*n*-sphere

In mathematics, an ** n-sphere** is a topological space that is homeomorphic to the set of points in (

*n*+ 1)-dimensional Euclidean space, that are situated at a constant distance r of 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**or simply

*n*-sphere**the**for brevity. In terms of the standard norm, the

*n*-sphere*n*-sphere is defined as

and an *n*-sphere of radius r can be defined as

The 0-sphere is a pair of points on the line at unit distance from the origin, the 1-sphere is a circle in the plane, and the 2-sphere is an ordinary sphere within 3-dimensional space.

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:

- the pair of points at the ends of a (one-dimensional) line segment is a 0-sphere,
- a circle, which is the one-dimensional circumference of a (two-dimensional) disk, is a 1-sphere,
- the two-dimensional surface of a (three-dimensional) ball in three-dimensional space is a 2-sphere, often simply called a sphere,
- the three-dimensional boundary of a (four-dimensional) 4-ball in four-dimensional Euclidean is a 3-sphere, also known as a
*glome*. - the
*n*– 1 dimensional boundary of a (n-dimensional) n-ball is a*n*– 1-sphere.

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.

## DescriptionEdit

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:

- a 0-sphere is a pair of points {
*c*−*r*,*c*+*r*}, and is the boundary of a line segment (1-ball). - a 1-sphere is a circle of radius
*r*centered at**c**, and is the boundary of a disk (2-ball). - a 2-sphere is an ordinary 3-dimensional sphere in 3-dimensional Euclidean space, and is the boundary of an ordinary ball (3-ball).
- a 3-sphere is a sphere in 4-dimensional Euclidean space.

### Euclidean coordinates in (*n* + 1)-spaceEdit

The set of points in (*n* + 1)-space, (*x*_{1}, *x*_{2}, ..., *x*_{n+1}), that define an *n*-sphere, , 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,

*n*-ballEdit

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:

- 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.

### Topological descriptionEdit

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]}

## Volume and surface areaEdit

*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*(for

_{n}*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.

*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

*n*th power of the radius,

*R*(with different constants of proportionality that vary with

*n*). We write

*V*(

_{n}*R*) =

*V*for the volume of the

_{n}R^{n}*n*-ball and

*S*(

_{n}*R*) =

*S*for the surface area of the

_{n}R^{n}*n*-sphere, both of radius

*R*, where

*V*=

_{n}*V*(1) and

_{n}*S*=

_{n}*S*(1) are the values for the unit-radius case.

_{n}In theory, one could compare the values of *S _{n}*(

*R*) and

*S*(

_{m}*R*) for

*n*≠

*m*. However, this is somewhat nonsensical. 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 lack of sense applies to a comparison of

*V*(

_{n}*R*) and

*V*(

_{m}*R*) for

*n*≠

*m*.

### ExamplesEdit

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 [−1,1] of length 2. So,

The 0-sphere consists of its two end-points, {−1,1}. 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

### RecurrencesEdit

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 tori, each the product 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:

### Closed formsEdit

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 2*k* + 1 by (2*k* + 1)!! = 1 × 3 × 5 × ... × (2*k* − 1) × (2*k* + 1) and similarly for even numbers (2*k*)!! = 2 × 4 × 6 × ... × (2*k* − 2) × (2*k*).

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*).

By multiplying *V _{n}* by

*R*, differentiating with respect to

^{n}*R*, and then setting R = 1, we get the closed form

### Other relationsEdit

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:

## Spherical coordinatesEdit

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*from

_{n}*r*,

*φ*

_{1}, ...

*φ*

_{n−1}with:

^{[2]}

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*are zero then

_{n}*φ*= 0 when

_{k}*x*> 0, and

_{k}*φ*= π (180 degrees) when

_{k}*x*< 0.

_{k}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*are zero; in this case

_{n}*φ*may be chosen to be zero.

_{k}### Spherical volume and area elementsEdit

Expressing the angular measures in radians, the volume element in *n*-dimensional Euclidean space will be found from the Jacobian of the transformation:

and the above equation for the volume of the *n*-ball can be recovered by integrating:

Similarly the surface area element of the (*n* − 1)-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 *j* = 1, 2,... *n* − 2, and the *e*^{isφj} for the angle *j* = *n* − 1 in concordance with the spherical harmonics.

## Stereographic projectionEdit

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−1} of radius 1 will map to the (*n* − 1)-dimensional hyperplane **R**^{n−1} perpendicular to the *x _{n}*-axis as

## Generating random pointsEdit

### Uniformly at random on the (*n* − 1)-sphereEdit

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/*r***x** 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/

*r*

**x**is uniformly distributed over the surface of the unit

*n*-ball.

### Uniformly at random within the *n*-ballEdit

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).^{[3]}

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 spheresEdit

- 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,
**R**P^{1}. Parallelizable. SO(2) = U(1). - 2-sphere
- Also known as the sphere. Complex structure; see Riemann sphere. Equivalent to the complex projective line,
**C**P^{1}. 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,
**H**P^{1}. SO(5)/SO(4). - 5-sphere
- Principal U(1)-bundle over
**C**P^{2}. 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) =
*G*_{2}/SU(3). The question of whether it has a complex structure is known as the*Hopf problem,*after Heinz Hopf.^{[4]} - 7-sphere
- Topological quasigroup structure as the set of unit octonions. Principal Sp(1)-bundle over
*S*^{4}. Parallelizable. SO(8)/SO(7) = SU(4)/SU(3) = Sp(2)/Sp(1) = Spin(7)/*G*_{2}= 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
**O**P^{1}. - 23-sphere
- A highly dense sphere-packing is possible in 24-dimensional space, which is related to the unique qualities of the Leech lattice.

## See alsoEdit

## NotesEdit

**^**James W. Vick (1994).*Homology theory*, p. 60. Springer**^**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.**^**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.**^**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.

## ReferencesEdit

- 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".
*Annals of Mathematical Statistics*.**43**(2): 645–646. doi:10.1214/aoms/1177692644. - Huber, Greg (1982). "Gamma function derivation of n-sphere volumes".
*Amer. Math. Monthly*.**89**(5): 301–302. doi:10.2307/2321716. JSTOR 2321716. MR 1539933. - Barnea, Nir (1999). "Hyperspherical functions with arbitrary permutational symmetry: Reverse construction".
*Phys. Rev. A*.**59**(2): 1135–1146. doi:10.1103/PhysRevA.59.1135.