Heptellated 8-simplex

8-simplex	Heptellated 8-simplex	Heptihexipentisteriruncicantitruncated 8-simplex (Omnitruncated 8-simplex)
Orthogonal projections in A8 Coxeter plane (A7 for omnitruncation)

In eight-dimensional geometry, a heptellated 8-simplex is a convex uniform 8-polytope, including 7th-order truncations (heptellation) from the regular 8-simplex.

There are 35 unique heptellations for the 8-simplex, including all permutations of truncations, cantellations, runcinations, sterications, and pentellations. The simplest heptellated 8-simplex is also called an expanded 8-simplex, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular 8-simplex. The highest form, the heptihexipentisteriruncicantitruncated 8-simplex is more simply called a omnitruncated 8-simplex with all of the nodes ringed.

Contents 1 Heptellated 8-simplex 1.1 Alternate names 1.2 Coordinates 1.3 Root vectors 1.4 Images 2 Omnitruncated 8-simplex 2.1 Alternate names 2.2 Coordinates 2.3 Images 2.4 Permutohedron and related tessellation 3 Related polytopes 4 Notes 5 References 6 External links

Heptellated 8-simplex

Heptellated 8-simplex
Type	uniform polyzetton
Schläfli symbol	t0,7{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	504
Vertices	72
Vertex figure	6-simplex antiprism
Coxeter group	A8, [[37]], order 725760
Properties	convex

Alternate names

• Expanded 8-simplex
• Small exated enneazetton (soxeb) (Jonathan Bowers)^[1]

Coordinates

The vertices of the heptellated 8-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,1,1,1,2). This construction is based on facets of the heptellated 9-orthoplex.

Root vectors

Its 72 vertices represent the root vectors of the simple Lie group A₈.

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]]

Omnitruncated 8-simplex

Omnitruncated 8-simplex
Type	uniform polyzetton
Schläfli symbol	t0,1,2,3,4,5,6,7{37}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	1451520
Vertices	362880
Vertex figure	irr. 7-simplex
Coxeter group	A8, [[37]], order 725760
Properties	convex

The symmetry order of an omnitruncated 9-simplex is 725760. The symmetry of a family of a uniform polytopes is equal to the number of vertices of the omnitruncation, being 362880 (9 factorial) in the case of the omnitruncated 8-simplex; but when the CD symbol is palindromic, the symmetry order is doubled, 725760 here, because the element corresponding to any element of the underlying 8-simplex can be exchanged with one of those corresponding to an element of its dual.

Alternate names

• Heptihexipentisteriruncicantitruncated 8-simplex
• Great exated enneazetton (goxeb) (Jonathan Bowers)^[2]

Coordinates

The Cartesian coordinates of the vertices of the omnitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,1,2,3,4,5,6,7,8). This construction is based on facets of the heptihexipentisteriruncicantitruncated 9-orthoplex, t_{0,1,2,3,4,5,6,7}{3⁷,4}

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]]

Permutohedron and related tessellation

The omnitruncated 8-simplex is the permutohedron of order 9. The omnitruncated 8-simplex is a zonotope, the Minkowski sum of nine line segments parallel to the nine lines through the origin and the nine vertices of the 8-simplex.

Like all uniform omnitruncated n-simplices, the omnitruncated 8-simplex can tessellate space by itself, in this case 8-dimensional space with three facets around each ridge. It has Coxeter-Dynkin diagram of .

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

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) x3o3o3o3o3o3o3x - soxeb, x3x3x3x3x3x3x3x - goxeb

External links

• Olshevsky, George, Cross polytope 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