Octeract
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Octeract (8-cube)
Orthogonal projection inside Petrie polygon
Type	Regular 8-polytope
Family	hypercube
Schläfli symbol	{4,36}
Coxeter-Dynkin diagram
7-faces	16 {4,35}
6-faces	112 {4,34}
5-faces	448 {4,33}
4-faces	1120 {4,32}
Cells	1792 {4,3}
Faces	1792 {4}
Edges	1024
Vertices	256
Vertex figure	7-simplex
Petrie polygon	hexadecagon
Coxeter group	C8, [36,4]
Dual	Octacross
Properties	convex

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

This 8-cube 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 1:8:28:56:70:56:28:8:1.

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

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

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