Hexateron

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Regular hexateron 5-simplex
(Orthographic projection)
Type	Regular 5-polytope
Family	simplex
Hypercells	6 {3,3,3}
Cells	15 {3,3}
Faces	20 {3}
Edges	15
Vertices	6
Vertex figure	{3,3,3}
Schläfli symbol	{3,3,3,3}
Coxeter-Dynkin diagram
Dual	Self-dual
Properties	convex

Stereographic projection 4D to 3D of Schlegel diagram 5D to 4D of hexateron.

A hexateron, or hexa-5-tope, is a 5-simplex, a self-dual regular 5-polytope with 6 vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, 6 5-cell hypercells.

The name hexateron is derived from hexa for six facets in Greek and -tera for having four-dimensional facets, and -on.

[edit] Cartesian coordinates

The hexateron can be constructed from a pentachoron (4-simplex) by adding a 6th vertex such that it is equidistant with all the other vertices of the pentachoron.

For example, the Cartesian coordinates for the vertices of a hexateron (not centered in the origin!), with edge length equal to $2 \sqrt{2}$ , may be:

$( 1, 1, 1, 0, 0),\,$
$(-1,-1, 1, 0, 0),\,$
$(-1, 1,-1, 0, 0),\,$
$( 1,-1,-1, 0, 0),\,$
$\left( 0, 0, 0,\sqrt{5}, 0\right),\,$
$\left( 0, 0, 0,{\sqrt{5} \over 5},2{\sqrt{30} \over 5}\right)\,$ ;

The xyz orthogonal projection of the first four coordinates corresponds to the coordinates of regular tetrahedron on alternate corners of the cube.

[edit] See also

Other regular 5-polytopes:
- Penteract - {4,3,3,3}
- Pentacross - {3,3,3,4}
Others in the simplex family
- Tetrahedron - {3,3}
- Pentachoron - {3,3,3}
- Hexateron - {3,3,3,3}
- Heptapeton - {3,3,3,3,3}
- Octaexon - {3,3,3,3,3,3}