10-cube

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10-cube
Dekeract
10-cube.svg
Orthogonal projection
inside Petrie polygon
Orange vertices are doubled, and central yellow one has four
Type Regular 10-polytope
Family hypercube
Schläfli symbol {4,38}
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
9-faces 20 {4,37}9-cube.svg
8-faces 180 {4,36}8-cube.svg
7-faces 960 {4,35}7-cube graph.svg
6-faces 3360 {4,34}6-cube graph.svg
5-faces 8064 {4,33}5-cube graph.svg
4-faces 13440 {4,3,3}4-cube graph.svg
Cells 15360 {4,3} 3-cube graph.svg
Faces 11520 squares 2-cube.svg
Edges 5120
Vertices 1024
Vertex figure 9-simplex 9-simplex graph.svg
Petrie polygon icosagon
Coxeter group C10, [38,4]
Dual 10-orthoplex 10-orthoplex.svg
Properties convex

In geometry, a 10-cube is a ten-dimensional hypercube. It has 1024 vertices, 5120 edges, 11520 square faces, 15360 cubic cells, 13440 tesseract 4-faces, 8064 5-cube 5-faces, 3360 6-cube 6-faces, 960 7-cube 7-faces, 180 8-cube 8-faces, and 20 9-cube 9-faces.

It can be named by its Schläfli symbol {4,38}, being composed of 3 9-cubes around each 8-face. It is sometimes called a dekeract, a portmanteau of tesseract (the 4-cube) and deka- for ten (dimensions) in Greek, It can also be called an icosaxennon or icosa-10-tope as a 10 dimensional polytope, constructed from 20 regular facets.

It is a part of an infinite family of polytopes, called hypercubes. The dual of a dekeract can be called a 10-orthoplex or decacross, and is a part of the infinite family of cross-polytopes.

Cartesian coordinates[edit]

Cartesian coordinates for the vertices of a dekeract centered at the origin and edge length 2 are

(±1,±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, x9) with −1 < xi < 1.

Other images[edit]

10-cube column graph.svg
This 10-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 1:10:45:120:210:252:210:120:45:10:1.
10cube ortho polygon.svg
Petrie polygon, skew orthogonal projection
Orthographic projections
B10 B9 B8
10-cube t0.svg 10-cube t0 B9.svg 10-cube t0 B8.svg
[20] [18] [16]
B7 B6 B5
10-cube t0 B7.svg 10-cube t0 B6.svg 10-cube t0 B5.svg
[14] [12] [10]
B4 B3 B2
10-cube t0 B4.svg 10-cube t0 B3.svg 10-cube t0 B2.svg
[8] [6] [4]

Derived polytopes[edit]

Applying an alternation operation, deleting alternating vertices of the dekeract, creates another uniform polytope, called a 10-demicube, (part of an infinite family called demihypercubes), which has 20 demiocteractic and 512 enneazettonic 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)
    • 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, 10D uniform polytopes (polyxenna), o3o3o3o3o3o3o3o3o4x - deker

External links[edit]