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

Euclidean spaces

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

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

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

Real projective space

The real projective spaces are a family of non-Euclidean spaces. The simplest definition of $\mathbb {P} ^{n}(\mathbb {R} )$ is that each point in real projective space is associated with exactly one line passing through the origin in $\mathbb {R} ^{n+1}$, 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

Complex polytopes are realized in complex coordinate space, for which the most natural coordinate system comprises vectors of complex numbers $\mathbb {C} ^{n}$.

Quaternionic polytopes reside in a "quaternionic space," which has coordinate locations belonging to the module $\mathbb {H} ^{n}$ over the quaternions.

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