The Sphere

Abstract

In classical geometry the sphere is viewed as a figure in three-dimensional euclidean space, analogous to the circle in the euclidean plane. The circle, however, is of interest only in relation to the plane. Its intrinsic structure is locally the same as the line because we have the map θ→eⁱθ which is a local isometry between the line and the unit circle. The sphere, on the other hand, is not locally isometric to the plane, hence it is of interest as a self-contained structure. This intrinsic structure makes the sphere the first example of a non-euclidean geometry.