Sphere

A sphere is, roughly speaking, a ball-shaped object. In mathematics, a sphere consists only of a surface and is therefore hollow. In non-mathematical usage a sphere is often considered to be solid.

More precisely, a sphere is the set of points in 3-dimensional Euclidean space which are at distance r from a fixed point of that space, where r is a positive real number called the radius of the sphere.

This can be generalized to other dimensions. For any natural number n, an n-sphere is the set of points in (n+1)-dimensional Euclidean space which are at distance r from a fixed point of that space, where r is, as before, a positive real number. A 3-sphere is therefore an ordinary sphere, while a 2-sphere is a circle and a 1-sphere is a pair of points. An n-sphere is an example of a compact n-manifold.