6-simplex

6-simplex
Type	uniform polypeton
Schläfli symbol	{35}
Coxeter diagrams	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.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	$\sqrt{\frac{3}{7}}$ 0.654654
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.

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.

$\left[\begin{array}{c}\begin{array}{cccccc}7& 6& 15& 20& 15& 6\\ 2& 21& 5& 10& 10& 5\\ 3& 3& 35& 4& 6& 4\\ 4& 6& 4& 35& 3& 3\\ 5& 10& 10& 5& 21& 2\\ 6& 15& 20& 15& 6& 7\end{array}\end{array}\right]$

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

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

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

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

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

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

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

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]

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.

References

• Coxeter, H.S.M.:
  • • (Paper 22)
    • (Paper 23)
    • (Paper 24)

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary