Runcinated 8-simplex


8-simplex

Runcinated 8-simplex

Biruncinated 8-simplex

Triruncinated 8-simplex

Runcitruncated 8-simplex

Biruncitruncated 8-simplex

Triruncitruncated 8-simplex

Runcicantellated 8-simplex

Biruncicantellated 8-simplex

Runcicantitruncated 8-simplex

Biruncicantitruncated 8-simplex

Triruncicantitruncated 8-simplex
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.

Contents

Runcinated 8-simplex

Runcinated 8-simplex
Type uniform polyzetton
Schläfli symbol t0,3{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 4536
Vertices 504
Vertex figure
Coxeter group A8, [37], order 362880
Properties convex

Alternate names

Coordinates

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

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]

Biruncinated 8-simplex

Biruncinated 8-simplex
Type uniform polyzetton
Schläfli symbol t1,4{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
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

Coordinates

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

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]

Triruncinated 8-simplex

Triruncinated 8-simplex
Type uniform polyzetton
Schläfli symbol t2,5{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 15120
Vertices 1680
Vertex figure
Coxeter group A8, [37], order 362880
Properties convex

Alternate names

Coordinates

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

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]]

Runcitruncated 8-simplex

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]]

Biruncitruncated 8-simplex

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]]

Triruncitruncated 8-simplex

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]]

Runcicantellated 8-simplex

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]]

Biruncicantellated 8-simplex

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]]

Runcicantitruncated 8-simplex

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]]

Biruncicantitruncated 8-simplex

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]]

Triruncicantitruncated 8-simplex

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]]

Related polytopes

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


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

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

References

References

External links