n-sphere

In mathematics, the n-sphere is the generalization of the ordinary sphere to spaces of arbitrary dimension. It is an n-dimensional manifold that can be embedded in Euclidean (n + 1)-space.

The 0-sphere is a pair of points, the 1-sphere is a circle, and the 2-sphere is an ordinary sphere. Generally, when embedded in an (n + 1)-dimensional Euclidean space, an n-sphere is the surface or boundary of an (n + 1)-dimensional ball. That is, for any natural number n, an n-sphere of radius r may be defined in terms of an embedding in (n + 1)-dimensional Euclidean space as the set of points that are at distance r from a central point, where the radius r may be any positive real number. Thus, the n-sphere would be defined by:

${\displaystyle S^{n}=\left\{x\in \mathbb {R} ^{n+1}:\left\|x\right\|=r\right\}.}$

In particular:

• the pair of points at the ends of a (one-dimensional) line segment is a 0-sphere,
• the circle, which is the one-dimensional circumference of a (two-dimensional) disk in the plane 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 general n-sphere which can be denoted as a "glone".

An n-sphere embedded in an (n + 1)-dimensional Euclidean space is called a hypersphere. The n-sphere of unit radius is called the unit n-sphere, denoted Sn, often referred to as the n-sphere.

When embedded as described, an n-sphere is the surface or boundary of an (n + 1)-dimensional ball. For n ≥ 2, the n-spheres are 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.

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 {cr, 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)-space

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

${\displaystyle r^{2}=\sum _{i=1}^{n+1}\left(x_{i}-c_{i}\right)^{2}}$

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

${\displaystyle \omega ={\frac {1}{r}}\sum _{j=1}^{n+1}(-1)^{j-1}x_{j}\,dx_{1}\wedge \cdots \wedge dx_{j-1}\wedge dx_{j+1}\wedge \cdots \wedge dx_{n+1}=*dr}$

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,

${\displaystyle dr\wedge \omega =dx_{1}\wedge \cdots \wedge dx_{n+1}.}$

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:

• 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 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 Rn. 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:

{\displaystyle {\begin{aligned}V_{0}&=1&V_{n+1}&={\frac {S_{n}}{n+1}}\\[6pt]S_{0}&=2&S_{n+1}&=2\pi V_{n}\end{aligned}}}

The surfaces and volumes can also be given in closed form:

{\displaystyle {\begin{aligned}S_{n}(R)&={\frac {2\,\pi ^{\frac {n+1}{2}}}{\Gamma \left({\frac {n+1}{2}}\right)}}R^{n}\\[6pt]V_{n}(R)&={\frac {\pi ^{\frac {n}{2}}}{\Gamma \left({\frac {n}{2}}+1\right)}}R^{n}\end{aligned}}}

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

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 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 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,

${\displaystyle V_{0}=1.}$

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

${\displaystyle V_{1}=2.}$

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

${\displaystyle S_{0}=2.}$

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

${\displaystyle S_{1}=2\pi .}$

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

${\displaystyle V_{2}=\pi .}$

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

${\displaystyle S_{2}=4\pi .}$

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

${\displaystyle V_{3}={\tfrac {4}{3}}\pi .}$

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

${\displaystyle S_{n}R^{n}={\frac {dV_{n+1}R^{n+1}}{dR}}={(n+1)V_{n+1}R^{n}},}$

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

${\displaystyle V_{n+1}=\int _{0}^{1}S_{n}r^{n}\,dr.}$

So,

${\displaystyle V_{n+1}={\frac {S_{n}}{n+1}}.}$

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

{\displaystyle {\begin{aligned}S_{n+2}&=\int _{0}^{\frac {\pi }{2}}S_{1}r\cdot S_{n}R^{n}\,d\theta \\[6pt]&=\int _{0}^{\frac {\pi }{2}}S_{1}\cdot S_{n}R^{n}\cos \theta \,d\theta \\[6pt]&=\int _{0}^{1}S_{1}\cdot S_{n}R^{n}\,dR\\[6pt]&=S_{1}\int _{0}^{1}S_{n}R^{n}\,dR\\[6pt]&=2\pi V_{n+1}.\end{aligned}}}

Since S1 = 2π V0, the equation

${\displaystyle S_{n+1}=2\pi V_{n}}$

holds for all n.

This completes the derivation of the recurrences:

{\displaystyle {\begin{aligned}V_{0}&=1&V_{n+1}&={\frac {S_{n}}{n+1}}\\[6pt]S_{0}&=2&S_{n+1}&=2\pi V_{n}\end{aligned}}}

Closed forms

Combining the recurrences, we see that

${\displaystyle V_{n+2}=2\pi {\frac {V_{n}}{n+2}}.}$

So it is simple to show by induction on k that,

{\displaystyle {\begin{aligned}V_{2k}&={\frac {\pi ^{k}}{k!}}\\[6pt]V_{2k+1}&={\frac {2\left(2\pi \right)^{k}}{(2k+1)!!}}={\frac {2k!\left(4\pi \right)^{k}}{(2k+1)!}}\end{aligned}}}

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

${\displaystyle V_{n}={\frac {\pi ^{\frac {n}{2}}}{\Gamma \left({\frac {n}{2}}+1\right)}}}$

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

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

${\displaystyle S_{n-1}={\frac {n\pi ^{\frac {n}{2}}}{\Gamma \left({\frac {n}{2}}+1\right)}}.}$

Other relations

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

${\displaystyle S_{n-1}={\frac {n}{2\pi }}S_{n+1}}$

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

{\displaystyle {\begin{aligned}V_{n}&={\frac {2\pi }{n}}V_{n-2}\\[6pt]S_{n-1}&={\frac {2\pi }{n-2}}S_{n-3}\end{aligned}}}

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:

{\displaystyle {\begin{aligned}V_{n}&=\int _{0}^{1}\int _{0}^{2\pi }V_{n-2}\left({\sqrt {1-r^{2}}}\right)^{n-2}\,r\,d\theta \,dr\\[6pt]&=\int _{0}^{1}\int _{0}^{2\pi }V_{n-2}\left(1-r^{2}\right)^{{\frac {n}{2}}-1}\,r\,d\theta \,dr\\[6pt]&=2\pi V_{n-2}\int _{0}^{1}\left(1-r^{2}\right)^{{\frac {n}{2}}-1}\,r\,dr\\[6pt]&=2\pi V_{n-2}\left[-{\frac {1}{n}}\left(1-r^{2}\right)^{\frac {n}{2}}\right]_{r=0}^{r=1}\\[6pt]&=2\pi V_{n-2}{\frac {1}{n}}={\frac {2\pi }{n}}V_{n-2}.\end{aligned}}}

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]

{\displaystyle {\begin{aligned}x_{1}&=r\cos(\varphi _{1})\\x_{2}&=r\sin(\varphi _{1})\cos(\varphi _{2})\\x_{3}&=r\sin(\varphi _{1})\sin(\varphi _{2})\cos(\varphi _{3})\\&\vdots \\x_{n-1}&=r\sin(\varphi _{1})\cdots \sin(\varphi _{n-2})\cos(\varphi _{n-1})\\x_{n}&=r\sin(\varphi _{1})\cdots \sin(\varphi _{n-2})\sin(\varphi _{n-1}).\end{aligned}}}

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

{\displaystyle {\begin{aligned}r&={\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}+\cdots +{x_{2}}^{2}+{x_{1}}^{2}}}\\[6pt]\varphi _{1}&=\operatorname {arccot} {\frac {x_{1}}{\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}+\cdots +{x_{2}}^{2}}}}&&=\arccos {\frac {x_{1}}{\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}+\cdots +{x_{1}}^{2}}}}\\[6pt]\varphi _{2}&=\operatorname {arccot} {\frac {x_{2}}{\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}+\cdots +{x_{3}}^{2}}}}&&=\arccos {\frac {x_{2}}{\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}+\cdots +{x_{2}}^{2}}}}\\[6pt]&\vdots &&\vdots \\[6pt]\varphi _{n-2}&=\operatorname {arccot} {\frac {x_{n-2}}{\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}}}}&&=\arccos {\frac {x_{n-2}}{\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}+{x_{n-2}}^{2}}}}\\[6pt]\varphi _{n-1}&=2\operatorname {arccot} {\frac {x_{n-1}+{\sqrt {x_{n}^{2}+x_{n-1}^{2}}}}{x_{n}}}&&={\begin{cases}\arccos {\frac {x_{n-1}}{\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}}}}&x_{n}\geq 0\\[6pt]2\pi -\arccos {\frac {x_{n-1}}{\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}}}}&x_{n}<0\end{cases}}\,.\end{aligned}}}

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 element

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

${\displaystyle {\begin{pmatrix}\cos(\varphi _{1})&-r\sin(\varphi _{1})&0&0&\cdots &0\\\sin(\varphi _{1})\cos(\varphi _{2})&r\cos(\varphi _{1})\cos(\varphi _{2})&-r\sin(\varphi _{1})\sin(\varphi _{2})&0&\cdots &0\\\vdots &\vdots &\vdots &\ddots &&\vdots \\&&&&&0\\\sin(\varphi _{1})\cdots \sin(\varphi _{n-2})\cos(\varphi _{n-1})&\cdots &\cdots &&&-r\sin(\varphi _{1})\cdots \sin(\varphi _{n-2})\sin(\varphi _{n-1})\\\sin(\varphi _{1})\cdots \sin(\varphi _{n-2})\sin(\varphi _{n-1})&r\cos(\varphi _{1})\cdots \sin(\varphi _{n-1})&\cdots &&&r\sin(\varphi _{1})\cdots \sin(\varphi _{n-2})\cos(\varphi _{n-1})\end{pmatrix}}}$
{\displaystyle {\begin{aligned}d^{n}V&=\left|\det {\frac {\partial (x_{i})}{\partial \left(r,\varphi _{j}\right)}}\right|dr\,d\varphi _{1}\,d\varphi _{2}\cdots d\varphi _{n-1}\\[6pt]&=r^{n-1}\sin ^{n-2}(\varphi _{1})\sin ^{n-3}(\varphi _{2})\cdots \sin(\varphi _{n-2})\,dr\,d\varphi _{1}\,d\varphi _{2}\cdots d\varphi _{n-1}\end{aligned}}}

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

${\displaystyle V_{n}=\int _{\varphi _{n-1}=0}^{2\pi }\int _{\varphi _{n-2}=0}^{\pi }\cdots \int _{\varphi _{1}=0}^{\pi }\int _{r=0}^{R}d^{n}V.}$

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

${\displaystyle d_{S^{n-1}}V=\sin ^{n-2}(\varphi _{1})\sin ^{n-3}(\varphi _{2})\cdots \sin(\varphi _{n-2})\,d\varphi _{1}\,d\varphi _{2}\cdots d\varphi _{n-1}.}$

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

{\displaystyle {\begin{aligned}&{}\quad \int _{0}^{\pi }\sin ^{n-j-1}\left(\varphi _{j}\right)C_{s}^{\left({\frac {n-j-1}{2}}\right)}\cos \left(\varphi _{j}\right)C_{s'}^{\left({\frac {n-j-1}{2}}\right)}\cos \left(\varphi _{j}\right)\,d\varphi _{j}\\[6pt]&={\frac {2^{3-n+j}\pi \Gamma (s+n-j-1)}{s!(2s+n-j-1)\Gamma ^{2}\left({\frac {n-j-1}{2}}\right)}}\delta _{s,s'}\end{aligned}}}

for j = 1, 2,... n − 2, and the eisφj 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 xy-plane. In other words,

${\displaystyle [x,y,z]\mapsto \left[{\frac {x}{1-z}},{\frac {y}{1-z}}\right].}$

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

${\displaystyle [x_{1},x_{2},\ldots ,x_{n}]\mapsto \left[{\frac {x_{1}}{1-x_{n}}},{\frac {x_{2}}{1-x_{n}}},\ldots ,{\frac {x_{n-1}}{1-x_{n}}}\right].}$

Generating random points

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

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:

${\displaystyle r={\sqrt {x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}}.}$

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.

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).[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 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
${\displaystyle \mathrm {Sp} (1)\cong \mathrm {SO} (4)/\mathrm {SO} (3)\cong \mathrm {SU} (2)\cong \mathrm {Spin} (3)}$.
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
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.[4]
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.

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. 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.
4. 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.

References

• 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.