Hexateron

Regular hexateron (5-simplex)
Orthogonal projection inside Petrie polygon
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}
Petrie polygon	hexagon
Schläfli symbol	{3,3,3,3}
Coxeter-Dynkin diagram
Coxeter group	A5 [3,3,3,3]
Dual	Self-dual
Properties	convex

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 having six facets and teron (with ter- being a corruption of tetra-) for having four-dimensional facets.

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.

The Cartesian coordinates for the vertices of an origin-centered hexateron having edge length 2 are:

Projected images

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

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}
    • Omnitruncated 5-simplex - t_0,1,2,3,4{3⁴}
  • 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

• Glossary for hyperspace, George Olshevsky.
• Polytopes of Various Dimensions, Jonathan Bowers
• Multi-dimensional Glossary

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