Octeract

An octeract is an eight-dimensional hypercube with 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 penteract 5-faces, 112 hexeract 6-faces, and 16 hepteract 7-faces.

The name "octeract" is derived from combining the name tesseract (the "4-cube") with "oct" for eight (dimensions) in Greek.

It can also be called a regular hexdeca-8-tope or hexadecazetton, being made of 16 regular facets.

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

Cartesian coordinates

Cartesian coordinates for the vertices of a penteract centered at the origin and edge length 2 are: (±1,±1,±1,±1,±1,±1,±1,±1)while the interior of the same consists of all points (x₀, x₁, x₂, x₃, x₄, x₅, x₆, x₇) with -1 < x_i < 1.

Projections

Derived polytopes

Applying an "alternation" operation, deleting alternating vertices of the hepteract, creates another uniform polytope, called a "demiocteract", (part of an infinite family called demihypercubes), which has 16 demihepteractic and 128 8-simplex facets.

See also

*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

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