Hexateron

In five dimensional geometry, 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".

Cartesian coordinates

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

For example, the Cartesian coordinates for the vertices of a "hexateron" (notcentered in the origin!), with edge length equal to $2\; sqrt\{2\}$, may be:

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

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

Projected images

See also

* Other regular 5-polytopes:
** Penteract - {4,3,3,3}
** Pentacross - {3,3,3,4}
* Others in the simplex family
** Tetrahedron - {3,3}
** 5-cell (pentachoron) - {3,3,3}
** 5-simplex hexateron - {3,3,3,3}
** 6-simplex - {3,3,3,3,3}
** 7-simplex - {3,3,3,3,3,3}
** 8-simplex - {3,3,3,3,3,3,3}
** 9-simplex - {3,3,3,3,3,3,3,3}
** 10-simplex - {3,3,3,3,3,3,3,3,3}

References

* T. Gosset: "On the Regular and Semi-Regular Figures in Space of n Dimensions", Messenger of Mathematics, Macmillan, 1900
* Norman Johnson "Uniform Polytopes", Manuscript (1991)
* Richard Klitzing 5D quasiregulars, (multi)prisms, non-prismatic Wythoffian polyterons

External links

* [http://members.aol.com/Polycell/glossary.html#simplex Glossary for hyperspace]
* [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions] , Jonathan Bowers
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]