In mathematics, an n-sphere or a hypersphere 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
and an n-sphere of radius r can be 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.
In particular:
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, which is not even connected, consisting of two points.
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 set of points in (n + 1)-space, (x_{1}, x_{2}, ..., x_{n+1}), that define an n-sphere, S^{n}(r), is represented by the equation:
where c = (c_{1}, c_{2}, ..., c_{n+1}) 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
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.
Specifically:
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} = ℝ^{n} ∪ {∞}, 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 ℝ^{n}. This forms the basis for stereographic projection.^{ [1]}
V_{n}(R) and S_{n}(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 V_{n} and S_{n} (for R = 1, the unit ball and sphere) are related by the recurrences:
The surfaces and volumes can also be given in closed form:
where Γ is the gamma function. Derivations of these equations are given in this section.
The volume of the unit n-ball is maximal in dimension five, where it begins to decrease, and tends to zero as n tends to infinity.^{ [2]} Furthermore, the sum of the volumes of even-dimensional n-balls of radius R can be expressed in closed form:^{ [2]}
For the odd-dimensional analogue,
where erf is the error function.^{ [3]}
The 0-ball consists of a single point. The 0-dimensional Hausdorff measure is the number of points in a set. So,
The 0-sphere consists of its two end-points, {−1, 1}. So,
The unit 1-ball is the interval [−1, 1] of length 2. 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,
So,
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 r^{2} + R^{2} = 1, so that R = sin θ and dR = cos θ dθ. Then,
Since S_{1} = 2π V_{0}, the equation
holds for all n.
This completes the derivation of the recurrences:
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 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) = xΓ(x), and so Γ(x + 1) = x!, and where we conversely define x! = Γ(x + 1) for every x.
By multiplying V_{n} by R^{n}, differentiating with respect to R, and then setting R = 1, we get the closed form
for the (n− 1)-dimensional surface of the sphere S^{n−1}.
The recurrences can be combined to give a "reverse-direction" recurrence relation for surface area, as depicted in the diagram:
Index-shifting n to n − 2 then yields the recurrence relations:
where S_{0} = 2, V_{1} = 2, S_{1} = 2π and V_{2} = π.
The recurrence relation for V_{n} 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 φ_{1}, φ_{2}, ..., φ_{n−1}, where the angles φ_{1}, φ_{2}, ..., φ_{n−2} range over [0, π] radians (or over [0, 180] degrees) and φ_{n−1} ranges over [0, 2π) radians (or over [0, 360) degrees). If x_{i} are the Cartesian coordinates, then we may compute x_{1}, ..., x_{n} from r, φ_{1}, ..., φ_{n−1} with:^{ [4]}
Except in the special cases described below, the inverse transformation is unique:
where if x_{k} ≠ 0 for some k but all of x_{k+1}, ... x_{n} are zero then φ_{k} = 0 when x_{k} > 0, and φ_{k} = π (180 degrees) when x_{k} < 0.
There are some special cases where the inverse transform is not unique; φ_{k} for any k will be ambiguous whenever all of x_{k}, x_{k+1}, ... x_{n} are zero; in this case φ_{k} 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 J_{n} can be constructed from J_{n−1} as follows. Except in column n, rows n − 1 and n of J_{n} are the same as row n − 1 of J_{n−1}, but multiplied by an extra factor of cos φ_{n−1} in row n − 1 and an extra factor of sin φ_{n−1} in row n. In column n, rows n − 1 and n of J_{n} are the same as column n − 1 of row n − 1 of J_{n−1}, but multiplied by extra factors of sin φ_{n−1} in row n − 1 and cos φ_{n−1} in row n, respectively. The determinant of J_{n} can be calculated by Laplace expansion in the final column. By the recursive description of J_{n}, the submatrix formed by deleting the entry at (n − 1, n) and its row and column almost equals J_{n−1}, except that its last row is multiplied by sin φ_{n−1}. Similarly, the submatrix formed by deleting the entry at (n, n) and its row and column almost equals J_{n−1}, except that its last row is multiplied by cos φ_{n−1}. Therefore the determinant of J_{n} 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 e^{isφj} for the angle j = n − 1 in concordance with the spherical harmonics.
The standard spherical coordinate system arises from writing ℝ^{n} as the product ℝ × ℝ^{n−1}. These two factors may be related using polar coordinates. For each point x of ℝ^{n}, the standard Cartesian coordinates
can be transformed into a mixed polar–Cartesian coordinate system:
This says that points in ℝ^{n} may be expressed by taking the ray starting at the origin and passing through , rotating it towards by , and traveling a distance along the ray. Repeating this decomposition eventually leads to the standard spherical coordinate system.
Polyspherical coordinate systems arise from a generalization of this construction.^{ [5]} The space ℝ^{n} 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 ℝ^{n} = ℝ^{p} × ℝ^{q}. Using this decomposition, a point x ∈ ℝ^{n} may be written as
This can be transformed into a mixed polar–Cartesian coordinate system by writing:
Here and are the unit vectors associated to y and z. This expresses x in terms of , , r ≥ 0, and an angle θ. It can be shown that the domain of θ is [0, 2π) if p = q = 1, [0, π] if exactly one of p and q is 1, and [0, π/2] if neither p nor q are 1. The inverse transformation is
These splittings may be repeated as long as one of the factors involved has dimension two or greater. A polyspherical coordinate system is the result of repeating these splittings until there are no Cartesian coordinates left. Splittings after the first do not require a radial coordinate because the domains of and are spheres, so the coordinates of a polyspherical coordinate system are a non-negative radius and n − 1 angles. The possible polyspherical coordinate systems correspond to binary trees with n leaves. Each non-leaf node in the tree corresponds to a splitting and determines an angular coordinate. For instance, the root of the tree represents ℝ^{n}, and its immediate children represent the first splitting into ℝ^{p} and ℝ^{q}. Leaf nodes correspond to Cartesian coordinates for S^{n−1}. The formulas for converting from polyspherical coordinates to Cartesian coordinates may be determined by finding the paths from the root to the leaf nodes. These formulas are products with one factor for each branch taken by the path. For a node whose corresponding angular coordinate is θ_{i}, taking the left branch introduces a factor of sin θ_{i} and taking the right branch introduces a factor of cos θ_{i}. The inverse transformation, from polyspherical coordinates to Cartesian coordinates, is determined by grouping nodes. Every pair of nodes having a common parent can be converted from a mixed polar–Cartesian coordinate system to a Cartesian coordinate system using the above formulas for a splitting.
Polyspherical coordinates also have an interpretation in terms of the special orthogonal group. A splitting ℝ^{n} = ℝ^{p} × ℝ^{q} determines a subgroup
This is the subgroup that leaves each of the two factors fixed. Choosing a set of coset representatives for the quotient is the same as choosing representative angles for this step of the polyspherical coordinate decomposition.
In polyspherical coordinates, the volume measure on ℝ^{n} and the area measure on S^{n−1} are products. There is one factor for each angle, and the volume measure on ℝ^{n} also has a factor for the radial coordinate. The area measure has the form:
where the factors F_{i} are determined by the tree. Similarly, the volume measure is
Suppose we have a node of the tree that corresponds to the decomposition ℝ^{n1+n2} = ℝ^{n1} × ℝ^{n2} and that has angular coordinate θ. The corresponding factor F depends on the values of n_{1} and n_{2}. When the area measure is normalized so that the area of the sphere is 1, these factors are as follows. If n_{1} = n_{2} = 1, then
If n_{1} > 1 and n_{2} = 1, and if B denotes the beta function, then
If n_{1} = 1 and n_{2} > 1, then
Finally, if both n_{1} and n_{2} are greater than one, 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 S^{n} of radius 1 will map to the (n − 1)-dimensional hyperplane ℝ^{n−1} perpendicular to the x_{n}-axis as
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 = (x_{1}, x_{2}, ..., x_{n}). Now calculate the "radius" of this point:
The vector 1/rx is uniformly distributed over the surface of the unit n-ball.
An alternative given by Marsaglia is to uniformly randomly select a point x = (x_{1}, x_{2}, ..., x_{n}) in the unit n-cube by sampling each x_{i} independently from the uniform distribution over (–1, 1), computing r as above, and rejecting the point and resampling if r ≥ 1 (i.e., if the point is not in the n-ball), and when a point in the ball is obtained scaling it up to the spherical surface by the factor 1/r; then again 1/rx is uniformly distributed over the surface of the unit n-ball. This method becomes very inefficient for higher dimensions, as a vanishingly small fraction of the unit cube is contained in the sphere. In ten dimensions, less than 2% of the cube is filled by the sphere, so that typically more than 50 attempts will be needed. In seventy dimensions, less than of the cube is filled, meaning typically a trillion quadrillion trials will be needed, far more than a computer could ever carry out.
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 u^{1/n} x 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 (x_{1}, x_{2}, ..., x_{n+2}) is a point selected uniformly from the unit (n + 1)-sphere, then (x_{1}, x_{2}, ..., x_{n}) is uniformly distributed within the unit n-ball (i.e., by simply discarding two coordinates).^{ [6]}
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.
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The octahedral n-sphere is defined similarly to the n-sphere but using the 1-norm
In general, it takes the shape of a cross-polytope.
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.^{ [9]} 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.
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