n-sphere

In mathematics, the n-sphere is the generalization of the ordinary sphere to spaces of arbitrary dimension. It is an n-dimensional manifold that can be embedded in Euclidean (n + 1)-space.Vn(R) and Sn(R) are the n-dimensional volume of the n-ball and the surface area of the n-sphere embedded in dimension n + 1, respectively, of radius R.In theory, one could compare the values of Sn(R) and Sm(R) for n ≠ m. However, this is somewhat nonsensical. For example, if n = 2 and m = 3 then the comparison is like comparing a number of square meters to a different number of cubic meters. The same lack of sense applies to a comparison of Vn(R) and Vm(R) for n ≠ m. In mathematics, the n-sphere is the generalization of the ordinary sphere to spaces of arbitrary dimension. It is an n-dimensional manifold that can be embedded in Euclidean (n + 1)-space. The 0-sphere is a pair of points, the 1-sphere is a circle, and the 2-sphere is an ordinary sphere. Generally, when embedded in an (n + 1)-dimensional Euclidean space, an n-sphere is the surface or boundary of an (n + 1)-dimensional ball. That is, for any natural number n, an n-sphere of radius r may be defined in terms of an embedding in (n + 1)-dimensional Euclidean space as the set of points that are at distance r from a central point, where the radius r may be any positive real number. Thus, the n-sphere would be defined by: