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. It is an n-dimensional manifold in Euclidean (n + 1)-space. In particular, a 0-sphere is a pair of points on a line, 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. In symbols:

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

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 manifold 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 which are at distance r from a fixed point, where r may be any positive real number. In particular:

a 0-sphere is a pair of points {p − r, p + r} containing a line segment.
a 1-sphere is a circle of radius r. These contain disks.
a 2-sphere is an ordinary sphere in 3-dimensional Euclidean space that contains a 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, (S^{n) 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 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

The space enclosed by 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.

Volume and surface area

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}=C_{n}R^{n}

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

C_{n}={\frac {\pi ^{\frac {n}{2}}}{\Gamma ({\frac {n}{2}}+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 for odd n,

C_{n}=2^{(n+1)/2}{\frac {\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 ^{\frac {n}{2}}R^{n-1} \over \Gamma ({\frac {n}{2}})}={nC_{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+1}/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}.

Conventionally, these formulas can also be proven directly by integrating by slabs, using:

V_{n}=\int _{-R}^{R}{V_{n-1}{\sqrt {R^{2}-x^{2}}}\,dx}.

Examples

For small values of n, the volumes, V_n, of the n-ball of radius R are:

$V_{0}\,$ (point)	${}=1\,$
$V_{1}\,$ (line segment)	${}=2\,R$
$V_{2}\,$ (disk)	${}=\pi \,R^{2}$	${}=3.14159\ldots \,R^{2}$
$V_{3}\,$ (ball)	${}={\frac {4\pi }{3}}\,R^{3}$	${}=4.18879\ldots \,R^{3}$
$V_{4}\,$	${}={\frac {\pi ^{2}}{2}}\,R^{4}$	${}=4.93480\ldots \,R^{4}$
$V_{5}\,$	${}={\frac {8\pi ^{2}}{15}}\,R^{5}$	${}=5.26379\ldots \,R^{5}$
$V_{6}\,$	${}={\frac {\pi ^{3}}{6}}\,R^{6}$	${}=5.16771\ldots \,R^{6}$
$V_{7}\,$	${}={\frac {16\pi ^{3}}{105}}\,R^{7}$	${}=4.72477\ldots \,R^{7}$
$V_{8}\,$	${}={\frac {\pi ^{4}}{24}}\,R^{8}$	${}=4.05871\ldots \,R^{8}$
$\lim _{n\rightarrow \infty }{\frac {V_{n}}{R^{n}}}\,$	${}=0.\,$

If the dimension n is not limited to integer values, the n-sphere volume is a continuous function of n with a global maximum for the unit sphere in "dimension" n = 5.2569464...^[1] where the "volume" is 5.277768... It has a hypervolume of 1 when n = 0 or when n = 12.76405...

The unit hypercube has an edge length of 1, and thus a volume of 1, regardless of the number of dimensions, while 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 n-sphere to the unit hypercube reaches a maximum between n=5 and n=6, while the ratio of the volume of the n-sphere 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 r, and n − 1 angular coordinates $\ \phi _{1},\phi _{2},\dots ,\phi _{n-1}$ . If $\ x_{i}$ are the Cartesian coordinates, then we may define

{\begin{aligned}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{aligned}}

While the inverse transformations can be derived from those above:

{\begin{aligned}\tan(\phi _{n-1})&={\frac {x_{n}}{x_{n-1}}}\\\tan(\phi _{n-2})&={\frac {\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}}}{x_{n-2}}}\\&{}\,\,\,\vdots \\\tan(\phi _{1})&={\frac {\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}+\cdots +{x_{2}}^{2}}}{x_{1}}}.\end{aligned}}

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

The volume element in n-dimensional Euclidean space will be found from the Jacobian of the transformation:

d_{\mathbb {R} ^{n}}V=\left|\det {\frac {\partial (x_{i})}{\partial (r,\phi _{j})}}\right|dr\,d\phi _{1}\,d\phi _{2}\ldots d\phi _{n-1}

=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}

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}\ldots d\phi _{n-1}.

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 points on the surface of the n-ball

To generate points on 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 n-ball.

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.

Another way to generate a random distribution on a hypersphere is to make a uniform one over a hypercube that includes the unit hypersphere, exclude those points that are outside the hypersphere, 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 hypersphere to the hypercube decreases very rapidly with dimension it will succeed with high probability only for fairly small numbers of dimensions.

Specific spheres

0-sphere: The pair of points {±1} with the discrete topology. The only sphere which is disconnected. Has a natural Lie group structure; isomorphic to O(1). Parallelizable.
1-sphere: Also known as the unit 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: Complex structure; see Riemann sphere. Equivalent to the complex projective line, CP¹. SO(3)/SO(2).
3-sphere: Lie group structure Sp(1). Principal U(1)-bundle over the 2-sphere. Parallelizable. SO(4)/SO(3) = SU(2) = Sp(1) = 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).
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

^ (sequence A074455 in the OEIS).

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 (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: 645–646.

External links

[1] (sequence A074455 in the OEIS).

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