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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. 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. 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. 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.

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Related concepts (5)

7-orthoplex

In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cells 4-faces, 448 5-faces, and 128 6-faces. It has two constructed forms, the first being regular with Schläfli symbol {35,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,3,31,1} or Coxeter symbol 411. It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes.

Uniform 7-polytope

In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets. A uniform 7-polytope is one whose symmetry group is transitive on vertices and whose facets are uniform 6-polytopes. Regular 7-polytopes are represented by the Schläfli symbol {p,q,r,s,t,u} with u {p,q,r,s,t} 6-polytopes facets around each 4-face. There are exactly three such convex regular 7-polytopes: {3,3,3,3,3,3} - 7-simplex {4,3,3,3,3,3} - 7-cube {3,3,3,3,3,4} - 7-orthoplex There are no nonconvex regular 7-polytopes.

Demihypercube

In geometry, demihypercubes (also called n-demicubes, n-hemicubes, and half measure polytopes) are a class of n-polytopes constructed from alternation of an n-hypercube, labeled as hγn for being half of the hypercube family, γn. Half of the vertices are deleted and new facets are formed. The 2n facets become 2n (n−1)-demicubes, and 2n (n−1)-simplex facets are formed in place of the deleted vertices. They have been named with a demi- prefix to each hypercube name: demicube, demitesseract, etc.