Space

In the study of polytopes, a space is an environment where a polytope realization resides. In the most general sense possible, a space can be any set, and its members are called points. 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, hyperbolic spaces, and real projective spaces. Spaces that are found in polytope studies are almost always s, sometimes s.

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.