It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets.
By Jonathan Bowers, a hexateron is given the acronym hix.
The rows and columns correspond to vertices, edges, faces, cells and 4-faces.
This self-dual simplex's matrix is identical to its 180 degree rotation.
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).
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.
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.
The vertex figure of the omnitruncated 5-simplex honeycomb, , is a 5-simplex with a petrie polygon cycle of 5 long edges.
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.
A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.
A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral dihedron.
(Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)