| 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 diagram | |
| 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 |
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, 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 a 7 dimensional polytope constructed from 14 regular facets.
Contents |
It is a part of an infinite family of polytopes, called hypercubes. The dual of a Hepteract can be 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.
Cartesian coordinates for the vertices of a hepteract centered at the origin and edge length 2 are
while the interior of the same consists of all points (x0, x1, x2, x3, x4, x5, x6) with -1 < xi < 1.
| 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] |
This hypercube graph is an orthogonal projection. This oriention 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. |
Petrie polygon, skew orthographic projection |
Another orthogonal projection |
Hepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D.