Nine-dimensional space

In mathematics, a sequence of n real numbers can be understood as a point in n-dimensional space. When n = 9, the set of all such locations is called 9-dimensional space. Often such spaces are studied as vector spaces, without any notion of distance. Nine-dimensional Euclidean space is nine-dimensional space equipped with a Euclidean metric, which is defined by the dot product.

More generally, the term may refer to a nine-dimensional vector space over any field, such as a nine-dimensional complex vector space, which has 18 real dimensions. It may also refer to a nine-dimensional manifold such as a 9-sphere, or any of a variety of other geometric constructions.



A polytope in nine dimensions is called a 9-polytope. The most studied are the regular polytopes, of which there are only three in nine dimensions: the 9-simplex, 9-cube, and 9-orthoplex. A broader family are the uniform 9-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 9-demicube is a unique polytope from the D9 family.

Regular and uniform polytopes in nine dimensions
(Displayed as orthogonal projections in each Coxeter plane of symmetry)
A9 B9 D9






  • 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).
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