Runcinated 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.png
8-simplex t03.svg
Runcinated 8-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.pngCDel 3.pngCDel node.png
8-simplex t14.svg
Biruncinated 8-simplex
CDel node.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.pngCDel 3.pngCDel node.png
8-simplex t25.svg
Triruncinated 8-simplex
CDel node.pngCDel 3.pngCDel node.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.png
8-simplex t013.svg
Runcitruncated 8-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.pngCDel 3.pngCDel node.png
8-simplex t124.svg
Biruncitruncated 8-simplex
CDel node.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.pngCDel 3.pngCDel node.png
8-simplex t235.svg
Triruncitruncated 8-simplex
CDel node.pngCDel 3.pngCDel node.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.png
8-simplex t023.svg
Runcicantellated 8-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.pngCDel 3.pngCDel node.png
8-simplex t134.svg
Biruncicantellated 8-simplex
CDel node.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.pngCDel 3.pngCDel node.png
8-simplex t0123.svg
Runcicantitruncated 8-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.pngCDel 3.pngCDel node.png
8-simplex t1234.svg
Biruncicantitruncated 8-simplex
CDel node.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.pngCDel 3.pngCDel node.png
8-simplex t2345.svg
Triruncicantitruncated 8-simplex
CDel node.pngCDel 3.pngCDel node.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.png
Orthogonal projections in A8 Coxeter plane

In eight-dimensional geometry, a runcinated 8-simplex is a convex uniform 8-polytope with 3rd order truncations (runcination) of the regular 8-simplex.

There are eleven unique runcinations of the 8-simplex, including permutations of truncation and cantellation. The triruncinated 8-simplex and triruncicantitruncated 8-simplex have a doubled symmetry, showing [18] order reflectional symmetry in the A8 Coxeter plane.

Runcinated 8-simplex[edit]

Runcinated 8-simplex
Type uniform 8-polytope
Schläfli symbol t0,3{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 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges 4536
Vertices 504
Vertex figure
Coxeter group A8, [37], order 362880
Properties convex

Alternate names[edit]

  • Runcinated enneazetton
  • Small prismated enneazetton (Acronym: spene) (Jonathan Bowers)[1]

Coordinates[edit]

The Cartesian coordinates of the vertices of the runcinated 8-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,1,1,2). This construction is based on facets of the runcinated 9-orthoplex.

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t03.svg 8-simplex t03 A7.svg 8-simplex t03 A6.svg 8-simplex t03 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t03 A4.svg 8-simplex t03 A3.svg 8-simplex t03 A2.svg
Dihedral symmetry [5] [4] [3]

Biruncinated 8-simplex[edit]

Biruncinated 8-simplex
Type uniform 8-polytope
Schläfli symbol t1,4{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram CDel node.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.pngCDel 3.pngCDel node.png
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 11340
Vertices 1260
Vertex figure
Coxeter group A8, [37], order 362880
Properties convex

Alternate names[edit]

  • Biruncinated enneazetton
  • Small biprismated enneazetton (Acronym: sabpene) (Jonathan Bowers)[2]

Coordinates[edit]

The Cartesian coordinates of the vertices of the biruncinated 8-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,2,2). This construction is based on facets of the biruncinated 9-orthoplex.

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t14.svg 8-simplex t14 A7.svg 8-simplex t14 A6.svg 8-simplex t14 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t14 A4.svg 8-simplex t14 A3.svg 8-simplex t14 A2.svg
Dihedral symmetry [5] [4] [3]

Triruncinated 8-simplex[edit]

Triruncinated 8-simplex
Type uniform 8-polytope
Schläfli symbol t2,5{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node.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.png
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 15120
Vertices 1680
Vertex figure
Coxeter group A8×2, [[37]], order 725760
Properties convex

Alternate names[edit]

  • Triruncinated enneazetton
  • Small triprismated enneazetton (Acronym: satpeb) (Jonathan Bowers)[3]

Coordinates[edit]

The Cartesian coordinates of the vertices of the triruncinated 8-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,2,2,2). This construction is based on facets of the triruncinated 9-orthoplex.

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t25.svg 8-simplex t25 A7.svg 8-simplex t25 A6.svg 8-simplex t25 A5.svg
Dihedral symmetry [[9]] = [18] [8] [[7]] = [14] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t25 A4.svg 8-simplex t25 A3.svg 8-simplex t25 A2.svg
Dihedral symmetry [[5]] = [10] [4] [[3]] = [6]

Runcitruncated 8-simplex[edit]

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.pngCDel 3.pngCDel node.png

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t013.svg 8-simplex t013 A7.svg 8-simplex t013 A6.svg 8-simplex t013 A5.svg
Dihedral symmetry [[9]] = [18] [8] [[7]] = [14] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t013 A4.svg 8-simplex t013 A3.svg 8-simplex t013 A2.svg
Dihedral symmetry [[5]] = [10] [4] [[3]] = [6]

Biruncitruncated 8-simplex[edit]

CDel node.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.pngCDel 3.pngCDel node.png

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t124.svg 8-simplex t124 A7.svg 8-simplex t124 A6.svg 8-simplex t124 A5.svg
Dihedral symmetry [[9]] = [18] [8] [[7]] = [14] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t124 A4.svg 8-simplex t124 A3.svg 8-simplex t124 A2.svg
Dihedral symmetry [[5]] = [10] [4] [[3]] = [6]

Triruncitruncated 8-simplex[edit]

CDel node.pngCDel 3.pngCDel node.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.png

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t235.svg 8-simplex t235 A7.svg 8-simplex t235 A6.svg 8-simplex t235 A5.svg
Dihedral symmetry [[9]] = [18] [8] [[7]] = [14] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t235 A4.svg 8-simplex t235 A3.svg 8-simplex t235 A2.svg
Dihedral symmetry [[5]] = [10] [4] [[3]] = [6]

Runcicantellated 8-simplex[edit]

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.pngCDel 3.pngCDel node.png

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t023.svg 8-simplex t023 A7.svg 8-simplex t023 A6.svg 8-simplex t023 A5.svg
Dihedral symmetry [[9]] = [18] [8] [[7]] = [14] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t023 A4.svg 8-simplex t023 A3.svg 8-simplex t023 A2.svg
Dihedral symmetry [[5]] = [10] [4] [[3]] = [6]

Biruncicantellated 8-simplex[edit]

CDel node.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.pngCDel 3.pngCDel node.png

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t134.svg 8-simplex t134 A7.svg 8-simplex t134 A6.svg 8-simplex t134 A5.svg
Dihedral symmetry [[9]] = [18] [8] [[7]] = [14] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t134 A4.svg 8-simplex t134 A3.svg 8-simplex t134 A2.svg
Dihedral symmetry [[5]] = [10] [4] [[3]] = [6]

Runcicantitruncated 8-simplex[edit]

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.pngCDel 3.pngCDel node.png

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t0123.svg 8-simplex t0123 A7.svg 8-simplex t0123 A6.svg 8-simplex t0123 A5.svg
Dihedral symmetry [[9]] = [18] [8] [[7]] = [14] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t0123 A4.svg 8-simplex t0123 A3.svg 8-simplex t0123 A2.svg
Dihedral symmetry [[5]] = [10] [4] [[3]] = [6]

Biruncicantitruncated 8-simplex[edit]

CDel node.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.pngCDel 3.pngCDel node.png

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t1234.svg 8-simplex t1234 A7.svg 8-simplex t1234 A6.svg 8-simplex t1234 A5.svg
Dihedral symmetry [[9]] = [18] [8] [[7]] = [14] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t1234 A4.svg 8-simplex t1234 A3.svg 8-simplex t1234 A2.svg
Dihedral symmetry [[5]] = [10] [4] [[3]] = [6]

Triruncicantitruncated 8-simplex[edit]

CDel node.pngCDel 3.pngCDel node.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.png

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t2345.svg 8-simplex t2345 A7.svg 8-simplex t2345 A6.svg 8-simplex t2345 A5.svg
Dihedral symmetry [[9]] = [18] [8] [[7]] = [14] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t2345 A4.svg 8-simplex t2345 A3.svg 8-simplex t2345 A2.svg
Dihedral symmetry [[5]] = [10] [4] [[3]] = [6]

Related polytopes[edit]

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

Notes[edit]

  1. ^ Klitzing (x3o3o3x3o3o3o3o - spene)
  2. ^ Klitzing (o3x3o3o3x3o3o3o - sabpene)
  3. ^ Klitzing (o3o3x3o3o3x3o3o - satpeb)

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)".  x3o3o3x3o3o3o3o - spene, o3x3o3o3x3o3o3o - sabpene, o3o3x3o3o3x3o3o - satpeb

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