section_path: In Euclidean geometry/Spheres and balls
section_name: Spheres and balls
paragraph_text: Spheres and balls
  
  A sphere in 3-space (also called a 2-sphere because it is a 2-dimensional object) consists of the set of all points in 3-space at a fixed distance r from a central point P. The solid enclosed by the sphere is called a ball (or, more precisely a 3-ball).
  
  The volume of the ball is given by
  
  and the surface area of the sphere is
  
  Another type of sphere arises from a 4-ball, whose three-dimensional surface is the 3-sphere: points equidistant to the origin of the euclidean space R. If a point has coordinates, P (x, y, z, w), then x + y + z + w = 1 characterizes those points on the unit 3-sphere centered at the origin.
  
  This 3-sphere is an example of a 3-manifold: a space which is 'looks locally' like 3-D space. In precise topological terms, each point of the 3-sphere has a neighborhood which is homeomorphic to an open subset of 3-D space.