3-sphereA 3-sphere (also called a glome by some) is a higher-dimensional analogue of a sphere. A sphere consists of all points an equal distance away from a single point in 3-dimensional Euclidean space. A 3-sphere similarly consists of all points an equal distance away from a single point, but in 4-dimensional Euclidean space.
In coordinate geometry a 3-sphere with centre (x0, y0, z0, w0) and radius r is the set of all points (x,y,z,w) in R4 such that
- (x − x0)2 + (y − y0)2 + (z − z0)2 + (w − w0)2 = r2
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