Main Page | See live article | Alphabetical index 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. See also: hypersphere simplex This article is from Wikipedia. All text is available under the terms of the GNU Free Documentation License.