Encyclopedia > 3-sphere

  Article Content

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 (x0y0z0w0) 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.


See also: hypersphere simplex



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Spectral type

... that some of the classes were actually duplicates and those classes were removed. It was only much later that it was discovered that the strength of the hydrogen line was ...