# 10-cube

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10-cube
Dekeract

Orthogonal projection
inside Petrie polygon
Orange vertices are doubled, and central yellow one has four
TypeRegular 10-polytope e
Family hypercube
Schläfli symbol {4,38}
Coxeter-Dynkin diagram
9-faces20 {4,37}
8-faces180 {4,36}
7-faces960 {4,35}
6-faces3360 {4,34}
5-faces8064 {4,33}
4-faces13440 {4,3,3}
Cells15360 {4,3}
Faces11520 squares
Edges5120 segments
Vertices1024 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.

## Contents

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

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

 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.
Orthographic projections
B10B9B8
[20][18][16]
B7B6B5
[14][12][10]
B4B3B2
[8][6][4]
A9A5
[10][6]
A7A3
[8][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 demienneractic and 512 enneazettonic facets.

## Related Research Articles

In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells. The tesseract is one of the six convex regular 4-polytopes.

In four-dimensional geometry, a cantellated tesseract is a convex uniform 4-polytope, being a cantellation of the regular tesseract.

In geometry, a truncated tesseract is a uniform 4-polytope formed as the truncation of the regular tesseract.

In geometry, the rectified tesseract, rectified 8-cell is a uniform 4-polytope bounded by 24 cells: 8 cuboctahedra, and 16 tetrahedra. It has half the vertices of a runcinated tesseract, with its construction, called a runcic tesseract.

In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.

In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.

In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices removed.

In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.

In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces.

In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell 4-faces, and 64 5-faces.

In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces.

In geometry, an 8-cube is an eight-dimensional hypercube. 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.

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.

In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells 4-faces, 1792 5-faces, 1024 6-faces, and 256 7-faces.

In geometry, a 9-orthoplex or 9-cross polytope, is a regular 9-polytope with 18 vertices, 144 edges, 672 triangle faces, 2016 tetrahedron cells, 4032 5-cells 4-faces, 5376 5-simplex 5-faces, 4608 6-simplex 6-faces, 2304 7-simplex 7-faces, and 512 8-simplex 8-faces.

In geometry, a 10-orthoplex or 10-cross polytope, is a regular 10-polytope with 20 vertices, 180 edges, 960 triangle faces, 3360 octahedron cells, 8064 5-cells 4-faces, 13440 5-faces, 15360 6-faces, 11520 7-faces, 5120 8-faces, and 1024 9-faces.

In geometry of five dimensions or higher, a cantic 5-cube, cantihalf 5-cube, truncated 5-demicube is a uniform 5-polytope, being a truncation of the 5-demicube. It has half the vertices of a cantellated 5-cube.

In six-dimensional geometry, a runcic 5-cube or is a convex uniform 5-polytope. There are 2 runcic forms for the 5-cube. Runcic 5-cubes have half the vertices of runcinated 5-cubes.

In seven-dimensional geometry, a cantic 7-cube or truncated 7-demicube as a uniform 7-polytope, being a truncation of the 7-demicube.

## 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, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, 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".
Family An Bn I2(p) / Dn E6 / E7 / E8 / / Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds