# 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.*

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.

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.^{[2]}^{[3]}

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:

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.

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 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. = ∩ .

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.

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.

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)