List of regular polytopes and compounds

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Example regular polytopes
Regular (2D) polygons
Convex Star
Regular pentagon.svg
{5}
Star polygon 5-2.svg
{5/2}
Regular (3D) polyhedra
Convex Star
Dodecahedron.png
{5,3}
Small stellated dodecahedron.png
{5/2,5}
Regular 2D tessellations
Euclidean Hyperbolic
Uniform tiling 44-t0.png
{4,4}
Uniform tiling 54-t0.png
{5,4}
Regular 4D polytopes
Convex Star
Schlegel wireframe 120-cell.png
{5,3,3}
Ortho solid 010-uniform polychoron p53-t0.png
{5/2,5,3}
Regular 3D tessellations
Euclidean Hyperbolic
Cubic honeycomb.png
{4,3,4}
Hyperbolic orthogonal dodecahedral honeycomb.png
{5,3,4}

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

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 a (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, 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.

Contents

Overview[edit]

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

Finite Euclidean Hyperbolic Compounds
Dim. Convex Star Skew Convex Compact Star Paracompact Convex Star
1 1 0 0 1 0 0 0 0 0
2 1 1 0 0
3 5 4 ? 3 5 0
4 6 10 ? 1 4 0 11 26 20
5 3 0 ? 3 5 4 2 0 0
6 3 0 ? 1 0 0 5 0 0
7 3 0 ? 1 0 0 0 3 0
8 3 0 ? 1 0 0 0 6 0
9+ 3 0 ? 1 0 0 0 * 0

* 1 if the number of dimensions is of the form 2k − 1; 2 if the number of dimensions is a power of two; 0 otherwise.

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

One dimension[edit]

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 ditel, { }, 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 { },[1][2] or a Coxeter diagram with a single ringed node, CDel node 1.png. Norman Johnson calls it a ditel and gives it the Schläfli symbol { }.[3]

Although trivial as a polytope, it appears as the edges of polygons and other higher dimensional polytopes.[4] 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.[5]

Two dimensions (polygons)[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)
Square
(2-orthoplex)
(2-cube)
Pentagon Hexagon Heptagon Octagon
Schläfli {3} {4} {5} {6} {7} {8}
Symmetry D3, [3] D4, [4] D5, [5] D6, [6] D7, [7] D8, [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 Decagon Hendecagon Dodecagon Tridecagon Tetradecagon
Schläfli {9} {10} {11} {12} {13} {14}
Symmetry D9, [9] D10, [10] D11, [11] D12, [12] D13, [13] D14, [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 Hexadecagon Heptadecagon Octadecagon Enneadecagon Icosagon ...p-gon
Schläfli {15} {16} {17} {18} {19} {20} {p}
Symmetry D15, [15] D16, [16] D17, [17] D18, [18] D19, [19] D20, [20] Dp, [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

Spherical[edit]

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.

Name Digon
Schläfli symbol {2}
Symmetry D2, [2]
Coxeter diagram CDel node 1.pngCDel 2.pngCDel node.png
Image Digon.svg

Stars[edit]

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.

Name Pentagram Heptagrams Octagram Enneagrams Decagram ...n-agrams
Schläfli {5/2} {7/2} {7/3} {8/3} {9/2} {9/4} {10/3} {p/q}
Symmetry D5, [5] D7, [7] D8, [8] D9, [9], D10, [10] Dp, [p]
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 Regular star polygon 5-2.svg Regular star polygon 7-2.svg Regular star polygon 7-3.svg Regular star polygon 8-3.svg Regular star polygon 9-2.svg Regular star polygon 9-4.svg Regular 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}

Skew polygons[edit]

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
Hexagon Octagon Decagons
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:

Pentagon Octagon Dodecagon Triacontagon
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)[edit]

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:

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

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

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

Convex[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)
{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
[3,3]
(*332)
(self)
Hexahedron
Cube
(3-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
[4,3]
(*432)
Octahedron
Octahedron
(3-orthoplex)
{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
[4,3]
(*432)
Cube
Dodecahedron {5,3} CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png Dodecahedron.svg Dodecahedron.png Uniform tiling 532-t0.png 12
{5}
30 20
{3}
Ih
[5,3]
(*532)
Icosahedron
Icosahedron {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
[5,3]
(*532)
Dodecahedron

Spherical[edit]

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.[6]

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

Name Schläfli
{p,q}
Coxeter
diagram
Image
(sphere)
Faces
{p}
Edges
{ }
Vertices
{q}
Symmetry Dual
Digonal dihedron
Digonal hosohedron
{2,2} CDel node 1.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png Digonal dihedron.png
Spherical digonal hosohedron.png
2
{2}
2
{ }
2
{2}
D2h
[2,2]
(*222)
Self
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
[3,2]
(*322)
Trigonal hosohedron
Trigonal hosohedron {2,3} CDel node 1.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png Spherical trigonal hosohedron.png 3
{2}
3
{ }
2
{3}
D3h
[2,3]
(*322)
Trigonal dihedron
Square dihedron {4,2} CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.png Tetragonal dihedron.png 2
{4}
4
{ }
4
{2}
D4h
[4,2]
(*422)
Square hosohedron
Square hosohedron {2,4} CDel node 1.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.png Spherical square hosohedron.png 4
{2}
4
{ }
2
{4}
D4h
[2,4]
(*422)
Square dihedron
Pentagonal dihedron {5,2} CDel node 1.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node.png Pentagonal dihedron.png 2
{5}
5
{ }
5
{2}
D5h
[5,2]
(*522)
Pentagonal hosohedron
Pentagonal hosohedron {2,5} CDel node 1.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.png Spherical pentagonal hosohedron.png 5
{2}
5
{ }
2
{5}
D5h
[2,5]
(*522)
Pentagonal 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
[6,2]
(*622)
Hexagonal hosohedron
Hexagonal hosohedron {2,6} CDel node 1.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.png Spherical hexagonal hosohedron.png 6
{2}
6
{ }
2
{6}
D6h
[2,6]
(*622)
Hexagonal dihedron

Star-dihedra and hosohedra also exist, such as {5/2,2} and {2,5/2}.

Stars[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 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.

Name Image
(transparent)
Image
(solid)
Image
(sphere)
Stellation
diagram
Schläfli
{p,q} and
Coxeter
Faces
{p}
Edges Vertices
{q}
verf.
χ Density Symmetry Dual
Small stellated dodecahedron 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
[5,3]
(*532)
Great dodecahedron
Great dodecahedron 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
[5,3]
(*532)
Small stellated dodecahedron
Great stellated dodecahedron 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
[5,3]
(*532)
Great icosahedron
Great icosahedron 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
[5,3]
(*532)
Great stellated dodecahedron

Skew polyhedra[edit]

A regular skew polyhedra is 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[edit]

Regular 4-polytopes 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 4-polytope) is a polyhedron, seen by the arrangement of neighboring vertices around a given vertex. For regular 4-polytopes, this vertex figure is a regular polyhedron.
  • An edge figure is a polygon, seen by the arrangement of faces around an edge. For regular 4-polytopes, this edge figure will always be a regular polygon.

The existence of a regular 4-polytope \{p,q,r\} is constrained by the existence of the regular polyhedra \{p,q\}, \{q,r\}. A suggested name for 4-polytopes is "polychoron".[7]

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-polytope
= 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 4-polytopes 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)
{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)
{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)
{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 {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 {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 {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

Spherical[edit]

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}. Polychora of the form {2,p,2} are both di-4-topes and hoso-4-topes. There are also the cases {p,2,q} which have dihedral cells and hosohedral vertex figures.

Stars[edit]

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)[3].

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

Name
Wireframe Solid 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)
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
[5,3,3]
Small stellated 120-cell
Small stellated 120-cell 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
[5,3,3]
Icosahedral 120-cell
Great 120-cell 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
[5,3,3]
Self-dual
Grand 120-cell 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
[5,3,3]
Great stellated 120-cell
Great stellated 120-cell 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
[5,3,3]
Grand 120-cell
Grand stellated 120-cell 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
[5,3,3]
Self-dual
Great grand 120-cell 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
[5,3,3]
Great icosahedral 120-cell
Great icosahedral 120-cell
(great faceted 600-cell)
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
[5,3,3]
Great grand 120-cell
Grand 600-cell 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
[5,3,3]
Great grand stellated 120-cell
Great grand stellated 120-cell 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
[5,3,3]
Grand 600-cell

There are 4 failed potential regular star 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.

Five and more dimensions[edit]

In five dimensions, a regular polytope can be named as \{p,q,r,s\} where \{p,q,r\} is the 4-face 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 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 \{p,q,r,s\} exists only if \{p,q,r\} and \{q,r,s\} are regular 4-polytopes.

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.

Convex[edit]

In dimensions 5 and higher, there are only three kinds of convex regular polytopes.[8]

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

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[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 {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 {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 {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 Vertices Edges Faces Cells 4-faces 5-faces χ
6-simplex {3,3,3,3,3} 7 21 35 35 21 7 0
6-cube {4,3,3,3,3} 64 192 240 160 60 12 0
6-orthoplex {3,3,3,3,4} 12 60 160 240 192 64 0
6-simplex t0.svg
6-simplex
6-cube graph.svg
6-cube
6-orthoplex.svg
6-orthoplex

7 dimensions[edit]

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

8 dimensions[edit]

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

9 dimensions[edit]

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

10 dimensions[edit]

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

Regular projective polytopes[edit]

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. [9]

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[edit]

3-dimensional regular hemi-polytopes
Name Coxeter
McMullen
Image Faces Edges Vertices χ
Hemi-cube {4,3}/2
{4,3}3
Hemicube2.PNG 3 6 4 1
Hemi-octahedron {3,4}/2
{3,4}3
Hemi-octahedron2.png 4 6 3 1
Hemi-dodecahedron {5,3}/2
{5,3}5
Hemi-dodecahedron.png 6 15 10 1
Hemi-icosahedron {3,5}/2
{3,5}5
Hemi-icosahedron2.png 10 15 6 1

Regular projective 4-polytopes[edit]

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
Name Coxeter
symbol
McMullen
Symbol
Cells Faces Edges Vertices χ
Hemi-tesseract {4,3,3}/2 {4,3,3}4 4 12 16 8 0
Hemi-16-cell {3,3,4}/2 {3,3,4}4 8 16 12 4 0
Hemi-24-cell {3,4,3}/2 {3,4,3}6 12 48 48 12 0
Hemi-120-cell {5,3,3}/2 {5,3,3}15 60 360 600 300 0
Hemi-600-cell {3,3,5}/2 {3,3,5}15 300 600 360 60 0

Regular projective 5-polytopes[edit]

There are only 2 convex regular projective hemi-polytopes in dimensions 5 or higher.

Name Schläfli 4-faces Cells Faces Edges Vertices χ
hemi-penteract {4,3,3,3}/2 5 20 40 40 16 1
hemi-pentacross {3,3,3,4}/2 16 40 40 20 5 1

Apeirotopes[edit]

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 apeirotope, is an infinite polygon, a 3-apeirotope, or apeirohedron, is an infinite polyhedron, etc.

There are two main geometric classes of apeirotope:[10]

  • Regular honeycombs in n dimensions, which completely fill an n-dimensional space.
  • Regular skew apeirotopes, comprising an n-dimensional manifold in a higher space.

One dimension (apeirogons)[edit]

The straight apeirogon is a regular tessellation of the line, subdviding 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.png...

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[edit]

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-dimensions 3-dimensions
Regular apeirogon zig-zag.png
Zig-zag apeirogon
Triangular helix.png
Helix apeirogon

Two dimensions (apeirohedra)[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)
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.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

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[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 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 (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.

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:

Spherical (Platonic)/Euclidean/hyperbolic (Poincaré disc: compact/paracompact/noncompact) tessellations with their Schläfli symbol
p \ q 3 4 5 6 7 8 ... ... iπ/λ
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
H2 tiling 23j12-4.png

{3,iπ/λ}
CDel node 1.pngCDel 3.pngCDel node.pngCDel ultra.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
{4,iπ/λ}
CDel node 1.pngCDel 4.pngCDel node.pngCDel ultra.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
{5,iπ/λ}
CDel node 1.pngCDel 5.pngCDel node.pngCDel ultra.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
{6,iπ/λ}
CDel node 1.pngCDel 6.pngCDel node.pngCDel ultra.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
{7,iπ/λ}
CDel node 1.pngCDel 7.pngCDel node.pngCDel ultra.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
{8,iπ/λ}
CDel node 1.pngCDel 8.pngCDel node.pngCDel ultra.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
{∞,iπ/λ}
CDel node 1.pngCDel infin.pngCDel node.pngCDel ultra.pngCDel node.png
...
iπ/λ H2 tiling 23j12-1.png
{iπ/λ,3}
CDel node 1.pngCDel ultra.pngCDel node.pngCDel 3.pngCDel node.png
{iπ/λ,4}
CDel node 1.pngCDel ultra.pngCDel node.pngCDel 4.pngCDel node.png
{iπ/λ,5}
CDel node 1.pngCDel ultra.pngCDel node.pngCDel 5.pngCDel node.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
{iπ/λ,∞}
CDel node 1.pngCDel ultra.pngCDel node.pngCDel infin.pngCDel node.png
{iπ/λ,iπ/λ}
CDel node 1.pngCDel ultra.pngCDel node.pngCDel ultra.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. 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 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
[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}
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}
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}
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}
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}
Hendecagon.svg
{11/2}
Star polygon 11-2.svg
3 *11.3.2
[11,3]
Order-11 hendecagrammic 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,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-p p-grammic tiling

Skew apeirohedra in Euclidean 3-space[edit]

There are three regular skew apeirohedra in Euclidean 3-space, with regular skew polygon vertex figures.[11][12][13] 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}.[14] 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.[15] 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[edit]

There are 31 regular skew apeirohedra in hyperbolic 3-space:[16]

  • 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)[edit]

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), {4, 3, 4}:[17]

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} 0 Self-dual

Improper tessellations of Euclidean 3-space[edit]

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[edit]

There are ten flat regular honeycombs of hyperbolic 3-space:[18] (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.

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

Noncompact solutions exist as Lorentzian Coxeter groups, and can be visualized with open domains in hyperbolic space (the fundamental tetrahedron having some parts inaccessible beyond infinity), and a few are drawn below showing their intersections with the ideal half-space plane. All honeycombs which are not shown in the set of tables below and do not have 2 in their Schläfli symbol are noncompact.

Spherical/Euclidean/hyperbolic(compact/paracompact/noncompact) honeycombs {p,3,r}
p \ r 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
Heptagonal tiling honeycomb.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
q = 4 q = 5 q = 6
p \ r 3 4 5
3
Uniform polyhedron-43-t2.png
Schlegel wireframe 24-cell.png
{3,4,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
H3 344 CC center.png
{3,4,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png

{3,4,5}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.png
4
Uniform tiling 44-t0.png
H3 443 FC boundary.png
{4,4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
H3 444 FC boundary.png
{4,4,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png

{4,4,5}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.png
5
H2 tiling 245-1.png

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

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

{5,4,5}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.png
p \ r 3 4
3
Uniform polyhedron-53-t2.png
H3 353 CC center.png
{3,5,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png

{3,5,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.png
4
H2 tiling 245-4.png

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

{4,5,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.png
5
H2 tiling 255-1.png

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

{5,5,4}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.png
p \ r 3 4
3
Uniform tiling 63-t2.png
H3 363 FC boundary.png
{3,6,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png

{3,6,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png
4
H2 tiling 246-4.png

{4,6,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png

{4,6,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png
5
H2 tiling 256-4.png

{5,6,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png

{5,6,4}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png

There are no regular hyperbolic star-honeycombs in H3: all forms with a regular star polyhedron as cell, vertex figure or both end up being spherical.

Four dimensions (5-apeirotopes)[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 {4,3,3,4} {4,3,3} {4,3} {4} {4} {3,4} {3,3,4} Self-dual
16-cell honeycomb {3,3,4,3} {3,3,4} {3,3} {3} {3} {4,3} {3,4,3} {3,4,3,3}
24-cell honeycomb {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)

There are also the two improper cases {4,3,4,2} and {2,4,3,4}. There are three flat regular honeycomb of Euclidean 4-space:[19]

  • {4,3,3,4}, {3,3,4,3}, and {3,4,3,3}.

There are seven flat regular convex honeycombs of hyperbolic 4-space:[20]

  • 5 are compact: {3,3,3,5}, {5,3,3,3}, {4,3,3,5}, {5,3,3,4}, {5,3,3,5}
  • 2 are paracompact: {3,4,3,4}, and {4,3,4,3}.

There are four flat regular star honeycombs of hyperbolic 4-space:[21]

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

Tessellations of hyperbolic 4-space[edit]

There are seven convex regular honeycombs and four star-honeycombs in H4 space.[22] 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}

Noncompact solutions exist as Lorentzian Coxeter groups, and can be visualized with open domains in hyperbolic space (the fundamental 5-cell having some parts inaccessible beyond infinity). All honeycombs which are not shown in the set of tables below and do not have 2 in their Schläfli symbol are noncompact.

Spherical/Euclidean/hyperbolic(compact/paracompact/noncompact) honeycombs {p,q,r,s}
q=3, s=3
p \ r 3 4 5
3 5-simplex t0.svg
{3,3,3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Demitesseractic tetra hc.png
{3,3,4,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png

{3,3,5,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
4 5-cube t0.svg
{4,3,3,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

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

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

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

{5,3,5,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
q=3, s=4
p \ r 3 4
3 5-cube t4.svg
{3,3,3,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

{3,3,4,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
4 Tesseractic tetracomb.png
{4,3,3,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

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

{5,3,4,4}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
q=3, s=5
p \ r 3 4
3
{3,3,3,5}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png

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

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

{5,3,4,5}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.png
q=4, s=3
p \ r 3 4
3 Icositetrachoronic tetracomb.png
{3,4,3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

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

{4,3,4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
q=4, s=4
p \ r 3 4
3
{3,4,3,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

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

{4,4,4,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
q=4, s=5
p \ r 3 4
3
{3,4,3,5}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png

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

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

Star tessellations of hyperbolic 4-space[edit]

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 Density
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} 5
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} 5
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} 10
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} 10

Five dimensions (6-apeirotopes)[edit]

There is only one flat regular honeycomb of Euclidean 5-space: (previously listed above as tessellations)[23]

  • {4,3,3,3,4}

There are five flat regular regular honeycombs of hyperbolic 5-space, all paracompact: (previously listed above as tessellations)[24]

  • {3,3,3,4,3}, {3,4,3,3,3}, {3,3,4,3,3}, {3,4,3,3,4}, and {4,3,3,4,3}

Tessellations of Euclidean 5-space[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 {4,4} {4} {4} Self-dual
Cubic honeycomb {4,3,4} {4,3} {3,4} Self-dual
Tesseractic honeycomb {4,32,4} {4,32} {32,4} Self-dual
5-cube honeycomb {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

In E5, there are also the improper cases {4,3,3,4,2}, {2,4,3,3,4}, {3,3,4,3,2}, {2,3,3,4,3}, {3,4,3,3,2}, and {2,3,4,3,3}. In En, {4,3n−3,4,2} and {2,4,3n−3,4} are always improper Euclidean tessellations.

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

Since there are no regular star n-polytopes for n ≥ 5, that could be potential cells or vertex figures, there are no more hyperbolic star honeycombs in Hn for n ≥ 5.

6 dimensions and higher (7-apeirotopes+)[edit]

Tessellations of hyperbolic 6-space and higher[edit]

There are no regular compact or paracompact tessellations of hyperbolic space of dimension 6 or higher. However, any Schläfli symbol of the form {p,q,r,s,...} not covered above (p,q,r,s,... natural numbers above 2, or infinity) will form a noncompact tessellation of hyperbolic n-space.

Compound polytopes[edit]

Main article: Polytope compound

Two dimensional compounds[edit]

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/(n−m)}) and m and n are coprime. When m and n are not coprime, the star polygon obtained will be a regular polygon with n/m sides. A new figure is obtained by rotating these regular n/m-gons one vertex to the left on the original polygon until the number of vertices rotated equals n/m minus one, and combining these figures. An extreme case of this is where n/m is 2, producing a figure consisting of n/2 straight line segments; this is called a degenerate star polygon.

In other cases where n and m have a common factor, a star polygon for a lower n is obtained, and rotated versions can be combined. These figures are called star figures, improper star polygons or compound polygons. The same notation {n/m} is often used for them, although authorities such as Grünbaum (1994) regard (with some justification) the form k{n} as being more correct, where usually k = m.

A further complication comes when we compound two or more star polygons, as for example two pentagrams, differing by a rotation of 36°, inscribed in a decagon. This is correctly written in the form k{n/m}, as 2{5/2}, rather than the commonly used {10/4}.

Coxeter's extended notation for compounds is of the form c{m,n,...}[d{p,q,...}]e{s,t,...}, indicating that d distinct {p,q,...}'s together together cover the vertices of {m,n,...} c times and the facets of {s,t,...} e times. If no regular {m,n,...} exists, the first part of the notation is removed, leaving [d{p,q,...}]e{s,t,...}; the opposite holds if no regular {s,t,...} exists. The dual of c{m,n,...}[d{p,q,...}]e{s,t,...} is e{t,s,...}[d{q,p,...}]c{n,m,...}. If c or e are 1, they may be omitted. For compound polygons, this notation reduces to {nk}[k{n/m}]{nk}: for example, the hexagram may be written thus as {6}[2{3}]{6}.

Examples for n=2..10, nk≤30
Regular star figure 2(2,1).svg
2{2}
Regular star figure 3(2,1).svg
3{2}
Regular star figure 4(2,1).svg
4{2}
Regular star figure 5(2,1).svg
5{2}
Regular star figure 6(2,1).svg
6{2}
Regular star figure 7(2,1).svg
7{2}
Regular star figure 8(2,1).svg
8{2}
Regular star figure 9(2,1).svg
9{2}
Regular star figure 10(2,1).svg
10{2}
Regular star figure 11(2,1).svg
11{2}
Regular star figure 12(2,1).svg
12{2}
Regular star figure 13(2,1).svg
13{2}
Regular star figure 14(2,1).svg
14{2}
Regular star figure 15(2,1).svg
15{2}
Regular star figure 2(3,1).svg
2{3}
Regular star figure 3(3,1).svg
3{3}
Regular star figure 4(3,1).svg
4{3}
Regular star figure 5(3,1).svg
5{3}
Regular star figure 6(3,1).svg
6{3}
Regular star figure 7(3,1).svg
7{3}
Regular star figure 8(3,1).svg
8{3}
Regular star figure 9(3,1).svg
9{3}
Regular star figure 10(3,1).svg
10{3}
Regular star figure 2(4,1).svg
2{4}
Regular star figure 3(4,1).svg
3{4}
Regular star figure 4(4,1).svg
4{4}
Regular star figure 5(4,1).svg
5{4}
Regular star figure 6(4,1).svg
6{4}
Regular star figure 7(4,1).svg
7{4}
Regular star figure 2(5,1).svg
2{5}
Regular star figure 3(5,1).svg
3{5}
Regular star figure 4(5,1).svg
4{5}
Regular star figure 5(5,1).svg
5{5}
Regular star figure 6(5,1).svg
6{5}
Regular star figure 2(5,2).svg
2{5/2}
Regular star figure 3(5,2).svg
3{5/2}
Regular star figure 4(5,2).svg
4{5/2}
Regular star figure 5(5,2).svg
5{5/2}
Regular star figure 6(5,2).svg
6{5/2}
Regular star figure 2(6,1).svg
2{6}
Regular star figure 3(6,1).svg
3{6}
Regular star figure 4(6,1).svg
4{6}
Regular star figure 5(6,1).svg
5{6}
Regular star figure 2(7,1).svg
2{7}
Regular star figure 3(7,1).svg
3{7}
Regular star figure 4(7,1).svg
4{7}
Regular star figure 2(7,2).svg
2{7/2}
Regular star figure 3(7,2).svg
3{7/2}
Regular star figure 4(7,2).svg
4{7/2}
Regular star figure 2(7,3).svg
2{7/3}
Regular star figure 3(7,3).svg
3{7/3}
Regular star figure 4(7,3).svg
4{7/3}
Regular star figure 2(8,1).svg
2{8}
Regular star figure 3(8,1).svg
3{8}
Regular star figure 2(8,3).svg
2{8/3}
Regular star figure 3(8,3).svg
3{8/3}
Regular star figure 2(9,1).svg
2{9}
Regular star figure 3(9,1).svg
3{9}
Regular star figure 2(9,2).svg
2{9/2}
Regular star figure 3(9,2).svg
3{9/2}
Regular star figure 2(9,4).svg
2{9/4}
Regular star figure 3(9,4).svg
3{9/4}
Regular star figure 2(10,1).svg
2{10}
Regular star figure 3(10,1).svg
3{10}
Regular star figure 2(10,3).svg
2{10/3}
Regular star figure 3(10,3).svg
3{10/3}
Regular star figure 2(11,1).svg
2{11}
Regular star figure 2(11,2).svg
2{11/2}
Regular star figure 2(11,3).svg
2{11/3}
Regular star figure 2(11,4).svg
2{11/4}
Regular star figure 2(11,5).svg
2{11/5}
Regular star figure 2(12,1).svg
2{12}
Regular star figure 2(12,5).svg
2{12/5}
Regular star figure 2(13,1).svg
2{13}
Regular star figure 2(13,2).svg
2{13/2}
Regular star figure 2(13,3).svg
2{13/3}
Regular star figure 2(13,4).svg
2{13/4}
Regular star figure 2(13,5).svg
2{13/5}
Regular star figure 2(13,6).svg
2{13/6}
Regular star figure 2(14,1).svg
2{14}
Regular star figure 2(14,3).svg
2{14/3}
Regular star figure 2(14,5).svg
2{14/5}
Regular star figure 2(15,1).svg
2{15}
Regular star figure 2(15,2).svg
2{15/2}
Regular star figure 2(15,4).svg
2{15/4}
Regular star figure 2(15,7).svg
2{15/7}

Regular skew polygons also create compounds, seen in the edges of prismatic compound of antiprisms, for instance:

Regular compound skew polygon
Compound
skew squares
Compound
skew hexagons
Compound
skew decagons
Two {2}#{ } Three {2}#{ } Two {3}#{ } Two {5/3}#{ }
Compound skew square in cube.png Skew tetragons in compound of three digonal antiprisms.png Compound skew hexagon in hexagonal prism.png Compound skew hexagon in pentagonal crossed antiprism.png

Three dimensional compounds[edit]

A regular polyhedron compound can be defined as a compound which, like a regular polyhedron, is vertex-transitive, edge-transitive, and face-transitive. With this definition there are 5 regular compounds.

Symmetry [4,3], Oh [5,3]+, I [5,3], Ih
Duality Self-dual Dual pairs
Image Compound of two tetrahedra.png Compound of five tetrahedra.png Compound of ten tetrahedra.png Compound of five cubes.png Compound of five octahedra.png
Spherical Spherical compound of two tetrahedra.png Spherical compound of five tetrahedra.png Spherical compound of ten tetrahedra.png Spherical compound of five cubes.png Spherical compound of five octahedra.png
Polyhedra 2 {3,3} 5 {3,3} 10 {3,3} 5 {4,3} 5 {3,4}
Coxeter {4,3}[2{3,3}]{3,4} {5,3}[5{3,3}]{3,5} 2{5,3}[10{3,3}]2{3,5} 2{5,3}[5{4,3}] [5{3,4}]2{3,5}

Euclidean and hyperbolic plane compounds[edit]

There are eighteen two-parameter families of regular compound tessellations of the Euclidean plane. In the hyperbolic plane, five one-parameter families and seventeen isolated cases are known, but the completeness of this listing has not yet been proven.

The Euclidean and hyperbolic compound families 2 {p,p} (4 ≤ p ≤ ∞, p an integer) are analogous to the spherical stella octangula, 2 {3,3}.

A few examples of Euclidean and hyperbolic regular compounds
Self-dual Duals Self-dual
2 {4,4} 2 {6,3} 2 {3,6} 2 {∞,∞}
Kah 4 4.png Compound 2 hexagonal tilings.png Compound 2 triangular tilings.png Infinite-order apeirogonal tiling and dual.png
{{4,4}} or a{4,4} or {4,4}[2{4,4}]{4,4}
CDel nodes 10ru.pngCDel split2-44.pngCDel node.png + CDel nodes 01rd.pngCDel split2-44.pngCDel node.png or CDel node h3.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
[2{6,3}]{3,6} a{6,3} or {6,3}[2{3,6}]
CDel branch 10ru.pngCDel split2.pngCDel node.png + CDel branch 01rd.pngCDel split2.pngCDel node.png or CDel node h3.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
{{∞,∞}} or a{∞,∞} or {4,∞}[2{∞,∞}]{∞,4}
CDel labelinfin.pngCDel branch 10ru.pngCDel split2-ii.pngCDel node.png + CDel labelinfin.pngCDel branch 01rd.pngCDel split2-ii.pngCDel node.png or CDel node h3.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.png
3 {6,3} 3 {3,6} 3 {∞,∞}
Compound 3 hexagonal tilings.png Compound 3 triangular tilings.png Iii symmetry 000.png
2{3,6}[3{6,3}]{6,3} {3,6}[3{3,6}]2{6,3}
CDel branch 10ru.pngCDel split2.pngCDel node.png + CDel branch 01rd.pngCDel split2.pngCDel node.png + CDel branch.pngCDel split2.pngCDel node 1.png

CDel labelinfin.pngCDel branch 10ru.pngCDel split2-ii.pngCDel node.png + CDel labelinfin.pngCDel branch 01rd.pngCDel split2-ii.pngCDel node.png + CDel labelinfin.pngCDel branch.pngCDel split2-ii.pngCDel node 1.png

Four dimensional compounds[edit]

Orthogonal projections
Regular compound 75 tesseracts.png Regular compound 75 16-cells.png
75 {4,3,3} 75 {3,3,4}

In 4-dimensions, there are thirty-two regular compounds of regular polytopes, which Coxeter lists in his book Regular Polytopes:[25]

Self-dual regular compounds
Compound Symmetry Vertex arrangement Cell arrangement
120 {3,3,3} [5,3,3], order 14400 {5,3,3} {3,3,5}
5 {3,4,3} [5,3,3], order 14400 {3,3,5} {5,3,3}
Regular compounds as dual pairs
Compound 1 Compound 2 Symmetry Vertex arrangement (1) Cell arrangement (1) Vertex arrangement (2) Cell arrangement (2)
3 {3,3,4}[26] 3 {4,3,3} [3,4,3], order 1152 {3,4,3} 2{3,4,3} 2{3,4,3} {3,4,3}
15 {3,3,4} 15 {4,3,3} [5,3,3], order 14400 {3,3,5} 2{5,3,3} 2{3,3,5} {5,3,3}
75 {3,3,4} 75 {4,3,3} [5,3,3], order 14400 5{3,3,5} 10{5,3,3} 10{3,3,5} 5{5,3,3}
75 {3,3,4} 75 {4,3,3} [5,3,3], order 14400 {5,3,3} 2{3,3,5} 2{5,3,3} {3,3,5}
300 {3,3,4} 300 {4,3,3} [5,3,3]+, order 7200 4{5,3,3} 8{3,3,5} 8{5,3,3} 4{3,3,5}
600 {3,3,4} 600 {4,3,3} [5,3,3], order 14400 8{5,3,3} 16{3,3,5} 16{5,3,3} 8{3,3,5}
25 {3,4,3} 25 {3,4,3} [5,3,3], order 14400 {5,3,3} 5{5,3,3} 5{3,3,5} {3,3,5}

There are two different compounds of 75 tesseracts: one shares the vertices of a 120-cell, while the other shares the vertices of a 600-cell. It immediately follows therefore that the corresponding dual compounds of 75 16-cells are also different.

Self-dual star compounds
Compound Symmetry Vertex arrangement Cell arrangement
5 {5,5/2,5} [5,3,3]+, order 7200 {5,3,3} {3,3,5}
10 {5,5/2,5} [5,3,3], order 14400 2{5,3,3} 2{3,3,5}
5 {5/2,5,5/2} [5,3,3]+, order 7200 {5,3,3} {3,3,5}
10 {5/2,5,5/2} [5,3,3], order 14400 2{5,3,3} 2{3,3,5}
Regular star compounds as dual pairs
Compound 1 Compound 2 Symmetry Vertex arrangement (1) Cell arrangement (1) Vertex arrangement (2) Cell arrangement (2)
5 {3,5,5/2} 5 {5/2,5,3} [5,3,3]+, order 7200 {5,3,3} {3,3,5} {5,3,3} {3,3,5}
10 {3,5,5/2} 10 {5/2,5,3} [5,3,3], order 14400 2{5,3,3} 2{3,3,5} 2{5,3,3} 2{3,3,5}
5 {5,5/2,3} 5 {3,5/2,5} [5,3,3]+, order 7200 {5,3,3} {3,3,5} {5,3,3} {3,3,5}
10 {5,5/2,3} 10 {3,5/2,5} [5,3,3], order 14400 2{5,3,3} 2{3,3,5} 2{5,3,3} 2{3,3,5}
5 {5/2,3,5} 5 {5,3,5/2} [5,3,3]+, order 7200 {5,3,3} {3,3,5} {5,3,3} {3,3,5}
10 {5/2,3,5} 10 {5,3,5/2} [5,3,3], order 14400 2{5,3,3} 2{3,3,5} 2{5,3,3} 2{3,3,5}

There are also fourteen partially regular compounds, that are either vertex-transitive or cell-transitive but not both. The seven vertex-transitive partially regular compounds are the duals of the seven cell-transitive partially regular compounds.

Partially regular compounds as dual pairs
Compound 1
Vertex-transitive
Compound 2
Cell-transitive
Symmetry
2 16-cells[27] 2 tesseracts [4,3,3], order 384
100 24-cell 100 24-cell [5,3,3]+, order 7200
200 24-cell 200 24-cell [5,3,3], order 14400
5 600-cell 5 120-cell [5,3,3]+, order 7200
10 600-cell 10 120-cell [5,3,3], order 14400
Partially regular star compounds as dual pairs
Compound 1
Vertex-transitive
Compound 2
Cell-transitive
Symmetry
5 {3,3,5/2} 5 {5/2,3,3} [5,3,3]+, order 7200
10 {3,3,5/2} 10 {5/2,3,3} [5,3,3], order 14400

Euclidean 3-space compounds[edit]

The only regular Euclidean compound honeycombs are an infinite family of compounds of cubic honeycombs, all sharing vertices and faces with another cubic honeycomb. This compound can have any number of cubic honeycombs. The Coxeter notation is {4,3,4}[d{4,3,4}]{4,3,4}.

Five dimensions and higher compounds[edit]

There are no regular compounds in five or six dimensions. There are three known seven-dimensional compounds (16, 240, or 480 7-simplexes), and six known eight-dimensional ones (16, 240, or 480 8-cubes or 8-orthoplexes). There is also one compound of n-simplexes in n-dimensional space provided that n is one less than a power of two, and also two compounds (one of n-cubes and a dual one of n-orthoplexes) in n-dimensional space if n is a power of two.

The Coxeter notation for these compounds are (using αn = {3n−1}, βn = {3n−2,4}, γn = {4,3n−2}:

  • 7-simplexes: cγ7[16cα7]cβ7, where c = 1, 15, or 30
  • 8-orthoplexes: cγ8[16cβ8]
  • 8-cubes: [16cγ8]cβ8

The general cases (where n = 2k and d = 22kk − 1, k = 2, 3, 4, ...):

  • Simplexes: γn−1[dαn−1n−1
  • Orthoplexes: γn[dβn]
  • Hypercubes: [dγnn

Euclidean honeycomb compounds[edit]

A known family of regular Euclidean compound honeycombs in five or more dimensions is an infinite family of compounds of hypercubic honeycombs, all sharing vertices and faces with another hypercubic honeycomb. This compound can have any number of hypercubic honeycombs. The Coxeter notation is δn[dδnn where δn = {∞} when n = 2 and {4,3n−3,4} when n ≥ 3.

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 regular polytopes that do not appear elsewhere in this list are the 11-cell, {3,5,3}, and the 57-cell, {5,3,5}, which have regular projective polyhedra as cells and vertex figures.

The elements of an abstract polyhedron are its body (the maximal element), its faces, edges, vertices and the null polytope or empty set. These abstract elements can be mapped into ordinary space or realised as geometrical figures. Some abstract polyhedra have well-formed or faithful realisations, others do not. A flag is a connected set of elements of each dimension - for a polyhedron that is the body, a face, an edge of the face, a vertex of the edge, and the null polytope. An abstract polytope is said to be regular if its combinatorial symmetries are transitive on its flags - that is to say, that any flag can be mapped onto any other under a symmetry of the polyhedron. Abstract regular polytopes remain an active area of research.

Five such regular abstract polyhedra, which can not be realised faithfully, were identified by H. S. M. Coxeter in his book Regular Polytopes (1977) and again by J. M. Wills in his paper "The combinatorially regular polyhedra of index 2" (1987).[28] They are all topologically equivalent to toroids. Their construction, by arranging n faces around each vertex, can be repeated indefinitely as tilings of the hyperbolic plane. In the diagrams below, the hyperbolic tiling images have colors corresponding to those of the polyhedra images.

Polyhedron DU36 medial rhombic triacontahedron.png
Medial rhombic triacontahedron
Dodecadodecahedron.png
Dodecadodecahedron
DU41 medial triambic icosahedron.png
Medial triambic icosahedron
Ditrigonal dodecadodecahedron.png
Ditrigonal dodecadodecahedron
Excavated dodecahedron.png
Excavated dodecahedron
Vertex figure {5}, {5/2}
Regular polygon 5.svgPentagram green.svg
(5.5/2)2
Dodecadodecahedron vertfig.png
{5}, {5/2}
Regular polygon 5.svgPentagram green.svg
(5.5/3)3
Ditrigonal dodecadodecahedron vertfig.png
Medial triambic icosahedron face.png
Faces 30 rhombi
Rhombus definition2.svg
12 pentagons
12 pentagrams
Regular polygon 5.svgPentagram green.svg
20 hexagons
Medial triambic icosahedron face.png
12 pentagons
12 pentagrams
Regular polygon 5.svgPentagram green.svg
20 hexagrams
Star hexagon face.png
Tiling Uniform tiling 45-t0.png
{4, 5}
Uniform tiling 552-t1.png
{5, 4}
Uniform tiling 65-t0.png
{6, 5}
Uniform tiling 553-t1.png
{5, 6}
Uniform tiling 66-t2.png
{6, 6}
χ −6 −6 −16 −16 −20

These occur as dual pairs as follows:

See also[edit]

Notes[edit]

  1. ^ Coxeter (1973), p. 129
  2. ^ McMullen & Schulte (2002), p. 30
  3. ^ Johnson (2012), p. 86
  4. ^ Coxeter (1973), p. 120
  5. ^ Coxeter (1973), p. 124
  6. ^ Coxeter (1973), pp. 66-67
  7. ^ http://www.mit.edu/~hlb/Associahedron/program.pdf
  8. ^ Coxeter (1973), Table I: Regular polytopes, (iii) The three regular polytopes in n dimensions (n>=5), pp. 294–295.
  9. ^ Abstract regular polytopes, p. 162-165 [1]
  10. ^ Grünbaum, B.; "Regular Polyhedra—Old and New", Aeqationes mathematicae, Vol. 16 (1977), pp 1–20.
  11. ^ Coxeter, H. S. M. Regular Skew Polyhedra in Three and Four Dimensions. Proc. London Math. Soc. 43, 33–62, 1937.
  12. ^ Coxeter, Regular and Semi-Regular Polytopes II 2.34)
  13. ^ The Symmetry of Things, 2008, Chapter 23 Objects with Primary Symmmetry, Infinite Platonic Polyhedra, pp. 333–335
  14. ^ McMullen & Schulte (2002, p.224)
  15. ^ McMullen & Schulte (2002, Section 7E)
  16. ^ Garner, C. W. L. Regular Skew Polyhedra in Hyperbolic Three-Space. Canad. J. Math. 19, 1179–1186, 1967. [2] Note: His paper says there are 32, but one is self-dual, leaving 31.
  17. ^ Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
  18. ^ The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space)
  19. ^ Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
  20. ^ The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space)
  21. ^ The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space)
  22. ^ Coxeter (1956), Table IV, p. 213.
  23. ^ Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
  24. ^ The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space)
  25. ^ Regular polytopes, Table VII, p. 305
  26. ^ Richard Klitzing, Uniform compound, stellated icositetrachoron
  27. ^ Richard Klitzing, Uniform compound, demidistesseract
  28. ^ The Regular Polyhedra (of index two), David A. Richter

References[edit]

External links[edit]