8-simplex

Last updated
Regular enneazetton
(8-simplex)
8-simplex t0.svg
Orthogonal projection
inside Petrie polygon
TypeRegular 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-faces9 7-simplex 7-simplex t0.svg
6-faces36 6-simplex 6-simplex t0.svg
5-faces84 5-simplex 5-simplex t0.svg
4-faces126 5-cell 4-simplex t0.svg
Cells126 tetrahedron 3-simplex t0.svg
Faces84 triangle 2-simplex t0.svg
Edges36
Vertices9
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°.

Contents

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 A8A7A6A5
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 planeA4A3A2
Graph 8-simplex t0 A4.svg 8-simplex t0 A3.svg 8-simplex t0 A2.svg
Dihedral symmetry[5][4][3]

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

Related Research Articles

5-cell Four-dimensional analogue of the tetrahedron

In geometry, the 5-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,3}. It is a four-dimensional object bounded by 5 tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is the 4-simplex (Coxeter's polytope), the simplest possible convex regular 4-polytope (four-dimensional analogue of a Platonic solid), and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The pentachoron is a four dimensional pyramid with a tetrahedral base.

In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(1/5), or approximately 78.46°.

7-cube 7-dimensional hypercube

In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces.

8-cube 8-dimensional hypercube

In geometry, an 8-cube is an eight-dimensional hypercube. It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 5-cube 5-faces, 112 6-cube 6-faces, and 16 7-cube 7-faces.

7-orthoplex

In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cells 4-faces, 448 5-faces, and 128 6-faces.

In geometry, a 6-simplex is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5-cell 4-faces, and 7 5-simplex 5-faces. Its dihedral angle is cos−1(1/6), or approximately 80.41°.

7-simplex

In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5-cell 5-faces, 28 5-simplex 6-faces, and 8 6-simplex 7-faces. Its dihedral angle is cos−1(1/7), or approximately 81.79°.

8-orthoplex

In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells 4-faces, 1792 5-faces, 1024 6-faces, and 256 7-faces.

9-simplex

In geometry, a 9-simplex is a self-dual regular 9-polytope. It has 10 vertices, 45 edges, 120 triangle faces, 210 tetrahedral cells, 252 5-cell 4-faces, 210 5-simplex 5-faces, 120 6-simplex 6-faces, 45 7-simplex 7-faces, and 10 8-simplex 8-faces. Its dihedral angle is cos−1(1/9), or approximately 83.62°.

10-simplex

In geometry, a 10-simplex is a self-dual regular 10-polytope. It has 11 vertices, 55 edges, 165 triangle faces, 330 tetrahedral cells, 462 5-cell 4-faces, 462 5-simplex 5-faces, 330 6-simplex 6-faces, 165 7-simplex 7-faces, 55 8-simplex 8-faces, and 11 9-simplex 9-faces. Its dihedral angle is cos−1(1/10), or approximately 84.26°.

2<sub> 31</sub> polytope

In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.

2<sub> 21</sub> polytope

In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 6-ic semi-regular figure. It is also called the Schläfli polytope.

3<sub> 21</sub> polytope

In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 7-ic semi-regular figure.

4<sub> 21</sub> polytope

In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 8-ic semi-regular figure.

In 7-dimensional geometry, the 331 honeycomb is a uniform honeycomb, also given by Schläfli symbol {3,3,3,33,1} and is composed of 321 and 7-simplex facets, with 56 and 576 of them respectively around each vertex.

Stericated 5-simplexes

In five-dimensional geometry, a stericated 5-simplex is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-simplex.

Pentellated 6-simplexes

In six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex.

In seven-dimensional geometry, a rectified 7-orthoplex is a convex uniform 7-polytope, being a rectification of the regular 7-orthoplex.

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

Simplectic honeycomb

In geometry, the simplectic honeycomb is a dimensional infinite series of honeycombs, based on the affine Coxeter group symmetry. It is given a Schläfli symbol {3[n+1]}, and is represented by a Coxeter-Dynkin diagram as a cyclic graph of n+1 nodes with one node ringed. It is composed of n-simplex facets, along with all rectified n-simplices. It can be thought of as an n-dimensional hypercubic honeycomb that has been subdivided along all hyperplanes , then stretched along its main diagonal until the simplices on the ends of the hypercubes become regular. The vertex figure of an n-simplex honeycomb is an expanded n-simplex.

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