A sphere is a perfectly symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface. In mathematics, a sphere is the set of all points in three-dimensional space (R³) 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. The fixed point is called the center or centre, and is not part of the sphere itself. The special case of r = 1 is called a unit sphere.

This article deals with the mathematical concept of a sphere. In physics, a sphere is an object (usually idealized for the sake of simplicity) capable of colliding or stacking with other objects which occupy space.

TerminologyPairs of points on a sphere that lie on a straight line through its center are called antipodal points. A great circle is a circle on the sphere that has the same center and radius as the sphere, and consequently divides it into two equal parts. The shortest distance between two distinct non-antipodal points on the surface and measured along the surface, is on the unique great circle passing through the two points.If a particular point on a sphere is designated as its north pole, then the corresponding antipodal point is called the south pole and the equator is the great circle that is equidistant to them. Great circles through the two poles are called lines (or meridians) of longitude, and the line connecting the two poles is called the axis of rotation. Circles on the sphere that are parallel to the equator are lines of latitude. This terminology is also used for astronomical bodies such as the planet Earth, even though it is neither spherical nor even spheroidal (see geoid).

A sphere is divided into two equal hemispheres by any plane that passes through its center. If two intersecting planes pass through its center, then they will subdivide the sphere into four lunes or biangles, the vertices of which all coincide with the antipodal points lying on the line of intersection of the planes.

Generalization to other dimensions

Spheres can be generalized to other dimensions. For any natural number n, an n-sphere, often written as Sⁿ, 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. For n> 0, the n-sphere is the simply connected n-dimensional manifold of constant, positive curvature, and can also be thought of embedded in an n+1-dimensional manifold, as the surface or boundary of a ball in the n+1-dimensional manifold.

• a 0-sphere is a pair of points on the line at ( − r,r)
• a 1-sphere is a circle of radius r
• a 2-sphere is an ordinary sphere
• a 3-sphere is a sphere in 4-dimensional Euclidean space.