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 a polytope to a point.

Classical polytope studies concern n-polytopes in n-dimensional. 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 s, sometimes s (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.