Definition and Construction
For any natural number n, an n-dimensional sphere, or n-sphere, can be defined as the set of points in an (n+1)-dimensional space which are a fixed distance from a central point. For concreteness, the central point can be taken to be the origin, and the distance of the points on the sphere from this origin can be assumed to be a unit length. With this convention, the n-sphere, Sn, consists of the points (x1, x2, …, xn+1) in Rn+1 with x12 + x22 + ⋯+ xn+12 = 1. For example, the 3-sphere consists of the points (x1, x2, x3, x4) in R4 with x12 + x22 + x32 + x42 = 1.
The Hopf fibration p: S3 → S2 of the 3-sphere over the 2-sphere can be defined in several ways.
Read more about this topic: Hopf Fibration
Famous quotes containing the words definition and/or construction:
“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)
“There is, I think, no point in the philosophy of progressive education which is sounder than its emphasis upon the importance of the participation of the learner in the formation of the purposes which direct his activities in the learning process, just as there is no defect in traditional education greater than its failure to secure the active cooperation of the pupil in construction of the purposes involved in his studying.”
—John Dewey (1859–1952)