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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.