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User:LittleWhole/Ten-dimensional space

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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

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10-polytope

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Main article: Uniform 10-polytope

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-simplex10-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, and 621261, and 162 polytopes from the E10 family.

Regular and uniform polytopes in eight dimensions
(Displayed as orthogonal projections in each Coxeter plane of symmetry)
A8 B8 D8
altN=8-simplex
10-simplex
altN=8-cube
10-cube
altN=8-orthoplex
10-orthoplex

10-demicube
E10
File:6 21 t0 E8.svg
621
File:2 61 t0 E8.svg
261
File:Gosset 1 62 polytope petrie.svg
162

References

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  • 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)

Category:Dimension Category:Mathematics Category:Multi-dimensional geometry Category:Ten-dimensional geometry