N sphere

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:

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 2-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)-space

The set of points in (n + 1)-space: (x₁,x₂,…,x_n+1) that define an n-sphere (Sⁿ), is represented by the equation:

r 2 = ∑ i = 1 n + 1 ( x i − c i ) 2 .

where c 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 ppl volume form ω of an n-sphere of radius r is given by

ω = 1 r ∑ j = 1 n + 1 ( − 1 ) j − 1 x j d x 1 ∧ ⋯ ∧ d x j − 1 ∧ d x j + 1 ∧ ⋯ ∧ d x n + 1 = ∗ d r

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,

d r ∧ ω = d x 1 ∧ ⋯ ∧ d x n + 1 .

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 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 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.

Volume and surface area

In general, the volumes of the n-ball in n-dimensional Euclidean space, and the n-sphere in (n + 1)-dimensional Euclidean, of radius R, are proportional to the nth power of the radius, R. We write V n ( R ) = V n R n for the volume of the n-ball and S n ( R ) = S n R n for the surface of the n-sphere, both of radius R .

Interestingly, given the radius R, the volume and the surface area of the n-sphere reaches a maximum and then decrease towards zero as the dimension n increases. In particular, the volume V n ( R ) of the n-sphere of constant radius R in n-dimensions reaches a maximum for dimension n ∗ = n if V n − 1 < R < V n and n ∗ = { n , n + 1 } if R = V n where V n = Γ ( n / 2 + 3 / 2 ) π Γ ( n / 2 + 1 ) for n = 1 , 2 , … . Similarly, defining the sequence { A n = V n − 2 } , the surface area S n ( R ) of the n-sphere of constant radius R in n dimensions reaches a maximum for dimension n ∗ = n if A n − 1 < R < A n and n ∗ = { n , n + 1 } if R = A n .

Examples

The 0-ball consists of a single point. The 0-dimensional Hausdorff measure is the number of points in a set, so

V 0 = 1 .

The unit 1-ball is the interval [ − 1 , 1 ] of length 2. So,

V 1 = 2.

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

S 0 = 2 .

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

S 1 = 2 π .

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

V 2 = π .

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

S 2 = 4 π .

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

V 3 = 4 3 π .

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

S n R n = d V n + 1 R n + 1 d R = ( n + 1 ) V n + 1 R n ,

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

V n + 1 = ∫ 0 1 S n r n d r

So,

V n + 1 = S n n + 1 .

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 d R = cos ⁡ θ d θ . Then,

S n + 2 = ∫ 0 π / 2 S 1 r . S n R n d θ = ∫ 0 π / 2 S 1 . S n R n cos ⁡ θ d θ = ∫ 0 1 S 1 . S n R n d R = S 1 ∫ 0 1 S n R n d R = 2 π V n + 1

Since S 1 = 2 π V 0 , the equation S n + 1 = 2 π V n holds for all n.

This completes our derivation of the recurrences:

V 0 = 1 V n + 1 = S n / ( n + 1 ) S 0 = 2 S n + 1 = 2 π V n

Closed forms

Combining the recurrences, we see that V n + 2 = 2 π V n / ( n + 2 ) . So it is simple to show by induction on k that,

V 2 k = π k k ! V 2 k + 1 = 2 ( 2 π ) k ( 2 k + 1 ) ! ! = 2 k ! ( 4 π ) k ( 2 k + 1 ) !

where ! ! denotes the double factorial, defined for odd integers 2k + 1 by (2k + 1)!! = 1 · 3 · 5 ··· (2k − 1) · (2k + 1).

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

V n = π n 2 Γ ( n 2 + 1 )

where Γ is the gamma function, which satisfies Γ ( 1 / 2 ) = π ; Γ ( 1 ) = 1 ; Γ ( x + 1 ) = x Γ ( x ) .

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

S n − 1 = 2 π n 2 Γ ( n 2 ) .

Other relations

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

S n − 1 = n 2 π S n − 1 + 2

Index-shifting n to n − 2 then yields the recurrence relations:

V n = 2 π n V n − 2 S n − 1 = 2 π n − 2 S n − 1 − 2

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

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

V n = ∫ 0 1 ∫ 0 2 π V n − 2 ( 1 − r 2 ) n − 2 r d θ d r = ∫ 0 1 ∫ 0 2 π V n − 2 ( 1 − r 2 ) n / 2 − 1 r d θ d r = 2 π V n − 2 ∫ 0 1 ( 1 − r 2 ) n / 2 − 1 r d r = 2 π V n − 2 [ − 1 n ( 1 − r 2 ) n / 2 ] r = 0 r = 1 = 2 π V n − 2 1 n = 2 π n V n − 2 .

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 ϕ 1 , ϕ 2 , … , ϕ n − 1 where ϕ n − 1 ranges over [ 0 , π ) radians (or over [0, 180) degrees) and the other angle ranges over [ 0 , 2 π ] 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:

x 1 = r cos ⁡ ( ϕ 1 ) x 2 = r sin ⁡ ( ϕ 1 ) cos ⁡ ( ϕ 2 ) x 3 = r sin ⁡ ( ϕ 1 ) sin ⁡ ( ϕ 2 ) cos ⁡ ( ϕ 3 ) ⋮ x n − 1 = r sin ⁡ ( ϕ 1 ) ⋯ sin ⁡ ( ϕ n − 2 ) cos ⁡ ( ϕ n − 1 ) x n = r sin ⁡ ( ϕ 1 ) ⋯ sin ⁡ ( ϕ n − 2 ) sin ⁡ ( ϕ n − 1 ) .

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

r = x n 2 + x n − 1 2 + ⋯ + x 2 2 + x 1 2 ϕ 1 = arccot ⁡ x 1 x n 2 + x n − 1 2 + ⋯ + x 2 2 = arccos ⁡ x 1 x n 2 + x n − 1 2 + ⋯ + x 1 2 ϕ 2 = arccot ⁡ x 2 x n 2 + x n − 1 2 + ⋯ + x 3 2 = arccos ⁡ x 2 x n 2 + x n − 1 2 + ⋯ + x 2 2 ⋮ ϕ n − 2 = arccot ⁡ x n − 2 x n 2 + x n − 1 2 = arccos ⁡ x n − 2 x n 2 + x n − 1 2 + x n − 2 2 ϕ n − 1 = 2 arccot ⁡ x n − 1 + x n 2 + x n − 1 2 x n = { arccos ⁡ x n − 1 x n 2 + x n − 1 2 x n ≥ 0 2 π − arccos ⁡ x n − 1 x n 2 + x n − 1 2 x n < 0 .

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 radians (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.

Spherical volume element

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

( cos ⁡ ( ϕ 1 ) − r sin ⁡ ( ϕ 1 ) 0 0     ⋅     ⋅     ⋅ 0 sin ⁡ ( ϕ 1 ) cos ⁡ ( ϕ 2 ) r cos ⁡ ( ϕ 1 ) cos ⁡ ( ϕ 2 ) − r sin ⁡ ( ϕ 1 ) sin ⁡ ( ϕ 2 ) 0     ⋅     ⋅     ⋅ 0 ⋮ ⋮ ⋮ ⋱ ⋮ 0 sin ⁡ ( ϕ 1 ) . . . sin ⁡ ( ϕ n − 2 ) cos ⁡ ( ϕ n − 1 ) ⋯ ⋯ − r sin ⁡ ( ϕ 1 ) . . . sin ⁡ ( ϕ n − 2 ) sin ⁡ ( ϕ n − 1 ) sin ⁡ ( ϕ 1 ) . . . sin ⁡ ( ϕ n − 2 ) sin ⁡ ( ϕ n − 1 ) r cos ⁡ ( ϕ 1 ) . . . sin ⁡ ( ϕ n − 1 ) ⋯ r sin ⁡ ( ϕ 1 ) . . . sin ⁡ ( ϕ n − 2 ) cos ⁡ ( ϕ n − 1 ) ) d n V = | det ∂ ( x i ) ∂ ( r , ϕ j ) | d r d ϕ 1 d ϕ 2 ⋯ d ϕ n − 1 = r n − 1 sin n − 2 ⁡ ( ϕ 1 ) sin n − 3 ⁡ ( ϕ 2 ) ⋯ sin ⁡ ( ϕ n − 2 ) d r d ϕ 1 d ϕ 2 ⋯ d ϕ n − 1

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

V n = ∫ ϕ n − 1 = 0 2 π ∫ ϕ n − 2 = 0 π ⋯ ∫ ϕ 1 = 0 π ∫ r = 0 R d n V .

The volume element of the (n-1)–sphere, which generalizes the area element of the 2-sphere, is given by

d S n − 1 V = sin n − 2 ⁡ ( ϕ 1 ) sin n − 3 ⁡ ( ϕ 2 ) ⋯ sin ⁡ ( ϕ n − 2 ) d ϕ 1 d ϕ 2 ⋯ d ϕ n − 1 .

The natural choice of an orthogonal basis over the angular coordinates is a product of ultraspherical polynomials,

∫ 0 π sin n − j − 1 ⁡ ( ϕ j ) C s ( ( n − j − 1 ) / 2 ) ( cos ⁡ ϕ j ) C s ′ ( ( n − j − 1 ) / 2 ) ( cos ⁡ ϕ j ) d ϕ j = π 2 3 − n + j Γ ( s + n − j − 1 ) s ! ( 2 s + n − j − 1 ) Γ 2 ( ( n − j − 1 ) / 2 ) δ s , s ′

for j = 1, 2, ..., n − 2, and the e ^isφ_j for the angle j = n − 1 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 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   x y plane. In other words,

  [ x , y , z ] ↦ [ x 1 − z , y 1 − z ] .

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

[ x 1 , x 2 , … , x n ] ↦ [ x 1 1 − x n , x 2 1 − x n , … , x n − 1 1 − x n ] .

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

To generate uniformly distributed random points on the (n − 1)-sphere (i.e., the surface of the 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, r = x 1 2 + x 2 2 + ⋯ + x n 2 .

The vector 1 r x is uniformly distributed over the surface of the unit n-ball.

Examples

For example, when n = 2 the normal distribution exp(−x₁²) when expanded over another axis exp(−x₂²) after multiplication takes the form exp(−x₁²−x₂²) or exp(−r²) and so is only dependent on distance from the origin.

Alternatives

Another way to generate a random distribution on a hypersphere is to make a uniform distribution over a hypercube that includes the unit hyperball, exclude those points that are outside the hyperball, then project the remaining interior points outward from the origin onto the surface. This will give a uniform distribution, but it is necessary to remove the exterior points. As the relative volume of the hyperball to the hypercube decreases very rapidly with dimension, this procedure will succeed with high probability only for fairly small numbers of dimensions.

Wendel's theorem gives the probability that all of the points generated will lie in the same half of the hypersphere.

Uniformly at random from the n-ball

Points may be sampled uniformly from the n-ball by a reduction to the (n - 1)-sphere. With a point selected from the (n - 1)-sphere uniformly at random (e.g., by using Marsaglia's algorithm), one needs only a radius to obtain a point uniformly at random within the 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 (n - 1)-shere then u^1/nx is uniformly distributed over the unit n-ball.

Points may also be sampled uniformly from the n-ball by a reduction to the (n + 1)-sphere. In particular, if x = ( x 1 , x 2 , … , x n + 2 ) is a point selected uniformly from the (n + 1)-sphere, then x = ( x 1 , x 2 , … , x n ) is uniformly distributed over the unit n-ball.

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 S p ( 1 ) ≅ S O ( 4 ) / S O ( 3 ) ≅ S U ( 2 ) ≅ S p i n ( 3 ) .

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 

Almost complex structure coming from the set of pure unit octonions. SO(7)/SO(6) = G₂/SU(3).

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.