10-cube

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10-cube
Orthogonal projection
Type	Regular 10-polytope
Family	hypercube
Schläfli symbol	{4,3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
9-faces	20 enneracts
8-faces	180 octeracts
7-faces	960 hepteracts
6-faces	3360 hexeracts
5-faces	8064 penteracts
4-faces	13440 tesseracts
Cells	15360 cubes
Faces	11520 squares
Edges	5120
Vertices	1024
Vertex figure	9-simplex
Symmetry group	B10, [3,3,3,3,3,3,3,3,4]
Dual	Decacross
Properties	convex

A 10-cube is a ten-dimensional hypercube with 1024 vertices, 5120 edges, 11520 square faces, 15360 cubic cells, 13440 tesseract 4-faces, 13440 penteract 5-faces, 3360 hexeract 6-faces, 960 hepteract 7-faces, and 180 octeract 8-faces, and 20 enneract 9-faces.

It can also be called a regular icosa-10-tope, being made of 20 regular facets.

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

Contents 1 Cartesian coordinates 2 Derived polytopes 3 See also 4 References 5 External links

[edit] Cartesian coordinates

Cartesian coordinates for the vertices of a penteract 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.

[edit] Derived polytopes

Applying an alternation operation, deleting alternating vertices of the enneract, creates another uniform polytope, called a 10-demihypercube, (part of an infinite family called demihypercubes), which has 20 demiocteractic and 512 enneazettonic facets.

[edit] See also

• Hypercubes family
  • Square - {4}
  • Cube - {4,3}
  • Tesseract - {4,3,3}
  • Penteract - {4,3,3,3}
  • Hexeract - {4,3,3,3,3}
  • Hepteract - {4,3,3,3,3,3}
  • Octeract - {4,3,3,3,3,3,3}
  • Enneract - {4,3,3,3,3,3,3,3}
  • 10-cube - {4,3,3,3,3,3,3,3,3}
  • ...

[edit] References

• Coxeter, H.S.M. 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)

[edit] External links

• Eric W. Weisstein, Hypercube at MathWorld.
• Olshevsky, George, Measure polytope at Glossary for Hyperspace.
• Multi-dimensional Glossary: hypercube Garrett Jones
• OEIS A135289 [[1]]