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.^{[dubious – discuss]}

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 an nine-dimensional manifold such as a 9-sphere, or any of a variety of other geometric constructions.

Geometry

9-polytope

A polytope in nine dimensions is called an 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
9-simplex			9-cube			9-orthoplex			9-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