n-sphere

In mathematics, an n-sphere is a topological space that is homeomorphic to a standard n-sphere, which is the set of points in (n + 1)-dimensional Euclidean space that are situated at a constant distance r 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 n-sphere or simply the n-sphere for brevity. In terms of the standard norm, the n-sphere is defined as

The dimension of n-sphere is n, and must not be confused with the dimension (n + 1) of the Euclidean space in which it is naturally embedded. An n-sphere is the surface or boundary of an (n + 1)-dimensional ball.

For n ≥ 2, the n-spheres that are differential manifolds can be characterized (up to a diffeomorphism) as 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. 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.

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:

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

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,

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.

Vn(R) and Sn(R) are the n-dimensional volume of the n-ball and the surface area of the n-sphere embedded in dimension n + 1, respectively, of radius R.

The constants Vn and Sn (for R = 1, the unit ball and sphere) are related by the recurrences:

where Γ is the gamma function. Derivations of these equations are given in this section.

In theory, one could compare the values of Sn(R) and Sm(R) for nm. However, this is not well-defined. For example, if n = 2 and m = 3 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 Vn(R) and Vm(R) for nm.

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

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

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

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

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

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

We can also represent the unit (n + 2)-sphere as a union of products of a circle (1-sphere) with an n-sphere. Let r = cos θ and r2 + R2 = 1, so that R = sin θ and dR = cos θ . Then,

where !! denotes the double factorial, defined for odd natural numbers 2k + 1 by (2k + 1)!! = 1 × 3 × 5 × ... × (2k − 1) × (2k + 1) and similarly for even numbers (2k)!! = 2 × 4 × 6 × ... × (2k − 2) × (2k).

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

where Γ is the gamma function, which satisfies Γ(1/2) = π, Γ(1) = 1, and Γ(x + 1) = (x), and so Γ(x + 1) = x!, and where we conversely define x! = Γ(x + 1) for any x.

By multiplying Vn by Rn, differentiating with respect to R, and then setting R = 1, we get the closed form

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

The recurrence relation for Vn can also be proved via integration with 2-dimensional polar 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 φ1, φ2, ... φn−1, where the angles φ1, φ2, ... φn−2 range over [0,π] radians (or over [0,180] degrees) and φn−1 ranges over [0,2π) radians (or over [0,360) degrees). If xi are the Cartesian coordinates, then we may compute x1, ... xn from r, φ1, ... φn−1 with:[2]

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

where if xk ≠ 0 for some k but all of xk+1, ... xn are zero then φk = 0 when xk > 0, and φk = π (180 degrees) when xk < 0.

There are some special cases where the inverse transform is not unique; φk for any k will be ambiguous whenever all of xk, xk+1, ... xn are zero; in this case φk may be chosen to be zero.

To express the volume element of n-dimensional Euclidean space in terms of spherical coordinates, first observe that the Jacobian matrix of the transformation is:

The determinant of this matrix can be calculated by induction. When n = 2, a straightforward computation shows that the determinant is r. For larger n, observe that Jn can be constructed from Jn − 1 as follows. Except in column n, rows n − 1 and n of Jn are the same as row n − 1 of Jn − 1, but multiplied by an extra factor of cos φn − 1 in row n − 1 and an extra factor of sin φn − 1 in row n. In column n, rows n − 1 and n of Jn are the same as column n − 1 of row n − 1 of Jn − 1, but multiplied by extra factors of sin φn − 1 in row n − 1 and cos φn − 1 in row n, respectively. The determinant of Jn can be calculated by Laplace expansion in the final column. By the recursive description of Jn, the submatrix formed by deleting the entry at (n − 1, n) and its row and column almost equals Jn − 1, except that its last row is multiplied by sin φn − 1. Similarly, the submatrix formed by deleting the entry at (n, n) and its row and column almost equals Jn − 1, except that its last row is multiplied by cos φn − 1. Therefore the determinant of Jn is

Induction then gives a closed-form expression for the volume element in spherical coordinates

The formula for the volume of the n-ball can be derived from this by integration.

Similarly the surface area element of the (n − 1)-sphere of radius R, 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 j = 1, 2,... n − 2, and the eisφj for the angle j = n − 1 in concordance with the spherical harmonics.

The standard spherical coordinate system arises from writing Rn as the product R × Rn − 1. These two factors may be related using polar coordinates. For each point x of Rn, the standard Cartesian coordinates

This says that points in Rn may be expressed by taking the ray starting at the origin and passing through zRn − 1, rotating it towards the first basis vector by θ, and traveling a distance r along the ray. Repeating this decomposition eventually leads to the standard spherical coordinate system.

Polyspherical coordinate systems arise from a generalization of this construction.[3] The space Rn is split as the product of two Euclidean spaces of smaller dimension, but neither space is required to be a line. Specifically, suppose that p and q are positive integers such that n = p + q. Then Rn = Rp × Rq. Using this decomposition, a point xRn may be written as

This can be transformed into a mixed polar–Cartesian coordinate system by writing:

Polyspherical coordinates also have an interpretation in terms of the special orthogonal group. A splitting Rn = Rp × Rq determines a subgroup

In polyspherical coordinates, the volume measure on Rn and the area measure on Sn − 1 are products. There is one factor for each angle, and the volume measure on Rn also has a factor for the radial coordinate. The area measure has the form:

where the factors Fi are determined by the tree. Similarly, the volume measure is

Suppose we have a node of the tree that corresponds to the decomposition Rn1 + n2 = Rn1 × Rn2 and that has angular coordinate θ. The corresponding factor F depends on the values of n1 and n2. When the area measure is normalized so that the area of the sphere is 1, these factors are as follows. If n1 = n2 = 1, then

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 [ x/1 − z, y/1 − z] on the xy-plane. In other words,

Likewise, the stereographic projection of an n-sphere Sn−1 of radius 1 will map to the (n − 1)-dimensional hyperplane Rn−1 perpendicular to the xn-axis as

A set of uniformly distributed points on the surface of a unit 2-sphere generated using Marsaglia's algorithm.

To generate uniformly distributed random points on the unit (n − 1)-sphere (that is, the surface of the unit 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), x = (x1, x2,... xn). Now calculate the "radius" of this point:

The vector 1/rx is uniformly distributed over the surface of the unit n-ball.

With a point selected uniformly at random from the surface of the unit (n - 1)-sphere (e.g., by using Marsaglia's algorithm), one needs only a radius to obtain a point uniformly at random from within the unit 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 unit (n - 1)-sphere, then u1nx is uniformly distributed within the unit n-ball.

Alternatively, points may be sampled uniformly from within the unit n-ball by a reduction from the unit (n + 1)-sphere. In particular, if (x1,x2,...,xn+2) is a point selected uniformly from the unit (n + 1)-sphere, then (x1,x2,...,xn) is uniformly distributed within the unit n-ball (i.e., by simply discarding two coordinates).[4]

If n is sufficiently large, most of the volume of the n-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.

The octahedral n-sphere is defined similarly to the n-sphere but using the 1-norm

The octahedral 1-sphere is a square (without its interior). The octahedral 2-sphere is a regular octahedron; hence the name. The octahedral n-sphere is the topological join of n+1 pairs of isolated points.[6] Intuitively, the topological join of two pairs is generated by drawing a segment between each point in one pair and each point in the other pair; this yields a square. To join this with a third pair, draw a segment between each point on the square and each point in the third pair; this gives a octahedron.