these polychora 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)
(Pentachoron)
(pen)
{3,3,3} CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 5
{3,3}
10
{3}
10
{3}
5
{3,3}
(self)
8-cell
(4-cube)
(Tesseract)
(tes)
{4,3,3} CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 8
{4,3}
24
{4}
32
{3}
16
{3,3}
16-cell
16-cell
(4-orthoplex)
(hex)
{3,3,4} CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 16
{3,3}
32
{3}
24
{4}
8
{3,4}
Tesseract
24-cell
(ico)
{3,4,3} CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 24
{3,4}
96
{3}
96
{3}
24
{4,3}
(self)
120-cell
(hi)
{5,3,3} CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 120
{5,3}
720
{5}
1200
{3}
600
{3,3}
600-cell
600-cell
(ex)
{3,3,5} CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png 600
{3,3}
1200
{3}
720
{5}
120
{3,5}
120-cell
5-cell 8-cell 16-cell 24-cell 120-cell 600-cell
{3,3,3} {4,3,3} {3,3,4} {3,4,3} {5,3,3} {3,3,5}
Wireframe (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

Degenerate (spherical)[edit]

Dichora and hosochora exist as regular tessellations of the 3-sphere.

Regular dichora (2 facets) include: {3,3,2}, {3,4,2}, {4,3,2}, {5,3,2}, {3,5,2}, {p,2,2}, and their hosochora duals (2 vertices): {2,3,3}, {2,4,3}, {2,3,4}, {2,3,5}, {2,5,3}, {2,2,p}. Polychora of the form {2,p,2} are both dichora and hosochora.

Non-convex[edit]

There are ten regular star polychora, which can be called Schläfli–Hess polychoron. 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)[1].

There are 4 unique edge arrangements and 7 unique face arrangements from these 10 nonconvex polychora, shown as orthogonal projections:

Name
Wireframe Solid Schläfli
{p, q, r}
Coxeter–Dynkin
Cells
{p, q}
Faces
{p}
Edges
{r}
Vertices
{q, r}
Density χ Symmetry group Dual
{r, q,p}
Icosahedral 120-cell
(faceted 600-cell)
(fix)
Schläfli-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}
Triangle.Equilateral.svg
720
{5/2}
Pentagram.svg
120
{5,5/2}
Great dodecahedron.png
4 480 H4 Small stellated 120-cell
Small stellated 120-cell
(sishi)
Schläfli-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}
Pentagram.svg
1200
{3}
Triangle.Equilateral.svg
120
{5,3}
Dodecahedron.png
4 −480 H4 Icosahedral 120-cell
Great 120-cell
(gohi)
Schläfli-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}
Pentagon.svg
720
{5}
Pentagon.svg
120
{5/2,5}
Small stellated dodecahedron.png
6 0 H4 Self-dual
Grand 120-cell
(gahi)
Schläfli-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}
Pentagon.svg
720
{5/2}
Pentagram.svg
120
{3,5/2}
Great icosahedron.png
20 0 H4 Great stellated 120-cell
Great stellated 120-cell
(gishi)
Schläfli-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}
Pentagram.svg
720
{5}
Pentagon.svg
120
{3,5}
Icosahedron.png
20 0 H4 Grand 120-cell
Grand stellated 120-cell
(gashi)
Schläfli-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}
Pentagram.svg
720
{5/2}
Pentagram.svg
120
{5,5/2}
Great dodecahedron.png
66 0 H4 Self-dual
Great grand 120-cell
(gaghi)
Schläfli-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}
Pentagon.svg
1200
{3}
Triangle.Equilateral.svg
120
{5/2,3}
Great stellated dodecahedron.png
76 −480 H4 Great icosahedral 120-cell
Great icosahedral 120-cell
(great faceted 600-cell)
(gofix)
Schläfli-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}
Triangle.Equilateral.svg
720
{5}
Pentagon.svg
120
{5/2,5}
Small stellated dodecahedron.png
76 480 H4 Great grand 120-cell
Grand 600-cell
(gax)
Schläfli-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}
Triangle.Equilateral.svg
720
{5/2}
Pentagram.svg
120
{3,5/2}
Great icosahedron.png
191 0 H4 Great grand stellated 120-cell
Great grand stellated 120-cell
(gogishi)
Schläfli-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}
Pentagram.svg
1200
{3}
Triangle.Equilateral.svg
600
{3,3}
Tetrahedron.png
191 0 H4 Grand 600-cell

There are 4 failed potential nonconvex regular polychora permutations: {3,5/2,3}, {4,3,5/2}, {5/2,3,4}, {5/2,3,5/2}. Their cells and vertex figures exist, but they do not cover a hypersphere with a finite number of repetitions.

Tessellations of Euclidean 3-space[edit]

Edge framework of cubic honeycomb, {4,3,4}

There is only one non-degenerate regular tessellation of 3-space (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
Cubic honeycomb
(chon)
{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} 0 Self-dual

Degenerate tessellations of Euclidean 3-space[edit]

Regular {2,4,4} honeycomb, seen projected into a sphere.

There are six degenerate 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 degenerate 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[edit]

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.

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} 0 Self-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} 0 Self-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}.

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} 0 Self-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} {3,4} 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} 0 {4,4,4}
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} {3,5} 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} 0 Self-dual

A subset of unform polychora and honeycombs are contained in the form {p,3,q} for values 3 to 6. Noncompact solutions exist as Lorentzian Coxeter groups for p or q greater than 6, and can be visualized with open domains in hyperbolic space, and a few are drawn below as tilings on the ideal half-space plane.

Spherical /Euclidean/hyperbolic(compact/paracompact/noncompact) honeycombs with their Schläfli symbol {p,3,q}
p \ q 3 4 5 6 7 8 ... ∞
3
Uniform polyhedron-33-t0.png
Schlegel wireframe 5-cell.png
{3,3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Schlegel wireframe 16-cell.png
{3,3,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Schlegel wireframe 600-cell vertex-centered.png
{3,3,5}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
H3 336 CC center.png
{3,3,6}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
H3 337 UHS plane at infinity.png
{3,3,7}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
H3 338 UHS plane at infinity.png
{3,3,8}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
H3 33inf UHS plane at infinity.png
{3,3,∞}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
4
Uniform polyhedron-43-t0.png
Schlegel wireframe 8-cell.png
{4,3,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Cubic honeycomb.png
{4,3,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
H3 435 CC center.png
{4,3,5}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
H3 436 CC center.png
{4,3,6}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png

{4,3,7}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png

{4,3,8}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png

{4,3,∞}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
5
Uniform polyhedron-53-t0.png
Schlegel wireframe 120-cell.png
{5,3,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
H3 534 CC center.png
{5,3,4}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
H3 535 CC center.png
{5,3,5}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
H3 536 CC center.png
{5,3,6}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png

{5,3,7}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png

{5,3,8}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
H3 53i UHS plane at infinity.png
{5,3,∞}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
6
Uniform tiling 63-t0.png
H3 633 FC boundary.png
{6,3,3}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
H3 634 FC boundary.png
{6,3,4}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
H3 635 FC boundary.png
{6,3,5}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
H3 636 FC boundary.png
{6,3,6}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
H3 637 UHS plane at infinity view 1.png
{6,3,7}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png

{6,3,8}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
H3 63i UHS plane at infinity.png
{6,3,∞}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
7
H2 tiling 237-1.png
{7,3,3}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{7,3,4}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{7,3,5}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{7,3,6}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
{7,3,7}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
{7,3,8}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
{7,3,∞}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
8
H2 tiling 238-1.png
{8,3,3}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{8,3,4}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{8,3,5}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{8,3,6}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
{8,3,7}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
{8,3,8}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
{8,3,∞}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
...
H2 tiling 23i-1.png
{∞,3,3}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{∞,3,4}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{∞,3,5}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{∞,3,6}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
{∞,3,7}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
{∞,3,8}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
{∞,3,∞}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png

Five-dimensional regular polytopes and higher[edit]

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 tetracomb.

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 5-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:

\frac{\cos^2\left(\frac{\pi}{q}\right)}{\sin^2\left(\frac{\pi}{p}\right)} + \frac{\cos^2\left(\frac{\pi}{r}\right)}{\sin^2\left(\frac{\pi}{s}\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. There are no non-convex regular polytopes in five dimensions or higher.

Higher-dimensional polytopes have sometimes received names. 6-polytopes have sometimes been called polypeta, 7-polytopes polyexa, 8-polytopes polyzetta, and 9-polytopes polyyotta.

Convex[edit]

In dimensions 5 and higher, there are only three kinds of convex regular polytopes.[1]

Name Schläfli
Symbol
{p1,...,pn−1}
Coxeter k-faces Facet
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 {{n+1} \choose {k+1}} {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 2^{n-k}{n \choose k} {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 2^{k+1}{n \choose {k+1}} {3n−2} {3n−3,4} n-cube

5 dimensions[edit]

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

{q,r,s}
5-simplex (hix) {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}
15 6 {3} {3,3} {3,3,3}
5-cube (pent) {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}
80 32 {3} {3,3} {3,3,3}
5-orthoplex (tac) {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}
40 10 {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[edit]

Name Schläfli
symbol
Vertices Edges Faces Cells 4-faces 5-faces χ
6-simplex (hop) {3,3,3,3,3} 7 21 35 35 21 7 0
6-cube (ax) {4,3,3,3,3} 64 192 240 160 60 12 0
6-orthoplex (gee) {3,3,3,3,4} 12 60 160 240 192 64 0
6-simplex t0.svg
6-simplex
6-cube graph.svg
6-cube
6-orthoplex.svg
6-orthoplex

7 dimensions[edit]

Name Schläfli
symbol
Vertices Edges Faces Cells 4-faces 5-faces 6-faces χ
7-simplex (oca) {3,3,3,3,3,3} 8 28 56 70 56 28 8 2
7-cube (hept) {4,3,3,3,3,3} 128 448 672 560 280 84 14 2
7-orthoplex (zee) {3,3,3,3,3,4} 14 84 280 560 672 448 128 2
7-simplex t0.svg
7-simplex
7-cube graph.svg
7-cube
7-orthoplex.svg
7-orthoplex

8 dimensions[edit]

Name Schläfli
symbol
Vertices Edges Faces Cells 4-faces 5-faces 6-faces 7-faces χ
8-simplex (ene) {3,3,3,3,3,3,3} 9 36 84 126 126 84 36 9 0
8-cube (octo) {4,3,3,3,3,3,3} 256 1024 1792 1792 1120 448 112 16 0
8-orthoplex (ek) {3,3,3,3,3,3,4} 16 112 448 1120 1792 1792 1024 256 0
8-simplex t0.svg
8-simplex
8-cube.svg
8-cube
8-orthoplex.svg
8-orthoplex

9 dimensions[edit]

Name Schläfli
symbol
Vertices Edges Faces Cells 4-faces 5-faces 6-faces 7-faces 8-faces χ
9-simplex (day) {38} 10 45 120 210 252 210 120 45 10 2
9-cube (enne) {4,37} 512 2304 4608 5376 4032 2016 672 144 18 2
9-orthoplex (vee) {37,4} 18 144 672 2016 4032 5376 4608 2304 512 2
9-simplex t0.svg
9-simplex
9-cube.svg
9-cube
9-orthoplex.svg
9-orthoplex

10 dimensions[edit]

Name Schläfli
symbol
Vertices Edges Faces Cells 4-faces 5-faces 6-faces 7-faces 8-faces 9-faces χ
10-simplex (ux) {39} 11 55 165 330 462 462 330 165 55 11 0
10-cube (deker) {4,38} 1024 5120 11520 15360 13440 8064 3360 960 180 20 0
10-orthoplex (ka) {38,4} 20 180 960 3360 8064 13440 15360 11520 5120 1024 0
10-simplex t0.svg
10-simplex
10-cube.svg
10-cube
10-orthoplex.svg
10-orthoplex

...

Non-convex[edit]

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

Tessellations of Euclidean space[edit]

Tessellations of Euclidean 4-space[edit]

There are three kinds of infinite regular tessellations (honeycombs) that can tessellate Euclidean 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
(test)
{4,3,3,4} {4,3,3} {4,3} {4} {4} {3,4} {3,3,4} Self-dual
16-cell honeycomb
(hext)
{3,3,4,3} {3,3,4} {3,3} {3} {3} {4,3} {3,4,3} {3,4,3,3}
24-cell honeycomb
(icot)
{3,4,3,3} {3,4,3} {3,4} {3} {3} {3,3} {4,3,3} {3,3,4,3}
Tesseractic tetracomb.png
Projected portion of {4,3,3,4}
(Tesseractic honeycomb)
Demitesseractic tetra hc.png
Projected portion of {3,3,4,3}
(16-cell honeycomb)
Icositetrachoronic tetracomb.png
Projected portion of {3,4,3,3}
(24-cell honeycomb)

Tessellations of Euclidean 5-space and higher[edit]

The hypercubic honeycomb is the only family of regular honeycomb 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 (squat) {4,4} {4} {4} Self-dual
Cubic honeycomb (chon) {4,3,4} {4,3} {3,4} Self-dual
Tesseractic honeycomb (test) {4,32,4} {4,32} {32,4} Self-dual
5-cube honeycomb (penth) {4,33,4} {4,33} {33,4} Self-dual
6-cube honeycomb {4,34,4} {4,34} {34,4} Self-dual
7-cube honeycomb {4,35,4} {4,35} {35,4} Self-dual
8-cube honeycomb {4,36,4} {4,36} {36,4} Self-dual
n-hypercubic honeycomb {4,3n−2,4} {4,3n−2} {3n−2,4} Self-dual

Tessellations of hyperbolic space[edit]

Tessellations of hyperbolic 4-space[edit]

There are seven convex regular honeycombs and four star-honeycombs in H4 space.[2] Five convex ones are compact, and two are paracompact.

Five compact 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 5-cell honeycomb {3,3,3,5} {3,3,3} {3,3} {3} {5} {3,5} {3,3,5} {5,3,3,3}
120-cell 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-4 120-cell honeycomb {5,3,3,4} {5,3,3} {5,3} {5} {4} {3,4} {3,3,4} {4,3,3,5}
Order-5 120-cell honeycomb {5,3,3,5} {5,3,3} {5,3} {5} {5} {3,5} {3,3,5} Self-dual

The two paracompact regular H4 honeycombs are: {3,4,3,4}, {4,3,4,3}.

Name Schläfli
Symbol
{p,q,r,s}
Facet
type
{p,q,r}
Cell
type
{p,q}
Face
type
{p}
Face
figure
{s}
Edge
figure
{r,s}
Vertex
figure

{q,r,s}
Dual
Order-4 24-cell honeycomb {3,4,3,4} {3,4,3} {3,4} {3} {4} {3,4} {4,3,4} {4,3,4,3}
Cubic honeycomb honeycomb {4,3,4,3} {4,3,4} {4,3} {4} {3} {4,3} {3,4,3} {3,4,3,4}

There are four regular star-honeycombs in H4 space:

Name Schläfli
Symbol
{p,q,r,s}
Facet
type
{p,q,r}
Cell
type
{p,q}
Face
type
{p}
Face
figure
{s}
Edge
figure
{r,s}
Vertex
figure

{q,r,s}
Dual
Small stellated 120-cell 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 600-cell 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 120-cell 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}
Great 120-cell 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}

Tessellations of hyperbolic 5-space[edit]

There are 5 regular honeycombs in H5, all paracompact, which include infinite (Euclidean) facets or vertex figures: {3,4,3,3,3}, {3,3,4,3,3}, {3,3,3,4,3}, {3,4,3,3,4}, and {4,3,3,4,3}.

There are no compact regular tessellations of hyperbolic space of dimension 5 or higher and no paracompact regular tessellations in hyperbolic space of dimension 6 or higher.

Name Schläfli
Symbol
{p,q,r,s,t}
Facet
type
{p,q,r,s}
4-face
type
{p,q,r}
Cell
type
{p,q}
Face
type
{p}
Cell
figure
{t}
Face
figure
{s,t}
Edge
figure
{r,s,t}
Vertex
figure

{q,r,s,t}
Dual
5-orthoplex honeycomb {3,3,3,4,3} {3,3,3,4} {3,3,3} {3,3} {3} {3} {4,3} {3,4,3} {3,3,4,3} {3,4,3,3,3}
24-cell honeycomb honeycomb {3,4,3,3,3} {3,4,3,3} {3,4,3} {3,4} {3} {3} {3,3} {3,3,3} {3,3,3,3} {3,3,3,4,3}
16-cell honeycomb honeycomb {3,3,4,3,3} {3,3,4,3} {3,3,4} {3,3} {3} {3} {3,3} {4,3,3} {3,4,3,3} self-dual
Order-4 24-cell honeycomb honeycomb {3,4,3,3,4} {3,4,3,3} {3,4,3} {3,4} {3} {4} {3,4} {3,3,4} {4,3,3,4} {4,3,3,4,3}
Tesseractic honeycomb honeycomb {4,3,3,4,3} {4,3,3,4} {4,3,3} {4,3} {4} {3} {4,3} {3,4,3} {3,3,4,3} {3,4,3,3,4}

Tessellations of hyperbolic 6-space and higher[edit]

Even allowing for infinite (Euclidean) facets and/or vertex figures, there are no regular compact or paracompact tessellations of hyperbolic space of dimension 6 or higher.

Apeirotopes[edit]

An apeirotope is, like any other polytope, an unbounded hyper-surface. The difference is that whereas a polytope's hyper-surface curls back on itself to close round a finite volume of hyperspace, an apeirotope does not curl back.

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

Two dimensions[edit]

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[edit]

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.[3] These include the tessellations of type {4,4}, {6,3}, {3,6} above, as well as (in the plane) polytopes of type: {∞,3}, {∞,4}, {∞,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)

See also regular skew polyhedron.

Four and higher dimensions[edit]

The apeirochora have not been completely classified.

Abstract polytopes[edit]

The abstract polytopes arose out of an attempt to study polytopes apart from the geometrical space they are embedded in. They include the tessellations of spherical, euclidean and hyperbolic space, tessellations of other manifolds, and many other objects that do not have a well-defined topology, but instead may be characterised by their "local" topology. There are infinitely many in every dimension. See this atlas for a sample. Some notable examples of abstract polytopes that do not appear elsewhere in this list are the 11-cell and the 57-cell.

See also[edit]

References[edit]

  1. ^ (Coxeter 1973, Table I: Regular polytopes, (iii) The three regular polytopes in n dimensions (n>=5), pp. 294–295)
  2. ^ Coxeter, The Beauty of Geometry: Twelve Essays, Chapter 10: Regular honeycombs in hyperbolic space, Summary tables IV p. 213
  3. ^ (McMullen & Schulte 2002, Section 7E)
  • 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, pp. 212–213) [2] PDF
  • D. M. Y. Sommerville, An Introduction to the Geometry of n Dimensions. New York, E. P. Dutton, 1930. 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes

External links[edit]