# 6-simplex

6-simplex :
| Type | uniform polypeton |
| Schläfli symbol | {3$^5$} |
| Coxeter diagrams | |
| Elements | f$_5$ = 7, f$_4$ = 21, C = 35, F = 35, E = 21, V = 7 (χ=0) |
| Coxeter group | A$_6$, [3$^5$], order 5040 |
| Bowers name and (acronym) | Heptapeton (hop) |
| Vertex figure | 5-simplex |
| Circumradius | ${\sqrt {\tfrac {3}{7}}}$ 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.
${\begin{bmatrix}{\begin{matrix}7&6&15&20&15&6\\2&21&5&10&10&5\\3&3&35&4&6&4\\4&6&4&35&3&3\\5&10&10&5&21&2\\6&15&20&15&6&7\end{matrix}}\end{bmatrix}}$

## Coordinates

The Cartesian coordinates for an origin-centered regular heptapeton having edge length 2 are:
    $\left({\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ {\sqrt {1/6}},\ {\sqrt {1/3}},\ \pm 1\right)$
    $\left({\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ {\sqrt {1/6}},\ -2{\sqrt {1/3}},\ 0\right)$
    $\left({\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ -{\sqrt {3/2}},\ 0,\ 0\right)$
    $\left({\sqrt {1/21}},\ {\sqrt {1/15}},\ -2{\sqrt {2/5}},\ 0,\ 0,\ 0\right)$
    $\left({\sqrt {1/21}},\ -{\sqrt {5/3}},\ 0,\ 0,\ 0,\ 0\right)$
    $\left(-{\sqrt {12/7}},\ 0,\ 0,\ 0,\ 0,\ 0\right)$
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 :
| A$_k$ Coxeter plane | A$_6$ | A$_5$ | A$_4$ |
| Graph | | | |
| Dihedral symmetry | [7] | [6] | [5] |
| A$_k$ Coxeter plane | A$_3$ | A$_2$ |
| Graph | | |
| 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 A$_6$ Coxeter plane orthographic projections.
A6 polytopes :
| t$_0$ | t$_1$ | t$_2$ | t$_{0,1}$ | t$_{0,2}$ | t$_{1,2}$ | t$_{0,3}$ | t$_{1,3}$ | t$_{2,3}$ |
| t$_{0,4}$ | t$_{1,4}$ | t$_{0,5}$ | t$_{0,1,2}$ | t$_{0,1,3}$ | t$_{0,2,3}$ | t$_{1,2,3}$ | t$_{0,1,4}$ | t$_{0,2,4}$ |
| t$_{1,2,4}$ | t$_{0,3,4}$ | t$_{0,1,5}$ | t$_{0,2,5}$ | t$_{0,1,2,3}$ | t$_{0,1,2,4}$ | t$_{0,1,3,4}$ | t$_{0,2,3,4}$ | t$_{1,2,3,4}$ |
| t$_{0,1,2,5}$ | t$_{0,1,3,5}$ | t$_{0,2,3,5}$ | t$_{0,1,4,5}$ | t$_{0,1,2,3,4}$ | t$_{0,1,2,3,5}$ | t$_{0,1,2,4,5}$ | t$_{0,1,2,3,4,5}$ |