7-cube

7-cube Hepteract
Orthogonal projection inside Petrie polygon The central orange vertex is doubled
Type	Regular 7-polytope
Family	hypercube
Schläfli symbol	{4,35}
Coxeter-Dynkin diagrams	Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD
6-faces	14 {4,34}
5-faces	84 {4,33}
4-faces	280 {4,3,3}
Cells	560 {4,3}
Faces	672 {4}
Edges	448
Vertices	128
Vertex figure	6-simplex
Petrie polygon	tetradecagon
Coxeter group	C7, [35,4]
Dual	7-orthoplex
Properties	convex, Hanner polytope

In geometry, a 7-cube 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.

It can be named by its Schläfli symbol {4,3⁵}, being composed of 3 6-cubes around each 5-face. It can be called a hepteract, a portmanteau of tesseract (the 4-cube) and hepta for seven (dimensions) in Greek. It can also be called a regular tetradeca-7-tope or tetradecaexon, being a 7 dimensional polytope constructed from 14 regular facets.

As a configuration

This configuration matrix represents the 7-cube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.Template:Sfnp

$\left[\begin{array}{c}\begin{array}{ccccccc}128& 7& 21& 35& 35& 21& 7\\ 2& 448& 6& 15& 20& 15& 6\\ 4& 4& 672& 5& 10& 10& 5\\ 8& 12& 6& 560& 4& 6& 4\\ 16& 32& 24& 8& 280& 3& 3\\ 32& 80& 80& 40& 10& 84& 2\\ 64& 192& 240& 160& 60& 12& 14\end{array}\end{array}\right]$

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. Template:-

Projections

This hypercube graph is an orthogonal projection. This orientation 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 numbers of vertices in each column are a row of Pascal's triangle: 1, 7, 21, 35, 35, 21, 7, 1.

Related polytopes

The 7-cube is 7th in a series of hypercube: Template:Hypercube polytopes

The dual of a 7-cube is called a 7-orthoplex, 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 6-simplex 6-faces.

References

• H.S.M. Coxeter:
  • H.S.M. Coxeter, Regular Polytopes, 1973, 3rd edition, Dover, New York, pp. 294–295, Table I (iii): Regular Polytopes, three regular polytopes in n dimensions (n ≥ 5), Template:Isbn
  • Template:Cite book
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, Template:Isbn
    • (Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• Template:KlitzingPolytopes

External links

• Template:MathWorld
• Template:MathWorld
• Template:GlossaryForHyperspace
• Multi-dimensional Glossary: hypercube Garrett Jones
• Rotation of 7D-Cube www.4d-screen.de