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Rectified 8-simplexes

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8-simplex

Rectified 8-simplex

Birectified 8-simplex

Trirectified 8-simplex
Orthogonal projections in A8 Coxeter plane

In eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.

There are unique 3 degrees of rectifications in regular 8-polytopes. Vertices of the rectified 8-simplex are located at the edge-centers of the 8-simplex. Vertices of the birectified 8-simplex are located in the triangular face centers of the 8-simplex. Vertices of the trirectified 8-simplex are located in the tetrahedral cell centers of the 8-simplex.

Rectified 8-simplex

[edit]
Rectified 8-simplex
Type uniform 8-polytope
Coxeter symbol 061
Schläfli symbol t1{37}
r{37} = {36,1}
or
Coxeter-Dynkin diagrams
or
7-faces 18
6-faces 108
5-faces 336
4-faces 630
Cells 756
Faces 588
Edges 252
Vertices 36
Vertex figure 7-simplex prism, {}×{3,3,3,3,3}
Petrie polygon enneagon
Coxeter group A8, [37], order 362880
Properties convex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S1
8
. It is also called 06,1 for its branching Coxeter-Dynkin diagram, shown as .

Coordinates

[edit]

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

Images

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

Birectified 8-simplex

[edit]
Birectified 8-simplex
Type uniform 8-polytope
Coxeter symbol 052
Schläfli symbol t2{37}
2r{37} = {35,2} or
Coxeter-Dynkin diagrams
or
7-faces 18
6-faces 144
5-faces 588
4-faces 1386
Cells 2016
Faces 1764
Edges 756
Vertices 84
Vertex figure {3}×{3,3,3,3}
Coxeter group A8, [37], order 362880
Properties convex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S2
8
. It is also called 05,2 for its branching Coxeter-Dynkin diagram, shown as .

The birectified 8-simplex is the vertex figure of the 152 honeycomb.

Coordinates

[edit]

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

Images

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

Trirectified 8-simplex

[edit]
Trirectified 8-simplex
Type uniform 8-polytope
Coxeter symbol 043
Schläfli symbol t3{37}
3r{37} = {34,3} or
Coxeter-Dynkin diagrams
or
7-faces 9 + 9
6-faces 36 + 72 + 36
5-faces 84 + 252 + 252 + 84
4-faces 126 + 504 + 756 + 504
Cells 630 + 1260 + 1260
Faces 1260 + 1680
Edges 1260
Vertices 126
Vertex figure {3,3}×{3,3,3}
Petrie polygon enneagon
Coxeter group A7, [37], order 362880
Properties convex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S3
8
. It is also called 04,3 for its branching Coxeter-Dynkin diagram, shown as .

Coordinates

[edit]

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

Images

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

This polytope is the vertex figure of the 9-demicube, and the edge figure of the uniform 261 honeycomb.

It is also one of 135 uniform 8-polytopes with A8 symmetry.

A8 polytopes

t0

t1

t2

t3

t01

t02

t12

t03

t13

t23

t04

t14

t24

t34

t05

t15

t25

t06

t16

t07

t012

t013

t023

t123

t014

t024

t124

t034

t134

t234

t015

t025

t125

t035

t135

t235

t045

t145

t016

t026

t126

t036

t136

t046

t056

t017

t027

t037

t0123

t0124

t0134

t0234

t1234

t0125

t0135

t0235

t1235

t0145

t0245

t1245

t0345

t1345

t2345

t0126

t0136

t0236

t1236

t0146

t0246

t1246

t0346

t1346

t0156

t0256

t1256

t0356

t0456

t0127

t0137

t0237

t0147

t0247

t0347

t0157

t0257

t0167

t01234

t01235

t01245

t01345

t02345

t12345

t01236

t01246

t01346

t02346

t12346

t01256

t01356

t02356

t12356

t01456

t02456

t03456

t01237

t01247

t01347

t02347

t01257

t01357

t02357

t01457

t01267

t01367

t012345

t012346

t012356

t012456

t013456

t023456

t123456

t012347

t012357

t012457

t013457

t023457

t012367

t012467

t013467

t012567

t0123456

t0123457

t0123467

t0123567

t01234567

Notes

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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)". o3x3o3o3o3o3o3o - rene, o3o3x3o3o3o3o3o - brene, o3o3o3x3o3o3o3o - trene
[edit]
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 polychoron Pentachoron 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