Regular enneazetton
(8-simplex)
8-simplex t0.svg

Orthogonal projection
inside Petrie polygon
Type Regular 8-polytope
Family simplex
Schläfli symbol {3,3,3,3,3,3,3}
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-faces 9 7-simplex
7-simplex t0.svg
6-faces 36 6-simplex
6-simplex t0.svg
5-faces 84 5-simplex
5-simplex t0.svg
4-faces 126 5-cell
4-simplex t0.svg
Cells 126 tetrahedron
3-simplex t0.svg
Faces 84 triangle
2-simplex t0.svg
Edges 36
Vertices 9
Vertex figure 7-simplex
Petrie polygon enneagon
Coxeter group A8 [3,3,3,3,3,3,3]
Dual Self-dual
Properties convex

In geometry, an 8-simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 126 5-cell 4-faces, 84 5-simplex 5-faces, 36 6-simplex 6-faces, and 9 7-simplex 7-faces. Its dihedral angle is cos−1(1/8), or approximately 82.82°.

It can also be called an enneazetton, or ennea-8-tope, as a 9-facetted polytope in eight-dimensions. The name enneazetton is derived from ennea for nine facets in Greek and -zetta for having seven-dimensional facets, and -on.

As a configuration

This configuration matrix represents the 8-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces and 7-faces. The diagonal numbers say how many of each element occur in the whole 8-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual simplex's matrix is identical to its 180 degree rotation.[1][2]

Coordinates

The Cartesian coordinates of the vertices of an origin-centered regular enneazetton having edge length 2 are:

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

Another origin-centered construction uses (1,1,1,1,1,1,1,1)/3 and permutations of (1,1,1,1,1,1,1,-11)/12 for edge length √2.

Images

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
8-simplex t0.svg
8-simplex t0 A7.svg
8-simplex t0 A6.svg
8-simplex t0 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
8-simplex t0 A4.svg
8-simplex t0 A3.svg
8-simplex t0 A2.svg
Dihedral symmetry [5] [4] [3]

Related polytopes and honeycombs

This polytope is a facet in the uniform tessellations: 251, and 521 with respective Coxeter-Dynkin diagrams:

CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png

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

A8 polytopes
8-simplex t0.svg

t0
8-simplex t1.svg

t1
8-simplex t2.svg

t2
8-simplex t3.svg

t3
8-simplex t01.svg

t01
8-simplex t02.svg

t02
8-simplex t12.svg

t12
8-simplex t03.svg

t03
8-simplex t13.svg

t13
8-simplex t23.svg

t23
8-simplex t04.svg

t04
8-simplex t14.svg

t14
8-simplex t24.svg

t24
8-simplex t34.svg

t34
8-simplex t05.svg

t05
8-simplex t15.svg

t15
8-simplex t25.svg

t25
8-simplex t06.svg

t06
8-simplex t16.svg

t16
8-simplex t07.svg

t07
8-simplex t012.svg

t012
8-simplex t013.svg

t013
8-simplex t023.svg

t023
8-simplex t123.svg

t123
8-simplex t014.svg

t014
8-simplex t024.svg

t024
8-simplex t124.svg

t124
8-simplex t034.svg

t034
8-simplex t134.svg

t134
8-simplex t234.svg

t234
8-simplex t015.svg

t015
8-simplex t025.svg

t025
8-simplex t125.svg

t125
8-simplex t035.svg

t035
8-simplex t135.svg

t135
8-simplex t235.svg

t235
8-simplex t045.svg

t045
8-simplex t145.svg

t145
8-simplex t016.svg

t016
8-simplex t026.svg

t026
8-simplex t126.svg

t126
8-simplex t036.svg

t036
8-simplex t136.svg

t136
8-simplex t046.svg

t046
8-simplex t056.svg

t056
8-simplex t017.svg

t017
8-simplex t027.svg

t027
8-simplex t037.svg

t037
8-simplex t0123.svg

t0123
8-simplex t0124.svg

t0124
8-simplex t0134.svg

t0134
8-simplex t0234.svg

t0234
8-simplex t1234.svg

t1234
8-simplex t0125.svg

t0125
8-simplex t0135.svg

t0135
8-simplex t0235.svg

t0235
8-simplex t1235.svg

t1235
8-simplex t0145.svg

t0145
8-simplex t0245.svg

t0245
8-simplex t1245.svg

t1245
8-simplex t0345.svg

t0345
8-simplex t1345.svg

t1345
8-simplex t2345.svg

t2345
8-simplex t0126.svg

t0126
8-simplex t0136.svg

t0136
8-simplex t0236.svg

t0236
8-simplex t1236.svg

t1236
8-simplex t0146.svg

t0146
8-simplex t0246.svg

t0246
8-simplex t1246.svg

t1246
8-simplex t0346.svg

t0346
8-simplex t1346.svg

t1346
8-simplex t0156.svg

t0156
8-simplex t0256.svg

t0256
8-simplex t1256.svg

t1256
8-simplex t0356.svg

t0356
8-simplex t0456.svg

t0456
8-simplex t0127.svg

t0127
8-simplex t0137.svg

t0137
8-simplex t0237.svg

t0237
8-simplex t0147.svg

t0147
8-simplex t0247.svg

t0247
8-simplex t0347.svg

t0347
8-simplex t0157.svg

t0157
8-simplex t0257.svg

t0257
8-simplex t0167.svg

t0167
8-simplex t01234.svg

t01234
8-simplex t01235.svg

t01235
8-simplex t01245.svg

t01245
8-simplex t01345.svg

t01345
8-simplex t02345.svg

t02345
8-simplex t12345.svg

t12345
8-simplex t01236.svg

t01236
8-simplex t01246.svg

t01246
8-simplex t01346.svg

t01346
8-simplex t02346.svg

t02346
8-simplex t12346.svg

t12346
8-simplex t01256.svg

t01256
8-simplex t01356.svg

t01356
8-simplex t02356.svg

t02356
8-simplex t12356.svg

t12356
8-simplex t01456.svg

t01456
8-simplex t02456.svg

t02456
8-simplex t03456.svg

t03456
8-simplex t01237.svg

t01237
8-simplex t01247.svg

t01247
8-simplex t01347.svg

t01347
8-simplex t02347.svg

t02347
8-simplex t01257.svg

t01257
8-simplex t01357.svg

t01357
8-simplex t02357.svg

t02357
8-simplex t01457.svg

t01457
8-simplex t01267.svg

t01267
8-simplex t01367.svg

t01367
8-simplex t012345.svg

t012345
8-simplex t012346.svg

t012346
8-simplex t012356.svg

t012356
8-simplex t012456.svg

t012456
8-simplex t013456.svg

t013456
8-simplex t023456.svg

t023456
8-simplex t123456.svg

t123456
8-simplex t012347.svg

t012347
8-simplex t012357.svg

t012357
8-simplex t012457.svg

t012457
8-simplex t013457.svg

t013457
8-simplex t023457.svg

t023457
8-simplex t012367.svg

t012367
8-simplex t012467.svg

t012467
8-simplex t013467.svg

t013467
8-simplex t012567.svg

t012567
8-simplex t0123456 A7.svg

t0123456
8-simplex t0123457 A7.svg

t0123457
8-simplex t0123467 A7.svg

t0123467
8-simplex t0123567 A7.svg

t0123567
8-simplex t01234567 A7.svg

t01234567

References

  1. ^ Coxeter 1973, §1.8 Configurations
  2. ^ Coxeter, H.S.M. (1991). Regular Complex Polytopes (2nd ed.). Cambridge University Press. p. 117. ISBN 9780521394901.
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