3-sphere
A 3-sphere (also calledglome by some) ishigher-dimensional analogue ofsphere. A sphere consistsall points an equal distance away fromsingle point3-dimensional Euclidean space. A 3-sphere similarly consistsall points an equal distance away fromsingle point, but4-dimensional Euclidean space.In coordinate geometry3-spherecentre (x0, y0, z0, w0)radius r issetall points (x,y,z,w)R4 such that
- (x − x0)2 + (y − y0)2 + (z − z0)2 + (w − w0)2 = r2
Whereassphere has dimension 2is therefore2-manifold (a surface),3-sphere has dimension 3is3-manifold.
Every non-empty intersection of3-sphere withthree space issphere (unlessspace merely touches3-sphere,which caseintersection issingle point).
The unit quaternions form3-sphere,sincearegroup under multiplication,3-sphere can be regarded astopological group, evenLie group, innatural fashion. This groupisomorphicSU(2),group2-by-2 complex unitary matricesdeterminant 1.
A major unsolved problem concerning 3-spheres isPoincaré conjecture.
See also: hypersphere simplex
