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7-simplex

Type of 7-polytope

Regular octaexon(7-simplex)
[[Image:7-simplex t0.svg	280px]]Orthogonal projectioninside Petrie polygon
Type
Family
Schläfli symbol
Coxeter-Dynkindiagram
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Petrie polygon
Coxeter group
Dual
Properties

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°.

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 an octaexon the acronym oca.

As a configuration

This configuration matrix represents the 7-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-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.

\begin{bmatrix}\begin{matrix}8 & 7 & 21 & 35 & 35 & 21 & 7 \ 2 & 28 & 6 & 15 & 20 & 15 & 6 \ 3 & 3 & 56 & 5 & 10 & 10 & 5 \ 4 & 6 & 4 & 70 & 4 & 6 & 4 \ 5 & 10 & 10 & 5 & 56 & 3 & 3 \ 6 & 15 & 20 & 15 & 6 & 28 & 2 \ 7 & 21 & 35 & 35 & 21 & 7 & 8 \end{matrix}\end{bmatrix}

Symmetry

[[File:Tetrahedral di-wedge.png	160px]]7-simplex as a join of two orthogonal tetrahedra in a symmetric 2D orthographic project: 2⋅{3,3} or {3,3}∨{3,3}, 6 red edges, 6 blue edges, and 16 yellow cross edges.	[[File:7-simplex-tetradisphenoid.png	160px]]7-simplex as a join of 4 orthogonal segments, projected into a 3D cube: 4⋅{ } = { }∨{ }∨{ }∨{ }. The 28 edges are shown as 12 yellow edges of the cube, 12 cube face diagonals in light green, and 4 full diagonals in red. This partition can be considered a tetradisphenoid, or a join of two disphenoid.

There are many lower symmetry constructions of the 7-simplex.

Some are expressed as join partitions of two or more lower simplexes. The symmetry order of each join is the product of the symmetry order of the elements, and raised further if identical elements can be interchanged.

Join		Symbol			Symmetry		Order		Extended f-vectors(factorization)
	Regular 7-simplex
	6-simplex-point join (pyramid)
	5-simplex-segment join
	5-cell-triangle join
	triangle-triangle-segment join
	Tetrahedron-tetrahedron join
	4 segment join
	8 point join

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

[[Image:Uniform polytope 3,3,3,3,3,3 t0.jpg	280px]]Ball and stick model in triakis tetrahedral envelope	[[File:Amplituhedron-0c.png	280px]]7-Simplex as an Amplituhedron Surface	[[File:Amplituhedron-0b.png	280px]]7-simplex to 3D with camera perspective showing hints of its 2D Petrie projection

Orthographic projections

Notes

References

{{KlitzingPolytopes. polyexa.htm. 7D uniform polytopes (polyexa). x3o3o3o3o3o3o — oca
Coxeter, H.S.M.. (1973). "Regular Polytopes". Dover.
Coxeter, H.S.M.. (1991). "Regular Complex Polytopes". Cambridge University Press.

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