N-sphere


In mathematics, an -sphere is a topological space that is homeomorphic to a standard -sphere, which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, called the center. It is the generalization of an ordinary sphere in the ordinary three-dimensional space. The "radius" of a sphere is the constant distance of its points to the center. When the sphere has unit radius, it is usual to call it the unit -sphere or simply the -sphere for brevity. In terms of the standard norm, the -sphere is defined as
and an -sphere of radius can be defined as
The 0-sphere is a pair of points on the line, the 1-sphere is a circle in the plane, and the 2-sphere is an ordinary sphere within 3-dimensional space.
The dimension of -sphere is, and must not be confused with the dimension of the Euclidean space in which it is naturally embedded. An -sphere is the surface or boundary of an -dimensional ball.
In particular:
For, the -spheres that are differential manifolds can be characterized as the simply connected -dimensional manifolds of constant, positive curvature. The -spheres admit several other topological descriptions: for example, they can be constructed by gluing two -dimensional Euclidean spaces together, by identifying the boundary of an -cube with a point, or by forming the suspension of an -sphere. The 1-sphere is the 1-manifold that is a circle, which is not simply connected. The 0-sphere is the 0-manifold consisting of two points, which is not even connected.

Description

For any natural number, an -sphere of radius is defined as the set of points in -dimensional Euclidean space that are at distance from some fixed point, where may be any positive real number and where may be any point in -dimensional space. In particular:
The set of points in -space,, that define an -sphere,, is represented by the equation:
where is a center point, and is the radius.
The above -sphere exists in -dimensional Euclidean space and is an example of an -manifold. The volume form of an -sphere of radius is given by
where is the Hodge star operator; see for a discussion and proof of this formula in the case. As a result,

-ball

The space enclosed by an -sphere is called an -ball. An -ball is closed if it includes the -sphere, and it is open if it does not include the -sphere.
Specifically:
, an -sphere can be constructed as a one-point compactification of -dimensional Euclidean space. Briefly, the -sphere can be described as, which is -dimensional Euclidean space plus a single point representing infinity in all directions.
In particular, if a single point is removed from an -sphere, it becomes homeomorphic to. This forms the basis for stereographic projection.

Volume and surface area


and are the -dimensional volume of the -ball and the surface area of the -sphere embedded in dimension, respectively, of radius.
The constants and are related by the recurrences:
The surfaces and volumes can also be given in closed form:
where is the gamma function. Derivations of these equations are given in this section.

In general, the volume of the -ball in -dimensional Euclidean space, and the surface area of the -sphere in -dimensional Euclidean space, of radius, are proportional to the th power of the radius, . We write for the volume of the -ball and for the surface area of the -sphere, both of radius, where and are the values for the unit-radius case.

In theory, one could compare the values of and for. However, this is not well-defined. For example, if and then the comparison is like comparing a number of square meters to a different number of cubic meters. The same applies to a comparison of and for.

Examples

The 0-ball consists of a single point. The 0-dimensional Hausdorff measure is the number of points in a set. So,
The unit 1-ball is the interval of length 2. So,
The 0-sphere consists of its two end-points,. So,
The unit 1-sphere is the unit circle in the Euclidean plane, and this has circumference
The region enclosed by the unit 1-sphere is the 2-ball, or unit disc, and this has area
Analogously, in 3-dimensional Euclidean space, the surface area of the unit 2-sphere is given by
and the volume enclosed is the volume of the unit 3-ball, given by

Recurrences

The surface area, or properly the -dimensional volume, of the -sphere at the boundary of the -ball of radius is related to the volume of the ball by the differential equation
or, equivalently, representing the unit -ball as a union of concentric -sphere shells,
So,
We can also represent the unit -sphere as a union of tori, each the product of a circle with an -sphere. Let and, so that and. Then,
Since, the equation
holds for all.
This completes the derivation of the recurrences:

Closed forms

Combining the recurrences, we see that
So it is simple to show by induction on k that,
where denotes the double factorial, defined for odd natural numbers by and similarly for even numbers.
In general, the volume, in -dimensional Euclidean space, of the unit -ball, is given by
where is the gamma function, which satisfies,, and, and so, and where we conversely define x! = for any x.
By multiplying by, differentiating with respect to, and then setting, we get the closed form
for the -dimensional volume of the sphere Sn-1.

Other relations

The recurrences can be combined to give a "reverse-direction" recurrence relation for surface area, as depicted in the diagram:
Index-shifting to then yields the recurrence relations:
where,, and.
The recurrence relation for can also be proved via integration with 2-dimensional polar coordinates:

Spherical coordinates

We may define a coordinate system in an -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, and angular coordinates, where the angles range over radians and ranges over radians. If are the Cartesian coordinates, then we may compute from with:
Except in the special cases described below, the inverse transformation is unique:
where if for some but all of are zero then when, and when.
There are some special cases where the inverse transform is not unique; for any will be ambiguous whenever all of are zero; in this case may be chosen to be zero.

Spherical volume and area elements

Expressing the angular measures in radians, the volume element in -dimensional Euclidean space will be found from the Jacobian of the transformation:
and the above equation for the volume of the -ball can be recovered by integrating:
Similarly the surface area element of the -sphere, which generalizes the area element of the 2-sphere, is given by
The natural choice of an orthogonal basis over the angular coordinates is a product of ultraspherical polynomials,
for, and the for the angle 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 -sphere can be mapped onto an -dimensional hyperplane by the -dimensional version of the stereographic projection. For example, the point on a two-dimensional sphere of radius 1 maps to the point on the -plane. In other words,
Likewise, the stereographic projection of an -sphere of radius 1 will map to the -dimensional hyperplane perpendicular to the -axis as

Generating random points

Uniformly at random on the -sphere

To generate uniformly distributed random points on the unit -sphere, gives the following algorithm.
Generate an -dimensional vector of normal deviates,. Now calculate the "radius" of this point:
The vector is uniformly distributed over the surface of the unit -ball.
An alternative given by Marsaglia is to uniformly randomly select a point in the unit -cube by sampling each independently from the uniform distribution over, computing as above, and rejecting the point and resampling if , and when a point in the ball is obtained scaling it up to the spherical surface by the factor ; then again is uniformly distributed over the surface of the unit -ball.

Uniformly at random within the -ball

With a point selected uniformly at random from the surface of the unit -sphere, one needs only a radius to obtain a point uniformly at random from within the unit -ball. If is a number generated uniformly at random from the interval and is a point selected uniformly at random from the unit -sphere, then is uniformly distributed within the unit -ball.
Alternatively, points may be sampled uniformly from within the unit -ball by a reduction from the unit -sphere. In particular, if is a point selected uniformly from the unit -sphere, then is uniformly distributed within the unit -ball.
If is sufficiently large, most of the volume of the -ball will be contained in the region very close to its surface, so a point selected from that volume will also probably be close to the surface. This is one of the phenomena leading to the so-called curse of dimensionality that arises in some numerical and other applications.

Specific spheres

; 0-sphere : The pair of points with the discrete topology for some. The only sphere that is not path-connected. Has a natural Lie group structure; isomorphic to O. Parallelizable.
; 1-sphere : Also known as the circle. Has a nontrivial fundamental group. Abelian Lie group structure U; the circle group. Topologically equivalent to the real projective line, RP1. Parallelizable. SO = U.
; 2-sphere : Also known as the sphere. Complex structure; see Riemann sphere. Equivalent to the complex projective line, CP1. SO/SO.
; 3-sphere : Also known as the glome. Parallelizable, principal U-bundle over the 2-sphere, Lie group structure Sp, where also
; 4-sphere : Equivalent to the quaternionic projective line, HP1. SO/SO.
; 5-sphere : Principal U-bundle over CP2. SO/SO = SU/SU.
; 6-sphere : Possesses an almost complex structure coming from the set of pure unit octonions. SO/SO = G2/SU. The question of whether it has a complex structure is known as the Hopf problem, after Heinz Hopf.
; 7-sphere : Topological quasigroup structure as the set of unit octonions. Principal Sp-bundle over S4. Parallelizable. SO/SO = SU/SU = Sp/Sp = Spin/G2 = Spin/SU. 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.