N-sphere

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2-sphere wireframe as an orthogonal projection Sphere wireframe.svg
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 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). Hypersphere coord.PNG
Just as a stereographic projection can project a sphere's surface to a plane, it can also project 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, an n-sphere is a topological space that is homeomorphic to a standardn-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

Contents

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 consisting of two points, which is not even connected.

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, , is represented by the equation:

where c = (c1, c2, ..., cn+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,

n-ball

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 Sn = ℝn ∪ {∞}, 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 n. This forms the basis for stereographic projection. [1]

Volume and surface area

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:

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.

Graphs of volumes (V) and surface areas (S) of n-balls of radius 1. In the SVG file, hover over a point to highlight it and its value. Hypersphere volume and surface area graphs.svg
Graphs of volumes  (V) and surface areas  (S) of n-balls of radius 1. In the SVG file, hover over a point to highlight it and its value.
In general, the volume of the n-ball in n-dimensional Euclidean space, and the surface area of the n-sphere in (n + 1)-dimensional Euclidean space, of radius R, are proportional to the nth power of the radius, R (with different constants of proportionality that vary with n). We write Vn(R) = VnRn for the volume of the n-ball and Sn(R) = SnRn for the surface area of the n-sphere, both of radius R, where Vn = Vn(1) and Sn = Sn(1) are the values for the unit-radius case.

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.

Examples

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

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

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 r2 + R2 = 1, so that R = sin θ and dR = cos θ. Then,

Since S1 = 2π V0, the equation

holds for all n.

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

for the (n  1)-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:

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 (where the n in the arrowhead refers to the n value that the arrowhead points to). If the direction of the bottom arrows were reversed, their arrowheads would say to multiply by
2p/n - 2. Alternatively said, the surface area Sn+1 of the sphere in n + 2 dimensions is exactly 2pR times the volume Vn enclosed by the sphere in n dimensions. N SpheresVolumeAndSurfaceArea.png
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 (where the n in the arrowhead refers to the n value that the arrowhead points to). If the direction of the bottom arrows were reversed, their arrowheads would say to multiply by /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:

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

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

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 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.

Spherical volume and area elements

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.

Polyspherical coordinates

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

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

This says that points in n may be expressed by taking the ray starting at the origin and passing through z ∈ ℝn 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 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 n = ℝp×q. Using this decomposition, a point x ∈ ℝn 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 n, and its immediate children represent the first splitting into p and q. Leaf nodes correspond to Cartesian coordinates for Sn 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 θi, taking the left branch introduces a factor of sin θi and taking the right branch introduces a factor of cos θi. 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 n = ℝp×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 n and the area measure on Sn 1 are products. There is one factor for each angle, and the volume measure on n 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 n1 + n2 = ℝn1×n2 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

If n1 > 1 and n2 = 1, and if B denotes the beta function, then

If n1 = 1 and n2 > 1, then

Finally, if both n1 and n2 are greater than one, then

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 [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 n−1 perpendicular to the xn-axis as

Generating random points

Uniformly at random on the (n − 1)-sphere

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

An alternative given by Marsaglia is to uniformly randomly select a point x = (x1, x2,... xn) in the unit n-cube by sampling each xi 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.

Uniformly at random within the 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.

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
.
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
Possesses an almost complex structure coming from the set of pure unit octonions. SO(7)/SO(6) = G2/SU(3). The question of whether it has a complex structure is known as the Hopf problem, after Heinz Hopf. [5]
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.

Octahedral sphere

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.

See also

Notes

  1. James W. Vick (1994). Homology theory, p. 60. Springer
  2. Blumenson, L. E. (1960). "A Derivation of n-Dimensional Spherical Coordinates". The American Mathematical Monthly. 67 (1): 63–66. doi:10.2307/2308932. JSTOR   2308932.
  3. N. Ja. Vilenkin and A. U. Klimyk, Representation of Lie groups and special functions, Vol. 2: Class I representations, special functions, and integral transforms, translated from the Russian by V. A. Groza and A. A. Groza, Math. Appl., vol. 74, Kluwer Acad. Publ., Dordrecht, 1992, ISBN   0-7923-1492-1, pp. 223–226.
  4. Voelker, Aaron R.; Gosmann, Jan; Stewart, Terrence C. (2017). Efficiently sampling vectors and coordinates from the n-sphere and n-ball (Report). Centre for Theoretical Neuroscience. doi:10.13140/RG.2.2.15829.01767/1.
  5. Agricola, Ilka; Bazzoni, Giovanni; Goertsches, Oliver; Konstantis, Panagiotis; Rollenske, Sönke (2018). "On the history of the Hopf problem". Differential Geometry and Its Applications . 57: 1–9. arXiv: 1708.01068 . doi:10.1016/j.difgeo.2017.10.014. S2CID   119297359.
  6. Meshulam, Roy (2001-01-01). "The Clique Complex and Hypergraph Matching". Combinatorica. 21 (1): 89–94. doi:10.1007/s004930170006. ISSN   1439-6912. S2CID   207006642.

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References