Hypersphere
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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 <math>\ R</math> consists of all points in n-dimensional Euclidean space for which the sum of the squares of every coordinate is constant. The constant is <math>\ R^2</math>, 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 <math>\ n-1</math>, so it is called the (n-1)-sphere and is denoted <math>\ \mathbb S^{n-1}</math>. It may be written as <math>\ (x_1,x_2,...,x_n)</math> where
- <math>\ R^2=\sum_{k=1}^n x_i^2.\,</math>
The above hypersphere in <math>\ n</math>-dimensional Euclidean space is an example of an <math>\ (n-1)</math>-manifold. For example, an ordinary sphere in three dimensions is a 2-sphere, denoted by <math>\mathbb{S}^2</math>; 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:
- <math>\ V_n={\pi^{n/2}R^n\over\Gamma(n/2+1)}</math>
where Γ is the gamma function. (For even <math>\ n</math> , <math>\ {\Gamma(n/2+1)= (n/2)!}</math>.)
The "surface area" of this sphere is
- <math>\ S_n=\frac{dV_n}{dR}=\frac{n V_n}{R}={2\pi^{n/2}R^{n-1}\over\Gamma(n/2)}</math>
The interior of a hypersphere, that is the set of all points whose distance from the centre is less than <math>\ R</math>, is called a hyperball, or if the hypersphere itself is included, a closed hyperball.
Hyperspherical volume - some examples
For small values of <math>\ n</math>, the volumes, <math>\ V_n</math> , of the unit n-ball (<math>\ R = 1 </math>) are:
<math>V_1\,</math> = <math>2\,</math> <math>V_2\,</math> = <math>\pi\,</math> = <math>3.14159\ldots\,</math> <math>V_3\,</math> = <math>\frac{4 \pi}{3}\,</math> = <math>4.18879\ldots\,</math> <math>V_4\,</math> = <math>\frac{\pi^2}{2}\,</math> = <math>4.93480\ldots\,</math> <math>V_5\,</math> = <math>\frac{8 \pi^2}{15}\,</math> = <math>5.26379\ldots\,</math> <math>V_6\,</math> = <math>\frac{\pi^3}{6}\,</math> = <math>5.16771\ldots\,</math> <math>V_7\,</math> = <math>\frac{16 \pi^3}{105}\,</math> = <math>4.72478\ldots\,</math> <math>V_8\,</math> = <math>\frac{\pi^4}{24}\,</math> = <math>4.05871\ldots\,</math> <math>\lim_{n\rightarrow\infty} V_n\,</math> = <math>0\,</math>
If the dimension, <math>\ n</math> , is not limited to integral values, the hypersphere volume is a continuous function of <math>\ n</math> with a global maximum for the unit sphere in "dimension" n = 5.2569464... where the "volume" is 5.277768... Note that 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
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 <math>\ r</math>, and <math>\ n-1</math> angular coordinates <math>\ \phi _1 , \phi _2 , ... , \phi _{n-1}</math>. If <math>\ x_i</math> are the Cartesian coordinates, then we may define
- <math>x_1=r\cos(\phi_1)\,</math>
- <math>x_2=r\sin(\phi_1)\cos(\phi_2)\,</math>
- <math>x_3=r\sin(\phi_1)\sin(\phi_2)\cos(\phi_3)\,</math>
- <math>\cdots\,</math>
- <math>x_{n-1}=r\sin(\phi_1)\cdots\sin(\phi_{n-2})\cos(\phi_{n-1})\,</math>
- <math>x_n~~\,=r\sin(\phi_1)\cdots\sin(\phi_{n-2})\sin(\phi_{n-1})\,</math>
The hyperspherical volume element will be found from the Jacobian of the transformation:
- <math>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}</math>
- <math>=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}</math>
and the above equation for the volume of the hypersphere can be recovered by integrating:
- <math>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. \,</math>
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 <math>\ [x,y,z]</math> on a two-dimensional sphere of radius 1 maps to the point <math>\ [x,y,z] \mapsto \left[\frac{x}{1-z},\frac{y}{1-z}\right]</math> on the <math>\ xy</math> plane. In other words:
- <math>\ [x,y,z] \mapsto \left[\frac{x}{1-z},\frac{y}{1-z}\right]</math>
Likewise, the stereographic projection of a hypersphere <math>\mathbb{S}^{n-1}</math> of radius 1 will map to the n-1 dimensional hyperplane <math>\mathbb{R}^{n-1}</math> perpendicular to the <math>\ x_n</math> axis as:
- <math>[x_1,x_2,\ldots,x_n] \mapsto \left[\frac{x_1}{1-x_n},\frac{x_2}{1-x_n},\ldots,\frac{x_{n-1}}{1-x_n}\right]</math>