Hypersphere

The $n$-hypersphere (often simply called the $n$-sphere) is a generalization of the Circle ($n=2$) and Sphere ($n=3$) to dimensions $n\geq 4$. It is therefore defined as the set of $n$-tuples of points ($x_1$, $x_2$, ..., $x_n$) such that

$${x_1}^2+{x_2}^2+\ldots+{x_n}^2=R^2,$$ (1)

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

$$V_n=\int_0^R S_n r^{n-1}\,dr = {S_nR^n\over n},$$ (2)

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

$$S_n \int_0^\infty e^{-r^2} r^{n-1}\,dr = \underbrace{\int_{-... ...s dx_m = \left({\int_{-\infty}^\infty e^{-x^2}\, dx}\right)^n.$$ (3)

But the Gamma Function can be defined by

$$\Gamma(m)=2\int_0^\infty e^{-r^2} r^{2m-1}\,dr,$$ (4)

so

$${\textstyle{1\over 2}}S_n\Gamma({\textstyle{1\over 2}}n)=[\Gamma({\textstyle{1\over 2}})]^n = (\pi^{1/2})^n$$ (5)

$$S_n={2\pi^{n/2}\over \Gamma({\textstyle{1\over 2}}n)}.$$ (6)

This gives the Recurrence Relation

$$S_{n+2}={2\pi S_n\over n}.$$ (7)

Using $\Gamma(n+1)=n\Gamma(n)$ then gives

$$V_n = {S_nR^n\over n} = {\pi^{n/2}R^n\over ({\textstyle{1\ov... ... 2}}n)} = {\pi^{n/2}R^n\over\Gamma(1+{\textstyle{1\over 2}}n)}$$ (8)

(Conway and Sloane 1993).

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

$${dS_n\over dn}={\pi^{n/2}[\ln\pi-\psi_0({\textstyle{1\over 2}}n)]\over\Gamma({\textstyle{1\over 2}}n)}=0,$$ (9)

where $\psi_0(x)\equiv \Psi(x)$ is the Digamma Function. The point of Maximal Content satisfies

$${dV_n\over dn} = {\pi^{n/2} [\ln\pi-\psi_0(1+{\textstyle{1\over 2}}n)]\over 2\Gamma(1+{\textstyle{1\over 2}}n)}=0.$$ (10)

Neither can be solved analytically for $n$, but the numerical solutions are $n=7.25695\ldots$ for hyper-Surface Area and $n=5.25695\ldots$ for Content. As a result, the 7-D and 5-D hyperspheres have Maximal hyper-Surface Area and Content, respectively (Le Lionnais 1983).

$n$	$V_n$	$V_{\rm sphere}/V_{\rm cube}$	$S_n$
0	1	1	0
1	2	1	2
2	$\pi$	${\textstyle{1\over 4}}\pi$	$2\pi$
3	${\textstyle{4\over 3}}\pi$	${\textstyle{1\over 6}}\pi$	$4\pi$
4	${\textstyle{1\over 2}}\pi^2$	${\textstyle{1\over 32}}\pi^2$	$2\pi^2$
5	${\textstyle{8\over 15}}\pi^2$	${\textstyle{1\over 60}}\pi^2$	${\textstyle{8\over 3}}\pi^2$
6	${\textstyle{1\over 6}}\pi^3$	${\textstyle{1\over 384}}\pi^3$	$\pi^3$
7	${\textstyle{16\over 105}}\pi^3$	${\textstyle{1\over 840}}\pi^3$	${\textstyle{16\over 15}}\pi^3$
8	${\textstyle{1\over 24}}\pi^4$	${\textstyle{1\over 6144}}\pi^4$	${\textstyle{1\over 3}}\pi^4$
9	${\textstyle{32\over 945}}\pi^4$	${\textstyle{1\over 15120}}\pi^4$	${\textstyle{32\over 105}}\pi^4$
10	${\textstyle{1\over 120}}\pi^5$	${\textstyle{1\over 122880}}\pi^5$	${\textstyle{1\over 12}}\pi^5$

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

$$x_1 = R\sin\psi\sin\phi\cos\theta$$ (11)

$$x_2 = R\sin\psi\sin\phi\sin\theta$$ (12)

$$x_3 = R\sin\psi\cos\phi$$ (13)

$$x_4 = R\cos\psi.$$ (14)

The equation for a 4-sphere is

$${x_1}^2+{x_2}^2+{x_3}^2+{x_4}^2 = R^2,$$ (15)

and the Line Element is

$$ds^2 = R^2[d\psi^2+\sin^2\psi (d\phi^2+\sin^2\phi\, d\theta^2)].$$ (16)

By defining $r \equiv R\sin \psi$, the Line Element can be rewritten

$$ds^2 = {dr^2\over\left({1 - {r^2\over R^2}}\right)} + r^2(d\phi^2+\sin^2 \phi \,d\theta^2).$$ (17)

The hyper-Surface Area is therefore given by

$$S_4 = \int^\pi_0 R\,d\psi \int^\pi_0 R\sin\psi\,d\phi\int^{2\pi}_0 R\sin\psi\sin\phi\,d\theta$$

$$= 2\pi^2R^3.$$ (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.