Hypersphere

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

(1)

Thus the circle

(2)

is a 1-sphere, while the ordinary sphere

(3)

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

(4)

The Gaussian integral gives

			(5)
			(6)

But the gamma function can be defined by

(7)

for , so

(8)

(9)

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

(10)

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

(11)

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

			(12)
			(13)
			(14)
			(15)

The equation for the 3-sphere is therefore

(16)

and the line element is

(17)

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

(18)

The hyper-surface area is therefore given by

(19)