# 5-simplex

5-simplex
Hexateron (hix)
Type uniform 5-polytope
Schläfli symbol {34}
Coxeter diagram         4-faces 6 6 {3,3,3} Cells 15 15 {3,3} Faces 20 20 {3} Edges 15
Vertices 6
Vertex figure 5-cell
Coxeter group A5, , order 720
Dual self-dual
Base point (0,0,0,0,0,1)
Circumradius 0.645497
Properties 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 construction can be seen as facets of the 6-orthoplex or rectified 6-cube respectively.

## Lower symmetry forms

A lower symmetry form is a 5-cell pyramid ( )v{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 { }v{3,3}, with [2,3,3] 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}v{3}, with [3,2,3] symmetry order 36, and extended symmetry [[3,2,3]], order 72. It represents joining of 2 orthogonal triangles, orthogonally offset, with all pairs of vertices connected between.

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.

Vertex figures for truncated 6-simplexes
( )v{3,3,3} { }v{3,3} {3}v{3}

truncated 6-simplex

truncated 6-cube

bitruncated 6-simplex

bitruncated 6-cube

tritruncated 6-simplex

## 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 13k series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral dihedron.

13k dimensional figures
Space Finite Euclidean Hyperbolic
n 4 5 6 7 8 9
Coxeter
group
A3A1 A5 D6 E7 ${\tilde {E}}_{7}$ =E7+ ${\bar {T}}_{8}$ =E7++
Coxeter
diagram

Symmetry [3−1,3,1] [30,3,1] [31,3,1] [32,3,1] [[33,3,1]] [34,3,1]
Order 48 720 23,040 2,903,040
Graph       - -
Name 13,-1 130 131 132 133 134

It is first in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3k1 series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.

3k1 dimensional figures
Space Finite Euclidean Hyperbolic
n 4 5 6 7 8 9
Coxeter
group
A3A1 A5 D6 E7 ${\tilde {E}}_{7}$ =E7+ ${\bar {T}}_{8}$ =E7++
Coxeter
diagram

Symmetry [3−1,3,1] [30,3,1] [[31,3,1]]
= [4,3,3,3,3]
[32,3,1] [33,3,1] [34,3,1]
Order 48 720 46,080 2,903,040
Graph       - -
Name 31,-1 310 311 321 331 341

The 5-simplex, as 220 polytope is first in dimensional series 22k.

22k figures of n dimensions
Space Finite Euclidean Hyperbolic
n 4 5 6 7 8
Coxeter
group
A2A2 A5 E6 ${\tilde {E}}_{6}$ =E6+ E6++
Coxeter
diagram

Graph
Name 22,-1 220 221 222 223

The regular 5-simplex is one of 19 uniform polytera based on the [3,3,3,3] Coxeter group, all shown here in A5 Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)