6-polytope - AbsoluteAstronomy.com

Graphs of three regular and related uniform polytope
Uniform polytope
A uniform polytope is a vertex-transitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons....

s
6-simplex	Truncated 6-simplex Truncated 6-simplex In six-dimensional geometry, a truncated 6-simplex is a convex uniform 6-polytope, being a truncation of the regular 6-simplex.There are unique 3 degrees of truncation. Vertices of the truncation 6-simplex are located as pairs on the edge of the 6-simplex. Vertices of the bitruncated 6-simplex are...		Rectified 6-simplex Rectified 6-simplex In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the rectified 6-simplex are located at the edge-centers of the...
Cantellated 6-simplex Cantellated 6-simplex In six-dimensional geometry, a cantellated 6-simplex is a convex uniform 6-polytope, being a cantellation of the regular 6-simplex.There are unique 4 degrees of cantellation for the 6-simplex, including truncations.- Cantellated 6-simplex:...		Runcinated 6-simplex Runcinated 6-simplex In six-dimensional geometry, a runcinated 6-simplex is a convex uniform 6-polytope constructed as a runcination of the regular 6-simplex....
Stericated 6-simplex Stericated 6-simplex In six-dimensional geometry, a stericated 6-simplex is a convex uniform 6-polytope with 4th order truncations of the regular 6-simplex....		Pentellated 6-simplex Pentellated 6-simplex In six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex.There are unique 10 degrees of pentellations of the 6-simplex with permutations of truncations, cantellations, runcinations, and sterications...
6-orthoplex	Truncated 6-orthoplex Truncated 6-orthoplex In six-dimensional geometry, a truncated 6-orthoplex is a convex uniform 6-polytope, being a truncation of the regular 6-orthoplex.There are 5 degrees of truncation for the 6-orthoplex. Vertices of the truncated 6-orthoplex are located as pairs on the edge of the 6-orthoplex. Vertices of the...		Rectified 6-orthoplex
Cantellated 6-orthoplex Cantellated 6-orthoplex In six-dimensional geometry, a cantellated 6-orthoplex is a convex uniform 6-polytope, being a cantellation of the regular 6-orthoplex.There are 8 cantellation for the 6-orthoplex including truncations...	Runcinated 6-orthoplex Runcinated 6-orthoplex In six-dimensional geometry, a runcinated 6-orthplex is a convex uniform 6-polytope with 3rd order truncations of the regular 6-orthoplex.There are 12 unique runcinations of the 6-orthoplex with permutations of truncations, and cantellations...		Stericated 6-orthoplex Stericated 6-orthoplex In six-dimensional geometry, a stericated 6-orthoplex is a convex uniform 6-polytope, constructed as a sterication of the regular 6-orthoplex....
Cantellated 6-cube Cantellated 6-cube In six-dimensional geometry, a cantellated 6-cube is a convex uniform 6-polytope, being a cantellation of the regular 6-cube.There are 8 cantellations for the 6-cube, including truncations...		Runcinated 6-cube Runcinated 6-cube In six-dimensional geometry, a runcinated 6-cube is a convex uniform 6-polytope with 3rd order truncations of the regular 6-cube.There are 12 unique runcinations of the 6-cube with permutations of truncations, and cantellations...
Stericated 6-cube Stericated 6-cube In six-dimensional geometry, a stericated 6-cube is a convex uniform 6-polytope, constructed as a sterication of the regular 6-cube....		Pentellated 6-cube Pentellated 6-cube In six-dimensional geometry, a pentellated 6-cube is a convex uniform 6-polytope with 5th order truncations of the regular 6-cube.There are unique 16 degrees of pentellations of the 6-cube with permutations of truncations, cantellations, runcinations, and sterications...
6-cube	Truncated 6-cube		Rectified 6-cube Rectified 6-cube In six-dimensional geometry, a rectified 6-cube is a convex uniform 6-polytope, being a rectification of the regular 6-cube.There are unique 6 degrees of rectifications, the zeroth being the 6-cube, and the 6th and last being the 6-orthoplex. Vertices of the rectified 6-cube are located at the...
6-demicube	Truncated 6-demicube		Cantellated 6-demicube Cantellated 6-demicube In six-dimensional geometry, a cantellated 6-demicube is a convex uniform 6-polytope, being a cantellation of the uniform 6-demicube. There are 2 unique cantellation for the 6-demicube including a truncation.- Cantellated 6-demicube:...
Runcinated 6-demicube Runcinated 6-demicube In six-dimensional geometry, a runcinated 6-demicube is a convex uniform 6-polytope with 3rd order truncations of the uniform 6-demicube.There are unique 4 runcinations of the 6-demicube, including permutations of truncations, and cantellations....		Stericated 6-demicube Stericated 6-demicube In six-dimensional geometry, a stericated 6-demicube is a convex uniform 6-polytope, constructed as a sterication of the 6-demicube....
2₂₁		1₂₂
Truncated 2₂₁		Truncated 1₂₂

In six-dimensional geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a uniform polypeton (or uniform 6-polytope

6-polytope

In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform polytera....

) is a six-dimensional uniform polytope

Uniform polytope

A uniform polytope is a vertex-transitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons....

. A uniform polypeton is vertex-transitive

Vertex-transitive

In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are the same...

, and all facets are uniform polytera

Uniform polyteron

In geometry, a uniform polyteron is a five-dimensional uniform polytope. By definition, a uniform polyteron is vertex-transitive and constructed from uniform polychoron facets....

.

The complete set of convex uniform polypeta has not been determined, but most can be made as Wythoff construction

Wythoff construction

In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction.- Construction process :...

s from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

s. Each combination of at least one ring on every connected group of nodes in the diagram produces a uniform 6-polytope.

The simplest uniform polypeta are regular polytope

Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of...

s: the 6-simplex {3,3,3,3,3}, the 6-cube (hexeract) {4,3,3,3,3}, and the 6-orthoplex (hexacross) {3,3,3,3,4}.

Uniform 6-polytopes by fundamental Coxeter groups

Uniform 6-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

s.

There are four fundamental reflective symmety groups which generate 153 unique uniform 6-polytopes.

#	Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...
1	A₆	[3⁵]
2	B₆	[4,3⁴]
3a	D₆	[3^3,1,1]
4	E₆	[3^2,2,1]

Uniform prismatic families

Uniform prism

There are 6 categorical uniform

Uniform polytope

A uniform polytope is a vertex-transitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons....

prisms based the uniform 5-polytopes.

#	Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...		Notes
1	A₅×A₁	[3,3,3,3] × [ ]	Prism family based on 6-simplex
2	B₅×A₁	[4,3,3,3] × [ ]	Prism family based on 6-cube
3a	D₅×A₁	[3^2,1,1] × [ ]	Prism family based on 6-demicube

#	Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...		Notes
4	A₃×I₂(p)×A₁	[3,3] × [p] × [ ]	Prism family based on tetrahedral Tetrahedron In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids... -p-gonal duoprism Duoprism In geometry of 4 dimensions or higher, a duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher... s
5	B₃×I₂(p)×A₁	[4,3] × [p] × [ ]	Prism family based on cubic Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... -p-gonal duoprism Duoprism In geometry of 4 dimensions or higher, a duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher... s
6	H₃×I₂(p)×A₁	[5,3] × [p] × [ ]	Prism family based on dodecahedral-p-gonal duoprism Duoprism In geometry of 4 dimensions or higher, a duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher... s

Uniform duoprism

There are 11 categorical uniform

Uniform polytope

A uniform polytope is a vertex-transitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons....

duoprism

Duoprism

In geometry of 4 dimensions or higher, a duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher...

atic families of polytopes based on Cartesian product

Cartesian product

In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...

s of lower dimensional uniform polytopes. Five are formed as the product of a uniform polychoron

Uniform polychoron

In geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra....

with a regular polygon

Regular polygon

A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

, and six are formed by the product of two uniform polyhedra

Uniform polyhedron

A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive...

#	Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...		Notes
1	A₄×I₂(p)	[3,3,3] × [p]	Family based on 5-cell-p-gonal duoprisms.
2	B₄×I₂(p)	[4,3,3] × [p]	Family based on tesseract Tesseract In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8... -p-gonal duoprisms.
3	F₄×I₂(p)	[3,4,3] × [p]	Family based on 24-cell-p-gonal duoprisms.
4	H₄×I₂(p)	[5,3,3] × [p]	Family based on 120-cell-p-gonal duoprisms.
5	D₄×I₂(p)	[3^1,1,1] × [p]	Family based on demitesseract-p-gonal duoprisms.

#	Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...		Notes
6	A₃²	[3,3]²	Family based on tetrahedral Tetrahedron In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids... duoprisms.
7	A₃×B₃	[3,3] × [4,3]	Family based on tetrahedral Tetrahedron In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids... -cubic Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... duoprisms.
8	A₃×H₃	[3,3] × [5,3]	Family based on tetrahedral Tetrahedron In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids... -dodecahedral duoprisms.
9	B₃²	[4,3]²	Family based on cubic Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... duoprisms.
10	B₃×H₃	[4,3] × [5,3]	Family based on cubic Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... -dodecahedral duoprisms.
11	H₃²	[5,3]²	Family based on dodecahedral duoprisms.

Uniform triaprism

There is one infinite family of uniform

Uniform polytope

A uniform polytope is a vertex-transitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons....

triaprism

Triaprism

In geometry of 6 dimensions or higher, a triaprism is a polytope resulting from the Cartesian product of three polytopes, each of two dimensions or higher...

atic families of polytopes constructed as a Cartesian product

Cartesian product

s of three regular polygons. Each combination of at least one ring on every connected group produces a uniform prismatic 6-polytope.

#	Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...		Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...	Notes
1	I₂(p)×I₂(q)×I₂(r)	[p] × [q] × [r]		Family based on p,q,r-gonal triprisms

Enumerating the convex uniform 6-polytopes

Simplex
Simplex
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...

family: A₆ [3⁴] -
- 35 uniform 6-polytopes as permutations of rings in the group diagram, including one regular:
  1. {3⁴} - 6-simplex -
Hypercube
Hypercube
In geometry, a hypercube is an n-dimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.An...

/orthoplex family: B₆ [4,3⁴] -
- 63 uniform 6-polytopes as permutations of rings in the group diagram, including two regular forms:
  1. {4,3³} — 6-cube (hexeract) -
  2. {3³,4} — 6-orthoplex, (hexacross) -
Demihypercube D₆ family: [3^3,1,1] -
- 47 uniform 6-polytopes (16 unique) as permutations of rings in the group diagram, including:
  1. {3,3^2,1}, 1₂₁ 6-demicube (demihexeract) - ; also as h{4,3³},
  2. {3,3,3^1,1}, 2₁₁ 6-orthoplex -
E₆ family: [3^3,1,1] -
- 39 uniform 6-polytopes (16 unique) as permutations of rings in the group diagram, including:
  1. {3,3,3^2,1}, 2₂₁
    Gosset 2 21 polytope
    In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 6-ic semi-regular figure....
    
    -
  2. {3,3^2,2}, 1₂₂
    Gosset 1 22 polytope
    In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V72 ....
    
    -

These fundamental families generate 153 nonprismatic convex uniform polypeta.

In addition, there are 105 uniform 6-polytope constructions based on prisms of the uniform polyteron

Uniform polyteron

In geometry, a uniform polyteron is a five-dimensional uniform polytope. By definition, a uniform polyteron is vertex-transitive and constructed from uniform polychoron facets....

s: [3,3,3,3]x[ ], [4,3,3,3]x[ ], [5,3,3,3]x[ ], [3^2,1,1]x[ ].

In addition, there are infinitely many uniform 6-polytope based on:

Duoprism prism families: [3,3]x[p]x[ ], [4,3]x[p]x[ ], [5,3]x[p]x[ ].
Duoprism families: [3,3,3]x[p], [4,3,3]x[p], [5,3,3]x[p].
Triaprism family: [p]x[q]x[r].

The A₆ family

There are 32+4−1=35 forms, derived by marking one or more nodes of the Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

.
All 35 are enumerated below. They are named by Norman Johnson from the Wythoff construction operations upon regular 6-simplex (heptapeton). Bowers-style acronym names are given in parentheses for cross-referencing.

The A₆ family has symmetry of order 5040 (7 factorial

Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

).

The coordinates of uniform 6-polytopes with 6-simplex symmetry can be generated as permutations of simple integers in 7-space, all in hyperplanes with normal vector (1,1,1,1,1,1,1).

See also list of A6 polytopes for graphs of these polytopes.

#	Johnson naming system Bowers name and (acronym)	Base point	Element counts
#	Johnson naming system Bowers name and (acronym)	Base point	5	4	3	2	1	0
1	6-simplex heptapeton (hop)	(0,0,0,0,0,0,1)	7	21	35	35	21	7
2	Rectified 6-simplex Rectified 6-simplex In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the rectified 6-simplex are located at the edge-centers of the... rectified heptapeton (ril)	(0,0,0,0,0,1,1)	14	63	140	175	105	21
3	Truncated 6-simplex Truncated 6-simplex In six-dimensional geometry, a truncated 6-simplex is a convex uniform 6-polytope, being a truncation of the regular 6-simplex.There are unique 3 degrees of truncation. Vertices of the truncation 6-simplex are located as pairs on the edge of the 6-simplex. Vertices of the bitruncated 6-simplex are... truncated heptapeton (til)	(0,0,0,0,0,1,2)	14	63	140	175	126	42
4	Birectified 6-simplex birectified heptapeton (bril)	(0,0,0,0,1,1,1)	14	84	245	350	210	35
5	Cantellated 6-simplex Cantellated 6-simplex In six-dimensional geometry, a cantellated 6-simplex is a convex uniform 6-polytope, being a cantellation of the regular 6-simplex.There are unique 4 degrees of cantellation for the 6-simplex, including truncations.- Cantellated 6-simplex:... small rhombated heptapeton (sril)	(0,0,0,0,1,1,2)	35	210	560	805	525	105
6	Bitruncated 6-simplex bitruncated heptapeton (batal)	(0,0,0,0,1,2,2)	14	84	245	385	315	105
7	Cantitruncated 6-simplex great rhombated heptapeton (gril)	(0,0,0,0,1,2,3)	35	210	560	805	630	210
8	Runcinated 6-simplex Runcinated 6-simplex In six-dimensional geometry, a runcinated 6-simplex is a convex uniform 6-polytope constructed as a runcination of the regular 6-simplex.... small prismated heptapeton (spil)	(0,0,0,1,1,1,2)	70	455	1330	1610	840	140
9	Bicantellated 6-simplex small prismated heptapeton (sabril)	(0,0,0,1,1,2,2)	70	455	1295	1610	840	140
10	Runcitruncated 6-simplex prismatotruncated heptapeton (patal)	(0,0,0,1,1,2,3)	70	560	1820	2800	1890	420
11	Tritruncated 6-simplex tetradecapeton (fe)	(0,0,0,1,2,2,2)	14	84	280	490	420	140
12	Runcicantellated 6-simplex prismatorhombated heptapeton (pril)	(0,0,0,1,2,2,3)	70	455	1295	1960	1470	420
13	Bicantitruncated 6-simplex great birhombated heptapeton (gabril)	(0,0,0,1,2,3,3)	49	329	980	1540	1260	420
14	Runcicantitruncated 6-simplex great prismated heptapeton (gapil)	(0,0,0,1,2,3,4)	70	560	1820	3010	2520	840
15	Stericated 6-simplex Stericated 6-simplex In six-dimensional geometry, a stericated 6-simplex is a convex uniform 6-polytope with 4th order truncations of the regular 6-simplex.... small cellated heptapeton (scal)	(0,0,1,1,1,1,2)	105	700	1470	1400	630	105
16	Biruncinated 6-simplex small biprismato-tetradecapeton (sibpof)	(0,0,1,1,1,2,2)	84	714	2100	2520	1260	210
17	Steritruncated 6-simplex cellitruncated heptapeton (catal)	(0,0,1,1,1,2,3)	105	945	2940	3780	2100	420
18	Stericantellated 6-simplex cellirhombated heptapeton (cral)	(0,0,1,1,2,2,3)	105	1050	3465	5040	3150	630
19	Biruncitruncated 6-simplex biprismatorhombated heptapeton (bapril)	(0,0,1,1,2,3,3)	84	714	2310	3570	2520	630
20	Stericantitruncated 6-simplex celligreatorhombated heptapeton (cagral)	(0,0,1,1,2,3,4)	105	1155	4410	7140	5040	1260
21	Steriruncinated 6-simplex celliprismated heptapeton (copal)	(0,0,1,2,2,2,3)	105	700	1995	2660	1680	420
22	Steriruncitruncated 6-simplex celliprismatotruncated heptapeton (captal)	(0,0,1,2,2,3,4)	105	945	3360	5670	4410	1260
23	Steriruncicantellated 6-simplex celliprismatorhombated heptapeton (copril)	(0,0,1,2,3,3,4)	105	1050	3675	5880	4410	1260
24	Biruncicantitruncated 6-simplex great biprismato-tetradecapeton (gibpof)	(0,0,1,2,3,4,4)	84	714	2520	4410	3780	1260
25	Steriruncicantitruncated 6-simplex great cellated heptapeton (gacal)	(0,0,1,2,3,4,5)	105	1155	4620	8610	7560	2520
26	Pentellated 6-simplex Pentellated 6-simplex In six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex.There are unique 10 degrees of pentellations of the 6-simplex with permutations of truncations, cantellations, runcinations, and sterications... small teri-tetradecapeton (staf)	(0,1,1,1,1,1,2)	126	434	630	490	210	42
27	Pentitruncated 6-simplex teracellated heptapeton (tocal)	(0,1,1,1,1,2,3)	126	826	1785	1820	945	210
28	Penticantellated 6-simplex teriprismated heptapeton (topal)	(0,1,1,1,2,2,3)	126	1246	3570	4340	2310	420
29	Penticantitruncated 6-simplex terigreatorhombated heptapeton (togral)	(0,1,1,1,2,3,4)	126	1351	4095	5390	3360	840
30	Pentiruncitruncated 6-simplex tericellirhombated heptapeton (tocral)	(0,1,1,2,2,3,4)	126	1491	5565	8610	5670	1260
31	Pentiruncicantellated 6-simplex teriprismatorhombi-tetradecapeton (taporf)	(0,1,1,2,3,3,4)	126	1596	5250	7560	5040	1260
32	Pentiruncicantitruncated 6-simplex terigreatoprismated heptapeton (tagopal)	(0,1,1,2,3,4,5)	126	1701	6825	11550	8820	2520
33	Pentisteritruncated 6-simplex tericellitrunki-tetradecapeton (tactaf)	(0,1,2,2,2,3,4)	126	1176	3780	5250	3360	840
34	Pentistericantitruncated 6-simplex tericelligreatorhombated heptapeton (tacogral)	(0,1,2,2,3,4,5)	126	1596	6510	11340	8820	2520
35	Omnitruncated 6-simplex great teri-tetradecapeton (gotaf)	(0,1,2,3,4,5,6)	126	1806	8400	16800	15120	5040

The B₆ family

There are 63 forms based on all permutations of the Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

s with one or more rings.

The B₆ family has symmetry of order 46080 (6 factorial

Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

x 2⁶).

They are named by Norman Johnson from the Wythoff construction operations upon the regular 6-cube and 6-orthoplex. Bowers names and acronym names are given for cross-referencing.

See also list of B6 polytopes for graphs of these polytopes.

#	Schläfli symbol	Names	Element counts
#	Schläfli symbol	Names	5	4	3	2	1	0
36	t₀{3,3,3,3,4}	6-orthoplex Hexacontatetrapeton (gee)	64	192	240	160	60	12
37	t₁{3,3,3,3,4}	Rectified 6-orthoplex Rectified hexacontatetrapeton (rag)	76	576	1200	1120	480	60
38	t₂{3,3,3,3,4}	Birectified 6-orthoplex Birectified hexacontatetrapeton (brag)	76	636	2160	2880	1440	160
39	t₂{4,3,3,3,3}	Birectified 6-cube Birectified hexeract (brox)	76	636	2080	3200	1920	240
40	t₁{4,3,3,3,3}	Rectified 6-cube Rectified 6-cube In six-dimensional geometry, a rectified 6-cube is a convex uniform 6-polytope, being a rectification of the regular 6-cube.There are unique 6 degrees of rectifications, the zeroth being the 6-cube, and the 6th and last being the 6-orthoplex. Vertices of the rectified 6-cube are located at the... Rectified hexeract (rax)	76	576	1200	1120	480	60
41	t₀{4,3,3,3,3}	6-cube Hexeract (ax)	12	60	160	240	192	64
42	t_0,1{3,3,3,3,4}	Truncated 6-orthoplex Truncated 6-orthoplex In six-dimensional geometry, a truncated 6-orthoplex is a convex uniform 6-polytope, being a truncation of the regular 6-orthoplex.There are 5 degrees of truncation for the 6-orthoplex. Vertices of the truncated 6-orthoplex are located as pairs on the edge of the 6-orthoplex. Vertices of the... Truncated hexacontatetrapeton (tag)	76	576	1200	1120	540	120
43	t_0,2{3,3,3,3,4}	Cantellated 6-orthoplex Cantellated 6-orthoplex In six-dimensional geometry, a cantellated 6-orthoplex is a convex uniform 6-polytope, being a cantellation of the regular 6-orthoplex.There are 8 cantellation for the 6-orthoplex including truncations... Small rhombated hexacontatetrapeton (srog)	136	1656	5040	6400	3360	480
44	t_1,2{3,3,3,3,4}	Bitruncated 6-orthoplex Bitruncated hexacontatetrapeton (botag)					1920	480
45	t_0,3{3,3,3,3,4}	Runcinated 6-orthoplex Runcinated 6-orthoplex In six-dimensional geometry, a runcinated 6-orthplex is a convex uniform 6-polytope with 3rd order truncations of the regular 6-orthoplex.There are 12 unique runcinations of the 6-orthoplex with permutations of truncations, and cantellations... Small prismated hexacontatetrapeton (spog)					7200	960
46	t_1,3{3,3,3,3,4}	Bicantellated 6-orthoplex Small birhombated hexacontatetrapeton (siborg)					8640	1440
47	t_2,3{4,3,3,3,3}	Tritruncated 6-cube Hexeractihexacontitetrapeton (xog)					3360	960
48	t_0,4{3,3,3,3,4}	Stericated 6-orthoplex Stericated 6-orthoplex In six-dimensional geometry, a stericated 6-orthoplex is a convex uniform 6-polytope, constructed as a sterication of the regular 6-orthoplex.... Small cellated hexacontatetrapeton (scag)					5760	960
49	t_1,4{4,3,3,3,3}	Biruncinated 6-cube Small biprismato-hexeractihexacontitetrapeton (sobpoxog)					11520	1920
50	t_1,3{4,3,3,3,3}	Bicantellated 6-cube Small birhombated hexeract (saborx)					9600	1920
51	t_1,2{4,3,3,3,3}	Bitruncated 6-cube Bitruncated hexeract (botox)					2880	960
52	t_0,5{4,3,3,3,3}	Pentellated 6-cube Pentellated 6-cube In six-dimensional geometry, a pentellated 6-cube is a convex uniform 6-polytope with 5th order truncations of the regular 6-cube.There are unique 16 degrees of pentellations of the 6-cube with permutations of truncations, cantellations, runcinations, and sterications... Small teri-hexeractihexacontitetrapeton (stoxog)					1920	384
53	t_0,4{4,3,3,3,3}	Stericated 6-cube Stericated 6-cube In six-dimensional geometry, a stericated 6-cube is a convex uniform 6-polytope, constructed as a sterication of the regular 6-cube.... Small cellated hexeract (scox)					5760	960
54	t_0,3{4,3,3,3,3}	Runcinated 6-cube Runcinated 6-cube In six-dimensional geometry, a runcinated 6-cube is a convex uniform 6-polytope with 3rd order truncations of the regular 6-cube.There are 12 unique runcinations of the 6-cube with permutations of truncations, and cantellations... Small prismated hexeract (spox)					7680	1280
55	t_0,2{4,3,3,3,3}	Cantellated 6-cube Cantellated 6-cube In six-dimensional geometry, a cantellated 6-cube is a convex uniform 6-polytope, being a cantellation of the regular 6-cube.There are 8 cantellations for the 6-cube, including truncations... Small rhombated hexeract (srox)					4800	960
56	t_0,1{4,3,3,3,3}	Truncated 6-cube Truncated hexeract (tox)	76	444	1120	1520	1152	384
57	t_0,1,2{3,3,3,3,4}	Cantitruncated 6-orthoplex Great rhombated hexacontatetrapeton (grog)					3840	960
58	t_0,1,3{3,3,3,3,4}	Runcitruncated 6-orthoplex Prismatotruncated hexacontatetrapeton (potag)					15840	2880
59	t_0,2,3{3,3,3,3,4}	Runcicantellated 6-orthoplex Prismatorhombated hexacontatetrapeton (prog)					11520	2880
60	t_1,2,3{3,3,3,3,4}	Bicantitruncated 6-orthoplex Great birhombated hexacontatetrapeton (gaborg)					10080	2880
61	t_0,1,4{3,3,3,3,4}	Steritruncated 6-orthoplex Cellitruncated hexacontatetrapeton (catog)					19200	3840
62	t_0,2,4{3,3,3,3,4}	Stericantellated 6-orthoplex Cellirhombated hexacontatetrapeton (crag)					28800	5760
63	t_1,2,4{3,3,3,3,4}	Biruncitruncated 6-orthoplex Biprismatotruncated hexacontatetrapeton (boprax)					23040	5760
64	t_0,3,4{3,3,3,3,4}	Steriruncinated 6-orthoplex Celliprismated hexacontatetrapeton (copog)					15360	3840
65	t_1,2,4{4,3,3,3,3}	Biruncitruncated 6-cube Biprismatotruncated hexeract (boprag)					23040	5760
66	t_1,2,3{4,3,3,3,3}	Bicantitruncated 6-cube Great birhombated hexeract (gaborx)					11520	3840
67	t_0,1,5{3,3,3,3,4}	Pentitruncated 6-orthoplex Teritruncated hexacontatetrapeton (tacox)					8640	1920
68	t_0,2,5{3,3,3,3,4}	Penticantellated 6-orthoplex Terirhombated hexacontatetrapeton (tapox)					21120	3840
69	t_0,3,4{4,3,3,3,3}	Steriruncinated 6-cube Celliprismated hexeract (copox)					15360	3840
70	t_0,2,5{4,3,3,3,3}	Penticantellated 6-cube Terirhombated hexeract (topag)					21120	3840
71	t_0,2,4{4,3,3,3,3}	Stericantellated 6-cube Cellirhombated hexeract (crax)					28800	5760
72	t_0,2,3{4,3,3,3,3}	Runcicantellated 6-cube Prismatorhombated hexeract (prox)					13440	3840
73	t_0,1,5{4,3,3,3,3}	Pentitruncated 6-cube Teritruncated hexeract (tacog)					8640	1920
74	t_0,1,4{4,3,3,3,3}	Steritruncated 6-cube Cellitruncated hexeract (catax)					19200	3840
75	t_0,1,3{4,3,3,3,3}	Runcitruncated 6-cube Prismatotruncated hexeract (potax)					17280	3840
76	t_0,1,2{4,3,3,3,3}	Cantitruncated 6-cube Great rhombated hexeract (grox)					5760	1920
77	t_0,1,2,3{3,3,3,3,4}	Runcicantitruncated 6-orthoplex Great prismated hexacontatetrapeton (gopog)					20160	5760
78	t_0,1,2,4{3,3,3,3,4}	Stericantitruncated 6-orthoplex Celligreatorhombated hexacontatetrapeton (cagorg)					46080	11520
79	t_0,1,3,4{3,3,3,3,4}	Steriruncitruncated 6-orthoplex Celliprismatotruncated hexacontatetrapeton (captog)					40320	11520
80	t_0,2,3,4{3,3,3,3,4}	Steriruncicantellated 6-orthoplex Celliprismatorhombated hexacontatetrapeton (coprag)					40320	11520
81	t_1,2,3,4{4,3,3,3,3}	Biruncicantitruncated 6-cube Great biprismato-hexeractihexacontitetrapeton (gobpoxog)					34560	11520
82	t_0,1,2,5{3,3,3,3,4}	Penticantitruncated 6-orthoplex Terigreatorhombated hexacontatetrapeton (togrig)					30720	7680
83	t_0,1,3,5{3,3,3,3,4}	Pentiruncitruncated 6-orthoplex Teriprismatotruncated hexacontatetrapeton (tocrax)					51840	11520
84	t_0,2,3,5{4,3,3,3,3}	Pentiruncicantellated 6-cube Teriprismatorhombi-hexeractihexacontitetrapeton (tiprixog)					46080	11520
85	t_0,2,3,4{4,3,3,3,3}	Steriruncicantellated 6-cube Celliprismatorhombated hexeract (coprix)					40320	11520
86	t_0,1,4,5{4,3,3,3,3}	Pentisteritruncated 6-cube Tericelli-hexeractihexacontitetrapeton (tactaxog)					30720	7680
87	t_0,1,3,5{4,3,3,3,3}	Pentiruncitruncated 6-cube Teriprismatotruncated hexeract (tocrag)					51840	11520
88	t_0,1,3,4{4,3,3,3,3}	Steriruncitruncated 6-cube Celliprismatotruncated hexeract (captix)					40320	11520
89	t_0,1,2,5{4,3,3,3,3}	Penticantitruncated 6-cube Terigreatorhombated hexeract (togrix)					30720	7680
90	t_0,1,2,4{4,3,3,3,3}	Stericantitruncated 6-cube Celligreatorhombated hexeract (cagorx)					46080	11520
91	t_0,1,2,3{4,3,3,3,3}	Runcicantitruncated 6-cube Great prismated hexeract (gippox)					23040	7680
92	t_0,1,2,3,4{3,3,3,3,4}	Steriruncicantitruncated 6-orthoplex Great cellated hexacontatetrapeton (gocog)					69120	23040
93	t_0,1,2,3,5{3,3,3,3,4}	Pentiruncicantitruncated 6-orthoplex Terigreatoprismated hexacontatetrapeton (tagpog)					80640	23040
94	t_0,1,2,4,5{3,3,3,3,4}	Pentistericantitruncated 6-orthoplex Tericelligreatorhombated hexacontatetrapeton (tecagorg)					80640	23040
95	t_0,1,2,4,5{4,3,3,3,3}	Pentistericantitruncated 6-cube Tericelligreatorhombated hexeract (tocagrax)					80640	23040
96	t_0,1,2,3,5{4,3,3,3,3}	Pentiruncicantitruncated 6-cube Terigreatoprismated hexeract (tagpox)					80640	23040
97	t_0,1,2,3,4{4,3,3,3,3}	Steriruncicantitruncated 6-cube Great cellated hexeract (gocax)					69120	23040
98	t_0,1,2,3,4,5{4,3,3,3,3}	Omnitruncated 6-cube Great teri-hexeractihexacontitetrapeton (gotaxog)					138240	46080

The D₆ family

The D₆ family has symmetry of order 23040 (6 factorial

Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

x 2⁵).

This family has 3×16−1=47 Wythoffian uniform polytopes, generated by marking one or more nodes of the D₆ Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

. Of these, 31 (2×16−1) are repeated from the B₆ family and 16 are unique to this family. The 16 unique forms are enumerated below. Bowers-style acronym names are given for cross-referencing.

See list of D6 polytopes for Coxeter plane graphs of these polytopes.

#	Names	Base point (Alternately signed)	Element counts						Circumrad
#	Names	Base point (Alternately signed)	5	4	3	2	1	0	Circumrad
99	6-demicube Hemihexeract (hax)	(1,1,1,1,1,1)	44	252	640	640	240	32	0.8660254
100	Truncated 6-demicube Truncated hemihexeract (thax)	(1,1,3,3,3,3)	76	636	2080	3200	2160	480	2.1794493
101	Cantellated 6-demicube Cantellated 6-demicube In six-dimensional geometry, a cantellated 6-demicube is a convex uniform 6-polytope, being a cantellation of the uniform 6-demicube. There are 2 unique cantellation for the 6-demicube including a truncation.- Cantellated 6-demicube:... Small rhombated hemihexeract (sirhax)	(1,1,1,3,3,3)					3840	640	1.9364916
102	Runcinated 6-demicube Runcinated 6-demicube In six-dimensional geometry, a runcinated 6-demicube is a convex uniform 6-polytope with 3rd order truncations of the uniform 6-demicube.There are unique 4 runcinations of the 6-demicube, including permutations of truncations, and cantellations.... Small prismated hemihexeract (sophax)	(1,1,1,1,3,3)					3360	480	1.6583123
103	Stericated 6-demicube Stericated 6-demicube In six-dimensional geometry, a stericated 6-demicube is a convex uniform 6-polytope, constructed as a sterication of the 6-demicube.... Small cellated demihexeract (sochax)	(1,1,1,1,1,3)					1440	192	1.3228756
104	Cantitruncated 6-demicube Great rhombated hemihexeract (girhax)	(1,1,3,5,5,5)					5760	1920	3.2787192
105	Runcitruncated 6-demicube Prismatotruncated hemihexeract (pithax)	(1,1,3,3,5,5)					12960	2880	2.95804
106	Runcicantellated 6-demicube Prismatorhombated hemihexeract (prohax)	(1,1,1,3,5,5)					7680	1920	2.7838821
107	Steritruncated 6-demicube Cellitruncated hemihexeract (cathix)	(1,1,3,3,3,5)					9600	1920	2.5980761
108	Stericantellated 6-demicube Cellirhombated hemihexeract (crohax)	(1,1,1,3,3,5)					10560	1920	2.3979158
109	Steriruncinated 6-demicube Celliprismated hemihexeract (cophix)	(1,1,1,1,3,5)					5280	960	2.1794496
110	Runcicantitruncated 6-demicube Great prismated hemihexeract (gophax)	(1,1,3,5,7,7)					17280	5760	4.0926762
111	Stericantitruncated 6-demicube Celligreatorhombated hemihexeract (cagrohax)	(1,1,3,5,5,7)					20160	5760	3.7080991
112	Steriruncitruncated 6-demicube Celliprismatotruncated hemihexeract (capthix)	(1,1,3,3,5,7)					23040	5760	3.4278274
113	Steriruncicantellated 6-demicube Celliprismatorhombated hemihexeract (caprohax)	(1,1,1,3,5,7)					15360	3840	3.2787192
114	Steriruncicantitruncated 6-demicube Great cellated hemihexeract (gochax)	(1,1,3,5,7,9)					34560	11520	4.5552168

The E₆ family

There are 39 forms based on all permutations of the Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

s with one or more rings. Bowers-style acronym names are given for cross-referencing. The E₆ family has symmetry of order 51,840.

See also list of E6 polytopes for graphs of these polytopes.

#	Names	Element counts
#	Names	5-faces	4-faces	Cells	Faces	Edges	Vertices
115	2₂₁ Icosiheptaheptacontidipeton (jak)	99	648	1080	720	216	27
116	Rectified 2₂₁ Rectified icosiheptaheptacontidipeton (rojak)	126	1350	4320	5040	2160	216
117	Rectified 1₂₂ Rectified pentacontatetrapeton (ram)	126	1566	6480	10800	6480	720
118	1₂₂ Pentacontatetrapeton (mo)	54	702	2160	2160	720	72
119	Truncated 2₂₁ Truncated icosiheptaheptacontidipeton (tojak)	126	1350	4320	5040	2376	432
120	Cantellated 2₂₁ Small rhombated icosiheptaheptacontidipeton (sirjak)	342	3942	15120	24480	15120	2160
121	Runcinated 2₂₁ Small demiprismated icosiheptaheptacontidipeton (shopjak)	342	4662	16200	19440	8640	1080
122	Stericated 2₂₁ Trirectified pentacontatetrapeton (trim)	558	4608	8640	6480	2160	270
123	Demified icosiheptaheptacontidipeton (hejak)	342	2430	7200	7920	3240	432
124	Bitruncated 2₂₁ Bitruncated icosiheptaheptacontidipeton (botajik)						2160
125	Bicantellated 2₂₁ Birectified pentacontatetrapeton (barm)					12960	2160
126	Demirectified icosiheptaheptacontidipeton (harjak)						1080
127	Truncated pentacontatetrapeton (tim)					13680	1440
128	Cantitruncated 2₂₁ Great rhombated icosiheptaheptacontidipeton (girjak)						4320
129	Runcitruncated 2₂₁ Demiprismatotruncated icosiheptaheptacontidipeton (hopitjak)						4320
130	Steritruncated 2₂₁ Cellitruncated icosiheptaheptacontidipeton (catjak)						2160
131	Demitruncated icosiheptaheptacontidipeton (hotjak)						2160
132	Runcicantellated 2₂₁ Demiprismatorhombated icosiheptaheptacontidipeton (haprojak)						6480
133	Stericantellated 2₂₁ Small birhombated pentacontatetrapeton (sabrim)						6480
134	Small demirhombated icosiheptaheptacontidipeton (shorjak)						4320
135	Small prismated icosiheptaheptacontidipeton (spojak)						4320
136	Small prismated pentacontatetrapeton (spam)						2160
137	Bitruncated pentacontatetrapeton (bitem)						6480
138	Tritruncated icosiheptaheptacontidipeton (titajak)						4320
139	Small rhombated pentacontatetrapeton (sram)						6480
140	Runcicantitruncated 2₂₁ Great demiprismated icosiheptaheptacontidipeton (ghopjak)						12960
141	Stericantitruncated 2₂₁ Celligreatorhombated icosiheptaheptacontidipeton (cograjik)						12960
142	Great demirhombated icosiheptaheptacontidipeton (ghorjak)						8640
143	Steriruncitruncated 2₂₁ Tritruncated pentacontatetrapeton (titam)						8640
144	Prismatotruncated icosiheptaheptacontidipeton (potjak)						12960
145	Demicellitruncated icosiheptaheptacontidipeton (hictijik)						8640
146	Prismatorhombated icosiheptaheptacontidipeton (projak)						12960
147	Prismatotruncated pentacontatetrapeton (patom)						12960
148	Great rhombated pentacontatetrapeton (gram)						12960
149	Steriruncicantitruncated 2₂₁ Great birhombated pentacontatetrapeton (gabrim)						25920
150	Great prismated icosiheptaheptacontidipeton (gapjak)						25920
151	Demicelligreatorhombated icosiheptaheptacontidipeton (hocgarjik)						25920
152	Prismatorhombated pentacontatetrapeton (prom)						25920
153	Great prismated pentacontatetrapeton (gopam)						51840

Regular and uniform honeycombs

There are four fundamental affine Coxeter groups and 27 prismatic groups that generate regular and uniform tessellations in 5-space:

#	Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...		Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...
1		[3^[6]]
2		h[4,3³,4] [4,3,3^1,1]
3		[4,3³,4]
4		q[4,3³,4] [3^1,1,3,3^1,1]

Prismatic groups
#	Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...
1	x	[3^[5]]x[∞]x[∞]
2	x	[4,3,3^1,1]x[∞]
3	x	[4,3,3,4]x[∞]
4	x	[3^1,1,1,1]x[∞]
5	x	[3,4,3,3]x[∞]
6	xx	[4,3,4]x[∞]x[∞]
7	xx	[4,3^1,1]x[∞]x[∞]
8	xx	[3^[4]]x[∞]x[∞]
9	xxx	[4,4]x[∞]x[∞]x[∞]
10	xxx	[6,3]x[∞]x[∞]x[∞]
11	xxx	[3^[3]]x[∞]x[∞]x[∞]
12	xxxx	[∞]x[∞]x[∞]x[∞]x[∞]
13	xx	[3^[3]]x[3^[3]]x[∞]
14	xx	[3^[3]]x[4,4]x[∞]
15	xx	[3^[3]]x[6,3]x[∞]
16	xx	[4,4]x[4,4]x[∞]
17	xx	[4,4]x[6,3]x[∞]
18	xx	[6,3]x[6,3]x[∞]
19	x	[3^[4]]x[3^[3]]
20	x	[4,3^1,1]x[3^[3]]
21	x	[4,3,4]x[3^[3]]
22	x	[3^[4]]x[4,4]
23	x	[4,3^1,1]x[4,4]
24	x	[4,3,4]x[4,4]
25	x	[3^[4]]x[6,3]
26	x	[4,3^1,1]x[6,3]
27	x	[4,3,4]x[6,3]

Regular and uniform honeycombs include:

- Regular hypercube honeycomb of Euclidean 5-space, the penteractic honeycomb, with symbols {4,3³,4}, =
- The uniform alternated hypercube honeycomb, demipenteractic honeycomb, with symbols h{4,3³,4}, = =
There are 12 unique uniform honeycombs, including:
- 5-simplex honeycomb
  5-simplex honeycomb
  In five-dimensional Euclidean geometry, the 5-simplex honeycomb or hexateric honeycomb is a space-filling tessellation . Each vertex is shared by 12 5-simplexes, 30 rectified 5-simplexes, and 20 birectified 5-simplexes...
- Truncated 5-simplex honeycomb
  Truncated 5-simplex honeycomb
  In five-dimensional Euclidean geometry, the truncated 5-simplex honeycomb or truncated hexateric honeycomb is a space-filling tessellation...
- Omnitruncated 5-simplex honeycomb
  Omnitruncated 5-simplex honeycomb
  In five-dimensional Euclidean geometry, the omnitruncated 5-simplex honeycomb or omnitruncated hexateric honeycomb is a space-filling tessellation . It is composed entirely of omnitruncated 5-simplex facets....

Regular and uniform hyperbolic honeycombs

There are no compact hyperbolic Coxeter groups of rank 6, groups that can generate honeycombs with all finite facets, and a finite vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

. However there are 12 noncompact hyperbolic Coxeter groups of rank 6, each generating uniform honeycombs in 5-space as permutations of rings of the Coxeter diagrams.

Hyperbolic noncompact groups
= [3,3^[5]]: = [(3,3,3,3,3,4)]: = [(3,3,4,3,3,4)]:	= [4,3,3^2,1]: = [3,4,3^1,1]: = [4,3,/3\,3,4]:	= [3,3,3,4,3]: = [3,3,4,3,3]: = [3,4,3,3,4]:	= [3^2,1,1,1]: = [4,3,3^1,1,1]: = [3^1,1,1,1,1]: