# 5-simplex

5-simplex Hexateron (hix) :
| Type | uniform 5-polytope | uniform 5-polytope |
| Schläfli symbol | {3$^4$} | {3$^4$} |
| Coxeter diagram | | |
| 4-faces | 6 | 6 {3,3,3} |
| Cells | 15 | 15 {3,3} |
| Faces | 20 | 20 {3} |
| Edges | 15 | 15 |
| Vertices | 6 | 6 |
| Vertex figure | 5-cell | 5-cell |
| Coxeter group | A$_5$, [3$^4$], order 720 | A$_5$, [3$^4$], order 720 |
| Dual | self-dual | self-dual |
| Base point | (0,0,0,0,0,1) | (0,0,0,0,0,1) |
| Circumradius | 0.645497 | 0.645497 |
| Properties | convex, isogonal regular, self-dual | convex, isogonal regular, self-dual |
In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos$^{−1}$(⁠1/5⁠), or approximately 78.46°.
The 5-simplex is a solution to the problem: Make 20 equilateral triangles using 15 matchsticks, where each side of every triangle is exactly one matchstick.

## Alternate names

It can also be called a hexateron, or hexa-5-tope, as a 6-facetted polytope in 5-dimensions. The name hexateron is derived from hexa- for having six facets and teron (with ter- being a corruption of tetra-) for having four-dimensional facets.
By Jonathan Bowers, a hexateron is given the acronym hix.

## As a configuration

This configuration matrix represents the 5-simplex. The rows and columns correspond to vertices, edges, faces, cells and 4-faces. The diagonal numbers say how many of each element occur in the whole 5-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.
${\begin{bmatrix}{\begin{matrix}6&5&10&10&5\\2&15&4&6&4\\3&3&20&3&3\\4&6&4&15&2\\5&10&10&5&6\end{matrix}}\end{bmatrix}}$

## Regular hexateron cartesian coordinates

The hexateron can be constructed from a 5-cell by adding a 6th vertex such that it is equidistant from all the other vertices of the 5-cell.
The Cartesian coordinates for the vertices of an origin-centered regular hexateron having edge length 2 are:
    ${\begin{aligned}&\left({\tfrac {1}{\sqrt {15}}},\ {\tfrac {1}{\sqrt {10}}},\ {\tfrac {1}{\sqrt {6}}},\ {\tfrac {1}{\sqrt {3}}},\ \pm 1\right)\\[5pt]&\left({\tfrac {1}{\sqrt {15}}},\ {\tfrac {1}{\sqrt {10}}},\ {\tfrac {1}{\sqrt {6}}},\ -{\tfrac {2}{\sqrt {3}}},\ 0\right)\\[5pt]&\left({\tfrac {1}{\sqrt {15}}},\ {\tfrac {1}{\sqrt {10}}},\ -{\tfrac {\sqrt {3}}{\sqrt {2}}},\ 0,\ 0\right)\\[5pt]&\left({\tfrac {1}{\sqrt {15}}},\ -{\tfrac {2{\sqrt {2}}}{\sqrt {5}}},\ 0,\ 0,\ 0\right)\\[5pt]&\left(-{\tfrac {\sqrt {5}}{\sqrt {3}}},\ 0,\ 0,\ 0,\ 0\right)\end{aligned}}$
The vertices of the 5-simplex can be more simply positioned on a hyperplane in 6-space as permutations of (0,0,0,0,0,1) or (0,1,1,1,1,1). These constructions can be seen as facets of the 6-orthoplex or rectified 6-cube respectively.

## Projected images

orthographic projections :
| A$_k$ Coxeter plane | A$_5$ | A$_4$ |
| Graph | | |
| Dihedral symmetry | [6] | [5] |
| A$_k$ Coxeter plane | A$_3$ | A$_2$ |
| Graph | | |
| Dihedral symmetry | [4] | [3] |
* Stereographic projection 4D to 3D of Schlegel diagram 5D to 4D of hexateron.

## Lower symmetry forms

A lower symmetry form is a 5-cell pyramid {3,3,3}∨( ), with [3,3,3] symmetry order 120, constructed as a 5-cell base in a 4-space hyperplane, and an apex point above the hyperplane. The five sides of the pyramid are made of 5-cell cells. These are seen as vertex figures of truncated regular 6-polytopes, like a truncated 6-cube.
Another form is {3,3}∨{ }, with [3,3,2,1] symmetry order 48, the joining of an orthogonal digon and a tetrahedron, orthogonally offset, with all pairs of vertices connected between. Another form is {3}∨{3}, with [3,2,3,1] symmetry order 36, and extended symmetry [[3,2,3],1], order 72. It represents joining of 2 orthogonal triangles, orthogonally offset, with all pairs of vertices connected between.
The form { }∨{ }∨{ } has symmetry [2,2,1,1], order 8, extended by permuting 3 segments as [3[2,2],1] or [4,3,1,1], order 48.
These are seen in the vertex figures of bitruncated and tritruncated regular 6-polytopes, like a bitruncated 6-cube and a tritruncated 6-simplex. The edge labels here represent the types of face along that direction, and thus represent different edge lengths.
The vertex figure of the omnitruncated 5-simplex honeycomb, , is a 5-simplex with a petrie polygon cycle of 5 long edges. Its symmetry is isomorphic to dihedral group Dih$_6$ or simple rotation group [6,2]$^+$, order 12.
Vertex figures for uniform 6-polytopes :
| Join | {3,3,3}∨( ) | {3,3}∨{ } | {3}∨{3} | { }∨{ }∨{ } | |
| Symmetry | [3,3,3,1] Order 120 | [3,3,2,1] Order 48 | [[3,2,3],1] Order 72 | [3[2,2],1,1]=[4,3,1,1] Order 48 | ~[6] or ~[6,2]$^+$ Order 12 |
| Diagram | | | | | |
| Polytope | truncated 6-simplex | bitruncated 6-simplex | tritruncated 6-simplex | 3-3-3 prism | Omnitruncated 5-simplex honeycomb |

## Compound

The compound of two 5-simplexes in dual configurations can be seen in this A6 Coxeter plane projection, with a red and blue 5-simplex vertices and edges. This compound has [[3,3,3,3]] symmetry, order 1440. The intersection of these two 5-simplexes is a uniform birectified 5-simplex. = ∩ .

## Related uniform 5-polytopes

It is first in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1$_{3k}$ series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.
1$_{3k}$ dimensional figures :
| Space | Finite | Finite | Finite | Finite | Euclidean | Hyperbolic |
| n | 4 | 5 | 6 | 7 | 8 | 9 |
| Coxeter group | A$_3$A$_1$ | A$_5$ | D$_6$ | E$_7$ | ${\tilde {E}}_{7}$=E$_7$$^+$ | ${\bar {T}}_{8}$=E$_7$$^{++}$ |
| Coxeter diagram | | | | | | |
| Symmetry | [3$^{−1,3,1}$] | [3$^{0,3,1}$] | [3$^{1,3,1}$] | [3$^{2,3,1}$] | [[3$^{3,3,1}$]] | [3$^{4,3,1}$] |
| Order | 48 | 720 | 23,040 | 2,903,040 | ∞ | ∞ |
| Graph | | | | | - | - |
| Name | 1$_{3,-1}$ | 1$_{30}$ | 1$_{31}$ | 1$_{32}$ | 1$_{33}$ | 1$_{34}$ |
It is first in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3$_{k1}$ series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral dihedron.
3$_{k1}$ dimensional figures :
| Space | Finite | Finite | Finite | Finite | Euclidean | Hyperbolic |
| n | 4 | 5 | 6 | 7 | 8 | 9 |
| Coxeter group | A$_3$A$_1$ | A$_5$ | D$_6$ | E$_7$ | ${\tilde {E}}_{7}$=E$_7$$^+$ | ${\bar {T}}_{8}$=E$_7$$^{++}$ |
| Coxeter diagram | | | | | | |
| Symmetry | [3$^{−1,3,1}$] | [3$^{0,3,1}$] | [[3$^{1,3,1}$]] = [4,3,3,3,3] | [3$^{2,3,1}$] | [3$^{3,3,1}$] | [3$^{4,3,1}$] |
| Order | 48 | 720 | 46,080 | 2,903,040 | ∞ | ∞ |
| Graph | | | | | - | - |
| Name | 3$_{1,-1}$ | 3$_{10}$ | 3$_{11}$ | 3$_{21}$ | 3$_{31}$ | 3$_{41}$ |
The 5-simplex, as 2$_{20}$ polytope is first in dimensional series 2$_{2k}$.
2$_{2k}$ figures of n dimensions :
| Space | Finite | Finite | Finite | Euclidean | Hyperbolic |
| n | 4 | 5 | 6 | 7 | 8 |
| Coxeter group | A$_2$A$_2$ | A$_5$ | E$_6$ | ${\tilde {E}}_{6}$=E$_6$$^+$ | E$_6$$^{++}$ |
| Coxeter diagram | | | | | |
| Graph | | | | ∞ | ∞ |
| Name | 2$_{2,-1}$ | 2$_{20}$ | 2$_{21}$ | 2$_{22}$ | 2$_{23}$ |
The regular 5-simplex is one of 19 uniform polytera based on the [3,3,3,3] Coxeter group, all shown here in A$_5$ Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)
A5 polytopes :
| t$_0$ | t$_1$ | t$_2$ | t$_{0,1}$ | t$_{0,2}$ | t$_{1,2}$ | t$_{0,3}$ |
| t$_{1,3}$ | t$_{0,4}$ | 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$_{0,1,2,3}$ | t$_{0,1,2,4}$ | t$_{0,1,3,4}$ | t$_{0,1,2,3,4}$ |