# Space

In the most general sense possible, a **space** is any set, and its members are called *points*. In the study of polytopes, a space, sometimes **containing space** or **ambient space**, is an environment where a polytope realization resides. A polytope realization maps each vertex of an abstract polytope to a point. If the interior of the polytope is defined and meaningful, it is a set of points.

Spaces that are found in polytope studies are almost always equipped with a metric, which allows for isometries and therefore symmetries to be defined.

## Examples[edit | edit source]

### Euclidean spaces[edit | edit source]

Euclidean spaces generalize the three-dimensional space defined in Euclid's *Elements*, which is a framework that describes the properties of everyday geometry, and is naturally represented with the Cartesian coordinate system.

Classical polytope studies concern *n*-polytopes in *n*-dimensional Euclidean space, and have a recursive property that every element of rank *r* is realized in an *r*-dimensional affine subspace. Euclidean tilings are *n*-polytopes in (*n* - 1)-dimensional Euclidean space. An *n*-polytope that requires at least (*n* + 1)-dimensional Euclidean space for a realization is one definition of a skew polytope.

### Hyperspheres[edit | edit source]

Hyperspheres are a type of non-Euclidean space that generalize the circle and sphere. A spherical polytope or *spherical tiling* is an *n*-polytope that completely covers the *n*-sphere.

### Hyperbolic space[edit | edit source]

Polytopes that cover the entirety of hyperbolic spaces are known as hyperbolic tilings.

### Real projective space[edit | edit source]

The real projective spaces are a family of non-Euclidean spaces. The simplest definition of is that each point in real projective space is associated with exactly one line passing through the origin in , and each line in real projective space is a plane passing through the origin. Visually, real projective space can be viewed as a hypersphere with opposite points considered identical.

Projective polytopes tile real projective spaces.

### Unusual spaces[edit | edit source]

Complex polytopes are realized in complex coordinate space, for which the most natural coordinate system comprises vectors of complex numbers .

Quaternionic polytopes reside in a "quaternionic space," which has coordinate locations belonging to the module over the quaternions.

The Hilbert cube may be defined as a polytope in a Hilbert space with a countably infinite number of dimensions.