In mathematics, a hypersphere is a sphere which has dimension 3 or higher. The term $n$-sphere is used for a sphere of dimension $n$, for any positive integer $n$. 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 $\backslash mathbb\; S^\{n-1\}$. It may be written as $(x\_1,x\_2,...,x\_n)$ where

$R^2=\backslash sum\_\{i=1\}^n\; x\_i^2.\backslash ,$

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 $\backslash mathbb\{S\}^2$; 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.

Hyperspherical volume

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

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

where $\backslash Gamma$ is the gamma function. (For even $n$, $\{\backslash Gamma(n/2+1)=\; (n/2)!\}$.)

The "surface area" of this sphere is

$S\_n=\backslash frac\{dV\_n\}\{dR\}=\backslash frac\{n\; V\_n\}\{R\}=\{2\backslash pi^\{n/2\}R^\{n-1\}\backslash over\backslash Gamma(n/2)\}$

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.

Hyperspherical volume - some examples

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

{| $V\_1\backslash ,$ = $2\backslash ,$     $V\_2\backslash ,$ = $\backslash pi\backslash ,$ = $3.14159\backslash ldots\backslash ,$ $V\_3\backslash ,$ = $\backslash frac\{4\; \backslash pi\}\{3\}\backslash ,$ = $4.18879\backslash ldots\backslash ,$ $V\_4\backslash ,$ = $\backslash frac\{\backslash pi^2\}\{2\}\backslash ,$ = $4.93480\backslash ldots\backslash ,$ $V\_5\backslash ,$ = $\backslash frac\{8\; \backslash pi^2\}\{15\}\backslash ,$ = $5.26379\backslash ldots\backslash ,$ $V\_6\backslash ,$ = $\backslash frac\{\backslash pi^3\}\{6\}\backslash ,$ = $5.16771\backslash ldots\backslash ,$ $V\_7\backslash ,$ = $\backslash frac\{16\; \backslash pi^3\}\{105\}\backslash ,$ = $4.72478\backslash ldots\backslash ,$ $V\_8\backslash ,$ = $\backslash frac\{\backslash pi^4\}\{24\}\backslash ,$ = $4.05871\backslash ldots\backslash ,$ $\backslash lim\_\{n\backslash rightarrow\backslash infty\}\; V\_n\backslash ,$ = $0\backslash ,$

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

The hypercube circumscribed around the unit n-sphere has an edge length of 2 and hence a volume of 2ⁿ; the ratio of the volume of the hypersphere to its circumscribed hypercube decreases monotonically as the dimension increases.

Hyperspherical coordinates

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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 $\backslash \; r$, and $\backslash \; n-1$ angular coordinates $\backslash \; \backslash phi\; \_1\; ,\; \backslash phi\; \_2\; ,\; ...\; ,\; \backslash phi\; \_\{n-1\}$. If $\backslash \; x\_i$ are the Cartesian coordinates, then we may define

$x\_1=r\backslash cos(\backslash phi\_1)\backslash ,$

$x\_2=r\backslash sin(\backslash phi\_1)\backslash cos(\backslash phi\_2)\backslash ,$

$x\_3=r\backslash sin(\backslash phi\_1)\backslash sin(\backslash phi\_2)\backslash cos(\backslash phi\_3)\backslash ,$

$\backslash cdots\backslash ,$

$x\_\{n-1\}=r\backslash sin(\backslash phi\_1)\backslash cdots\backslash sin(\backslash phi\_\{n-2\})\backslash cos(\backslash phi\_\{n-1\})\backslash ,$

$x\_n~~\backslash ,=r\backslash sin(\backslash phi\_1)\backslash cdots\backslash sin(\backslash phi\_\{n-2\})\backslash sin(\backslash phi\_\{n-1\})\backslash ,$

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}

$=r^\{n-1\}\backslash sin^\{n-2\}(\backslash phi\_1)\backslash sin^\{n-3\}(\backslash phi\_2)\backslash cdots\; \backslash sin(\backslash phi\_\{n-2\})\backslash ,$

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:

$V\_n=\backslash int\_\{r=0\}^R\; \backslash int\_\{\backslash phi\_1=0\}^\backslash pi$

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

Stereographic projection

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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

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

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

$\backslash mapsto\; \backslash left[\backslash frac\{x\_1\}\{1-x\_n\},\backslash frac\{x\_2\}\{1-x\_n\},\backslash ldots,\backslash frac\{x\_\{n-1\}\}\{1-x\_n\}\backslash right$

See also

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• Hypercube
• 3-sphere

Multi-dimensional geometry

Hyperkoule | Sphäre (Mathematik) | Ipersfera