6-simplex
Type uniform polypeton
Schläfli symbol {35}
Coxeter diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
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
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]

Coordinates

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

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

Related uniform 6-polytopes

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
6-simplex t0.svg

t0
6-simplex t1.svg

t1
6-simplex t2.svg

t2
6-simplex t01.svg

t0,1
6-simplex t02.svg

t0,2
6-simplex t12.svg

t1,2
6-simplex t03.svg

t0,3
6-simplex t13.svg

t1,3
6-simplex t23.svg

t2,3
6-simplex t04.svg

t0,4
6-simplex t14.svg

t1,4
6-simplex t05.svg

t0,5
6-simplex t012.svg

t0,1,2
6-simplex t013.svg

t0,1,3
6-simplex t023.svg

t0,2,3
6-simplex t123.svg

t1,2,3
6-simplex t014.svg

t0,1,4
6-simplex t024.svg

t0,2,4
6-simplex t124.svg

t1,2,4
6-simplex t034.svg

t0,3,4
6-simplex t015.svg

t0,1,5
6-simplex t025.svg

t0,2,5
6-simplex t0123.svg

t0,1,2,3
6-simplex t0124.svg

t0,1,2,4
6-simplex t0134.svg

t0,1,3,4
6-simplex t0234.svg

t0,2,3,4
6-simplex t1234.svg

t1,2,3,4
6-simplex t0125.svg

t0,1,2,5
6-simplex t0135.svg

t0,1,3,5
6-simplex t0235.svg

t0,2,3,5
6-simplex t0145.svg

t0,1,4,5
6-simplex t01234.svg

t0,1,2,3,4
6-simplex t01235.svg

t0,1,2,3,5
6-simplex t01245.svg

t0,1,2,4,5
6-simplex t012345.svg

t0,1,2,3,4,5

Notes

  1. ^ Klitzing, Richard. "heptapeton". bendwavy.org.
  2. ^ Klitzing, Richard. "6D uniform polytopes (polypeta) x3o3o3o3o3o — hop".
  3. ^ Coxeter 1973, §1.8 Configurations
  4. ^ Coxeter, H.S.M. (1991). Regular Complex Polytopes (2nd ed.). Cambridge University Press. p. 117. ISBN 9780521394901.

References

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds