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 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: (x_{1}, x_{2}, ..., x_{n+1}) that define an n-sphere (S^{n}), is represented by the equation:
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
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 , 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 . This forms the basis for stereographic projection.^{[1]}
Volume and surface area
and 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
The constants and (for the unit ball and sphere) are related by the recurrences:
The surfaces and volumes can also be given in closed form:
where 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 enclosed by the (n–1)-sphere of constant radius R embedded in n dimensions reaches a maximum at the dimension satisfying and where is given in the sidebar to the right; if the last (weak) inequality holds with equality, then the same maximum also occurs at 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 of the n-sphere of constant radius R embedded in n+1 dimensions reaches a maximum for dimension that satisfies and where is given in the sidebar to the right; if the last (weak) inequality holds with equality, then the same maximum also occurs at ^{[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 of length 2. So,
The 0-sphere consists of its two end-points, . 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 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 and , so that and . Then,
Since , 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 is the gamma function, which satisfies , , and .
By multiplying by , differentiating with respect to , and then setting , 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 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, , and n − 1 angular coordinates where the angles range over radians (or over [0, 180] degrees) and ranges over radians (or over [0, 360] degrees). If are the Cartesian coordinates, then we may compute from with:
Except in the special cases described below, the inverse transformation is unique:
where if for some but all of are zero then when , and radians (180 degrees) when .
There are some special cases where the inverse transform is not unique; for any will be ambiguous whenever all of are zero; in this case 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 on a two-dimensional sphere of radius 1 maps to the point on the plane. In other words,
Likewise, the stereographic projection of an n-sphere of radius 1 will map to the dimensional hyperplane perpendicular to the 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 (i.e., 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),
Now calculate the "radius" of this point,
The vector is uniformly distributed over the surface of the unit n-ball.
An alternative given by Marsaglia is to uniformly randomly select a point 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 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 is a point selected uniformly from the unit (n + 1)-sphere, then 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.