Hexicated 7-simplexes

  (Redirected from Hexicantellated 7-simplex)
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.

Hexicated 7-simplexEdit

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 A7×2, [[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 vectorsEdit

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

Alternate namesEdit

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

CoordinatesEdit

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, CDel node.pngCDel 4.pngCDel 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.

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)

ImagesEdit

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-simplexEdit

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 namesEdit

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

CoordinatesEdit

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, CDel node.pngCDel 4.pngCDel 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.

ImagesEdit

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-simplexEdit

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 namesEdit

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

CoordinatesEdit

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, CDel node.pngCDel 4.pngCDel 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.

ImagesEdit

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-simplexEdit

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 namesEdit

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

CoordinatesEdit

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, CDel node.pngCDel 4.pngCDel 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.

ImagesEdit

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-simplexEdit

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 namesEdit

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

CoordinatesEdit

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, CDel node.pngCDel 4.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.pngCDel 3.pngCDel node 1.png.

ImagesEdit

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-simplexEdit

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 namesEdit

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

CoordinatesEdit

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, CDel node.pngCDel 4.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.pngCDel 3.pngCDel node 1.png.

ImagesEdit

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-simplexEdit

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 namesEdit

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

CoordinatesEdit

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, CDel node.pngCDel 4.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.pngCDel 3.pngCDel node 1.png.

ImagesEdit

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-simplexEdit

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 namesEdit

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

CoordinatesEdit

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, CDel node.pngCDel 4.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.pngCDel 3.pngCDel node 1.png.

ImagesEdit

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-simplexEdit

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 namesEdit

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

CoordinatesEdit

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, CDel node.pngCDel 4.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.pngCDel 3.pngCDel node 1.png.

ImagesEdit

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-simplexEdit

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 namesEdit

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

CoordinatesEdit

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, CDel node.pngCDel 4.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.pngCDel 3.pngCDel node 1.png.

ImagesEdit

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-simplexEdit

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 namesEdit

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

CoordinatesEdit

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, 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.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png.

ImagesEdit

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-simplexEdit

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 namesEdit

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

CoordinatesEdit

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, CDel node.pngCDel 4.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.pngCDel 3.pngCDel node 1.png.

ImagesEdit

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-simplexEdit

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 namesEdit

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

CoordinatesEdit

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, CDel node.pngCDel 4.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.pngCDel 3.pngCDel node 1.png.

ImagesEdit

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-simplexEdit

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 namesEdit

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

CoordinatesEdit

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, CDel node.pngCDel 4.pngCDel 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.

ImagesEdit

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-simplexEdit

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 namesEdit

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

CoordinatesEdit

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, CDel node.pngCDel 4.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.pngCDel 3.pngCDel node 1.png.

ImagesEdit

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-simplexEdit

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 namesEdit

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

CoordinatesEdit

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, 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.pngCDel 3.pngCDel node 1.png.

ImagesEdit

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-simplexEdit

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 namesEdit

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

CoordinatesEdit

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, 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.pngCDel 3.pngCDel node 1.png.

ImagesEdit

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-simplexEdit

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 namesEdit

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

CoordinatesEdit

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, 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.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png.

ImagesEdit

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-simplexEdit

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 namesEdit

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

CoordinatesEdit

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, CDel node.pngCDel 4.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.pngCDel 3.pngCDel node 1.png.

ImagesEdit

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-simplexEdit

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 tessellationEdit

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 namesEdit

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

CoordinatesEdit

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.

ImagesEdit

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 polytopesEdit

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

NotesEdit

  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)

ReferencesEdit

  • 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". 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 linksEdit

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds