| 释义 |
HypersphereThe  -hypersphere (often simply called the -sphere) is a generalization of the Circle ( ) andSphere ( ) to dimensions  . It is therefore defined as the set of -tuples of points( ,  , ...,  ) such that
 | (1) |
 is the Radius of the hypersphere. The Content (i.e., -D Volume) of an -hypersphere ofRadius is given by
 | (2) |
 is the hyper-Surface Area of an -sphere of unit radius. But, for a unit hypersphere, it must be true that
 | (3) |
 | (4) |
 | (5) |
 | (6) |
 | (7) |
 then gives
 | (8) |
Strangely enough, the hyper-Surface Area and Content reach Maximaand then decrease towards 0 as increases. The point of Maximal hyper-Surface Area satisfies
 | (9) |
 is the Digamma Function. The point of Maximal Content satisfies
 | (10) |
 for hyper-Surface Area and for Content. As a result, the 7-D and 5-D hyperspheres have Maximalhyper-Surface Area and Content, respectively (Le Lionnais 1983).
In 4-D, the generalization of Spherical Coordinates is defined by
The equation for a 4-sphere is
 | (15) |
 | (16) |
 , the Line Element can be rewritten
 | (17) |
See also Circle, Hypercube, Hypersphere Packing, Mazur's Theorem, Sphere, Tesseract
References
Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, p. 9, 1993.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 58, 1983. Peterson, I. The Mathematical Tourist: Snapshots of Modern Mathematics. New York: W. H. Freeman, pp. 96-101, 1988. |
