# 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.

## Geometry[]

### 9-polytope[]

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 9-simplex                  9-cube                  9-orthoplex                  9-demicube               