Hexeract

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Hexeract 6-cube
Vertex-Edge graph.
Type	Regular 6-polytope
Family	hypercube
Schläfli symbol	{4,3,3,3,3}
Coxeter-Dynkin diagram
Facets	12 penteracts
Hypercells	60 tesseracts
Cells	160 cubes
Faces	240 squares
Edges	192
Vertices	64
Vertex figure	5-simplex
Symmetry group	B6, [3,3,3,3,4]
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.

Contents 1 Cartesian coordinates 2 Projections 3 See also 4 References 5 External links

[edit] 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.

[edit] Projections

An orthogonal projection viewed along the axes of two opposite vertices and the average plane of one edge path between.

[edit] 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}
  • ...

[edit] 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)

[edit] External links

• Eric W. Weisstein, Hypercube at MathWorld.
• Olshevsky, George, Measure polytope at Glossary for Hyperspace.
• Multi-dimensional Glossary: hypercube Garrett Jones