Hexicated 7-simplexes

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.

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)

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)

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)

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)

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)

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, .

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)

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)

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)

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)

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)

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, .

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)

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, .

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)

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, .

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)

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, .

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)

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)

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, .

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)

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)

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, .

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)

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, .

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)

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, .

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)

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, .

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)

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}, .

Related polytopes

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

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". https://bendwavy.org/klitzing/dimensions/polyexa.htm.  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