Hexicated 7-simplexes

In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-simplex.

7-simplex	Hexicated 7-simplex	Hexitruncated 7-simplex	Hexicantellated 7-simplex
Hexiruncinated 7-simplex	Hexicantitruncated 7-simplex	Hexiruncitruncated 7-simplex	Hexiruncicantellated 7-simplex
Hexisteritruncated 7-simplex	Hexistericantellated 7-simplex	Hexipentitruncated 7-simplex	Hexiruncicantitruncated 7-simplex
Hexistericantitruncated 7-simplex	Hexisteriruncitruncated 7-simplex	Hexisteriruncicantellated 7-simplex	Hexipenticantitruncated 7-simplex
Hexipentiruncitruncated 7-simplex	Hexisteriruncicantitruncated 7-simplex	Hexipentiruncicantitruncated 7-simplex	Hexipentistericantitruncated 7-simplex
Hexipentisteriruncicantitruncated 7-simplex (Omnitruncated 7-simplex)
Orthogonal projections in A7 Coxeter plane

There are 20 unique hexications for the 7-simplex, including all permutations of truncations, cantellations, runcinations, sterications, and pentellations.

The simple hexicated 7-simplex is also called an expanded 7-simplex, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular 7-simplex. The highest form, the hexipentisteriruncicantitruncated 7-simplex is more simply called a omnitruncated 7-simplex with all of the nodes ringed.

Hexicated 7-simplex

Hexicated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,6{36}
Coxeter-Dynkin diagrams
6-faces	254: 8+8 {35} 28+28 {}x{34} 56+56 {3}x{3,3,3} 70 {3,3}x{3,3}
5-faces
4-faces
Cells
Faces
Edges	336
Vertices	56
Vertex figure	5-simplex antiprism
Coxeter group	A7×2, [[36]], order 80640
Properties	convex

In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, a hexication (6th order truncation) of the regular 7-simplex, or alternately can be seen as an expansion operation.

The vertices of the A₇ 2D orthogonal projection are seen in the Ammann–Beenker tiling.

Root vectors

Its 56 vertices represent the root vectors of the simple Lie group A₇.

Alternate names

• Expanded 7-simplex
• Small petated hexadecaexon (acronym: suph) (Jonathan Bowers)^[1]

Coordinates

The vertices of the hexicated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,1,1,2). This construction is based on facets of the hexicated 8-orthoplex, .

A second construction in 8-space, from the center of a rectified 8-orthoplex is given by coordinate permutations of:

(1,-1,0,0,0,0,0,0)

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[[7]]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[[5]]	[4]	[[3]]

Hexitruncated 7-simplex

hexitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	1848
Vertices	336
Vertex figure
Coxeter group	A7, [36], order 40320
Properties	convex

Alternate names

• Petitruncated octaexon (acronym: puto) (Jonathan Bowers)^[2]

Coordinates

The vertices of the hexitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,1,2,3). This construction is based on facets of the hexitruncated 8-orthoplex, .

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[5]	[4]	[3]

Hexicantellated 7-simplex

Hexicantellated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,2,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	5880
Vertices	840
Vertex figure
Coxeter group	A7, [36], order 40320
Properties	convex

Alternate names

• Petirhombated octaexon (acronym: puro) (Jonathan Bowers)^[3]

Coordinates

The vertices of the hexicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,2,2,3). This construction is based on facets of the hexicantellated 8-orthoplex, .

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[5]	[4]	[3]

Hexiruncinated 7-simplex

Hexiruncinated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,3,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	8400
Vertices	1120
Vertex figure
Coxeter group	A7×2, [[36]], order 80640
Properties	convex

Alternate names

• Petiprismated hexadecaexon (acronym: puph) (Jonathan Bowers)^[4]

Coordinates

The vertices of the hexiruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,2,2,2,3). This construction is based on facets of the hexiruncinated 8-orthoplex, .

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[[7]]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[[5]]	[4]	[[3]]

Hexicantitruncated 7-simplex

Hexicantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	8400
Vertices	1680
Vertex figure
Coxeter group	A7, [36], order 40320
Properties	convex

Alternate names

• Petigreatorhombated octaexon (acronym: pugro) (Jonathan Bowers)^[5]

Coordinates

The vertices of the hexicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,2,3,4). This construction is based on facets of the hexicantitruncated 8-orthoplex, .

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[5]	[4]	[3]

Hexiruncitruncated 7-simplex

Hexiruncitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,3,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	20160
Vertices	3360
Vertex figure
Coxeter group	A7, [36], order 40320
Properties	convex

Alternate names

• Petiprismatotruncated octaexon (acronym: pupato) (Jonathan Bowers)^[6]

Coordinates

The vertices of the hexiruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,2,2,3,4). This construction is based on facets of the hexiruncitruncated 8-orthoplex, .

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[5]	[4]	[3]

Hexiruncicantellated 7-simplex

Hexiruncicantellated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,2,3,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	16800
Vertices	3360
Vertex figure
Coxeter group	A7, [36], order 40320
Properties	convex

In seven-dimensional geometry, a hexiruncicantellated 7-simplex is a uniform 7-polytope.

Alternate names

• Petiprismatorhombated octaexon (acronym: pupro) (Jonathan Bowers)^[7]

Coordinates

The vertices of the hexiruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,2,3,3,4). This construction is based on facets of the hexiruncicantellated 8-orthoplex, .

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[5]	[4]	[3]

Hexisteritruncated 7-simplex

hexisteritruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,4,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	20160
Vertices	3360
Vertex figure
Coxeter group	A7, [36], order 40320
Properties	convex

Alternate names

• Peticellitruncated octaexon (acronym: pucto) (Jonathan Bowers)^[8]

Coordinates

The vertices of the hexisteritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,2,3,4). This construction is based on facets of the hexisteritruncated 8-orthoplex, .

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[5]	[4]	[3]

Hexistericantellated 7-simplex

hexistericantellated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,2,4,6{36}
Coxeter-Dynkin diagrams
6-faces	t0,2,4{3,3,3,3,3} {}xt0,2,4{3,3,3,3} {3}xt0,2{3,3,3} t0,2{3,3}xt0,2{3,3}
5-faces
4-faces
Cells
Faces
Edges	30240
Vertices	5040
Vertex figure
Coxeter group	A7×2, [[36]], order 80640
Properties	convex

Alternate names

• Peticellirhombihexadecaexon (acronym: pucroh) (Jonathan Bowers)^[9]

Coordinates

The vertices of the hexistericantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,3,4). This construction is based on facets of the hexistericantellated 8-orthoplex, .

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[[7]]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[[5]]	[4]	[[3]]

Hexipentitruncated 7-simplex

Hexipentitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,5,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	8400
Vertices	1680
Vertex figure
Coxeter group	A7×2, [[36]], order 80640
Properties	convex

Alternate names

• Petiteritruncated hexadecaexon (acronym: putath) (Jonathan Bowers)^[10]

Coordinates

The vertices of the hexipentitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,2,2,3,4). This construction is based on facets of the hexipentitruncated 8-orthoplex, .

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[[7]]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[[5]]	[4]	[[3]]

Hexiruncicantitruncated 7-simplex

Hexiruncicantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,3,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	30240
Vertices	6720
Vertex figure
Coxeter group	A7, [36], order 40320
Properties	convex

Alternate names

• Petigreatoprismated octaexon (acronym: pugopo) (Jonathan Bowers)^[11]

Coordinates

The vertices of the hexiruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,4,5). This construction is based on facets of the hexiruncicantitruncated 8-orthoplex, .

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[[7]]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[[5]]	[4]	[[3]]

Hexistericantitruncated 7-simplex

Hexistericantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,4,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	50400
Vertices	10080
Vertex figure
Coxeter group	A7, [36], order 40320
Properties	convex

Alternate names

• Peticelligreatorhombated octaexon (acronym: pucagro) (Jonathan Bowers)^[12]

Coordinates

The vertices of the hexistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,4,5). This construction is based on facets of the hexistericantitruncated 8-orthoplex, .

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[[7]]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[[5]]	[4]	[[3]]

Hexisteriruncitruncated 7-simplex

Hexisteriruncitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,3,4,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	45360
Vertices	10080
Vertex figure
Coxeter group	A7, [36], order 40320
Properties	convex

Alternate names

• Peticelliprismatotruncated octaexon (acronym: pucpato) (Jonathan Bowers)^[13]

Coordinates

The vertices of the hexisteriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,3,3,4,5). This construction is based on facets of the hexisteriruncitruncated 8-orthoplex, .

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[5]	[4]	[3]

Hexisteriruncicantellated 7-simplex

Hexisteriruncicantellated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,2,3,4,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	45360
Vertices	10080
Vertex figure
Coxeter group	A7×2, [[36]], order 80640
Properties	convex

Alternate names

• Peticelliprismatorhombihexadecaexon (acronym: pucproh) (Jonathan Bowers)^[14]

Coordinates

The vertices of the hexisteriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,3,4,4,5). This construction is based on facets of the hexisteriruncitruncated 8-orthoplex, .

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[[7]]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[[5]]	[4]	[[3]]

Hexipenticantitruncated 7-simplex

hexipenticantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,5,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	30240
Vertices	6720
Vertex figure
Coxeter group	A7, [36], order 40320
Properties	convex

Alternate names

• Petiterigreatorhombated octaexon (acronym: putagro) (Jonathan Bowers)^[15]

Coordinates

The vertices of the hexipenticantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,2,3,4,5). This construction is based on facets of the hexipenticantitruncated 8-orthoplex, .

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[5]	[4]	[3]

Hexipentiruncitruncated 7-simplex

Hexipentiruncitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,3,5,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices	10080
Vertex figure
Coxeter group	A7×2, [[36]], order 80640
Properties	convex

Alternate names

• Petiteriprismatotruncated hexadecaexon (acronym: putpath) (Jonathan Bowers)^[16]

Coordinates

The vertices of the hexipentiruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,3,4,4,5). This construction is based on facets of the hexipentiruncitruncated 8-orthoplex, .

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[[7]]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[[5]]	[4]	[[3]]

Hexisteriruncicantitruncated 7-simplex

Hexisteriruncicantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,3,4,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	80640
Vertices	20160
Vertex figure
Coxeter group	A7, [36], order 40320
Properties	convex

Alternate names

• Petigreatocellated octaexon (acronym: pugaco) (Jonathan Bowers)^[17]

Coordinates

The vertices of the hexisteriruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,3,4,5,6). This construction is based on facets of the hexisteriruncicantitruncated 8-orthoplex, .

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[[7]]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[[5]]	[4]	[[3]]

Hexipentiruncicantitruncated 7-simplex

Hexipentiruncicantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,3,5,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	80640
Vertices	20160
Vertex figure
Coxeter group	A7, [36], order 40320
Properties	convex

Alternate names

• Petiterigreatoprismated octaexon (acronym: putgapo) (Jonathan Bowers)^[18]

Coordinates

The vertices of the hexipentiruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,3,4,5,6). This construction is based on facets of the hexipentiruncicantitruncated 8-orthoplex, .

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[[7]]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[[5]]	[4]	[[3]]

Hexipentistericantitruncated 7-simplex

Hexipentistericantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,4,5,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	80640
Vertices	20160
Vertex figure
Coxeter group	A7×2, [[36]], order 80640
Properties	convex

Alternate names

• Petitericelligreatorhombihexadecaexon (acronym: putcagroh) (Jonathan Bowers)^[19]

Coordinates

The vertices of the hexipentistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,3,3,4,5,6). This construction is based on facets of the hexipentistericantitruncated 8-orthoplex, .

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[[7]]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[[5]]	[4]	[[3]]

Omnitruncated 7-simplex

Omnitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,3,4,5,6{36}
Coxeter-Dynkin diagrams
6-faces	254
5-faces	5796
4-faces	40824
Cells	126000
Faces	191520
Edges	141120
Vertices	40320
Vertex figure	Irr. 6-simplex
Coxeter group	A7×2, [[36]], order 80640
Properties	convex

The omnitruncated 7-simplex is composed of 40320 (8 factorial) vertices and is the largest uniform 7-polytope in the A₇ symmetry of the regular 7-simplex. It can also be called the hexipentisteriruncicantitruncated 7-simplex which is the long name for the omnitruncation for 7 dimensions, with all reflective mirrors active.

Permutohedron and related tessellation

The omnitruncated 7-simplex is the permutohedron of order 8. The omnitruncated 7-simplex is a zonotope, the Minkowski sum of eight line segments parallel to the eight lines through the origin and the eight vertices of the 7-simplex.

Like all uniform omnitruncated n-simplices, the omnitruncated 7-simplex can tessellate space by itself, in this case 7-dimensional space with three facets around each ridge. It has Coxeter-Dynkin diagram of .

Alternate names

• Great petated hexadecaexon (Acronym: guph) (Jonathan Bowers)^[20]

Coordinates

The vertices of the omnitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,3,4,5,6,7). This construction is based on facets of the hexipentisteriruncicantitruncated 8-orthoplex, t_{0,1,2,3,4,5,6}{3⁶,4}, .

Images

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[[7]]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[[5]]	[4]	[[3]]

Related polytopes

These polytope are a part of 71 uniform 7-polytopes with A₇ symmetry.

A7 polytopes
t0	t1	t2	t3	t0,1	t0,2	t1,2	t0,3
t1,3	t2,3	t0,4	t1,4	t2,4	t0,5	t1,5	t0,6
t0,1,2	t0,1,3	t0,2,3	t1,2,3	t0,1,4	t0,2,4	t1,2,4	t0,3,4
t1,3,4	t2,3,4	t0,1,5	t0,2,5	t1,2,5	t0,3,5	t1,3,5	t0,4,5
t0,1,6	t0,2,6	t0,3,6	t0,1,2,3	t0,1,2,4	t0,1,3,4	t0,2,3,4	t1,2,3,4
t0,1,2,5	t0,1,3,5	t0,2,3,5	t1,2,3,5	t0,1,4,5	t0,2,4,5	t1,2,4,5	t0,3,4,5
t0,1,2,6	t0,1,3,6	t0,2,3,6	t0,1,4,6	t0,2,4,6	t0,1,5,6	t0,1,2,3,4	t0,1,2,3,5
t0,1,2,4,5	t0,1,3,4,5	t0,2,3,4,5	t1,2,3,4,5	t0,1,2,3,6	t0,1,2,4,6	t0,1,3,4,6	t0,2,3,4,6
t0,1,2,5,6	t0,1,3,5,6	t0,1,2,3,4,5	t0,1,2,3,4,6	t0,1,2,3,5,6	t0,1,2,4,5,6	t0,1,2,3,4,5,6

Notes

1. Klitzing, (x3o3o3o3o3o3x - suph)
2. Klitzing, (x3x3o3o3o3o3x- puto)
3. Klitzing, (x3o3x3o3o3o3x - puro)
4. Klitzing, (x3o3o3x3o3o3x - puph)
5. Klitzing, (x3o3o3o3x3o3x - pugro)
6. Klitzing, (x3x3x3o3o3o3x - pupato)
7. Klitzing, (x3o3x3x3o3o3x - pupro)
8. Klitzing, (x3x3o3o3x3o3x - pucto)
9. Klitzing, (x3o3x3o3x3o3x - pucroh)
10. Klitzing, (x3x3o3o3o3x3x - putath)
11. Klitzing, (x3x3x3x3o3o3x - pugopo)
12. Klitzing, (x3x3x3o3x3o3x - pucagro)
13. Klitzing, (x3x3o3x3x3o3x - pucpato)
14. Klitzing, (x3o3x3x3x3o3x - pucproh)
15. Klitzing, (x3x3x3o3o3x3x - putagro)
16. Klitzing, (x3x3o3x3o3x3x - putpath)
17. Klitzing, (x3x3x3x3x3o3x - pugaco)
18. Klitzing, (x3x3x3x3o3x3x - putgapo)
19. Klitzing, (x3x3x3o3x3x3x - putcagroh)
20. Klitzing, (x3x3x3x3x3x3x - guph)

References

• H.S.M. Coxeter:
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6, wiley.com
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, PhD (1966)
• Klitzing, Richard. "7D". x3o3o3o3o3o3x - suph, x3x3o3o3o3o3x- puto, x3o3x3o3o3o3x - puro, x3o3o3x3o3o3x - puph, x3o3o3o3x3o3x - pugro, x3x3x3o3o3o3x - pupato, x3o3x3x3o3o3x - pupro, x3x3o3o3x3o3x - pucto, x3o3x3o3x3o3x - pucroh, x3x3o3o3o3x3x - putath, x3x3x3x3o3o3x - pugopo, x3x3x3o3x3o3x - pucagro, x3x3o3x3x3o3x - pucpato, x3o3x3x3x3o3x - pucproh, x3x3x3o3o3x3x - putagro, x3x3x3x3o3x3x - putpath, x3x3x3x3x3o3x - pugaco, x3x3x3x3o3x3x - putgapo, x3x3x3o3x3x3x - putcagroh, x3x3x3x3x3x3x - guph

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	An	Bn	I2(p) / Dn	E6 / E7 / E8 / F4 / G2	Hn
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	122 • 221
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	132 • 231 • 321
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	142 • 241 • 421
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds