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°.

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**.

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:

(
1
/
15
,
1
/
10
,
1
/
6
,
1
/
3
,
±
1
)
(
1
/
15
,
1
/
10
,
1
/
6
,
−
2
1
/
3
,
0
)
(
1
/
15
,
1
/
10
,
−
3
/
2
,
0
,
0
)
(
1
/
15
,
−
2
2
/
5
,
0
,
0
,
0
)
(
−
5
/
3
,
0
,
0
,
0
,
0
)
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.

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 dihedron.

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 hosohedron.

The 5-simplex, as 2_{20} polytope is first in dimensional series 2_{2k}.

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)

The *5-simplex* can also be considered a 5-cell pyramid, 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.