# Hypersphere

A hypersphere is a higher-dimensional analogue of a sphere. A hypersphere of radius R in n-dimensional Euclidean space consists of all points at distance R from a given fixed point (the centre of the hypersphere).

The "volume" it encloses is

[itex]V_n={\pi^{n/2}R^n\over\Gamma(n/2+1)}[itex]

where Γ is the gamma function.

The "surface area" of this hypersphere is

[itex]S_n=\frac{dV_n}{dR}={2\pi^{n/2}R^{n-1}\over\Gamma(n/2)}[itex]

The above hypersphere in n-dimensional Euclidean space is an example of an (n−1)-manifold. It is called an (n−1)-sphere and is denoted Sn−1. For example, an ordinary sphere in three dimensions is a 2-sphere.

The interior of a hypersphere, that is the set of all points whose distance from the centre is less than or equal to R, is called an hyperball.

## Hyperspherical volume - some examples

For a unit sphere ([itex] R = 1 [itex] ) the volumes for some values of n are:

 [itex]V_1\,[itex] = [itex]2\,[itex] [itex]V_2\,[itex] = [itex]\pi\,[itex] = [itex]3.14159\ldots\,[itex] [itex]V_3\,[itex] = [itex]\frac{4 \pi}{3}\,[itex] = [itex]4.18879\ldots\,[itex] [itex]V_4\,[itex] = [itex]\frac{\pi^2}{2}\,[itex] = [itex]4.93480\ldots\,[itex] [itex]V_5\,[itex] = [itex]\frac{8 \pi^2}{15}\,[itex] = [itex]5.26379\ldots\,[itex] [itex]V_6\,[itex] = [itex]\frac{\pi^3}{6}\,[itex] = [itex]5.16771\ldots\,[itex] [itex]V_7\,[itex] = [itex]\frac{16 \pi^3}{105}\,[itex] = [itex]4.72478\ldots\,[itex] [itex]V_8\,[itex] = [itex]\frac{\pi^4}{24}\,[itex] = [itex]4.05871\ldots\,[itex]

If the dimension, n, is not limited to integral values, the hypersphere volume is a continuous function of n with a global maximum for the unit sphere in dimension n  = 5.2569464... where the "volume" is 5.277768...

## Hyperspherical coordinates

We may define a coordinate system in an n-dimensional Euclidean space which is analogous to the spherical coordinate system defined for 3-dimensional Euclidean space, in which the coordinates consist of a radial coordinate r, and n-1 angular coordinates {φ12...φn-1}. If xi are the Cartesian coordinates, then we may define

[itex]x_1=r\cos(\phi_1)\,[itex]
[itex]x_2=r\sin(\phi_1)\cos(\phi_2)\,[itex]
[itex]x_3=r\sin(\phi_1)\sin(\phi_2)\cos(\phi_3)\,[itex]
[itex]\cdots\,[itex]
[itex]x_{n-1}=r\sin(\phi_1)\cdots\sin(\phi_{n-2})\cos(\phi_{n-1})\,[itex]
[itex]x_n~~\,=r\sin(\phi_1)\cdots\sin(\phi_{n-2})\sin(\phi_{n-1})\,[itex]

The hyperspherical volume element will be found from the Jacobian of the transformation:

[itex]d^nr =

\left|\det\frac{\partial (x_i)}{\partial(r,\phi_i)}\right| dr\,d\phi_1 \, d\phi_2\ldots d\phi_{n-1}[itex]

[itex]=r^{n-1}\sin^{n-2}(\phi_1)\sin^{n-3}(\phi_2)\cdots \sin(\phi_{n-2})\,

dr\,d\phi_1 \, d\phi_2\cdots d\phi_{n-1}[itex]

and the above equation for the volume of the hypersphere can be recovered by integrating:

[itex]V_n=\int_{r=0}^R \int_{\phi_1=0}^\pi

\cdots \int_{\phi_{n-2}=0}^\pi\int_{\phi_{n-1}=0}^{2\pi}d^nr. \,[itex]

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