9-cube

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9-cube
Enneract
9-cube.svg
Orthogonal projection
inside Petrie polygon
Orange vertices are doubled, yellow have 4, and the green center has 8
Type Regular 9-polytope
Family hypercube
Schläfli symbol {4,37}
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.pngCDel 3.pngCDel node.png
8-faces 18 {4,36}8-cube.svg
7-faces 144 {4,35}7-cube graph.svg
6-faces 672 {4,34}6-cube graph.svg
5-faces 2016 {4,33}5-cube graph.svg
4-faces 4032 {4,3,3}4-cube graph.svg
Cells 5376 {4,3}3-cube graph.svg
Faces 4608 {4}2-cube.svg
Edges 2304
Vertices 512
Vertex figure 8-simplex 8-simplex graph.svg
Petrie polygon octadecagon
Coxeter group C9, [37,4]
Dual 9-orthoplex 9-orthoplex.svg
Properties convex

In geometry, a 9-cube is a nine-dimensional hypercube with 512 vertices, 2304 edges, 4608 square faces, 5376 cubic cells, 4032 tesseract 4-faces, 2016 5-cube 5-faces, 672 6-cube 6-faces, 144 7-cube 7-faces, and 18 8-cube 8-faces.

It can be named by its Schläfli symbol {4,37}, being composed of three 8-cubes around each 7-face. It is also called an enneract, a portmanteau of tesseract (the 4-cube) and enne for nine (dimensions) in Greek. It can also be called a regular octadeca-9-tope or octadecayotton, as a nine-dimensional polytope constructed with 18 regular facets.

It is a part of an infinite family of polytopes, called hypercubes. The dual of a 9-cube can be called a 9-orthoplex, and is a part of the infinite family of cross-polytopes.

Cartesian coordinates[edit]

Cartesian coordinates for the vertices of a 9-cube centered at the origin and edge length 2 are

(±1,±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, x8) with −1 < xi < 1.

Projections[edit]

9-cube column graph.svg
This 9-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 1:9:36:84:126:126:84:36:9:1.
Enneract ortho petrie.svg
Petrie polygon, skew orthographic projection

Images[edit]

orthographic projections
B9 B8 B7
9-cube t0.svg 9-cube t0 B8.svg 9-cube t0 B7.svg
[18] [16] [14]
B6 B5
9-cube t0 B6.svg 9-cube t0 B5.svg
[12] [10]
B4 B3 B2
4-cube t0.svg 4-cube t0 B3.svg 9-cube t0 B2.svg
[8] [6] [4]

Derived polytopes[edit]

Applying an alternation operation, deleting alternating vertices of the 9-cube, creates another uniform polytope, called a 9-demicube, (part of an infinite family called demihypercubes), which has 18 8-demicube and 256 8-simplex facets.

Notes[edit]

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, 9D uniform polytopes (polyyotta), o3o3o3o3o3o3o3o4x - enne

External links[edit]