6-simplex

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Template:Short description

6-simplex
Type uniform polypeton
Schläfli symbol {35}
Coxeter diagrams Template:CDD
Elements

f5 = 7, f4 = 21, C = 35, F = 35, E = 21, V = 7
(χ=0)

Coxeter group A6, [35], order 5040
Bowers name
and (acronym)		 Heptapeton
(hop)
Vertex figure 5-simplex
Circumradius 37
0.654654[1]
Properties convex, isogonal self-dual

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°.

Alternate names

It can also be called a heptapeton, or hepta-6-tope, as a 7-facetted polytope in 6-dimensions. The name heptapeton is derived from hepta for seven facets in Greek and -peta for having five-dimensional facets, and -on. Jonathan Bowers gives a heptapeton the acronym hop.[2]

As a configuration

This configuration matrix represents the 6-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-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.[3][4]

[76152015622151010533354644643533510105212615201567]

Coordinates

The Cartesian coordinates for an origin-centered regular heptapeton having edge length 2 are:

(1/21, 1/15, 1/10, 1/6, 1/3, ±1)
(1/21, 1/15, 1/10, 1/6, −21/3, 0)
(1/21, 1/15, 1/10, −3/2, 0, 0)
(1/21, 1/15, −22/5, 0, 0, 0)
(1/21, −5/3, 0, 0, 0, 0)
(−12/7, 0, 0, 0, 0, 0)

The vertices of the 6-simplex can be more simply positioned in 7-space as permutations of:

(0,0,0,0,0,0,1)

This construction is based on facets of the 7-orthoplex.

Images

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

The regular 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.

A6 polytopes
File:6-simplex t0.svg
t0 		File:6-simplex t1.svg
t1 		File:6-simplex t2.svg
t2 		File:6-simplex t01.svg
t0,1 		File:6-simplex t02.svg
t0,2 		File:6-simplex t12.svg
t1,2 		File:6-simplex t03.svg
t0,3 		File:6-simplex t13.svg
t1,3 		File:6-simplex t23.svg
t2,3
File:6-simplex t04.svg
t0,4 		File:6-simplex t14.svg
t1,4 		File:6-simplex t05.svg
t0,5 		File:6-simplex t012.svg
t0,1,2 		File:6-simplex t013.svg
t0,1,3 		File:6-simplex t023.svg
t0,2,3 		File:6-simplex t123.svg
t1,2,3 		File:6-simplex t014.svg
t0,1,4 		File:6-simplex t024.svg
t0,2,4
File:6-simplex t124.svg
t1,2,4 		File:6-simplex t034.svg
t0,3,4 		File:6-simplex t015.svg
t0,1,5 		File:6-simplex t025.svg
t0,2,5 		File:6-simplex t0123.svg
t0,1,2,3 		File:6-simplex t0124.svg
t0,1,2,4 		File:6-simplex t0134.svg
t0,1,3,4 		File:6-simplex t0234.svg
t0,2,3,4 		File:6-simplex t1234.svg
t1,2,3,4
File:6-simplex t0125.svg
t0,1,2,5 		File:6-simplex t0135.svg
t0,1,3,5 		File:6-simplex t0235.svg
t0,2,3,5 		File:6-simplex t0145.svg
t0,1,4,5 		File:6-simplex t01234.svg
t0,1,2,3,4 		File:6-simplex t01235.svg
t0,1,2,3,5 		File:6-simplex t01245.svg
t0,1,2,4,5 		File:6-simplex t012345.svg
t0,1,2,3,4,5

Notes

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References

  • Coxeter, H.S.M.:
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Family AnScript error: No such module "Check for unknown parameters". BnScript error: No such module "Check for unknown parameters". I2(p)Script error: No such module "Check for unknown parameters". / DnScript error: No such module "Check for unknown parameters". E6Script error: No such module "Check for unknown parameters". / E7Script error: No such module "Check for unknown parameters". / E8Script error: No such module "Check for unknown parameters". / F4Script error: No such module "Check for unknown parameters". / G2Script error: No such module "Check for unknown parameters". HnScript error: No such module "Check for unknown parameters".
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 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations
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