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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

\end{displaymath} (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},
\end{displaymath} (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.
\end{displaymath} (3)

But the Gamma Function can be defined by
\Gamma(m)=2\int_0^\infty e^{-r^2} r^{2m-1}\,dr,
\end{displaymath} (4)

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

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

This gives the Recurrence Relation
S_{n+2}={2\pi S_n\over n}.
\end{displaymath} (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)}
\end{displaymath} (8)

(Conway and Sloane 1993).

\begin{figure}\begin{center}\BoxedEPSF{hypersphere_volume.epsf scaled 900}\end{center}\end{figure}

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,
\end{displaymath} (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.
\end{displaymath} (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

$\displaystyle x_1$ $\textstyle =$ $\displaystyle R\sin\psi\sin\phi\cos\theta$ (11)
$\displaystyle x_2$ $\textstyle =$ $\displaystyle R\sin\psi\sin\phi\sin\theta$ (12)
$\displaystyle x_3$ $\textstyle =$ $\displaystyle R\sin\psi\cos\phi$ (13)
$\displaystyle x_4$ $\textstyle =$ $\displaystyle R\cos\psi.$ (14)

The equation for a 4-sphere is
{x_1}^2+{x_2}^2+{x_3}^2+{x_4}^2 = R^2,
\end{displaymath} (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)].
\end{displaymath} (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).
\end{displaymath} (17)

The hyper-Surface Area is therefore given by
$\displaystyle S_4$ $\textstyle =$ $\displaystyle \int^\pi_0 R\,d\psi \int^\pi_0 R\sin\psi\,d\phi\int^{2\pi}_0 R\sin\psi\sin\phi\,d\theta$  
  $\textstyle =$ $\displaystyle 2\pi^2R^3.$ (18)

See also Circle, Hypercube, Hypersphere Packing, Mazur's Theorem, Sphere, Tesseract


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.

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© 1996-9 Eric W. Weisstein