Hepteract

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Hepteract 7-cube
Vertex-Edge graph.
Type	Regular 7-polytope
Family	hypercube
Schläfli symbol	{4,3,3,3,3,3}
Coxeter-Dynkin diagram
6-faces	14 hexeracts
5-faces	84 penteracts
4-faces	280 tesseracts
Cells	560 cubes
Faces	672 squares
Edges	448
Vertices	128
Vertex figure|6-simplex
Symmetry group	B7, [3,3,3,3,3,4]
Dual	Heptacross
Properties	convex

A hepteract is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces.

The name hepteract is derived from combining the name tesseract (the 4-cube) with hepta for seven (dimensions) in Greek.

It can also be called a regular tetradeca-7-tope or tetradecaexon, being made of 14 regular facets.

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

Applying an alternation operation, deleting alternating vertices of the hepteract, creates another uniform polytope, called a demihepteract, (part of an infinite family called demihypercubes), which has 14 demihexeractic and 64 heptaexonic 6-faces.

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

[edit] Cartesian coordinates

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

(±1,±1,±1,±1,±1,±1,±1)

while the interior of the same consists of all points (x₀, x₁, x₂, x₃, x₄, x₅, x₆) 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

• 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