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                                                                                           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 Coxeter group C7, [35,4]
Dual 7-orthoplex
Properties convex

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

It is a part of an infinite family of polytopes, called hypercubes. 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.

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

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 < xi < 1.

Images 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.