Hexicated 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

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

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.

Contents 1 Hexicated 7-simplex 1.1 Root vectors 1.2 Alternate names 1.3 Coordinates 1.4 Images 2 Hexitruncated 7-simplex 2.1 Alternate names 2.2 Coordinates 2.3 Images 3 Hexicantellated 7-simplex 3.1 Alternate names 3.2 Coordinates 3.3 Images 4 Hexiruncinated 7-simplex 4.1 Alternate names 4.2 Coordinates 4.3 Images 5 Hexicantitruncated 7-simplex 5.1 Alternate names 5.2 Coordinates 5.3 Images 6 Hexiruncitruncated 7-simplex 6.1 Alternate names 6.2 Coordinates 6.3 Images 7 Hexiruncicantellated 7-simplex 7.1 Alternate names 7.2 Coordinates 7.3 Images 8 Hexisteritruncated 7-simplex 8.1 Alternate names 8.2 Coordinates 8.3 Images 9 Hexistericantellated 7-simplex 9.1 Alternate names 9.2 Coordinates 9.3 Images 10 Hexipentitruncated 7-simplex 10.1 Alternate names 10.2 Coordinates 10.3 Images 11 Hexiruncicantitruncated 7-simplex 11.1 Alternate names 11.2 Coordinates 11.3 Images 12 Hexistericantitruncated 7-simplex 12.1 Alternate names 12.2 Coordinates 12.3 Images 13 Hexisteriruncitruncated 7-simplex 13.1 Alternate names 13.2 Coordinates 13.3 Images 14 Hexisteriruncicantellated 7-simplex 14.1 Alternate names 14.2 Coordinates 14.3 Images 15 Hexipenticantitruncated 7-simplex 15.1 Alternate names 15.2 Coordinates 15.3 Images 16 Hexipentiruncitruncated 7-simplex 16.1 Alternate names 16.2 Coordinates 16.3 Images 17 Hexisteriruncicantitruncated 7-simplex 17.1 Alternate names 17.2 Coordinates 17.3 Images 18 Hexipentiruncicantitruncated 7-simplex 18.1 Alternate names 18.2 Coordinates 18.3 Images 19 Hexipentistericantitruncated 7-simplex 19.1 Alternate names 19.2 Coordinates 19.3 Images 20 Omnitruncated 7-simplex 20.1 Permutohedron and related tessellation 20.2 Alternate names 20.3 Coordinates 20.4 Images 21 Related polytopes 22 Notes 23 References 24 External links

Hexicated 7-simplex

Hexicated 7-simplex
Type	uniform polyexon
Schläfli symbol	t0,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	336
Vertices	56
Vertex figure	5-simplex antiprism
Coxeter group	A7, [[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.

Root vectors

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

Alternate names

• 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.

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 polyexon
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 polyexon
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 polyexon
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, [[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 (0,0,1,1,2,2,2,3). 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 polyexon
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 polyexon
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 polyexon
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 polyexon
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 polyexon
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, [[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 polyexon
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, [[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 polyexon
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 polyexon
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 polyexon
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

Hexisteriruncitruncated 7-simplex
Type	uniform polyexon
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, [[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 polyexon
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

Hexisteriruncicantitruncated 7-simplex
Type	uniform polyexon
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 80640
Properties	convex

Alternate names

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

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]]

Hexisteriruncicantitruncated 7-simplex

Hexisteriruncicantitruncated 7-simplex
Type	uniform polyexon
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 polyexon
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 polyexon
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, [[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 polyexon
Schläfli symbol	t0,1,2,3,4,5,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	141120
Vertices	40320
Vertex figure	Irr. 6-simplex
Coxeter group	A7, [[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.

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

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, editied 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)
• Richard Klitzing, , 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

• Olshevsky, George, Cross polytope at Glossary for Hyperspace.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary

Family	An	BCn	Dn	E6 / E7 / E8 / F4 / G2	Hn
Regular polygon	Triangle	Square		Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	5-cell	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
n-polytopes	n-simplex	n-orthoplex • n-cube	n-demicube	1k2 • 2k1 • k21	pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes