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Hexicated 7-simplexes

October 15, 2023


7-simplex
Hexicated 7-simplex
Hexitruncated 7-simplex
Hexicantellated 7-simplex

Hexiruncinated 7-simplex
Hexicantitruncated 7-simplex
Hexiruncitruncated 7-simplex
Hexiruncicantellated 7-simplex

Hexisteritruncated 7-simplex
Hexistericantellated 7-simplex
Hexipentitruncated 7-simplex
Hexiruncicantitruncated 7-simplex

Hexistericantitruncated 7-simplex
Hexisteriruncitruncated 7-simplex
Hexisteriruncicantellated 7-simplex
Hexipenticantitruncated 7-simplex

Hexipentiruncitruncated 7-simplex
Hexisteriruncicantitruncated 7-simplex
Hexipentiruncicantitruncated 7-simplex
Hexipentistericantitruncated 7-simplex

Hexipentisteriruncicantitruncated 7-simplex
(Omnitruncated 7-simplex)
Orthogonal projections in A7 Coxeter plane

In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-simplex.

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

The simple hexicated 7-simplex is also called an expanded 7-simplex, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular 7-simplex. The highest form, the hexipentisteriruncicantitruncated 7-simplex is more simply called a omnitruncated 7-simplex with all of the nodes ringed.

Hexicated 7-simplex Edit

Hexicated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,6{36}
Coxeter-Dynkin diagrams
6-faces 254:
8+8 {35}
28+28 {}x{34}
56+56 {3}x{3,3,3}
70 {3,3}x{3,3}
5-faces
4-faces
Cells
Faces
Edges 336
Vertices 56
Vertex figure 5-simplex antiprism
Coxeter group A7×2, [[36]], order 80640
Properties convex

In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, a hexication (6th order truncation) of the regular 7-simplex, or alternately can be seen as an expansion operation.

The vertices of the A7 2D orthogonal projection are seen in the Ammann–Beenker tiling.

Root vectors 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
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
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
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
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
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
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
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
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
6-faces
5-faces
4-faces
Cells
Faces
Edges 8400
Vertices 1120
Vertex figure
Coxeter group A7×2, [[36]], order 80640
Properties convex

Alternate names 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 (0,1,1,1,2,2,2,3). This construction is based on facets of the hexiruncinated 8-orthoplex, .

Images Edit

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
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
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
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
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
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
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
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
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
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
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
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
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
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
6-faces t0,2,4{3,3,3,3,3}

{}xt0,2,4{3,3,3,3}
{3}xt0,2{3,3,3}
t0,2{3,3}xt0,2{3,3}

5-faces
4-faces
Cells
Faces
Edges 30240
Vertices 5040
Vertex figure
Coxeter group A7×2, [[36]], order 80640
Properties convex

Alternate names 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
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
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
6-faces
5-faces
4-faces
Cells
Faces
Edges 8400
Vertices 1680
Vertex figure
Coxeter group A7×2, [[36]], order 80640
Properties convex

Alternate names 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
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
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
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
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
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
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
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
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
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
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Hexisteriruncicantellated 7-simplex Edit

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

Alternate names 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
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
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
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
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Hexipentiruncitruncated 7-simplex Edit

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

Alternate names Edit

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

Coordinates Edit

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, .

Images Edit

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 Edit

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

Alternate names 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
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
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
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
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
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
6-faces
5-faces
4-faces
Cells
Faces
Edges 80640
Vertices 20160
Vertex figure
Coxeter group A7×2, [[36]], order 80640
Properties convex

Alternate names 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
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
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
6-faces 254
5-faces 5796
4-faces 40824
Cells 126000
Faces 191520
Edges 141120
Vertices 40320
Vertex figure Irr. 6-simplex
Coxeter group A7×2, [[36]], order 80640
Properties convex

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

Permutohedron and related tessellation 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 .

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}, .

Images Edit

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]]

Related polytopes Edit

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

Notes Edit

  1. ^ Klitzing, (x3o3o3o3o3o3x - suph)
  2. ^ Klitzing, (x3x3o3o3o3o3x- puto)
  3. ^ Klitzing, (x3o3x3o3o3o3x - puro)
  4. ^ Klitzing, (x3o3o3x3o3o3x - puph)
  5. ^ Klitzing, (x3o3o3o3x3o3x - pugro)
  6. ^ Klitzing, (x3x3x3o3o3o3x - pupato)
  7. ^ Klitzing, (x3o3x3x3o3o3x - pupro)
  8. ^ Klitzing, (x3x3o3o3x3o3x - pucto)
  9. ^ Klitzing, (x3o3x3o3x3o3x - pucroh)
  10. ^ Klitzing, (x3x3o3o3o3x3x - putath)
  11. ^ Klitzing, (x3x3x3x3o3o3x - pugopo)
  12. ^ Klitzing, (x3x3x3o3x3o3x - pucagro)
  13. ^ Klitzing, (x3x3o3x3x3o3x - pucpato)
  14. ^ Klitzing, (x3o3x3x3x3o3x - pucproh)
  15. ^ Klitzing, (x3x3x3o3o3x3x - putagro)
  16. ^ Klitzing, (x3x3o3x3o3x3x - putpath)
  17. ^ Klitzing, (x3x3x3x3x3o3x - pugaco)
  18. ^ Klitzing, (x3x3x3x3o3x3x - putgapo)
  19. ^ Klitzing, (x3x3x3o3x3x3x - putcagroh)
  20. ^ Klitzing, (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)
  • 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 links Edit

  • Multi-dimensional Glossary
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 Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

October 15, 2023
hexicated, simplexes, simplex, hexicated, simplex, hexitruncated, simplex, hexicantellated, simplex, hexiruncinated, simplex, hexicantitruncated, simplex, hexiruncitruncated, simplex, hexiruncicantellated, simplex, hexisteritruncated, simplex, hexistericantell. 7 simplex Hexicated 7 simplex Hexitruncated 7 simplex Hexicantellated 7 simplex Hexiruncinated 7 simplex Hexicantitruncated 7 simplex Hexiruncitruncated 7 simplex Hexiruncicantellated 7 simplex Hexisteritruncated 7 simplex Hexistericantellated 7 simplex Hexipentitruncated 7 simplex Hexiruncicantitruncated 7 simplex Hexistericantitruncated 7 simplex Hexisteriruncitruncated 7 simplex Hexisteriruncicantellated 7 simplex Hexipenticantitruncated 7 simplex Hexipentiruncitruncated 7 simplex Hexisteriruncicantitruncated 7 simplex Hexipentiruncicantitruncated 7 simplex Hexipentistericantitruncated 7 simplex Hexipentisteriruncicantitruncated 7 simplex Omnitruncated 7 simplex Orthogonal projections in A7 Coxeter plane In seven dimensional geometry a hexicated 7 simplex is a convex uniform 7 polytope including 6th order truncations hexication from the regular 7 simplex There are 20 unique hexications for the 7 simplex including all permutations of truncations cantellations runcinations sterications and pentellations The simple hexicated 7 simplex is also called an expanded 7 simplex with only the first and last nodes ringed is constructed by an expansion operation applied to the regular 7 simplex The highest form the hexipentisteriruncicantitruncated 7 simplex is more simply called a omnitruncated 7 simplex with all of the nodes ringed Contents 1 Hexicated 7 simplex 1 1 Root vectors 1 2 Alternate names 1 3 Coordinates 1 4 Images 2 Hexitruncated 7 simplex 2 1 Alternate names 2 2 Coordinates 2 3 Images 3 Hexicantellated 7 simplex 3 1 Alternate names 3 2 Coordinates 3 3 Images 4 Hexiruncinated 7 simplex 4 1 Alternate names 4 2 Coordinates 4 3 Images 5 Hexicantitruncated 7 simplex 5 1 Alternate names 5 2 Coordinates 5 3 Images 6 Hexiruncitruncated 7 simplex 6 1 Alternate names 6 2 Coordinates 6 3 Images 7 Hexiruncicantellated 7 simplex 7 1 Alternate names 7 2 Coordinates 7 3 Images 8 Hexisteritruncated 7 simplex 8 1 Alternate names 8 2 Coordinates 8 3 Images 9 Hexistericantellated 7 simplex 9 1 Alternate names 9 2 Coordinates 9 3 Images 10 Hexipentitruncated 7 simplex 10 1 Alternate names 10 2 Coordinates 10 3 Images 11 Hexiruncicantitruncated 7 simplex 11 1 Alternate names 11 2 Coordinates 11 3 Images 12 Hexistericantitruncated 7 simplex 12 1 Alternate names 12 2 Coordinates 12 3 Images 13 Hexisteriruncitruncated 7 simplex 13 1 Alternate names 13 2 Coordinates 13 3 Images 14 Hexisteriruncicantellated 7 simplex 14 1 Alternate names 14 2 Coordinates 14 3 Images 15 Hexipenticantitruncated 7 simplex 15 1 Alternate names 15 2 Coordinates 15 3 Images 16 Hexipentiruncitruncated 7 simplex 16 1 Alternate names 16 2 Coordinates 16 3 Images 17 Hexisteriruncicantitruncated 7 simplex 17 1 Alternate names 17 2 Coordinates 17 3 Images 18 Hexipentiruncicantitruncated 7 simplex 18 1 Alternate names 18 2 Coordinates 18 3 Images 19 Hexipentistericantitruncated 7 simplex 19 1 Alternate names 19 2 Coordinates 19 3 Images 20 Omnitruncated 7 simplex 20 1 Permutohedron and related tessellation 20 2 Alternate names 20 3 Coordinates 20 4 Images 21 Related polytopes 22 Notes 23 References 24 External links Hexicated 7 simplex Edit Hexicated 7 simplex Type uniform 7 polytope Schlafli symbol t0 6 36 Coxeter Dynkin diagrams 6 faces 254 8 8 35 28 28 x 34 56 56 3 x 3 3 3 70 3 3 x 3 3 5 faces 4 faces Cells Faces Edges 336 Vertices 56 Vertex figure 5 simplex antiprism Coxeter group A7 2 36 order 80640 Properties convex In seven dimensional geometry a hexicated 7 simplex is a convex uniform 7 polytope a hexication 6th order truncation of the regular 7 simplex or alternately can be seen as an expansion operation The vertices of the A7 2D orthogonal projection are seen in the Ammann Beenker tiling Root vectors 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 91 1 93 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 Dihedral symmetry 8 7 6 Ak Coxeter plane A4 A3 A2 Graph Dihedral symmetry 5 4 3 Hexitruncated 7 simplex Edit hexitruncated 7 simplex Type uniform 7 polytope Schlafli symbol t0 1 6 36 Coxeter Dynkin diagrams 6 faces 5 faces 4 faces Cells Faces Edges 1848 Vertices 336 Vertex figure Coxeter group A7 36 order 40320 Properties convex Alternate names Edit Petitruncated octaexon acronym puto Jonathan Bowers 91 2 93 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 Dihedral symmetry 8 7 6 Ak Coxeter plane A4 A3 A2 Graph Dihedral symmetry 5 4 3 Hexicantellated 7 simplex Edit Hexicantellated 7 simplex Type uniform 7 polytope Schlafli symbol t0 2 6 36 Coxeter Dynkin diagrams 6 faces 5 faces 4 faces Cells Faces Edges 5880 Vertices 840 Vertex figure Coxeter group A7 36 order 40320 Properties convex Alternate names Edit Petirhombated octaexon acronym puro Jonathan Bowers 91 3 93 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 Dihedral symmetry 8 7 6 Ak Coxeter plane A4 A3 A2 Graph Dihedral symmetry 5 4 3 Hexiruncinated 7 simplex Edit Hexiruncinated 7 simplex Type uniform 7 polytope Schlafli symbol t0 3 6 36 Coxeter Dynkin diagrams 6 faces 5 faces 4 faces Cells Faces Edges 8400 Vertices 1120 Vertex figure Coxeter group A7 2 36 order 80640 Properties convex Alternate names Edit Petiprismated hexadecaexon acronym puph Jonathan Bowers 91 4 93 Coordinates Edit 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 Images Edit 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 Edit Hexicantitruncated 7 simplex Type uniform 7 polytope Schlafli symbol t0 1 2 6 36 Coxeter Dynkin diagrams 6 faces 5 faces 4 faces Cells Faces Edges 8400 Vertices 1680 Vertex figure Coxeter group A7 36 order 40320 Properties convex Alternate names Edit Petigreatorhombated octaexon acronym pugro Jonathan Bowers 91 5 93 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 Dihedral symmetry 8 7 6 Ak Coxeter plane A4 A3 A2 Graph Dihedral symmetry 5 4 3 Hexiruncitruncated 7 simplex Edit Hexiruncitruncated 7 simplex Type uniform 7 polytope Schlafli symbol t0 1 3 6 36 Coxeter Dynkin diagrams 6 faces 5 faces 4 faces Cells Faces Edges 20160 Vertices 3360 Vertex figure Coxeter group A7 36 order 40320 Properties convex Alternate names Edit Petiprismatotruncated octaexon acronym pupato Jonathan Bowers 91 6 93 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 Dihedral symmetry 8 7 6 Ak Coxeter plane A4 A3 A2 Graph Dihedral symmetry 5 4 3 Hexiruncicantellated 7 simplex Edit Hexiruncicantellated 7 simplex Type uniform 7 polytope Schlafli symbol t0 2 3 6 36 Coxeter Dynkin diagrams 6 faces 5 faces 4 faces Cells Faces Edges 16800 Vertices 3360 Vertex figure Coxeter group A7 36 order 40320 Properties convex In seven dimensional geometry a hexiruncicantellated 7 simplex is a uniform 7 polytope Alternate names Edit Petiprismatorhombated octaexon acronym pupro Jonathan Bowers 91 7 93 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 Dihedral symmetry 8 7 6 Ak Coxeter plane A4 A3 A2 Graph Dihedral symmetry 5 4 3 Hexisteritruncated 7 simplex Edit hexisteritruncated 7 simplex Type uniform 7 polytope Schlafli symbol t0 1 4 6 36 Coxeter Dynkin diagrams 6 faces 5 faces 4 faces Cells Faces Edges 20160 Vertices 3360 Vertex figure Coxeter group A7 36 order 40320 Properties convex Alternate names Edit Peticellitruncated octaexon acronym pucto Jonathan Bowers 91 8 93 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 Dihedral symmetry 8 7 6 Ak Coxeter plane A4 A3 A2 Graph Dihedral symmetry 5 4 3 Hexistericantellated 7 simplex Edit hexistericantellated 7 simplex Type uniform 7 polytope Schlafli symbol t0 2 4 6 36 Coxeter Dynkin diagrams 6 faces t0 2 4 3 3 3 3 3 xt0 2 4 3 3 3 3 3 xt0 2 3 3 3 t0 2 3 3 xt0 2 3 3 5 faces 4 faces Cells Faces Edges 30240 Vertices 5040 Vertex figure Coxeter group A7 2 36 order 80640 Properties convex Alternate names Edit Peticellirhombihexadecaexon acronym pucroh Jonathan Bowers 91 9 93 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 Dihedral symmetry 8 7 6 Ak Coxeter plane A4 A3 A2 Graph Dihedral symmetry 5 4 3 Hexipentitruncated 7 simplex Edit Hexipentitruncated 7 simplex Type uniform 7 polytope Schlafli symbol t0 1 5 6 36 Coxeter Dynkin diagrams 6 faces 5 faces 4 faces Cells Faces Edges 8400 Vertices 1680 Vertex figure Coxeter group A7 2 36 order 80640 Properties convex Alternate names Edit Petiteritruncated hexadecaexon acronym putath Jonathan Bowers 91 10 93 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 Dihedral symmetry 8 7 6 Ak Coxeter plane A4 A3 A2 Graph Dihedral symmetry 5 4 3 Hexiruncicantitruncated 7 simplex Edit Hexiruncicantitruncated 7 simplex Type uniform 7 polytope Schlafli symbol t0 1 2 3 6 36 Coxeter Dynkin diagrams 6 faces 5 faces 4 faces Cells Faces Edges 30240 Vertices 6720 Vertex figure Coxeter group A7 36 order 40320 Properties convex Alternate names Edit Petigreatoprismated octaexon acronym pugopo Jonathan Bowers 91 11 93 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 Dihedral symmetry 8 7 6 Ak Coxeter plane A4 A3 A2 Graph Dihedral symmetry 5 4 3 Hexistericantitruncated 7 simplex Edit Hexistericantitruncated 7 simplex Type uniform 7 polytope Schlafli symbol t0 1 2 4 6 36 Coxeter Dynkin diagrams 6 faces 5 faces 4 faces Cells Faces Edges 50400 Vertices 10080 Vertex figure Coxeter group A7 36 order 40320 Properties convex Alternate names Edit Peticelligreatorhombated octaexon acronym pucagro Jonathan Bowers 91 12 93 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 Dihedral symmetry 8 7 6 Ak Coxeter plane A4 A3 A2 Graph Dihedral symmetry 5 4 3 Hexisteriruncitruncated 7 simplex Edit Hexisteriruncitruncated 7 simplex Type uniform 7 polytope Schlafli symbol t0 1 3 4 6 36 Coxeter Dynkin diagrams 6 faces 5 faces 4 faces Cells Faces Edges 45360 Vertices 10080 Vertex figure Coxeter group A7 36 order 40320 Properties convex Alternate names Edit Peticelliprismatotruncated octaexon acronym pucpato Jonathan Bowers 91 13 93 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 Dihedral symmetry 8 7 6 Ak Coxeter plane A4 A3 A2 Graph Dihedral symmetry 5 4 3 Hexisteriruncicantellated 7 simplex Edit Hexisteriruncicantellated 7 simplex Type uniform 7 polytope Schlafli symbol t0 2 3 4 6 36 Coxeter Dynkin diagrams 6 faces 5 faces 4 faces Cells Faces Edges 45360 Vertices 10080 Vertex figure Coxeter group A7 2 36 order 80640 Properties convex Alternate names Edit Peticelliprismatorhombihexadecaexon acronym pucproh Jonathan Bowers 91 14 93 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 Dihedral symmetry 8 7 6 Ak Coxeter plane A4 A3 A2 Graph Dihedral symmetry 5 4 3 Hexipenticantitruncated 7 simplex Edit hexipenticantitruncated 7 simplex Type uniform 7 polytope Schlafli symbol t0 1 2 5 6 36 Coxeter Dynkin diagrams 6 faces 5 faces 4 faces Cells Faces Edges 30240 Vertices 6720 Vertex figure Coxeter group A7 36 order 40320 Properties convex Alternate names Edit Petiterigreatorhombated octaexon acronym putagro Jonathan Bowers 91 15 93 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 Dihedral symmetry 8 7 6 Ak Coxeter plane A4 A3 A2 Graph Dihedral symmetry 5 4 3 Hexipentiruncitruncated 7 simplex Edit Hexipentiruncitruncated 7 simplex Type uniform 7 polytope Schlafli symbol t0 1 3 5 6 36 Coxeter Dynkin diagrams 6 faces 5 faces 4 faces Cells Faces Edges Vertices 10080 Vertex figure Coxeter group A7 2 36 order 80640 Properties convex Alternate names Edit Petiteriprismatotruncated hexadecaexon acronym putpath Jonathan Bowers 91 16 93 Coordinates Edit 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 Images Edit 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 Edit Hexisteriruncicantitruncated 7 simplex Type uniform 7 polytope Schlafli symbol t0 1 2 3 4 6 36 Coxeter Dynkin diagrams 6 faces 5 faces 4 faces Cells Faces Edges 80640 Vertices 20160 Vertex figure Coxeter group A7 36 order 40320 Properties convex Alternate names Edit Petigreatocellated octaexon acronym pugaco Jonathan Bowers 91 17 93 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 Dihedral symmetry 8 7 6 Ak Coxeter plane A4 A3 A2 Graph Dihedral symmetry 5 4 3 Hexipentiruncicantitruncated 7 simplex Edit Hexipentiruncicantitruncated 7 simplex Type uniform 7 polytope Schlafli symbol t0 1 2 3 5 6 36 Coxeter Dynkin diagrams 6 faces 5 faces 4 faces Cells Faces Edges 80640 Vertices 20160 Vertex figure Coxeter group A7 36 order 40320 Properties convex Alternate names Edit Petiterigreatoprismated octaexon acronym putgapo Jonathan Bowers 91 18 93 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 Dihedral symmetry 8 7 6 Ak Coxeter plane A4 A3 A2 Graph Dihedral symmetry 5 4 3 Hexipentistericantitruncated 7 simplex Edit Hexipentistericantitruncated 7 simplex Type uniform 7 polytope Schlafli symbol t0 1 2 4 5 6 36 Coxeter Dynkin diagrams 6 faces 5 faces 4 faces Cells Faces Edges 80640 Vertices 20160 Vertex figure Coxeter group A7 2 36 order 80640 Properties convex Alternate names Edit Petitericelligreatorhombihexadecaexon acronym putcagroh Jonathan Bowers 91 19 93 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 Dihedral symmetry 8 7 6 Ak Coxeter plane A4 A3 A2 Graph Dihedral symmetry 5 4 3 Omnitruncated 7 simplex Edit Omnitruncated 7 simplex Type uniform 7 polytope Schlafli symbol t0 1 2 3 4 5 6 36 Coxeter Dynkin diagrams 6 faces 254 5 faces 5796 4 faces 40824 Cells 126000 Faces 191520 Edges 141120 Vertices 40320 Vertex figure Irr 6 simplex Coxeter group A7 2 36 order 80640 Properties convex The omnitruncated 7 simplex is composed of 40320 8 factorial vertices and is the largest uniform 7 polytope in the A7 symmetry of the regular 7 simplex It can also be called the hexipentisteriruncicantitruncated 7 simplex which is the long name for the omnitruncation for 7 dimensions with all reflective mirrors active Permutohedron and related tessellation 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 Alternate names Edit Great petated hexadecaexon Acronym guph Jonathan Bowers 91 20 93 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 Images Edit 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 Related polytopes Edit These polytope are a part of 71 uniform 7 polytopes with A7 symmetry 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 Notes Edit Klitzing x3o3o3o3o3o3x suph Klitzing x3x3o3o3o3o3x puto Klitzing x3o3x3o3o3o3x puro Klitzing x3o3o3x3o3o3x puph Klitzing x3o3o3o3x3o3x pugro Klitzing x3x3x3o3o3o3x pupato Klitzing x3o3x3x3o3o3x pupro Klitzing x3x3o3o3x3o3x pucto Klitzing x3o3x3o3x3o3x pucroh Klitzing x3x3o3o3o3x3x putath Klitzing x3x3x3x3o3o3x pugopo Klitzing x3x3x3o3x3o3x pucagro Klitzing x3x3o3x3x3o3x pucpato Klitzing x3o3x3x3x3o3x pucproh Klitzing x3x3x3o3o3x3x putagro Klitzing x3x3o3x3o3x3x putpath Klitzing x3x3x3x3x3o3x pugaco Klitzing x3x3x3x3o3x3x putgapo Klitzing x3x3x3o3x3x3x putcagroh Klitzing 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 160 978 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 links Edit Polytopes of Various Dimensions Multi dimensional Glossary vteFundamental 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 Octahedron Cube Demicube Dodecahedron Icosahedron Uniform polychoron Pentachoron 16 cell Tesseract Demitesseract 24 cell 120 cell 600 cell Uniform 5 polytope 5 simplex 5 orthoplex 5 cube 5 demicube Uniform 6 polytope 6 simplex 6 orthoplex 6 cube 6 demicube 122 221 Uniform 7 polytope 7 simplex 7 orthoplex 7 cube 7 demicube 132 231 321 Uniform 8 polytope 8 simplex 8 orthoplex 8 cube 8 demicube 142 241 421 Uniform 9 polytope 9 simplex 9 orthoplex 9 cube 9 demicube Uniform 10 polytope 10 simplex 10 orthoplex 10 cube 10 demicube Uniform n polytope n simplex n orthoplex n cube n demicube 1k2 2k1 k21 n pentagonal polytope Topics Polytope families Regular polytope List of regular polytopes and compounds Retrieved from https en wikipedia org w index php title Hexicated 7 simplexes amp oldid 1105152094 Hexisteritruncated 7 simplex, wikipedia, wiki, book, books, library,

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