7-cube

From Wikipedia, the free encyclopedia
Jump to: navigation, search
7-cube
Hepteract
7-cube t0.svg
Orthogonal projection
inside Petrie polygon
The central orange vertex is doubled
Type Regular 7-polytope
Family hypercube
Schläfli symbol {4,35}
Coxeter-Dynkin diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-faces 14 {4,34} 6-cube graph.svg
5-faces 84 {4,33} 5-cube graph.svg
4-faces 280 {4,3,3} 4-cube graph.svg
Cells 560 {4,3} 3-cube graph.svg
Faces 672 {4} 2-cube.svg
Edges 448
Vertices 128
Vertex figure 6-simplex 6-simplex graph.svg
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, 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[edit]

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.

Cartesian coordinates[edit]

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[edit]

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph 7-cube t0.svg 7-cube t0 B6.svg 7-cube t0 B5.svg
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph 7-cube t0 B4.svg 7-cube t0 B3.svg 7-cube t0 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph 7-cube t0 A5.svg 7-cube t0 A3.svg
Dihedral symmetry [6] [4]

Projections[edit]

7-cube column graph.svg
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.
7-cubePetrie.svg
Petrie polygon, skew orthographic projection
Hepteract graph.svg
Another orthogonal projection


Hepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D.

References[edit]

  • 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)
  • Richard Klitzing, 7D uniform polytopes (polyexa), o3o3o3o3o3o4x - hept

External links[edit]