Hypersphere

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In mathematics, a hypersphere is a higher dimensional sphere.

The term <math>n</math>-sphere, or <math>\mathbb S^n</math>, is a higher dimensional sphere, with n surface dimensions and embedded in (n+1)-space.

Specifically:

1. A 0-sphere represents two points on a line.
2. A 1-sphere is a circle on a plane.
3. A 2-sphere is an ordinary sphere in 3 dimensional space.
4. And higher, a 3-sphere exists in 4 dimensional space, etc.

[edit] Euclidean coordinates in (n+1)-space

The set of points in (n+1)-space: <math>(x_1,x_2,x_3,...,x_{n+1})</math> that define an n-sphere, (<math>\mathbb S^n</math>) is represented by the equation:

<math>r^2=\sum_{i=1}^{n+1} (x_i - C_i)^2.\,</math>

where C is a center point, and r is the radius.

The above hypersphere exists in <math>n+1</math>-dimensional Euclidean space is an example of an <math>n</math>-manifold.

[edit] n-ball

The interior of an n-sphere is called an (n+1)-ball. An (n+1)-ball is closed if it included the equality, and open otherwise.

Specifically:

• A 1-ball, a line segment, is the interior of a (0-sphere).
• A 2-ball, a disk, is the interior of a circle (1-sphere).
• A 3-ball, an ordinary ball, is the interior of a sphere (2-sphere).
• A 4-ball, is the interior of a 3-sphere, etc.

[edit] Hyperspherical volume

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

<math>V_n={\pi^\frac{n}{2}R^n\over\Gamma(\frac{n}{2} + 1)}</math>

where <math>\Gamma</math> is the gamma function. (For even <math>n</math>, <math>\Gamma\left(\frac{n}{2}+1\right)= \left(\frac{n}{2}\right)!</math>; for odd <math>n</math>, <math>\Gamma\left(\frac{n}{2}+1\right)= \sqrt{\pi} \frac{n!!}{2^{(n+1)/2}}</math>, where <math>n!!</math> denotes the double factorial.)

The "surface area" of this (n-1)-sphere is

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

The following relationships hold between the hyperspherical surface area and volume:

<math>V_n/S_n = R/n\,</math>

<math>S_{n+2}/V_n = 2\pi R\,</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.

[edit] Hyperspherical volume - some examples

For small values of <math>n</math>, the volumes, <math>V_n</math> , of the unit <math>n</math>-ball (<math>R=1</math>) are:

<math>V_0\,</math>	=	<math>1\,</math>
<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...

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.

[edit] 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>

While the inverse transformations can be derived from those above:

<math>\tan(\phi_{n-1})=\frac{x_n}{x_{n-1}}</math>

<math>\tan(\phi_{n-2})=\frac{\sqrt{{x_n}^2+{x_{n-1}}^2}}{x_{n-2}}</math>

<math>\cdots\,</math>

<math>\tan(\phi_{1})=\frac{\sqrt{{x_n}^2+{x_{n-1}}^2+\cdots+{x_2}^2}}{x_{1}}</math>

Note that last angle <math>\phi _{n-1}</math> has a range of <math>2\pi</math> while the other angles have a range of <math>\pi</math>. This range covers the whole sphere.

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>

[edit] 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>

[edit] See also

• Hypercube
• 3-sphere

[edit] References

• David W. Henderson, Experiencing Geometry: In Euclidean, Spherical, and Hyperbolic Spaces, second edition, 2001, [1] (Chapter 20: 3-spheres and hyperbolic 3-spaces.)
• Jeffrey R. Weeks, The Shape of Space: How to Visualize Surfaces and Three-dimensional Manifolds, 1985, (Chapter 14: The Hypersphere)cs:Hyperkoule

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