Hypersphere

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In mathematics, a hypersphere is a sphere which has dimension 3 or higher. The term n-sphere is often used for a sphere of any dimension. An origin-centered sphere of radius R consists of all points in n-dimensional Euclidean space for which the sum of the squares of every coordinate is constant. The constant is $R 2$ , and its square root is the Euclidean distance of every point on the sphere from the origin. The set of all points on this sphere has dimension n-1, so it is called the (n-1)-sphere and is denoted $\mathbb{S}^{n-1}$ . It may be written as $[x_1,x_2,\ldots,x_n]$ where

$R^2=\sum_{k=1}^n x_i^2.\,$

The above hypersphere in n-dimensional Euclidean space is an example of an (n−1)-manifold. For example, an ordinary sphere in three dimensions is a 2-sphere, denoted by $\mathbb{S}^2$ ; the 1-sphere being a circle, and the 0-sphere is the end points of an interval. Of course, translating the coordinates (i.e. moving the center around) doesn't change the analytic or geometric properties of the sphere.

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Hyperspherical volume

The hyperdimensional volume of the space which a (n-1)-sphere encloses (the n-ball) is:

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

where Γ is the gamma function. (For even n, $Γ(n / 2 + 1) = (n / 2)!$ .)

The "surface area" of this sphere is

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

The interior of a hypersphere, that is the set of all points whose distance from the centre is less than R, is called a hyperball, or if the hypersphere itself is included, a closed hyperball.

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Hyperspherical volume - some examples

For small values of n, the volumes, V_n , of the unit n-ball ( $R = 1$ ) are:

$V_1\,$	=	$2\,$
$V_2\,$	=	$\pi\,$	=	$3.14159\ldots\,$
$V_3\,$	=	$\frac{4 \pi}{3}\,$	=	$4.18879\ldots\,$
$V_4\,$	=	$\frac{\pi^2}{2}\,$	=	$4.93480\ldots\,$
$V_5\,$	=	$\frac{8 \pi^2}{15}\,$	=	$5.26379\ldots\,$
$V_6\,$	=	$\frac{\pi^3}{6}\,$	=	$5.16771\ldots\,$
$V_7\,$	=	$\frac{16 \pi^3}{105}\,$	=	$4.72478\ldots\,$
$V_8\,$	=	$\frac{\pi^4}{24}\,$	=	$4.05871\ldots\,$
$\lim_{n\rightarrow\infty} V_n\,$			=	$0\,$

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...

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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 {φ₁,φ₂...φ_n-1}. If x_i are the Cartesian coordinates, then we may define

$x_1=r\cos(\phi_1)\,$

$x_2=r\sin(\phi_1)\cos(\phi_2)\,$

$x_3=r\sin(\phi_1)\sin(\phi_2)\cos(\phi_3)\,$

$\cdots\,$

$x_{n-1}=r\sin(\phi_1)\cdots\sin(\phi_{n-2})\cos(\phi_{n-1})\,$

$x_n~~\,=r\sin(\phi_1)\cdots\sin(\phi_{n-2})\sin(\phi_{n-1})\,$

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

$d^nr = \left|\det\frac{\partial (x_i)}{\partial(r,\phi_i)}\right| dr\,d\phi_1 \, d\phi_2\ldots d\phi_{n-1}$

$=r^{n-1}\sin^{n-2}(\phi_1)\sin^{n-3}(\phi_2)\cdots \sin(\phi_{n-2})\, dr\,d\phi_1 \, d\phi_2\cdots d\phi_{n-1}$

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

$V_n=\int_{r=0}^R \int_{\phi_1=0}^\pi \cdots \int_{\phi_{n-2}=0}^\pi\int_{\phi_{n-1}=0}^{2\pi}d^nr. \,$

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Stereographic projection

Just as a two dimensional sphere embedded in three dimensions can be mapped onto a two-dimensional plane by a stereographic projection, an n-dimensional hypersphere can be mapped onto an n-dimensional hyperplane by the n-dimensional version of the stereographic projection. For example, the point $[x, y, z]$ on a two-dimensional sphere of radius 1 maps to the point $[x / (1 - z), y / (1 - z)]$ on the $x y$ plane. In other words:

$[x,y,z] \mapsto \left[\frac{x}{1-z},\frac{y}{1-z}\right]$

Likewise, the stereographic projection of a hypersphere $\mathbb{S}^{n-1}$ of radius 1 will map to the n-1 dimensional hyperplane $\mathbb{R}^{n-1}$ perpendicular to the $x n$ axis as: