10-cube

10-cube Dekeract
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
9-faces	20 {4,37}
8-faces	180 {4,36}
7-faces	960 {4,35}
6-faces	3360 {4,34}
5-faces	8064 {4,33}
4-faces	13440 {4,3,3}
Cells	15360 {4,3}
Faces	11520 squares
Edges	5120
Vertices	1024
Vertex figure	9-simplex
Petrie polygon	icosagon
Coxeter group	C10, [38,4]
Dual	10-orthoplex
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, the name derived from combining the name tesseract (the 4-cube) with 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 an dekeract can be called a 10-orthoplex or decacross, and is a part of the infinite family of cross-polytopes.

Cartesian coordinates

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 < x_i < 1.

Other images

This 10-cube graph is an orthogonal projection. This oriention 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.

Petrie polygon, skew orthogonal projection

orthographic projections

B10	B9	B8
[20]	[18]	[16]
B7	B6	B5
[14]	[12]	[10]
B4	B3	B2
[8]	[6]	[4]

Derived polytopes

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

• 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, editied 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

• Weisstein, Eric W., "Hypercube" from MathWorld.
• Olshevsky, George, Measure polytope at Glossary for Hyperspace.
• Multi-dimensional Glossary: hypercube Garrett Jones
• Sloane's A135289 : Hypercubes:10-cube. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.