7-simplex

Regular octaexon (7-simplex)
File: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	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
6-faces	8 6-simplex File:6-simplex t0.svg
5-faces	28 5-simplex File:5-simplex t0.svg
4-faces	56 5-cell File:4-simplex t0.svg
Cells	70 tetrahedron File:3-simplex t0.svg
Faces	56 triangle File:2-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°.

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.^[1]

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.^[2]^[3]

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

Symmetry

File:Tetrahedral di-wedge.png 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 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	{3,3,3,3,3,3}	[3,3,3,3,3,3]	8! = 40320	(1,8,28,56,70,56,28,8,1)
6-simplex-point join (pyramid)	{3,3,3,3,3}∨( )	[3,3,3,3,3,1]	7!×1! = 5040	(1,7,21,35,35,21,7,1)*(1,1)
5-simplex-segment join	{3,3,3,3}∨{ }	[3,3,3,3,2,1]	6!×2! = 1440	(1,6,15,20,15,6,1)*(1,2,1)
5-cell-triangle join	{3,3,3}∨{3}	[3,3,3,2,3,1]	5!×3! = 720	(1,5,10,10,5,1)*(1,3,3,1)
triangle-triangle-segment join	{3}∨{3}∨{ }	[[3,2,3],2,1,1]	((3!)²×2!)×2! = 144	(1,3,3,1)²*(1,2,1)
Tetrahedron-tetrahedron join	2⋅{3,3} = {3,3}∨{3,3}	[[3,3,2,3,3],1]	(4!)²×2! = 1052	(1,4,6,4,1)²
4 segment join	4⋅{ } = { }∨{ }∨{ }∨{ }	[4[2,2,2],1,1,1]	(2!)⁴×4! = 384	(1,2,1)⁴
8 point join	8⋅( )	[8[1,1,1,1,1,1]]	(1!)⁸×8! = 40320	(1,1)⁸

Coordinates

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

(\sqrt{1 / 28}, \sqrt{1 / 21}, \sqrt{1 / 15}, \sqrt{1 / 10}, \sqrt{1 / 6}, \sqrt{1 / 3}, \pm 1)

(\sqrt{1 / 28}, \sqrt{1 / 21}, \sqrt{1 / 15}, \sqrt{1 / 10}, \sqrt{1 / 6}, - 2 \sqrt{1 / 3}, 0)

(\sqrt{1 / 28}, \sqrt{1 / 21}, \sqrt{1 / 15}, \sqrt{1 / 10}, - \sqrt{3 / 2}, 0, 0)

(\sqrt{1 / 28}, \sqrt{1 / 21}, \sqrt{1 / 15}, - 2 \sqrt{2 / 5}, 0, 0, 0)

(\sqrt{1 / 28}, \sqrt{1 / 21}, - \sqrt{5 / 3}, 0, 0, 0, 0)

(\sqrt{1 / 28}, - \sqrt{12 / 7}, 0, 0, 0, 0, 0)

(- \sqrt{7 / 4}, 0, 0, 0, 0, 0, 0)

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

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

Orthographic projections

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

Related polytopes

This polytope is a facet in the uniform tessellation 3₃₁ with Coxeter-Dynkin diagram:

File:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png

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

A7 polytopes
File:7-simplex t0.svg t₀	File:7-simplex t1.svg t₁	File:7-simplex t2.svg t₂	File:7-simplex t3.svg t₃	File:7-simplex t01.svg t_0,1	File:7-simplex t02.svg t_0,2	File:7-simplex t12.svg t_1,2	File:7-simplex t03.svg t_0,3
File:7-simplex t13.svg t_1,3	File:7-simplex t23.svg t_2,3	File:7-simplex t04.svg t_0,4	File:7-simplex t14.svg t_1,4	File:7-simplex t24.svg t_2,4	File:7-simplex t05.svg t_0,5	File:7-simplex t15.svg t_1,5	File:7-simplex t06.svg t_0,6
File:7-simplex t012.svg t_0,1,2	File:7-simplex t013.svg t_0,1,3	File:7-simplex t023.svg t_0,2,3	File:7-simplex t123.svg t_1,2,3	File:7-simplex t014.svg t_0,1,4	File:7-simplex t024.svg t_0,2,4	File:7-simplex t124.svg t_1,2,4	File:7-simplex t034.svg t_0,3,4
File:7-simplex t134.svg t_1,3,4	File:7-simplex t234.svg t_2,3,4	File:7-simplex t015.svg t_0,1,5	File:7-simplex t025.svg t_0,2,5	File:7-simplex t125.svg t_1,2,5	File:7-simplex t035.svg t_0,3,5	File:7-simplex t135.svg t_1,3,5	File:7-simplex t045.svg t_0,4,5
File:7-simplex t016.svg t_0,1,6	File:7-simplex t026.svg t_0,2,6	File:7-simplex t036.svg t_0,3,6	File:7-simplex t0123.svg t_0,1,2,3	File:7-simplex t0124.svg t_0,1,2,4	File:7-simplex t0134.svg t_0,1,3,4	File:7-simplex t0234.svg t_0,2,3,4	File:7-simplex t1234.svg t_1,2,3,4
File:7-simplex t0125.svg t_0,1,2,5	File:7-simplex t0135.svg t_0,1,3,5	File:7-simplex t0235.svg t_0,2,3,5	File:7-simplex t1235.svg t_1,2,3,5	File:7-simplex t0145.svg t_0,1,4,5	File:7-simplex t0245.svg t_0,2,4,5	File:7-simplex t1245.svg t_1,2,4,5	File:7-simplex t0345.svg t_0,3,4,5
File:7-simplex t0126.svg t_0,1,2,6	File:7-simplex t0136.svg t_0,1,3,6	File:7-simplex t0236.svg t_0,2,3,6	File:7-simplex t0146.svg t_0,1,4,6	File:7-simplex t0246.svg t_0,2,4,6	File:7-simplex t0156.svg t_0,1,5,6	File:7-simplex t01234.svg t_0,1,2,3,4	File:7-simplex t01235.svg t_0,1,2,3,5
File:7-simplex t01245.svg t_0,1,2,4,5	File:7-simplex t01345.svg t_0,1,3,4,5	File:7-simplex t02345.svg t_0,2,3,4,5	File:7-simplex t12345.svg t_1,2,3,4,5	File:7-simplex t01236.svg t_0,1,2,3,6	File:7-simplex t01246.svg t_0,1,2,4,6	File:7-simplex t01346.svg t_0,1,3,4,6	File:7-simplex t02346.svg t_0,2,3,4,6
File:7-simplex t01256.svg t_0,1,2,5,6	File:7-simplex t01356.svg t_0,1,3,5,6	File:7-simplex t012345.svg t_0,1,2,3,4,5	File:7-simplex t012346.svg t_0,1,2,3,4,6	File:7-simplex t012356.svg t_0,1,2,3,5,6	File:7-simplex t012456.svg t_0,1,2,4,5,6	File:7-simplex t0123456.svg t_{0,1,2,3,4,5,6}

Notes

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External links

Glossary for hyperspace, George Olshevsky.
Polytopes of Various Dimensions
Multi-dimensional Glossary

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	$A n$	$B n$	$I 2 (p)$ / $D n$	$E 6$ / $E 7$ / $E 8$ / $F 4$ / $G 2$	$H n$
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	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
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations

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7-simplex

Contents

Alternate names

As a configuration

Symmetry

Coordinates

Images

Orthographic projections

Related polytopes

Notes

External links

Navigation menu

7-simplex

Alternate names

As a configuration

Symmetry

Coordinates

Images

Orthographic projections

Related polytopes

Notes

External links

Navigation menu

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