Heptellated 8-simplexes

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8-simplex t0.svg
8-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.pngCDel 3.pngCDel node 1.png
8-simplex t07.svg
Heptellated 8-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
8-simplex t01234567 A7.svg
Heptihexipentisteriruncicantitruncated 8-simplex
(Omnitruncated 8-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.pngCDel 3.pngCDel node 1.png
Orthogonal projections in A8 Coxeter plane (A7 for omnitruncation)

In eight-dimensional geometry, a heptellated 8-simplex is a convex uniform 8-polytope, including 7th-order truncations (heptellation) from the regular 8-simplex.

There are 35 unique heptellations for the 8-simplex, including all permutations of runcations, cantellations, runcinations, sterications, pentellations, and hexications. The simplest heptellated 8-simplex is also called an expanded 8-simplex, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular 8-simplex. The highest form, the heptihexipentisteriruncicantitruncated 8-simplex is more simply called a omnitruncated 8-simplex with all of the nodes ringed.

Heptellated 8-simplex[edit]

Heptellated 8-simplex
Type uniform 8-polytope
Schläfli symbol t0,7{3,3,3,3,3,3,3}
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.pngCDel 3.pngCDel node 1.png
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 504
Vertices 72
Vertex figure 6-simplex antiprism
Coxeter group A8×2, [[37]], order 725760
Properties convex

Alternate names[edit]

  • Expanded 8-simplex
  • Small exated enneazetton (soxeb) (Jonathan Bowers)[1]

Coordinates[edit]

The vertices of the heptellated 8-simplex can bepositioned in 8-space as permutations of (0,1,1,1,1,1,1,1,2). This construction is based on facets of the heptellated 9-orthoplex.

A second construction in 9-space, from the center of a rectified 9-orthoplex is given by coordinate permutations of:

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

Root vectors[edit]

Its 72 vertices represent the root vectors of the simple Lie group A8.

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t07.svg 8-simplex t07 A7.svg 8-simplex t07 A6.svg 8-simplex t07 A5.svg
Dihedral symmetry [[9]] = [18] [8] [[7]] = [14] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t07 A4.svg 8-simplex t07 A3.svg 8-simplex t07 A2.svg
Dihedral symmetry [[5]] = [10] [4] [[3]] = [6]

Omnitruncated 8-simplex[edit]

Omnitruncated 8-simplex
Type uniform 8-polytope
Schläfli symbol t0,1,2,3,4,5,6,7{37}
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.pngCDel 3.pngCDel node 1.png
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 1451520
Vertices 362880
Vertex figure irr. 7-simplex
Coxeter group A8, [[37]], order 725760
Properties convex

The symmetry order of an omnitruncated 9-simplex is 725760. The symmetry of a family of a uniform polytopes is equal to the number of vertices of the omnitruncation, being 362880 (9 factorial) in the case of the omnitruncated 8-simplex; but when the CD symbol is palindromic, the symmetry order is doubled, 725760 here, because the element corresponding to any element of the underlying 8-simplex can be exchanged with one of those corresponding to an element of its dual.

Alternate names[edit]

  • Heptihexipentisteriruncicantitruncated 8-simplex
  • Great exated enneazetton (goxeb) (Jonathan Bowers)[2]

Coordinates[edit]

The Cartesian coordinates of the vertices of the omnitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,1,2,3,4,5,6,7,8). This construction is based on facets of the heptihexipentisteriruncicantitruncated 9-orthoplex, t0,1,2,3,4,5,6,7{37,4}

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t01234567.svg 8-simplex t01234567 A7.svg 8-simplex t01234567 A6.svg 8-simplex t01234567 A5.svg
Dihedral symmetry [[9]] = [18] [8] [[7]] = [14] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t01234567 A4.svg 8-simplex t01234567 A3.svg 8-simplex t01234567 A2.svg
Dihedral symmetry [[5]] = [10] [4] [[3]] = [6]

Permutohedron and related tessellation[edit]

The omnitruncated 8-simplex is the permutohedron of order 9. The omnitruncated 8-simplex is a zonotope, the Minkowski sum of nine line segments parallel to the nine lines through the origin and the nine vertices of the 8-simplex.

Like all uniform omnitruncated n-simplices, the omnitruncated 8-simplex can tessellate space by itself, in this case 8-dimensional space with three facets around each ridge. It has Coxeter-Dynkin diagram of CDel branch 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png.

Related polytopes[edit]

This polytope is one of 135 uniform 8-polytopes with A8 symmetry.

Notes[edit]

  1. ^ Klitzing, (x3o3o3o3o3o3o3x - soxeb)
  2. ^ Klitzing, (x3x3x3x3x3x3x3x - goxeb)

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 [1]
      • (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, Ph.D.
  • Klitzing, Richard. "8D uniform polytopes (polyzetta)".  x3o3o3o3o3o3o3x - soxeb, x3x3x3x3x3x3x3x - goxeb

External links[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / 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