Rectified 8-simplex

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. 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.

Contents 1 Rectified 8-simplex 1.1 Coordinates 1.2 Images 2 Rectified 8-simplex 2.1 Coordinates 2.2 Images 3 Trirectified 8-simplex 3.1 Coordinates 3.2 Images 4 Related polytopes 5 Notes 6 References 7 External links

Rectified 8-simplex

Rectified 8-simplex
Type	uniform polyzetton
Schläfli symbol	t1{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces	18
6-faces	108
5-faces	336
4-faces	630
Cells	576
Faces	588
Edges	252
Vertices	36
Vertex figure	7-simplex prism
Petrie polygon	enneagon
Coxeter group	A8, [37], order 362880
Properties	convex

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]

Rectified 8-simplex

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

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
Type	uniform polyzetton
Schläfli symbol	t3{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces	18
6-faces
5-faces
4-faces
Cells
Faces
Edges	1260
Vertices	126
Vertex figure	{3,3}x{3,3,3}
Petrie polygon	enneagon
Coxeter group	A7, [37], order 362880
Properties	convex

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]

Related polytopes

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.

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, editied 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) o3x3o3o3o3o3o3o - rene, o3o3x3o3o3o3o3o - brene, o3o3o3x3o3o3o3o - trene

External links

• Olshevsky, George, Simplex at Glossary for Hyperspace.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary

Family	An	BCn	Dn	E6 / E7 / E8 / F4 / G2	Hn
Regular polygon	Triangle	Square		Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	5-cell	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	122 • 221
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	132 • 231 • 321
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	142 • 241 • 421
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
n-polytopes	n-simplex	n-orthoplex • n-cube	n-demicube	1k2 • 2k1 • k21	pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes