# Penteract

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Penteract

In five dimensional geometry, a penteract is a name for a five dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract hypercells.

The name "penteract" is derived from combining the name tesseract (the "4-cube") with "pente" for five (dimensions) in Greek.

It can also be called a regular deca-5-tope or decateron, being made of 10 regular facets.

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

Applying an "alternation" operation, deleting alternating vertices of the penteract, creates another uniform polytope, called a demipenteract, which is also part of an infinite family called the demihypercubes.

Cartesian coordinates

Cartesian coordinates for the vertices of a penteract centered at the origin and edge length 2 are: (±1,±1,±1,±1,±1)while the interior of the same consists of all points (x0, x1, x2, x3, x4) with -1 < xi < 1.

Projections

* Other Regular 5-polytopes:
** 5-simplex (hexateron) - {3,3,3,3}
** 5-orthoplex (pentacross) - {3,3,3,4}
** 5-demicube (demipenteract) - {31,2,1}
* Others in the hypercube 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)

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*GlossaryForHyperspace | anchor=Measure | title=Measure polytope
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary: hypercube] Garrett Jones

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