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A hypersphere is the set of points at a fixed positive distance from a center in Euclidean space, generalizing the circle and ordinary sphere to higher dimensions. In the standard convention, an -sphere has intrinsic dimension and lies in -dimensional Euclidean space (Hocking and Young 1988, p. 17; Maunder 1997, p. 21). For an integer , the -sphere of radius centered at the origin is the set of -tuples satisfying

Thus the circle

is a 1-sphere, while the ordinary sphere

is a 2-sphere. A 3-sphere lies in four-dimensional Euclidean space and is also called a glome. Coxeter (1973, p. 125) instead indexes spheres by the dimension of the ambient Euclidean space, calling a circle a 2-sphere and an ordinary sphere a 3-sphere. This alternative convention must be considered when consulting older literature.

In the following formulas, the positive integer denotes the dimension of the ambient Euclidean space, so the hypersphere is an -sphere. Let denote its hyper-surface area when the radius is 1, and let denote the content (i.e., -dimensional volume) of the enclosed ball of radius . Then

The Gaussian integral gives

But the gamma function can be defined by

for , so

Special forms of for an integer allow the above expression to be written as

where is a factorial and is a double factorial (OEIS A072478 and A072479).

Strangely enough, for the unit hypersphere, the hyper-surface area reaches a maximum and then decreases towards 0 as increases. Extending the gamma function expression for to real , the point of maximal hyper-surface area satisfies

where is the digamma function. The numerical solution is (OEIS A074457; Wells 1986, p. 67). As a result, among integer ambient dimensions, the unit hypersphere in (a 6-sphere) has maximum hyper-surface area (Le Lionnais 1983; Wells 1986, p. 60).

In four dimensions, the generalization of spherical coordinates is given by

The equation for the 3-sphere is therefore

and the line element is

By defining , the line element can be rewritten locally for as

The hyper-surface area is therefore given by

See also

Ball, Circle, Glome, Hypercube, Hypersphere Packing, Hypersphere Point Picking, Mazur's Theorem, Peg, Sphere, Tesseract Explore this topic in the MathWorld classroom

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References

Collins, G. P. "The Shapes of Space." Sci. Amer. 291, 94-103, July 2004.Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, p. 9, 1993.Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.Hocking, J. G. and Young, G. S. Topology. New York: Dover, 1988.Le Lionnais, F. Les nombres remarquables. Paris, France: Hermann, p. 58, 1983.Maunder, C. M. C. Algebraic Topology. New York: Dover, 1997.Numberphile. "Strange Spheres in Higher Dimensions." Sep. 18, 2017. https://www.youtube.com/watch?v=mceaM2_zQd8.Peterson, I. The Mathematical Tourist: Snapshots of Modern Mathematics. New York: W. H. Freeman, pp. 96-101, 1988.Sloane, N. J. A. Sequences A072478, A072479, and A074457 in "The On-Line Encyclopedia of Integer Sequences."Sommerville, D. M. Y. An Introduction to the Geometry of n Dimensions. New York: Dover, p. 136, 1958.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986.

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Hypersphere

Cite this as:

Weisstein, Eric W. "Hypersphere." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Hypersphere.html

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