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.

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 an omnitruncated 7-simplex with all of the nodes ringed.

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

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.

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

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

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)

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
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

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

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, .

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
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

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

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, .

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
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

• Petaprismated hexadecaexon (Acronym: puph) (Jonathan Bowers)^[4]

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, .

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
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

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

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, .

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
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

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

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, .

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
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.

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

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, .

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
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

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

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, .

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
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

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

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, .

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
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

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

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, .

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
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

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

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, .

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
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

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

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, .

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
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

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

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, .

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
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

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

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, .

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
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

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

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, .

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
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

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

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, .

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
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

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

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, .

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
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

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

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, .

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
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

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

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, .

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
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.

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 .

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

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}, .

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]]

The 20 polytopes presented in this article are a part of 71 uniform 7-polytopes with A₇ symmetry shown in the table below.

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

• 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, wiley.com, ISBN 978-0-471-01003-6
    • (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 uniform polytopes (polyexa) with acronyms". 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

• Polytopes of Various Dimensions
• Multi-dimensional Glossary