Ten-dimensional space

In mathematics, a sequence of n real numbers can be understood as a location in n-dimensional space. When n = 10, the set of all such locations is called 10-dimensional space. Often such spaces are studied as vector spaces, without any notion of distance. Ten-dimensional Euclidean space is ten-dimensional space equipped with a Euclidean metric, which is defined by the dot product.^{[Dubious – discuss]}

More generally the term may refer to a ten-dimensional vector space over any field, such as an ten-dimensional complex vector space, which has 20 real dimensions. It may also refer to an Ten-dimensional manifold such as an 10-sphere, or a variety of other geometric constructions.

Geometry

10-polytope

A polytope in ten dimensions is called a 10-polytope. The most studied are the regular polytopes, of which there are only three in ten dimensions: the 10-simplex, 10-cube, and 10-orthoplex. A wider family are the uniform 10-polytopes, constructed from fundamental symmetry domains of reflection, each domain defined by a Coxeter group. Each uniform polytope is defined by a ringed Coxeter-Dynkin diagram. The 10-demicube is a unique polytope from the D₁₀ family.

Regular and uniform polytopes in ten dimensions
(Displayed as orthogonal projections in each Coxeter plane of symmetry)

A10			B10						D10
10-simplex			10-cube			10-orthoplex			10-demicube

References

• H.S.M. Coxeter:
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• 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 Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter
  • (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]
• Table of the Highest Kissing Numbers Presently Known maintained by Gabriele Nebe and Neil Sloane (lower bounds)
• . (Review).
• (Second printing)

  Dimension
  Dimensional spaces	Vector space Euclidean space Affine space Projective space Free module Manifold Algebraic variety Spacetime
  Other dimensions	Krull Homological Lebesgue covering Inductive Hausdorff Minkowski Fractal Degrees of freedom
  Polytopes and shapes	Hyperplane Hypersurface Hypercube Hypersphere Hyperrectangle Demihypercube Cross-polytope Simplex
  Dimensions by number	Zero One Two Three Four Five Six (degrees of freedom) Seven Eight Nine Ten n-dimensions Negative dimensions
  Category