7-cube

Short description: 7-dimensional hypercube

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
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.

Related polytopes

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

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.

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.

[math]\displaystyle{ \begin{bmatrix}\begin{matrix} 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{matrix}\end{bmatrix} }[/math]

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.

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 number of vertices in each column represents rows in Pascal's triangle, being 1:7:21:35:35:21:7:1.

orthographic projections

Coxeter plane	B7 / A6	B6 / D7	B5 / D6 / A4
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

References

• H.S.M. Coxeter:
  • Coxeter, 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)
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN:978-0-471-01003-6 [1]
    • (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)
• Klitzing, Richard. "7D uniform polytopes (polyexa) o3o3o3o3o3o4x - hept". https://bendwavy.org/klitzing/dimensions/polyexa.htm. 

External links

• Weisstein, Eric W.. "Hypercube". http://mathworld.wolfram.com/Hypercube.html. 
• Weisstein, Eric W.. "Hypercube graph". http://mathworld.wolfram.com/HypercubeGraph.html. 
• Olshevsky, George. "Measure polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007. https://web.archive.org/web/20070204075028/members.aol.com/Polycell/glossary.html#Measure. 
• Multi-dimensional Glossary: hypercube Garrett Jones
• Rotation of 7D-Cube www.4d-screen.de