Hexeract (6-cube)
Orthogonal projection inside Petrie polygon
Type	Regular 6-polytope
Family	hypercube
Schläfli symbol	{4,34}
Coxeter-Dynkin diagram
Facets	12 penteracts
Hypercells	60 tesseracts
Cells	160 cubes
Faces	240 squares
Edges	192
Vertices	64
Vertex figure	5-simplex
Petrie polygon	dodecagon
Coxeter group	C6, [34,4]
Dual	Hexacross
Properties	convex

A hexeract is a name for a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 penteract 5-faces.

The name hexeract is derived from combining the name tesseract (the 4-cube) with hex for six (dimensions) in Greek.

It can also be called a regular dodeca-6-tope or dodecapeton, being made of 12 regular facets.

It is a part of an infinite family of polytopes, called hypercubes. The dual of a penteract can be called a hexacross, and is a part of the infinite family of cross-polytopes.

Applying an alternation operation, deleting alternating vertices of the hexeract, creates another uniform polytope, called a demihexeract, (part of an infinite family called demihypercubes), which has 12 demipenteractic and 32 hexateronic facets.

Cartesian coordinates

Cartesian coordinates for the vertices of a hexeract centered at the origin and edge length 2 are

(±1,±1,±1,±1,±1,±1)

while the interior of the same consists of all points (x₀, x₁, x₂, x₃, x₄, x₅) with -1 < x_i < 1.

Projections

This hypercube graph is an orthogonal projection. This oriention shows columns of vertices positioned a vertex-edge-vertex distance from one vertex on the left to one vertex on the right, and edges attaching adjacent columns of vertices. The number of vertices in each column represents rows in Pascal's triangle, being 1:6:15:20:15:6:1.

Another orthogonal projection

See also

• Other Regular 6-polytopes:
  • Heptapeton (6-simplex) - {3,3,3,3,3}
  • Hexacross (6-Cross polytope) - {3,3,3,3,4}
• Others in the Hypercubes family
  • Square - {4}
  • Cube - {4,3}
  • Tesseract - {4,3,3}
  • Penteract - {4,3,3,3}
  • Hexeract - {4,3,3,3,3}
  • Hepteract - {4,3,3,3,3,3}
  • Octeract - {4,3,3,3,3,3,3}
  • Enneract - {4,3,3,3,3,3,3,3}
  • 10-cube - {4,3,3,3,3,3,3,3,3}
  • ...

References

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n>=5)

External links

• Multi-dimensional Glossary: hypercube Garrett Jones

it:Esseratto