7cube
7cube Hepteract  

Orthogonal projection inside Petrie polygon The central orange vertex is doubled  
Type  Regular 7polytope 
Family  hypercube 
Schläfli symbol  {4,3^{5}} 
CoxeterDynkin diagrams 

6faces  14 {4,3^{4}} 
5faces  84 {4,3^{3}} 
4faces  280 {4,3,3} 
Cells  560 {4,3} 
Faces  672 {4} 
Edges  448 
Vertices  128 
Vertex figure  6simplex 
Petrie polygon  tetradecagon 
Coxeter group  C_{7}, [3^{5},4] 
Dual  7orthoplex 
Properties  convex, Hanner polytope 
In geometry, a 7cube is a sevendimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4faces, 84 penteract 5faces, and 14 hexeract 6faces.
It can be named by its Schläfli symbol {4,3^{5}}, being composed of 3 6cubes around each 5face. It can be called a hepteract, a portmanteau of tesseract (the 4cube) and hepta for seven (dimensions) in Greek. It can also be called a regular tetradeca7tope or tetradecaexon, being a 7 dimensional polytope constructed from 14 regular facets.
Related polytopes
The 7cube is 7th in a series of hypercube:
The dual of a 7cube is called a 7orthoplex, and is a part of the infinite family of crosspolytopes.
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 6simplex 6faces.
As a configuration
This configuration matrix represents the 7cube. The rows and columns correspond to vertices, edges, faces, cells, 4faces, 5faces and 6faces. The diagonal numbers say how many of each element occur in the whole 7cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^{[1]}^{[2]}
[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 (x_{0}, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}) with 1 < x_{i} < 1.
Projections
This hypercube graph is an orthogonal projection. This orientation shows columns of vertices positioned a vertexedgevertex 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. 
Coxeter plane  B_{7} / A_{6}  B_{6} / D_{7}  B_{5} / D_{6} / A_{4} 

Graph  
Dihedral symmetry  [14]  [12]  [10] 
Coxeter plane  B_{4} / D_{5}  B_{3} / D_{4} / A_{2}  B_{2} / D_{3} 
Graph  
Dihedral symmetry  [8]  [6]  [4] 
Coxeter plane  A_{5}  A_{3}  
Graph  
Dihedral symmetry  [6]  [4] 
References
 H.S.M. Coxeter:
 Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN:0486614808, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in ndimensions (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 ndimensions (n≥5)
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN:9780471010036 [1]
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10]
 (Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559591]
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
 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.
 Multidimensional Glossary: hypercube Garrett Jones
 Rotation of 7DCube www.4dscreen.de
Original source: https://en.wikipedia.org/wiki/7cube.
Read more 