In coordinate geometry a 3-sphere with centre (*x*_{0}, *y*_{0}, *z*_{0}, *w*_{0}) and radius *r* is the set of all points (*x*,*y*,*z*,w) in **R**^{4} such that

- (
*x*−*x*_{0})^{2}+ (*y*−*y*_{0})^{2}+ (*z*−*z*_{0})^{2}+ (*w*−*w*_{0})^{2}=*r*^{2}

Whereas a sphere has dimension 2 and is therefore a 2-manifold (a surface), a 3-sphere has dimension 3 and is a 3-manifold.

Every non-empty intersection of a 3-sphere with a three space is a sphere (unless the space merely touches the 3-sphere, in which case the intersection is a single point).

The unit quaternions form a 3-sphere, and since they are a group under multiplication, the 3-sphere can be regarded as a topological group, even a Lie group, in a natural fashion. This group is isomorphic to SU(2), the group of 2-by-2 complex unitary matrices with determinant 1.

A major unsolved problem concerning 3-spheres is the Poincaré conjecture.

See also: hypersphere simplex