3-sphere

A 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

The 3-sphere with radius 1 and center (0,0,0,0) is also denoted by S³.

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.

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See also: hypersphere simplex