In seven-dimensionalgeometry, ahexicated 7-simplex is a convexuniform 7-polytope, including 6th-ordertruncations (hexication) from the regular7-simplex.

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

The simplehexicated 7-simplex is also called anexpanded 7-simplex, with only the first and last nodes ringed, is constructed by anexpansion operation applied to the regular7-simplex. The highest form, thehexipentisteriruncicantitruncated 7-simplex is more simply called anomnitruncated 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 A7Coxeter 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-dimensionalgeometry, ahexicated 7-simplex is a convexuniform 7-polytope, ahexication (6th order truncation) of the regular7-simplex, or alternately can be seen as anexpansion operation.

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

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

The vertices of thehexicated 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 onfacets of thehexicated 8-orthoplex,.

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

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

orthographic projections

AkCoxeter 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 thehexitruncated 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 onfacets of thehexitruncated 8-orthoplex,.

orthographic projections

AkCoxeter 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 thehexicantellated 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 onfacets of thehexicantellated 8-orthoplex,.

orthographic projections

AkCoxeter 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 thehexiruncinated 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 onfacets of thehexiruncinated 8-orthoplex,.

orthographic projections

AkCoxeter 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 thehexicantitruncated 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 onfacets of thehexicantitruncated 8-orthoplex,.

orthographic projections

AkCoxeter 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 thehexiruncitruncated 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 onfacets of thehexiruncitruncated 8-orthoplex,.

orthographic projections

AkCoxeter 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-dimensionalgeometry, ahexiruncicantellated 7-simplex is auniform 7-polytope.

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

The vertices of thehexiruncicantellated 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 onfacets of thehexiruncicantellated 8-orthoplex,.

orthographic projections

AkCoxeter 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 thehexisteritruncated 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 onfacets of thehexisteritruncated 8-orthoplex,.

orthographic projections

AkCoxeter 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 thehexistericantellated 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 onfacets of thehexistericantellated 8-orthoplex,.

orthographic projections

AkCoxeter 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 thehexipentitruncated 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 onfacets of thehexipentitruncated 8-orthoplex,.

orthographic projections

AkCoxeter 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 thehexiruncicantitruncated 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 onfacets of thehexiruncicantitruncated 8-orthoplex,.

orthographic projections

AkCoxeter 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 thehexistericantitruncated 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 onfacets of thehexistericantitruncated 8-orthoplex,.

orthographic projections

AkCoxeter 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 thehexisteriruncitruncated 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 onfacets of thehexisteriruncitruncated 8-orthoplex,.

orthographic projections

AkCoxeter 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 thehexisteriruncitruncated 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 onfacets of thehexisteriruncitruncated 8-orthoplex,.

orthographic projections

AkCoxeter 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 thehexipenticantitruncated 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 onfacets of thehexipenticantitruncated 8-orthoplex,.

orthographic projections

AkCoxeter 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 thehexipentiruncitruncated 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 onfacets of thehexipentiruncitruncated 8-orthoplex,.

orthographic projections

AkCoxeter 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 thehexisteriruncicantitruncated 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 onfacets of thehexisteriruncicantitruncated 8-orthoplex,.

orthographic projections

AkCoxeter 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 thehexipentiruncicantitruncated 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 onfacets of thehexipentiruncicantitruncated 8-orthoplex,.

orthographic projections

AkCoxeter 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 thehexipentistericantitruncated 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 onfacets of thehexipentistericantitruncated 8-orthoplex,.

orthographic projections

AkCoxeter 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

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

The omnitruncated 7-simplex is thepermutohedron of order 8. The omnitruncated 7-simplex is azonotope, theMinkowski 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, theomnitruncated 7-simplex cantessellate space by itself, in this case 7-dimensional space with three facets around eachridge. It hasCoxeter-Dynkin diagram of.

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

The vertices of theomnitruncated 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 onfacets of thehexipentisteriruncicantitruncated 8-orthoplex, t_{0,1,2,3,4,5,6}{3⁶,4},.

orthographic projections

AkCoxeter 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 71uniform 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 JohnsonUniform 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

• 7-polytopes

• Articles with short description
• Short description matches Wikidata