n-sphere

2-sphere wireframe as an orthogonal projection
Just as a stereographic projection can project a sphere's surface to a plane, it can also project the surface of a 3-sphere into 3-space. This image shows three coordinate directions projected to 3-space: parallels (red), meridians (blue) and hypermeridians (green). Due to the conformal property of the stereographic projection, the curves intersect each other orthogonally (in the yellow points) as in 4D. All of the curves are circles: the curves that intersect <0,0,0,1> have an infinite radius (= straight line).

In mathematics, the n-sphere is the generalization of the ordinary sphere to spaces of 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. Thus, the n-sphere centred at the origin is defined by:

S^n = \left\{ x \in \mathbb{R}^{n+1} : \left\| x \right\| = r \right\} .

It is an n-dimensional manifold in Euclidean (n + 1)-space.

In particular:

a 0-sphere is the pair of points at the ends of a (one-dimensional) line segment,
a 1-sphere is the circle, which is the one-dimensional circumference of a (two-dimensional) disk in the plane,
a 2-sphere is the two-dimensional surface of a (three-dimensional) ball in three-dimensional space.

Spheres of dimension n > 2 are sometimes called hyperspheres, with a 3-sphere sometimes known as a glome. 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.

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:

Euclidean coordinates in (n + 1)-space

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

r^2=\sum_{i=1}^{n+1} (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+1} (-1)^{j-1} x_j \,dx_1 \wedge \cdots \wedge dx_{j-1} \wedge dx_{j+1}\wedge \cdots \wedge dx_{n+1} = * 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+1}}.

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:

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 \mathbb{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

V_n (R) and S_n (R) are the n-dimensional volume and surface area of the n-ball and n-sphere of radius R, respectively.

The constants V_n and S_n (for the unit ball and sphere) are related by the recurrences:

V_0=1\qquad V_{n+1}=S_n/(n+1)
S_0=2\qquad S_{n+1}=2\pi V_n

The surfaces and volumes can also be given in closed form:

\begin{array}{ll} S_{n}(R) &= \displaystyle{\frac{2\pi^{(n + 1)/2}}{\Gamma(\frac{n+1}{2})}R^{n}} \\[1 em] V_n(R) &= \displaystyle{\frac{\pi^{n/2}}{\Gamma(\frac{n}{2} + 1)}}R^n \end{array}

where \Gamma \, is the gamma function. Derivations of these equations are given in this section.

In general, the volumes of the n-ball in n-dimensional Euclidean space, and the n-sphere in (n + 1)-dimensional Euclidean, of radius R, are proportional to the nth power of the radius, R. We write V_n(R) = V_n R^n for the volume of the n-ball and S_n(R) = S_n R^n for the surface of the n-sphere, both of radius R.

Interestingly, given the radius R, the volume and the surface area of the n-sphere reaches a maximum and then decrease towards zero as the dimension n increases. In particular, the volume V_n(R) of the n-sphere of constant radius R in n-dimensions reaches a maximum for dimension n^{\ast}=n if V_{n-1} <R< V_n and n^{\ast}=\{n,n+1\} if R=V_n where V_n = \frac{\Gamma(n/2+3/2)}{\sqrt{\pi}\Gamma(n/2+1)} for n = 1,2,\ldots. Similarly, defining the sequence \{A_n=V_{n-2}\}, the surface area S_n(R) of the n-sphere of constant radius R in n dimensions reaches a maximum for dimension n^{\ast}=n if A_{n-1} < R < A_n and n^{\ast}= \{n,n+1\} if R = A_n.[2]

Examples

The 0-ball consists of a single point. The 0-dimensional Hausdorff measure is the number of points in a set, so

V_0=1.

The unit 1-ball is the interval [-1,1] of length 2. So,

V_1 = 2.

The 0-sphere consists of its two end-points, \{-1,1\}. So

S_0 = 2.

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

S_1 = 2\pi. \,

The region enclosed by the unit 1-sphere is the 2-ball, or unit disc, and this has area (2-dimensional measure)

V_2 = \pi. \,

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

S_2 = 4\pi. \,

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

V_3 = \frac{4}{3} \pi. \,

Recurrences

The surface area, or properly the n-dimensional volume, of the n-sphere at the boundary of the (n + 1)-ball of radius R is related to the volume of the ball by the differential equation

S_{n}R^{n}=\frac{dV_{n+1}R^{n+1}}{dR}={(n+1)V_{n+1}R^{n}},

or, equivalently, representing the unit n-ball as a union of concentric (n − 1)-sphere shells,

V_{n+1} = \int_0^1 S_{n}r^{n}\,dr

So,

V_{n+1} = \frac{S_n}{n+1}.

We can also represent the unit (n + 2)-sphere as a union of tori, each the product of a circle (1-sphere) with an n-sphere. Let r = \cos\theta and r^2 + R^2 = 1, so that R = \sin\theta and dR = \cos\theta\,d\theta. Then,

\begin{align} S_{n+2} &= \int_0^{\pi/2}S_1 r . S_n R^n\, d\theta =\int_0^{\pi/2}S_1 . S_n R^n\cos\theta\,d\theta\\ &=\int_0^1 S_1 . S_n R^n \,dR= S_1 \int_0^1 S_n R^n \,dR\\ &= 2\pi V_{n+1} \end{align}

Since S_1 = 2\pi V_0, the equation S_{n+1} = 2\pi V_{n} holds for all n.

This completes our derivation of the recurrences:

V_0=1\qquad V_{n+1}=S_n/(n+1)
S_0=2\qquad S_{n+1}=2\pi V_n

Closed forms

Combining the recurrences, we see that V_{n+2}=2\pi V_n/(n+2). So it is simple to show by induction on k that,

V_{2k} = \frac{\pi^k}{k!}
V_{2k+1} = \frac{2(2\pi)^k}{(2k+1)!!} = \frac{2 k! (4\pi)^k}{(2k+1)!}

where !! denotes the double factorial, defined for odd integers 2k + 1 by (2k + 1)!! = 1 · 3 · 5 ··· (2k − 1) · (2k + 1).

In general, the volume, in n-dimensional Euclidean space, of the unit n-ball, is given by

V_n = \frac{\pi^\frac{n}{2}}{\Gamma(\frac{n}{2} + 1)}

where \Gamma \, is the gamma function, which satisfies \Gamma(1/2) = \sqrt{\pi}; \Gamma(1) = 1; \Gamma(x + 1) = x\Gamma(x).

By multiplying V_n by R^n, differentiating with respect to R, and then setting R = 1, we get the closed form

S_{n-1} = \frac{2\pi^\frac{n}{2}}{\Gamma(\frac{n}{2} )}.

Other relations

The recurrences can be combined to give a "reverse-direction" recurrence relation for surface area, as depicted in the diagram:

S_{n-1} = \frac{n}{2 \pi} S_{n-1+2}
(Note that n refers to the dimension of the ambient Euclidean space, which is also the intrinsic dimension of the solid whose volume is listed here, but which is 1 more than the intrinsic dimension of the sphere whose surface area is listed here.) The curved red arrows show the relationship between formulas for different n. The formula coefficient at each arrow's tip equals the formula coefficient at that arrow's tail times the factor in the arrowhead. If the direction of the bottom arrows were reversed, their arrowheads would say to multiply by 2π/n − 2. Alternatively said, the surface area Sn+1 of the sphere in n + 2 dimensions is exactly 2πR times the volume Vn enclosed by the sphere in n dimensions.

Index-shifting n to n − 2 then yields the recurrence relations:

V_n = \frac{2 \pi}{n} V_{n-2}
S_{n-1} = \frac{2 \pi}{n-2} S_{n-1-2}

where S0 = 2, V1 = 2, S1 = 2π and V2 = π.

The recurrence relation for V_n can also be proved via integration with 2-dimensional polar coordinates:

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

Spherical 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:

\begin{align} 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) \\ &\vdots\\ 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}) \,. \end{align}

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

\begin{align} r &= \sqrt{{x_n}^2 + {x_{n-1}}^2 + \cdots + {x_2}^2 + {x_1}^2} \\ \phi_1 &= \arccot \frac{x_{1}}{\sqrt{{x_n}^2+{x_{n-1}}^2+\cdots+{x_2}^2}} = \arccos \frac{x_{1}}{\sqrt{{x_n}^2+{x_{n-1}}^2+\cdots+{x_1}^2}} \\ \phi_2 &= \arccot \frac{x_{2}}{\sqrt{{x_n}^2+{x_{n-1}}^2+\cdots+{x_3}^2}} = \arccos \frac{x_{2}}{\sqrt{{x_n}^2+{x_{n-1}}^2+\cdots+{x_2}^2}} \\ &\vdots\\ \phi_{n-2} &= \arccot \frac{x_{n-2}}{\sqrt{{x_n}^2+{x_{n-1}}^2}} = \arccos \frac{x_{n-2}}{\sqrt{{x_n}^2+{x_{n-1}}^2+{x_{n-2}}^2}} \\ \phi_{n-1} &= 2\arccot \frac{x_{n-1}+\sqrt{x_n^2+x_{n-1}^2}}{x_n} = \begin{cases} \arccos \frac{x_{n-1}}{\sqrt{{x_n}^2+{x_{n-1}}^2}} & x_n\geq 0 \\ 2 \pi - \arccos \frac{x_{n-1}}{\sqrt{{x_n}^2+{x_{n-1}}^2}} & x_n < 0 \end{cases} \,. \end{align}

where if x_k \ne 0 for some k but all of x_{k+1},\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+1},\ldots,x_n are zero; in this case \phi_k may be chosen to be zero.

Spherical 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:

\left(\begin{smallmatrix} \cos(\phi_{1}) &-r\sin(\phi_{1}) &0 &0&\ \ \cdot \ \ \cdot \ \ \cdot &0 \\ \sin(\phi_{1})\cos(\phi_{2}) &r\cos(\phi_{1})\cos(\phi_{2}) &-r\sin(\phi_{1})\sin(\phi_{2})&0&\ \ \cdot \ \ \cdot \ \ \cdot &0 \\ \vdots & \vdots & \vdots &\ddots & &\vdots\\ & & & & &0 \\ \sin(\phi_{1})...\sin(\phi_{n-2})\cos(\phi_{n-1})& \cdots &\cdots & & &-r\sin(\phi_{1})....\sin(\phi_{n-2})\sin(\phi_{n-1}) \\ \sin(\phi_{1})...\sin(\phi_{n-2})\sin(\phi_{n-1})& r\cos(\phi_{1})...\sin(\phi_{n-1})& \cdots & & &r\sin(\phi_{1})...\sin(\phi_{n-2})\cos(\phi_{n-1}) \end{smallmatrix}\right)
\begin{align} d^nV & = \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_{\phi_{n-1}=0}^{2\pi} \int_{\phi_{n-2}=0}^\pi \cdots \int_{\phi_1=0}^\pi\int_{r=0}^R d^nV. \,

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+j}\Gamma(s+n-j-1)}{s!(2s+n-j-1)\Gamma^2((n-j-1)/2)}\delta_{s,s'} \end{align}

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

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+x_2^2+\cdots+x_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 u1/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 not path-connected. 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 
Also known as the glome. 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.

See also

Notes

  1. ↑ James W. Vick (1994). Homology theory, p. 60. Springer
  2. ↑ Loskot, Pavel (November 2007). "On Monotonicity of the Hypersphere Volume and Area". Journal of Geometry 87 (1-2): 96–98. doi:10.1007/s00022-007-1891-1.

References

External links

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