5-simplex

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{align}

&\left(\tfrac{1}\sqrt{15},\ \tfrac{1}\sqrt{10},\ \tfrac{1}\sqrt{6},\ \tfrac{1}\sqrt{3},\ \pm1\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{align}$$

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 {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 isomophic to dihedral group Dih6 or simple rotation group [6,2]+, order 12.

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. = ∩.
 * Compound two 5-simplexes.png

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 hosohedron.

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 dihedron.

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

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)