Hexicated 7-simplexes

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7-simplex t0.svg
7-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-simplex t06.svg
Hexicated 7-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
7-simplex t016.svg
Hexitruncated 7-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
7-simplex t026.svg
Hexicantellated 7-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
7-simplex t036.svg
Hexiruncinated 7-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
7-simplex t0126.svg
Hexicantitruncated 7-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
7-simplex t0136.svg
Hexiruncitruncated 7-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
7-simplex t0236.svg
Hexiruncicantellated 7-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
7-simplex t0146.svg
Hexisteritruncated 7-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
7-simplex t0246.svg
Hexistericantellated 7-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
7-simplex t0156.svg
Hexipentitruncated 7-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
7-simplex t01236.svg
Hexiruncicantitruncated 7-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
7-simplex t01246.svg
Hexistericantitruncated 7-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
7-simplex t01346.svg
Hexisteriruncitruncated 7-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
7-simplex t02346.svg
Hexisteriruncicantellated 7-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
7-simplex t01256.svg
Hexipenticantitruncated 7-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
7-simplex t01356.svg
Hexipentiruncitruncated 7-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
7-simplex t012346.svg
Hexisteriruncicantitruncated 7-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
7-simplex t012356.svg
Hexipentiruncicantitruncated 7-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
7-simplex t012456.svg
Hexipentistericantitruncated 7-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
7-simplex t0123456.svg
Hexipentisteriruncicantitruncated 7-simplex
(Omnitruncated 7-simplex)
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
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

Hexicated 7-simplex[edit]

Hexicated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,6{36}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-faces 254:
8+8 {35} 6-simplex t0.svg
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, [[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[edit]

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

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t06.svg 7-simplex t06 A6.svg 7-simplex t06 A5.svg
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t06 A4.svg 7-simplex t06 A3.svg 7-simplex t06 A2.svg
Dihedral symmetry [[5]] [4] [[3]]

Hexitruncated 7-simplex[edit]

hexitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,1,6{36}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 1848
Vertices 336
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t016.svg 7-simplex t016 A6.svg 7-simplex t016 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t016 A4.svg 7-simplex t016 A3.svg 7-simplex t016 A2.svg
Dihedral symmetry [5] [4] [3]

Hexicantellated 7-simplex[edit]

Hexicantellated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,2,6{36}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 5880
Vertices 840
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t026.svg 7-simplex t026 A6.svg 7-simplex t026 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t026 A4.svg 7-simplex t026 A3.svg 7-simplex t026 A2.svg
Dihedral symmetry [5] [4] [3]

Hexiruncinated 7-simplex[edit]

Hexiruncinated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,3,6{36}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 8400
Vertices 1120
Vertex figure
Coxeter group A7, [[36]], order 80640
Properties convex

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t036.svg 7-simplex t036 A6.svg 7-simplex t036 A5.svg
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t036 A4.svg 7-simplex t036 A3.svg 7-simplex t036 A2.svg
Dihedral symmetry [[5]] [4] [[3]]

Hexicantitruncated 7-simplex[edit]

Hexicantitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,1,2,6{36}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 8400
Vertices 1680
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t0126.svg 7-simplex t0126 A6.svg 7-simplex t0126 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t0126 A4.svg 7-simplex t0126 A3.svg 7-simplex t0126 A2.svg
Dihedral symmetry [5] [4] [3]

Hexiruncitruncated 7-simplex[edit]

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

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t0136.svg 7-simplex t0136 A6.svg 7-simplex t0136 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t0136 A4.svg 7-simplex t0136 A3.svg 7-simplex t0136 A2.svg
Dihedral symmetry [5] [4] [3]

Hexiruncicantellated 7-simplex[edit]

Hexiruncicantellated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,2,3,6{36}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
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[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t0236.svg 7-simplex t0236 A6.svg 7-simplex t0236 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t0236 A4.svg 7-simplex t0236 A3.svg 7-simplex t0236 A2.svg
Dihedral symmetry [5] [4] [3]

Hexisteritruncated 7-simplex[edit]

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

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t0146.svg 7-simplex t0146 A6.svg 7-simplex t0146 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t0146 A4.svg 7-simplex t0146 A3.svg 7-simplex t0146 A2.svg
Dihedral symmetry [5] [4] [3]

Hexistericantellated 7-simplex[edit]

hexistericantellated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,2,4,6{36}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
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[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t0246.svg 7-simplex t0246 A6.svg 7-simplex t0246 A5.svg
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t0246 A4.svg 7-simplex t0246 A3.svg 7-simplex t0246 A2.svg
Dihedral symmetry [[5]] [4] [[3]]

Hexipentitruncated 7-simplex[edit]

Hexipentitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,1,5,6{36}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 8400
Vertices 1680
Vertex figure
Coxeter group A7, [[36]], order 80640
Properties convex

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t0156.svg 7-simplex t0156 A6.svg 7-simplex t0156 A5.svg
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t0156 A4.svg 7-simplex t0156 A3.svg 7-simplex t0156 A2.svg
Dihedral symmetry [[5]] [4] [[3]]

Hexiruncicantitruncated 7-simplex[edit]

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

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t01236.svg 7-simplex t01236 A6.svg 7-simplex t01236 A5.svg
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t01236 A4.svg 7-simplex t01236 A3.svg 7-simplex t01236 A2.svg
Dihedral symmetry [[5]] [4] [[3]]

Hexistericantitruncated 7-simplex[edit]

Hexistericantitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,1,2,4,6{36}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 50400
Vertices 10080
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t01246.svg 7-simplex t01246 A6.svg 7-simplex t01246 A5.svg
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t01246 A4.svg 7-simplex t01246 A3.svg 7-simplex t01246 A2.svg
Dihedral symmetry [[5]] [4] [[3]]

Hexisteriruncitruncated 7-simplex[edit]

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

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t01346.svg 7-simplex t01346 A6.svg 7-simplex t01346 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t01346 A4.svg 7-simplex t01346 A3.svg 7-simplex t01346 A2.svg
Dihedral symmetry [5] [4] [3]

Hexisteriruncicantellated 7-simplex[edit]

Hexisteriruncitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,2,3,4,6{36}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 45360
Vertices 10080
Vertex figure
Coxeter group A7, [[36]], order 80640
Properties convex

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t02346.svg 7-simplex t02346 A6.svg 7-simplex t02346 A5.svg
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t02346 A4.svg 7-simplex t02346 A3.svg 7-simplex t02346 A2.svg
Dihedral symmetry [[5]] [4] [[3]]

Hexipenticantitruncated 7-simplex[edit]

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

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t01256.svg 7-simplex t01256 A6.svg 7-simplex t01256 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t01256 A4.svg 7-simplex t01256 A3.svg 7-simplex t01256 A2.svg
Dihedral symmetry [5] [4] [3]

Hexipentiruncitruncated 7-simplex[edit]

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

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t012346.svg 7-simplex t012346 A6.svg 7-simplex t012346 A5.svg
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t012346 A4.svg 7-simplex t012346 A3.svg 7-simplex t012346 A2.svg
Dihedral symmetry [[5]] [4] [[3]]

Hexisteriruncicantitruncated 7-simplex[edit]

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

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t012346.svg 7-simplex t012346 A6.svg 7-simplex t012346 A5.svg
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t012346 A4.svg 7-simplex t012346 A3.svg 7-simplex t012346 A2.svg
Dihedral symmetry [[5]] [4] [[3]]

Hexipentiruncicantitruncated 7-simplex[edit]

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

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t012356.svg 7-simplex t012356 A6.svg 7-simplex t012356 A5.svg
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t012356 A4.svg 7-simplex t012356 A3.svg 7-simplex t012356 A2.svg
Dihedral symmetry [[5]] [4] [[3]]

Hexipentistericantitruncated 7-simplex[edit]

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

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t012456.svg 7-simplex t012456 A6.svg 7-simplex t012456 A5.svg
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t012456 A4.svg 7-simplex t012456 A3.svg 7-simplex t012456 A2.svg
Dihedral symmetry [[5]] [4] [[3]]

Omnitruncated 7-simplex[edit]

Omnitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,1,2,3,4,5,6{36}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
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[edit]

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 CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png.

Alternate names[edit]

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

Coordinates[edit]

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, t0,1,2,3,4,5,6{36,4}, CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png.

Images[edit]

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph 7-simplex t0123456.svg 7-simplex t0123456 A6.svg 7-simplex t0123456 A5.svg
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t0123456 A4.svg 7-simplex t0123456 A3.svg 7-simplex t0123456 A2.svg
Dihedral symmetry [[5]] [4] [[3]]

Related polytopes[edit]

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

7-simplex t0.svg
t0
7-simplex t1.svg
t1
7-simplex t2.svg
t2
7-simplex t3.svg
t3
7-simplex t01.svg
t0,1
7-simplex t02.svg
t0,2
7-simplex t12.svg
t1,2
7-simplex t03.svg
t0,3
7-simplex t13.svg
t1,3
7-simplex t23.svg
t2,3
7-simplex t04.svg
t0,4
7-simplex t14.svg
t1,4
7-simplex t24.svg
t2,4
7-simplex t05.svg
t0,5
7-simplex t15.svg
t1,5
7-simplex t06.svg
t0,6
7-simplex t012.svg
t0,1,2
7-simplex t013.svg
t0,1,3
7-simplex t023.svg
t0,2,3
7-simplex t123.svg
t1,2,3
7-simplex t014.svg
t0,1,4
7-simplex t024.svg
t0,2,4
7-simplex t124.svg
t1,2,4
7-simplex t034.svg
t0,3,4
7-simplex t134.svg
t1,3,4
7-simplex t234.svg
t2,3,4
7-simplex t015.svg
t0,1,5
7-simplex t025.svg
t0,2,5
7-simplex t125.svg
t1,2,5
7-simplex t035.svg
t0,3,5
7-simplex t135.svg
t1,3,5
7-simplex t045.svg
t0,4,5
7-simplex t016.svg
t0,1,6
7-simplex t026.svg
t0,2,6
7-simplex t036.svg
t0,3,6
7-simplex t0123.svg
t0,1,2,3
7-simplex t0124.svg
t0,1,2,4
7-simplex t0134.svg
t0,1,3,4
7-simplex t0234.svg
t0,2,3,4
7-simplex t1234.svg
t1,2,3,4
7-simplex t0125.svg
t0,1,2,5
7-simplex t0135.svg
t0,1,3,5
7-simplex t0235.svg
t0,2,3,5
7-simplex t1235.svg
t1,2,3,5
7-simplex t0145.svg
t0,1,4,5
7-simplex t0245.svg
t0,2,4,5
7-simplex t1245.svg
t1,2,4,5
7-simplex t0345.svg
t0,3,4,5
7-simplex t0126.svg
t0,1,2,6
7-simplex t0136.svg
t0,1,3,6
7-simplex t0236.svg
t0,2,3,6
7-simplex t0146.svg
t0,1,4,6
7-simplex t0246.svg
t0,2,4,6
7-simplex t0156.svg
t0,1,5,6
7-simplex t01234.svg
t0,1,2,3,4
7-simplex t01235.svg
t0,1,2,3,5
7-simplex t01245.svg
t0,1,2,4,5
7-simplex t01345.svg
t0,1,3,4,5
7-simplex t02345.svg
t0,2,3,4,5
7-simplex t12345.svg
t1,2,3,4,5
7-simplex t01236.svg
t0,1,2,3,6
7-simplex t01246.svg
t0,1,2,4,6
7-simplex t01346.svg
t0,1,3,4,6
7-simplex t02346.svg
t0,2,3,4,6
7-simplex t01256.svg
t0,1,2,5,6
7-simplex t01356.svg
t0,1,3,5,6
7-simplex t012345.svg
t0,1,2,3,4,5
7-simplex t012346.svg
t0,1,2,3,4,6
7-simplex t012356.svg
t0,1,2,3,5,6
7-simplex t012456.svg
t0,1,2,4,5,6
7-simplex t0123456.svg
t0,1,2,3,4,5,6

Notes[edit]

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

References[edit]

  • 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)
  • 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[edit]