*For other uses, see sphere (disambiguation).*

A **sphere** is, roughly speaking, a ball-shaped object. In non-mathematical usage, the term *sphere* is often used for something "solid" (which mathematicians call **ball**). But in mathematics, *sphere* refers to the boundary of a ball, which is "hollow". This article deals with the mathematical concept of sphere.

## Geometry

In three-dimensional Euclidean geometry, a **sphere** is the set of points in **R**^{3} 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**.

### Equations

In analytic geometry, a sphere with center (*x*_{0}, *y*_{0}, *z*_{0}) and radius *r* is the set of all points (*x*, *y*, *z*) such that

The points on the sphere with radius *r* can be parametrized via

(see also trigonometric functions and spherical coordinates).

A sphere of any radius centered at the origin is described by the following differential equation:

This equation reflects the fact that the position and velocity vectors of a point travelling on the sphere are always orthogonal to each other.

The surface area of a sphere of radius *r* is:

and its enclosed volume is:

The sphere has the smallest surface area among all surfaces enclosing a given volume and it encloses the largest volume among all closed surfaces with a given surface area. For this reason, the sphere appears in nature: for instance bubbles and small water drops are roughly spherical, because the surface tension minimizes surface area.

The circumscribed cylinder for a given sphere has a volume which is 3/2 times the volume of the sphere, and also a surface area which is 3/2 times the surface area of the sphere. This fact, along with the volume and surface formulas given above, was already known to Archimedes.

A sphere can also be defined as the surface formed by rotating a circle about its diameter. If the circle is replaced by an ellipse, the shape becomes a spheroid.

### Generalization to higher dimensions

Spheres can be generalized to higher 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 0-sphere is a pair of points ( -
*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

Spheres for *n* > 2 are sometimes called hyperspheres. The *n*-sphere of unit radius centred at the origin is denoted *S*^{n} and is often referred to as "the" *n*-sphere.

### See also

## Topology

In topology, an *n*-sphere is defined as a space homeomorphic to the boundary of an (n+1)-ball; thus, it is homeomorphic to the Euclidean *n*-sphere described above under *Geometry*, but perhaps lacking its metric.

The *n*-sphere is denoted *S*^{n}. It is an example of a compact *n*-manifold without boundary. A sphere need not be smooth; if it is smooth, it need not be diffeomorphic to the Euclidean sphere.

The Heine-Borel theorem is used in a short proof that an *n*-sphere is compact. The sphere is the inverse image of a one-point set under the continuous function ||*x*||. Therefore the sphere is closed. *S*^{n} is also bounded. Therefore it is compact.

### See also

## External links

Last updated: 10-24-2005 09:26:53