
related topics 
{math, number, function} 
{math, energy, light} 
{line, north, south} 
{black, white, people} 
{group, member, jewish} 
{style, bgcolor, rowspan} 

In mathematics, a 3sphere is a higherdimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4dimensional Euclidean space. Just as an ordinary sphere (or 2sphere) is a two dimensional surface that forms the boundary of a ball in three dimensions, a 3sphere is an object with three dimensions that forms the boundary of a ball in four dimensions. A 3sphere is an example of a 3manifold.
A 3sphere is also called a "glome" or a hypersphere, although the term hypersphere can in general describe any nsphere for n ≥ 3.
Contents
Definition
In coordinates, a 3sphere with center (C_{0}, C_{1}, C_{2}, C_{3}) and radius r is the set of all points (x_{0}, x_{1}, x_{2}, x_{3}) in real, 4dimensional space (R^{4}) such that
The 3sphere centered at the origin with radius 1 is called the unit 3sphere and is usually denoted S^{3}:
It is often convenient to regard R^{4} as the space with 2 complex dimensions (C^{2}) or the quaternions (H). The unit 3sphere is then given by
or
The last description is often the most useful. It describes the 3sphere as the set of all unit quaternions—quaternions with absolute value equal to unity. Just as the unit circle is important for planar polar coordinates, so the 3sphere is important in the polar view of 4space involved in quaternion multiplication. See polar decomposition of a quaternion for details of this development of the threesphere.
Full article ▸


related documents 
Wave function 
Euclidean geometry 
Differential calculus 
Perturbation theory 
Shor's algorithm 
Plane (geometry) 
Dihedral group 
Rasterisation 
Vector field 
Cellular automaton 
Euler characteristic 
Wiener process 
Least squares 
Fuzzy logic 
Quantum harmonic oscillator 
Algebraic structure 
Linear combination 
Abstract interpretation 
Ideal class group 
Gaussian quadrature 
Natural transformation 
Axiom schema of replacement 
Heine–Borel theorem 
Presentation of a group 
Absolute convergence 
Factorization 
Partition (number theory) 
Galois theory 
Linear independence 
Type theory 
