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.

Classical polytope studies concern n-polytopes in n-dimensional Euclidean space. Common non-Euclidean spaces include hyperspheres (spherical polytopes), hyperbolic spaces (hyperbolic tilings), and real projective spaces (projective polytopes). Spaces that are found in polytope studies are almost always manifolds, sometimes complex manifolds (see complex polytope). An (n + 1)-polytope in an n-dimensional manifold is often called a tiling.

Curvature

Spaces can be divided into three categories based on curvature:

Spherical

Spherical space is finite and has positive curvature everywhere. The circumference of a circle is always less than 2πr. The angles in a triangle add up to more than 180°. The sum of the squares of the legs of a right triangle is always greater than the square of the hypotenuse.

Euclidean

Euclidean space is infinite and has zero curvature everywhere. The circumference of a circle is equal to 2πr. The angles in a triangle add up to exactly 180°, a direct consequence of the parallel postulate. and the sum of the squares of the legs of a right triangle is always equal to the square of the hypotenuse; in other words, the Pythagoream Theorem holds.

Hyperbolic

Hyperbolic space is infinite and has negative curvature everywhere. The circumference of a circle is always greater than 2πr. The angles in a triangle add up to less than 180°. The sum of the squares of the legs of a right triangle is always less than the square of the hypotenuse.