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Regular octaexon
(7-simplex)
7-simplex graph.png
Orthogonal projection
inside Petrie polygon
Type Regular 7-polytope
Family simplex
6-faces 8 6-simplexComplete graph K7.svg
5-faces 28 5-simplexComplete graph K6.svg
4-faces 56 5-cellComplete graph K5.svg
Cells 70 tetrahedronComplete graph K4.svg
Faces 56 triangleComplete graph K3.svg
Edges 28
Vertices 8
Vertex figure 6-simplex
Petrie polygon octagon
Schläfli symbol {3,3,3,3,3,3}
Coxeter-Dynkin diagram CDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
Coxeter group A7 [3,3,3,3,3,3]
Dual Self-dual
Properties convex

In geometry, an octexon, or octa-7-tope is a 7-simplex, a self-dual regular 7-polytope with 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.

The name octaexon is derived from octa for eight facets in Greek and -ex for having six-dimensional facets, and -on.

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)

External links

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Regular octaexon
(7-simplex)

Orthogonal projection
inside Petrie polygon
TypeRegular 7-polytope
Familysimplex
6-faces8 6-simplex
5-faces28 5-simplex
4-faces56 5-cell
Cells70 tetrahedron
Faces56 triangle
Edges28
Vertices8
Vertex figure6-simplex
Petrie polygonoctagon
Schläfli symbol {3,3,3,3,3,3}
Coxeter-Dynkin diagram
Coxeter group A7 [3,3,3,3,3,3]
DualSelf-dual
Propertiesconvex

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
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Related polytopes

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


t0

t1

t2

t3

t0,1

t0,2

t1,2

t0,3

t1,3

t2,3

t0,4

t1,4

t2,4

t0,5

t1,5

t0,6

t0,1,2

t0,1,3

t0,2,3

t1,2,3

t0,1,4

t0,2,4

t1,2,4

t0,3,4

t1,3,4

t2,3,4

t0,1,5

t0,2,5

t1,2,5

t0,3,5

t1,3,5

t0,4,5

t0,1,6

t0,2,6

t0,3,6

t0,1,2,3

t0,1,2,4

t0,1,3,4

t0,2,3,4

t1,2,3,4

t0,1,2,5

t0,1,3,5

t0,2,3,5

t1,2,3,5

t0,1,4,5

t0,2,4,5

t1,2,4,5

t0,3,4,5

t0,1,2,6

t0,1,3,6

t0,2,3,6

t0,1,4,6

t0,2,4,6

t0,1,5,6

t0,1,2,3,4

t0,1,2,3,5

t0,1,2,4,5

t0,1,3,4,5

t0,2,3,4,5

t1,2,3,4,5

t0,1,2,3,6

t0,1,2,4,6

t0,1,3,4,6

t0,2,3,4,6

t0,1,2,5,6

t0,1,3,5,6

t0,1,2,3,4,5

t0,1,2,3,4,6

t0,1,2,3,5,6

t0,1,2,4,5,6

t0,1,2,3,4,5,6

Notes

  1. ^ Richard Klitzing 5D quasiregulars, (multi)prisms, non-prismatic Wythoffian polyterons x3o3o3o3o3o - oca

External links


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