n-sphere

In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an n-sphere of radius r is defined as the set of points in (n + 1)-dimensional Euclidean space which are at distance r from a central point, where the radius r may be any positive real number. In symbols:

$S^n = \left\{ x \in \mathbb{R}^{n%2B1}�: \|x\| = r\right\}.$

It is an n-dimensional manifold in Euclidean (n + 1)-space. In particular, a 0-sphere is a pair of points that are the ends of a line segment, a 1-sphere is a circle in the plane, and a 2-sphere is an ordinary sphere in three-dimensional space. Spheres of dimension n > 2 are sometimes called hyperspheres, with 3-spheres sometimes known as glomes. The n-sphere of unit radius centered at the origin is called the unit n-sphere, denoted Sⁿ. The unit n-sphere is often referred to as the n-sphere. An n-sphere is the surface or boundary of an (n + 1)-dimensional ball, and is an n-dimensional manifold. For n ≥ 2, the n-spheres are the simply connected n-dimensional manifolds of constant, positive curvature. The n-spheres admit several other topological descriptions: for example, they can be constructed by gluing two n-dimensional Euclidean spaces together, by identifying the boundary of an n-cube with a point, or (inductively) by forming the suspension of an (n − 1)-sphere.

1 Description
2 Volume and surface area
3 Hyperspherical coordinates
- 3.1 Hyperspherical volume element
4 Stereographic projection
5 Generating random points
- 5.1 Uniformly at random from the (n − 1)-sphere
  - 5.1.1 Examples
  - 5.1.2 Alternatives
- 5.2 Uniformly at random from the n-ball
6 Specific spheres
7 See also
8 Notes
9 References
10 External links

Description

For any natural number n, an n-sphere of radius r is defined as the set of points in (n + 1)-dimensional Euclidean space that are at distance r from some fixed point c, where r may be any positive real number and where c may be any point in (n + 1)-dimensional space. In particular:

a 0-sphere is a pair of points {c − r, c + r}, and is the boundary of a line segment (1-ball).
a 1-sphere is a circle of radius r centered at c, and is the boundary of a disk (2-ball).
a 2-sphere is an ordinary 2-dimensional sphere in 3-dimensional Euclidean space, and is the boundary of an ordinary ball (3-ball).
a 3-sphere is a sphere in 4-dimensional Euclidean space.

Euclidean coordinates in (n + 1)-space

The set of points in (n + 1)-space: (x₁,x₁,x₂,…,x_n+1) that define an n-sphere, (Sⁿ) is represented by the equation:

$r^2=\sum_{i=1}^{n%2B1} (x_i - c_i)^2.\,$

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

The above n-sphere exists in (n + 1)-dimensional Euclidean space and is an example of an n-manifold. The volume form ω of an n-sphere of radius r is given by

$\omega = {1 \over r} \sum_{j=1}^{n%2B1} (-1)^{j-1} x_j \,dx_1 \wedge \cdots \wedge dx_{j-1} \wedge dx_{j%2B1}\wedge \cdots \wedge dx_{n%2B1} = * dr$

where * is the Hodge star operator; see Flanders (1989, §6.1) for a discussion and proof of this formula in the case r = 1. As a result, $\scriptstyle{dr \wedge \omega = dx_1 \wedge \cdots \wedge dx_{n%2B1}}.$

n-ball

Main article: ball (mathematics)

The space enclosed by an n-sphere is called an (n + 1)-ball. An (n + 1)-ball is closed if it includes the n-sphere, and it is open if it does not include the n-sphere.

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.

Topological description

Topologically, an n-sphere can be constructed as a one-point compactification of n-dimensional Euclidean space. Briefly, the n-sphere can be described as $S^n = \mathbb{R}^n \cup \{ \infty \}$ , which is n-dimensional Euclidean space plus a single point representing infinity in all directions. In particular, if a single point is removed from an n-sphere, it becomes homeomorphic to $\mathbb{R}^n$ . This forms the basis for stereographic projection. ^[1]

Volume and surface area

The $n$ -volume of an $n$ -sphere of radius $R$ or, equivalently, the surface area of an $(n%2B1)$ -ball of radius $R$ is:

$S_n(R) = \frac{ 2 \, \pi^{\frac{n%2B1}{2}}}{\Gamma(\frac{n%2B1}{2})} R^n$

The $n$ -volume of a $n$ -ball of radius $R$ :

$V_n(R) = \frac{\pi^\frac{n}{2}}{\Gamma(\frac{n}{2} %2B 1)} R^n$

The 1-sphere of radius R is the circle of radius R in the Euclidean plane, and this has circumference (1-dimensional measure)

$S_1(R) = 2\pi R. \,$

The region enclosed by the 1-sphere is the 2-ball, or disk of radius R, and this has area (2-dimensional measure)

$V_2(R) = \pi R^2. \,$

Analogously, in 3-dimensional Euclidean space, the surface area (2-dimensional measure) of the 2-sphere of radius R is given by

$S_2(R) = 4\pi R^2 \,$

and the volume enclosed is the volume (3-dimensional measure) of the 3-ball, and is given by

$V_3(R) = \frac{4}{3} \pi R^3. \,$

In general, the volume, in n-dimensional Euclidean space, of the n-ball of radius R is proportional to the n^th power of the R:

$V_n (R) = C_n R^n \,$

where the constant of proportionality, the volume of the unit n-ball, is given by

$C_n = \frac{\pi^\frac{n}{2}}{\Gamma(\frac{n}{2} %2B 1)}$

where $\Gamma \,$ is the gamma function. For even n, this reduces to

$C_n = \frac{\pi^\frac{n}{2}}{\left(\frac{n}{2}\right)!} \,,$

and since

$\Gamma\left(\frac{1}{2}\right) = \sqrt{\pi},$

for odd n,

$C_n = \frac{2^\frac{n%2B1}{2} \pi^\frac{n-1}{2}}{n!!}$

where $n!!$ denotes the double factorial.

The "surface area", or properly the (n − 1)-dimensional volume, of the (n−1)-sphere at the boundary of the n-ball is

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

The following relationships hold between the n-spherical surface area and volume:

$V_n/S_{n-1} = R/n\,$

$S_{n%2B1}/V_n = 2\pi R.\,$

This leads to the recurrence relations:

$V_n = \frac{2 \pi R^2}{n} V_{n-2}$

$S_n= \frac{2 \pi R^2}{n-1} S_{n-2}.$

The recurrence relation for $V_n(R)$ can be proved via integration with 2-dimensional polar coordinates:

$\begin{align} V_n (R) & = \int_0^R \int_0^{2\pi} V_{n-2}(\sqrt{R^2-r^2}) \, r \, d\theta \, dr \\[6pt] & = \int_0^R \int_0^{2\pi} C_{n-2} (R^2-r^2)^{n/2-1}\, r \, d\theta \, dr \\[6pt] & = 2 \pi C_{n-2} \int_{0}^{R} (R^2-r^2)^{n/2-1}\, r \, dr \\[6pt] & = 2 \pi C_{n-2} \left[ -\frac{1}{n}(R^2-r^2)^{n/2} \right]^{r=R}_{r=0} \\[6pt] & = 2 \pi C_{n-2} \frac{R^n}{n} = \frac{2 \pi R^2}{n} V_{n-2}(R). \end{align}$

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 $\phi _1 , \phi _2 , \dots , \phi _{n-1} \,$ where $\phi_{n-1} \,$ ranges over $[0, 2 \pi) \,$ radians (or over [0, 360) degrees) and the other angles range over $[0, \pi] \,$ radians (or over [0, 180] degrees). If $\ x_i$ are the Cartesian coordinates, then we may compute $x_1,\ldots,x_n$ from $r, \phi_1,\ldots,\phi_{n-1}$ with:

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

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

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

$\begin{align} {}\,\,\,\vdots\\ \end{align}$

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

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

Except in the special cases described below, the inverse transformation is unique:

$r = \sqrt{{x_n}^2 %2B {x_{n-1}}^2 %2B \cdots %2B {x_2}^2 %2B {x_1}^2} \,$

$\phi_1 = \arccot \frac{x_{1}}{\sqrt{{x_n}^2%2B{x_{n-1}}^2%2B\cdots%2B{x_2}^2}} \,$

$\phi_2 = \arccot \frac{x_{2}}{\sqrt{{x_n}^2%2B{x_{n-1}}^2%2B\cdots%2B{x_3}^2}} \,$

$\begin{align} {}\,\,\,\vdots\\ \end{align}$

$\phi_{n-2} = \arccot \frac{x_{n-2}}{\sqrt{{x_n}^2%2B{x_{n-1}}^2}} \,$

$\phi_{n-1} = 2 \arccot \frac{\sqrt{{x_n}^2%2B{x_{n-1}}^2} %2B x_{n-1}}{x_{n}} \,.$

where if $x_k \ne 0$ for some $k$ but all of $x_{k%2B1},\ldots,x_n$ are zero then $\phi_k = 0$ when $x_k > 0$ , and $\phi_k = \pi$ radians (180 degrees) when $x_k < 0$ .

There are some special cases where the inverse transform is not unique; $\phi_k$ for any $k$ will be ambiguous whenever all of $x_k,x_{k%2B1},\ldots,x_n$ are zero; in this case $\phi_k$ may be chosen to be zero.

Note that a half-angle formula is used for $\phi_{n-1}$ because the more straightforward $\phi_{n-1} = \arccot \frac{x_{n-1}}{x_n}$ is too small by an addend of π when $x_n$ < 0.

Hyperspherical volume element

Expressing the angular measures in radians, the volume element in n-dimensional Euclidean space will be found from the Jacobian of the transformation:

$\begin{align} d_{\mathbb{R}^n}V & = \left|\det\frac{\partial (x_i)}{\partial(r,\phi_j)}\right| dr\,d\phi_1 \, d\phi_2\cdots d\phi_{n-1} \\[6pt] & = r^{n-1}\sin^{n-2}(\phi_1)\sin^{n-3}(\phi_2)\cdots \sin(\phi_{n-2})\, dr\,d\phi_1 \, d\phi_2\cdots d\phi_{n-1} \end{align}$

and the above equation for the volume of the n-ball can be recovered by integrating:

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

The volume element of the (n-1)–sphere, which generalizes the area element of the 2-sphere, is given by

$d_{S^{n-1}}V = \sin^{n-2}(\phi_1)\sin^{n-3}(\phi_2)\cdots \sin(\phi_{n-2})\, d\phi_1 \, d\phi_2\cdots d\phi_{n-1}.$

The natural choice of an orthogonal basis over the angular coordinates is a product of ultraspherical polynomials,

$\begin{align} & {} \quad \int_0^\pi \sin^{n-j-1}(\phi_j) C_s^{((n-j-1)/2)}(\cos \phi_j)C_{s'}^{((n-j-1)/2)}(\cos\phi_j) \, d\phi_j \\[6pt] & = \frac{\pi 2^{3-n%2Bj}\Gamma(s%2Bn-j-1)}{s!(2s%2Bn-j-1)\Gamma^2((n-j-1)/2)}\delta_{s,s'} \end{align}$

for j = 1, 2, ..., n − 2, and the e^isφ_j for the angle j = n − 1 in concordance with the spherical harmonics.

Stereographic projection

Main article: 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-sphere 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 $\left[\frac{x}{1-z},\frac{y}{1-z}\right]$ on the $\ xy$ plane. In other words,

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

Likewise, the stereographic projection of an n-sphere $\mathbf{S}^{n-1}$ of radius 1 will map to the $n-1$ dimensional hyperplane $\mathbf{R}^{n-1}$ perpendicular to the $\ x_n$ axis as

$[x_1,x_2,\ldots,x_n] \mapsto \left[\frac{x_1}{1-x_n},\frac{x_2}{1-x_n},\ldots,\frac{x_{n-1}}{1-x_n}\right].$

Generating random points

Uniformly at random from the (n − 1)-sphere

To generate uniformly distributed random points on the (n − 1)-sphere (i.e., the surface of the n-ball), Marsaglia (1972) gives the following algorithm.

Generate an n-dimensional vector of normal deviates (it suffices to use N(0, 1), although in fact the choice of the variance is arbitrary), $\mathbf{x}=(x_1,x_2,\ldots,x_n)$ .

Now calculate the "radius" of this point, $r=\sqrt{x_1^2%2Bx_2^2%2B\cdots%2Bx_n^2}$ .

The vector $\frac{1}{r} \mathbf{x}$ is uniformly distributed over the surface of the unit n-ball.

Examples

For example, when n = 2 the normal distribution exp(−x₁²) when expanded over another axis exp(−x₂²) after multiplication takes the form exp(−x₁²−x₂²) or exp(−r²) and so is only dependent on distance from the origin.

Alternatives

Another way to generate a random distribution on a hypersphere is to make a uniform distribution over a hypercube that includes the unit hyperball, exclude those points that are outside the hyperball, then project the remaining interior points outward from the origin onto the surface. This will give a uniform distribution, but it is necessary to remove the exterior points. As the relative volume of the hyperball to the hypercube decreases very rapidly with dimension, this procedure will succeed with high probability only for fairly small numbers of dimensions.

Wendel's theorem gives the probability that all of the points generated will lie in the same half of the hypersphere.

Uniformly at random from the n-ball

With a point selected from the surface of the n-ball uniformly at random, one needs only a radius to obtain a point uniformly at random within the n-ball. If u is a number generated uniformly at random from the interval [0, 1] and x is a point selected uniformly at random from the surface of the n-ball then u^1/nx is uniformly distributed over the entire unit n-ball.

Specific spheres

0-sphere: The pair of points {±R} with the discrete topology for some R > 0. The only sphere that is disconnected. Has a natural Lie group structure; isomorphic to O(1). Parallelizable.
1-sphere: Also known as the circle. Has a nontrivial fundamental group. Abelian Lie group structure U(1); the circle group. Topologically equivalent to the real projective line, RP¹. Parallelizable. SO(2) = U(1).
2-sphere: Also known as the sphere. Complex structure; see Riemann sphere. Equivalent to the complex projective line, CP¹. SO(3)/SO(2).
3-sphere: Parallelizable, Principal U(1)-bundle over the 2-sphere, Lie group structure Sp(1), where also; $\mathrm{Sp}(1) \cong \mathrm{SO}(4)/\mathrm{SO}(3) \cong \mathrm{SU}(2) \cong \mathrm{Spin}(3)$ .
4-sphere: Equivalent to the quaternionic projective line, HP¹. SO(5)/SO(4).
5-sphere: Principal U(1)-bundle over CP². SO(6)/SO(5) = SU(3)/SU(2).
6-sphere: Almost complex structure coming from the set of pure unit octonions. SO(7)/SO(6) = G₂/SU(3).
7-sphere: Topological quasigroup structure as the set of unit octonions. Principal Sp(1)-bundle over S⁴. Parallelizable. SO(8)/SO(7) = SU(4)/SU(3) = Sp(2)/Sp(1) = Spin(7)/G₂ = Spin(6)/SU(3). The 7-sphere is of particular interest since it was in this dimension that the first exotic spheres were discovered.
8-sphere: Equivalent to the octonionic projective line OP¹.
23-sphere: A highly dense sphere-packing is possible in 24 dimensional space, which is related to the unique qualities of the Leech lattice.

Notes

^ James W. Vick (1994). Homology theory, p. 60. Springer

References

Flanders, Harley (1989). Differential forms with applications to the physical sciences. New York: Dover Publications. ISBN 978-0-486-66169-8. .
Moura, Eduarda; Henderson, David G. (1996). Experiencing geometry: on plane and sphere. Prentice Hall. ISBN 978-0-13-373770-7. http://www.math.cornell.edu/~henderson/books/eg00 (Chapter 20: 3-spheres and hyperbolic 3-spaces.)
Weeks, Jeffrey R. (1985). The Shape of Space: how to visualize surfaces and three-dimensional manifolds. Marcel Dekker. ISBN 978-0-8247-7437-0 (Chapter 14: The Hypersphere)
Marsaglia, G. (1972). "Choosing a Point from the Surface of a Sphere". Ann. Math. Stat. 43 (2): 645–646. doi:10.1214/aoms/1177692644.
Huber, Greg (1982). "Gamma function derivation of n-sphere volumes". Am. Math. Monthly 89 (5): 301–302. doi:10.2307/2321716. JSTOR 2321716. MR 1539933.

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