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 polyzetton
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 polyzetton
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: spene) (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 polyzetton
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: sabpene) (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.

8-simplex t0.svg
t0
8-simplex t1.svg
t1
8-simplex t2.svg
t2
8-simplex t3.svg
t3
8-simplex t01.svg
t01
8-simplex t02.svg
t02
8-simplex t12.svg
t12
8-simplex t03.svg
t03
8-simplex t13.svg
t13
8-simplex t23.svg
t23
8-simplex t04.svg
t04
8-simplex t14.svg
t14
8-simplex t24.svg
t24
8-simplex t34.svg
t34
8-simplex t05.svg
t05
8-simplex t15.svg
t15
8-simplex t25.svg
t25
8-simplex t06.svg
t06
8-simplex t16.svg
t16
8-simplex t07.svg
t07
8-simplex t012.svg
t012
8-simplex t013.svg
t013
8-simplex t023.svg
t023
8-simplex t123.svg
t123
8-simplex t014.svg
t014
8-simplex t024.svg
t024
8-simplex t124.svg
t124
8-simplex t034.svg
t034
8-simplex t134.svg
t134
8-simplex t234.svg
t234
8-simplex t015.svg
t015
8-simplex t025.svg
t025
8-simplex t125.svg
t125
8-simplex t035.svg
t035
8-simplex t135.svg
t135
8-simplex t235.svg
t235
8-simplex t045.svg
t045
8-simplex t145.svg
t145
8-simplex t016.svg
t016
8-simplex t026.svg
t026
8-simplex t126.svg
t126
8-simplex t036.svg
t036
8-simplex t136.svg
t136
8-simplex t046.svg
t046
8-simplex t056.svg
t056
8-simplex t017.svg
t017
8-simplex t027.svg
t027
8-simplex t037.svg
t037
8-simplex t0123.svg
t0123
8-simplex t0124.svg
t0124
8-simplex t0134.svg
t0134
8-simplex t0234.svg
t0234
8-simplex t1234.svg
t1234
8-simplex t0125.svg
t0125
8-simplex t0135.svg
t0135
8-simplex t0235.svg
t0235
8-simplex t1235.svg
t1235
8-simplex t0145.svg
t0145
8-simplex t0245.svg
t0245
8-simplex t1245.svg
t1245
8-simplex t0345.svg
t0345
8-simplex t1345.svg
t1345
8-simplex t2345.svg
t2345
8-simplex t0126.svg
t0126
8-simplex t0136.svg
t0136
8-simplex t0236.svg
t0236
8-simplex t1236.svg
t1236
8-simplex t0146.svg
t0146
8-simplex t0246.svg
t0246
8-simplex t1246.svg
t1246
8-simplex t0346.svg
t0346
8-simplex t1346.svg
t1346
8-simplex t0156.svg
t0156
8-simplex t0256.svg
t0256
8-simplex t1256.svg
t1256
8-simplex t0356.svg
t0356
8-simplex t0456.svg
t0456
8-simplex t0127.svg
t0127
8-simplex t0137.svg
t0137
8-simplex t0237.svg
t0237
8-simplex t0147.svg
t0147
8-simplex t0247.svg
t0247
8-simplex t0347.svg
t0347
8-simplex t0157.svg
t0157
8-simplex t0257.svg
t0257
8-simplex t0167.svg
t0167
8-simplex t01234.svg
t01234
8-simplex t01235.svg
t01235
8-simplex t01245.svg
t01245
8-simplex t01345.svg
t01345
8-simplex t02345.svg
t02345
8-simplex t12345.svg
t12345
8-simplex t01236.svg
t01236
8-simplex t01246.svg
t01246
8-simplex t01346.svg
t01346
8-simplex t02346.svg
t02346
8-simplex t12346.svg
t12346
8-simplex t01256.svg
t01256
8-simplex t01356.svg
t01356
8-simplex t02356.svg
t02356
8-simplex t12356.svg
t12356
8-simplex t01456.svg
t01456
8-simplex t02456.svg
t02456
8-simplex t03456.svg
t03456
8-simplex t01237.svg
t01237
8-simplex t01247.svg
t01247
8-simplex t01347.svg
t01347
8-simplex t02347.svg
t02347
8-simplex t01257.svg
t01257
8-simplex t01357.svg
t01357
8-simplex t02357.svg
t02357
8-simplex t01457.svg
t01457
8-simplex t01267.svg
t01267
8-simplex t01367.svg
t01367
8-simplex t012345.svg
t012345
8-simplex t012346.svg
t012346
8-simplex t012356.svg
t012356
8-simplex t012456.svg
t012456
8-simplex t013456.svg
t013456
8-simplex t023456.svg
t023456
8-simplex t123456.svg
t123456
8-simplex t012347.svg
t012347
8-simplex t012357.svg
t012357
8-simplex t012457.svg
t012457
8-simplex t013457.svg
t013457
8-simplex t023457.svg
t023457
8-simplex t012367.svg
t012367
8-simplex t012467.svg
t012467
8-simplex t013467.svg
t013467
8-simplex t012567.svg
t012567
8-simplex t0123456 A7.svg
t0123456
8-simplex t0123457 A7.svg
t0123457
8-simplex t0123467 A7.svg
t0123467
8-simplex t0123567 A7.svg
t0123567
8-simplex t01234567 A7.svg
t01234567

Notes[edit]

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

References[edit]

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.
  • Richard Klitzing, 8D, uniform polytopes (polyzetta) x3o3o3x3o3o3o3o - spene, o3x3o3o3x3o3o3o - sabpene, o3o3x3o3o3x3o3o - satpeb

External links[edit]