Rectified 8-simplexes

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


8-simplex

Rectified 8-simplex

Birectified 8-simplex

Trirectified 8-simplex
Orthogonal projections in A8 Coxeter plane

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

Rectified 8-simplex
Typeuniform 8-polytope
Coxeter symbol061
Schläfli symbolt1{37}
r{37} = {36,1}
or
Coxeter-Dynkin diagrams
or
7-faces18
6-faces108
5-faces336
4-faces630
Cells756
Faces588
Edges252
Vertices36
Vertex figure7-simplex prism, {}×{3,3,3,3,3}
Petrie polygonenneagon
Coxeter groupA8, [37], order 362880
Propertiesconvex

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

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

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

Birectified 8-simplex
Typeuniform 8-polytope
Coxeter symbol052
Schläfli symbolt2{37}
2r{37} = {35,2} or
Coxeter-Dynkin diagrams
or
7-faces18
6-faces144
5-faces588
4-faces1386
Cells2016
Faces1764
Edges756
Vertices84
Vertex figure{3}×{3,3,3,3}
Coxeter groupA8, [37], order 362880
Propertiesconvex

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

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

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

Trirectified 8-simplex
Typeuniform 8-polytope
Coxeter symbol043
Schläfli symbolt3{37}
3r{37} = {34,3} or
Coxeter-Dynkin diagrams
or
7-faces9 + 9
6-faces36 + 72 + 36
5-faces84 + 252 + 252 + 84
4-faces126 + 504 + 756 + 504
Cells630 + 1260 + 1260
Faces1260 + 1680
Edges1260
Vertices126
Vertex figure{3,3}×{3,3,3}
Petrie polygonenneagon
Coxeter groupA7, [37], order 362880
Propertiesconvex

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

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

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]

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

    References

    • 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
        • (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
    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
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