8-cube

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8-cube
Octeract
8-cube.svg
Orthogonal projection
inside Petrie polygon
Type Regular 8-polytope
Family hypercube
Schläfli symbol {4,36}
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.pngCDel 3.pngCDel node.png
7-faces 16 {4,35}7-cube graph.svg
6-faces 112 {4,34}6-cube graph.svg
5-faces 448 {4,33}5-cube graph.svg
4-faces 1120 {4,32}4-cube graph.svg
Cells 1792 {4,3}3-cube.png
Faces 1792 {4}2-cube.svg
Edges 1024
Vertices 256
Vertex figure 7-simplex 7-simplex graph.svg
Petrie polygon hexadecagon
Coxeter group C8, [36,4]
Dual 8-orthoplex 8-orthoplex.svg
Properties convex

In geometry, an 8-cube is an eight-dimensional hypercube (8-cube). It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 5-cube 5-faces, 112 6-cube 6-faces, and 16 7-cube 7-faces.

It is represented by Schläfli symbol {4,36}, being composed of 3 7-cubes around each 6-face. It is called an octeract, a portmanteau of tesseract (the 4-cube) and oct for eight (dimensions) in Greek. It can also be called a regular hexdeca-8-tope or hexadecazetton, being an 8-dimensional polytope constructed from 16 regular facets.

Related polytopes[edit]

It is a part of an infinite family of polytopes, called hypercubes. The dual of an 8-cube can be called a 8-orthoplex, and is a part of the infinite family of cross-polytopes.

Cartesian coordinates[edit]

Cartesian coordinates for the vertices of an 8-cube centered at the origin and edge length 2 are

(±1,±1,±1,±1,±1,±1,±1,±1)

while the interior of the same consists of all points (x0, x1, x2, x3, x4, x5, x6, x7) with -1 < xi < 1.

Projections[edit]

orthographic projections
B8 B7
8-cube t0.svg 8-cube t0 B7.svg
[16] [14]
B6 B5
8-cube t0 B6.svg 8-cube t0 B5.svg
[12] [10]
B4 B3 B2
4-cube t0.svg 8-cube t0 B3.svg 8-cube t0 B2.svg
[8] [6] [4]
A7 A5 A3
8-cube t0 A7.svg 8-cube t0 A5.svg 8-cube t0 A3.svg
[8] [6] [4]
8-cube column graph.svg
This 8-cube 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 1:8:28:56:70:56:28:8:1.

Derived polytopes[edit]

Applying an alternation operation, deleting alternating vertices of the hepteract, creates another uniform polytope, called a 8-demicube, (part of an infinite family called demihypercubes), which has 16 demihepteractic and 128 8-simplex facets.

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)
    • 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, 8D uniform polytopes (polyzetta), o3o3o3o3o3o3o4x - octo

External links[edit]