10-cube

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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 segments
Vertices	1024 points
Vertex figure	9-simplex
Petrie polygon	icosagon
Coxeter group	C10, [38,4]
Dual	10-orthoplex
Properties	convex, Hanner polytope

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 icosaronnon 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 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.

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 numbers of vertices in each column are a row of Pascal's triangle: 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1.

Orthographic projections

B10	B9	B8
[20]	[18]	[16]
B7	B6	B5
[14]	[12]	[10]
B4	B3	B2
[8]	[6]	[4]
A9		A5
[10]		[6]
A7		A3
[8]		[4]

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 demienneractic and 512 enneazettonic facets.

• H.S.M. Coxeter:
  • H.S.M. Coxeter, Regular Polytopes, 1973, 3rd edition, Dover, New York, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n dimensions (n ≥ 5), ISBN 0-486-61480-8
  • 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, wiley.com, ISBN 978-0-471-01003-6
    • (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)
• Klitzing, Richard. "10D uniform polytopes (polyxenna) o3o3o3o3o3o3o3o3o4x - deker".

• Weisstein, Eric W. "Hypercube". MathWorld.
• Olshevsky, George. "Measure polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• Multi-dimensional Glossary: hypercube Garrett Jones
• OEIS sequence A135289 (Hypercubes:10-cube)