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:
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.
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 S^{n}, 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.
Contents
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 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, (x_{1}, x_{2}, ..., x_{n+1}), that define an n-sphere, S^{n}, 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-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 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 area
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.
Given the radius R, the enclosed volume and the surface area of the n-sphere reach a maximum and then decrease towards zero as the dimension n increases. In particular, the volume V_{n}(R) enclosed by the (n – 1)-sphere of constant radius R embedded in n dimensions reaches a maximum at the dimension n satisfying V_{n−1} < V_{n}R and V_{n} ≥ V_{n+1}R where V_{n} is given in the sidebar to the right; if the last (weak) inequality holds with equality, then the same maximum also occurs at n + 1. Specifically, for unit radius the largest enclosed volume is that enclosed by a 4-dimensional sphere bounding a 5-dimensional ball.
Similarly, the surface area S_{n}(R) of the n-sphere of constant radius R embedded in n + 1 dimensions reaches a maximum for dimension n that satisfies S_{n−1} < S_{n}R and S_{n} ≥ S_{n+1}R where S_{n} is given in the sidebar to the right; if the last (weak) inequality holds with equality, then the same maximum also occurs at n + 1.^{[2]} Specifically, for unit radius the largest surface area occurs for the 6-dimensional sphere bounding a 7-dimensional ball.
Examples
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
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 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 our 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 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
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^{n}, differentiating with respect to R, and then setting R = 1, we get the closed form
Other relations
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 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:
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.
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:
and the above equation for the volume of the n-ball can be recovered by integrating:
The volume 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 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 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 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 = (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.
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 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]}
Note that 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, RP^{1}. Parallelizable. SO(2) = U(1).
- 2-sphere
- Also known as the sphere. Complex structure; see Riemann sphere. Equivalent to the complex projective line, CP^{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, HP^{1}. SO(5)/SO(4).
- 5-sphere
- Principal U(1)-bundle over CP^{2}. 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) = G_{2}/SU(3).
- 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 OP^{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 also
Notes
- ^ James W. Vick (1994). Homology theory, p. 60. Springer
- ^ Loskot, Pavel (November 2007). "On Monotonicity of the Hypersphere Volume and Area". Journal of Geometry. 87 (1–2): 96–98. doi:10.1007/s00022-007-1891-1.
- ^ 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.
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.
External links
- Exploring Hyperspace with the Geometric Product
- Properties of Spherical Coordinates in n Dimensions
- Weisstein, Eric W. "Hypersphere". MathWorld.