## Hypersphere

The -hypersphere (often simply called the -sphere) is a generalization of the Circle () and Sphere () to dimensions . It is therefore defined as the set of -tuples of points (, , ..., ) such that

 (1)

where is the Radius of the hypersphere. The Content (i.e., -D Volume) of an -hypersphere of Radius is given by
 (2)

where is the hyper-Surface Area of an -sphere of unit radius. But, for a unit hypersphere, it must be true that

 (3)

But the Gamma Function can be defined by
 (4)

so
 (5)

 (6)

This gives the Recurrence Relation
 (7)

Using then gives
 (8)

(Conway and Sloane 1993).

Strangely enough, the hyper-Surface Area and Content reach Maxima and then decrease towards 0 as increases. The point of Maximal hyper-Surface Area satisfies

 (9)

where is the Digamma Function. The point of Maximal Content satisfies
 (10)

Neither can be solved analytically for , but the numerical solutions are for hyper-Surface Area and for Content. As a result, the 7-D and 5-D hyperspheres have Maximal hyper-Surface Area and Content, respectively (Le Lionnais 1983).

 0 1 1 0 1 2 1 2 2 3 4 5 6 7 8 9 10

In 4-D, the generalization of Spherical Coordinates is defined by

 (11) (12) (13) (14)

The equation for a 4-sphere is
 (15)

and the Line Element is
 (16)

By defining , the Line Element can be rewritten
 (17)

The hyper-Surface Area is therefore given by
 (18)

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.

© 1996-9 Eric W. Weisstein
1999-05-25