In coordinate geometry a 3sphere 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
Whereas a sphere has dimension 2 and is therefore a 2manifold (a surface), a 3sphere has dimension 3 and is a 3manifold.
Every nonempty intersection of a 3sphere with a three space is a sphere (unless the space merely touches the 3sphere, in which case the intersection is a single point).
The unit quaternions form a 3sphere, and since they are a group under multiplication, the 3sphere 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 2by2 complex unitary matrices with determinant 1.
A major unsolved problem concerning 3spheres is the Poincaré conjecture.
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