7-simplex

7-simplex
Regular octaexon
(7-simplex)
7-simplex t0.svg
Orthogonal projection
inside Petrie polygon
Type Regular 7-polytope
Family simplex
Schläfli symbol {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.png
6-faces 8 6-simplex6-simplex t0.svg
5-faces 28 5-simplex5-simplex t0.svg
4-faces 56 5-cell4-simplex t0.svg
Cells 70 tetrahedron4-simplex t0.svg
Faces 56 triangle3-simplex t0.svg
Edges 28
Vertices 8
Vertex figure 6-simplex
Petrie polygon octagon
Coxeter group A7 [3,3,3,3,3,3]
Dual Self-dual
Properties convex

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°.

Contents

Alternate names

It can also be called an octaexon, or octa-7-tope, as an 8-facetted polytope in 7-dimensions. The name octaexon is derived from octa for eight facets in Greek and -ex for having six-dimensional facets, and -on. Jonathan Bowers gives a octaexon the acronym oca.[1]

Coordinates

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

\left(\sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ \sqrt{1/3},\ \pm1\right)
\left(\sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ -2\sqrt{1/3},\ 0\right)
\left(\sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ -\sqrt{3/2},\ 0,\ 0\right)
\left(\sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right)
\left(\sqrt{1/28},\ \sqrt{1/21},\ -\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right)
\left(\sqrt{1/28},\ -\sqrt{12/7},\ 0,\ 0,\ 0,\ 0,\ 0\right)
\left(-\sqrt{7/4},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)

More simply, the vertices of the 7-simplex can be positioned in 8-space as permutations of (0,0,0,0,0,0,0,1). This construction is based on facets of the 8-orthoplex.

Images

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

Related polytopes

This polytope is one of 71 uniform 7-polytopes with A7 symmetry.

7-simplex t0.svg
t0		 7-simplex t1.svg
t1		 7-simplex t2.svg
t2		 7-simplex t3.svg
t3		 7-simplex t01.svg
t0,1		 7-simplex t02.svg
t0,2		 7-simplex t12.svg
t1,2		 7-simplex t03.svg
t0,3
7-simplex t13.svg
t1,3		 7-simplex t23.svg
t2,3		 7-simplex t04.svg
t0,4		 7-simplex t14.svg
t1,4		 7-simplex t24.svg
t2,4		 7-simplex t05.svg
t0,5		 7-simplex t15.svg
t1,5		 7-simplex t06.svg
t0,6
7-simplex t012.svg
t0,1,2		 7-simplex t013.svg
t0,1,3		 7-simplex t023.svg
t0,2,3		 7-simplex t123.svg
t1,2,3		 7-simplex t014.svg
t0,1,4		 7-simplex t024.svg
t0,2,4		 7-simplex t124.svg
t1,2,4		 7-simplex t034.svg
t0,3,4
7-simplex t134.svg
t1,3,4		 7-simplex t234.svg
t2,3,4		 7-simplex t015.svg
t0,1,5		 7-simplex t025.svg
t0,2,5		 7-simplex t125.svg
t1,2,5		 7-simplex t035.svg
t0,3,5		 7-simplex t135.svg
t1,3,5		 7-simplex t045.svg
t0,4,5
7-simplex t016.svg
t0,1,6		 7-simplex t026.svg
t0,2,6		 7-simplex t036.svg
t0,3,6		 7-simplex t0123.svg
t0,1,2,3		 7-simplex t0124.svg
t0,1,2,4		 7-simplex t0134.svg
t0,1,3,4		 7-simplex t0234.svg
t0,2,3,4		 7-simplex t1234.svg
t1,2,3,4
7-simplex t0125.svg
t0,1,2,5		 7-simplex t0135.svg
t0,1,3,5		 7-simplex t0235.svg
t0,2,3,5		 7-simplex t1235.svg
t1,2,3,5		 7-simplex t0145.svg
t0,1,4,5		 7-simplex t0245.svg
t0,2,4,5		 7-simplex t1245.svg
t1,2,4,5		 7-simplex t0345.svg
t0,3,4,5
7-simplex t0126.svg
t0,1,2,6		 7-simplex t0136.svg
t0,1,3,6		 7-simplex t0236.svg
t0,2,3,6		 7-simplex t0146.svg
t0,1,4,6		 7-simplex t0246.svg
t0,2,4,6		 7-simplex t0156.svg
t0,1,5,6		 7-simplex t01234.svg
t0,1,2,3,4		 7-simplex t01235.svg
t0,1,2,3,5
7-simplex t01245.svg
t0,1,2,4,5		 7-simplex t01345.svg
t0,1,3,4,5		 7-simplex t02345.svg
t0,2,3,4,5		 7-simplex t12345.svg
t1,2,3,4,5		 7-simplex t01236.svg
t0,1,2,3,6		 7-simplex t01246.svg
t0,1,2,4,6		 7-simplex t01346.svg
t0,1,3,4,6		 7-simplex t02346.svg
t0,2,3,4,6
7-simplex t01256.svg
t0,1,2,5,6		 7-simplex t01356.svg
t0,1,3,5,6		 7-simplex t012345.svg
t0,1,2,3,4,5		 7-simplex t012346.svg
t0,1,2,3,4,6		 7-simplex t012356.svg
t0,1,2,3,5,6		 7-simplex t012456.svg
t0,1,2,4,5,6		 7-simplex t0123456.svg
t0,1,2,3,4,5,6

Notes

  1. ^ Richard Klitzing, 7D uniform polytopes (polyexa), x3o3o3o3o3o - oca

External links

v · d · eFundamental convex regular and uniform polytopes in dimensions 2–10
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
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