In mathematics, an n-sphere is a generalization of an ordinary sphere to arbitrary dimension. For any natural number n, an n-sphere of radius r is defined the set of points in (n+1)-dimensional Euclidean space which are at distance r from a central point, where the radius r may be any positive real number. It is an n-dimensional manifold in Euclidean (n+1)-space. In particular, a 0-sphere is a pair of points on a line, a 1-sphere is a circle in the plane, and a 2-sphere is an ordinary sphere in three dimensional space. Spheres of dimension n > 2 are sometimes called hyperspheres. The n-sphere of unit radius centered at the origin is called the unit n-sphere, denoted Sn. The unit n-sphere is often referred to as the n-sphere. In symbols:
- <math>S^n = \left\{ x \in \mathbb{R}^{n+1} : \|x\| = 1\right\}.</math>
An n-sphere is the surface or boundary of an (n+1)-dimensional ball (mathematics), and is an n-dimensional manifold. For n ≥ 2, the n-spheres are the simply connected n-dimensional manifold 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 the set of points in (n+1)-dimensional Euclidean space which are at distance r from a fixed point, where r may be any positive real number. In particular:
- a 0-sphere is a pair of points {p − r, p + r}.
- a 1-sphere is a circle of radius r.
- a 2-sphere is an ordinary sphere in 3-dimensional Euclidean space.
- 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: <math>(x_1,x_2,x_3,\dots,x_{n+1})</math> that define an n-sphere, (<math>\mathbf S^n</math>) is represented by the equation:
- <math>r^2=\sum_{i=1}^{n+1} (x_i - C_i)^2.\,</math>
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 volume element <math>\omega</math> of n-sphere of radius <math>r</math> is given by:
- <math>\omega = {1 \over r} \sum_{j=1}^{n+1} (-1)^{j-1} x_j dx_1 \wedge ... dx_{j-1} \wedge dx_j ... dx_{n+1}</math>
In fact, <math>dr \wedge \omega = dx_1 \wedge ... dx_{n+1}</math>
n-ball
The space enclosed by an n-sphere is called an (n+1)-ball. An (n+1)-ball is closed if it included the equality, and open otherwise. 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.
Notation
Labelling n-spheres with the dimensionality of the surface (as used in this article) is the convention common in mathematical use. Potentially confusingly, some authors use the dimensionality of the containing space to label n-spheres.[1] Thus what most call a 1-sphere (a regular circle in a plane), others term a 2-sphere (reflecting the dimensionality of the plane in which it lies).
n-spherical volume
The hyperdimensional volume of the space which a <math>(n-1)</math>-sphere encloses (the <math>n</math>-ball) is given by
- <math>V_n={\pi^\frac{n}{2}R^n\over\Gamma(\frac{n}{2} + 1)}={C_n R^n}</math>,
where <math>\Gamma</math> is the gamma function. (For even <math>n</math>, <math>\Gamma\left(\frac{n}{2}+1\right)= \left(\frac{n}{2}\right)!</math>; for odd <math>n</math>, <math>\Gamma\left(\frac{n}{2}+1\right)= \sqrt{\pi} \frac{n!!}{2^{(n+1)/2}}</math>, where <math>n!!</math> denotes the double factorial.) From this, it follows that the value of the constant <math>C_n</math> for a given <math>n</math> is:
- <math>C_n={\frac{\pi^r}{r!}}</math>, for even <math>n</math> such that <math>n={2r}</math>, and
- <math>C_n={\frac{2^{(n+1)/2}\pi^{(n-1)/2}}{n!!}}</math>, for odd <math>n</math>.
The "surface area" of this (n-1)-sphere is
- <math>S_n=\frac{dV_n}{dR}=\frac{nV_n}{R}={2\pi^\frac{n}{2}R^{n-1}\over\Gamma(\frac{n}{2})}={n C_n R^{n-1}}</math>
The following relationships hold between the n-spherical surface area and volume:
- <math>V_n/S_n = R/n\,</math>
- <math>S_{n+2}/V_n = 2\pi R\,</math>
The interior of an n-sphere, the set of all points whose distance from the center is less than <math>R</math>, is called a hyperball, or if the n-sphere itself is included (that is, the set of all points whose distance from the center is less than or equal to <math>R</math>), a closed hyperball.
n-spherical volume - some examples
For small values of <math>n</math>, the volumes, <math>V_n</math> , of the unit <math>n</math>-ball (<math>R=1</math>) are:
-
<math>V_0\,</math> = <math>1\,</math> <math>V_1\,</math> (line segment) = <math>2\,</math> <math>V_2\,</math> (disk) = <math>\pi\,</math> = <math>3.14159\ldots\,</math> <math>V_3\,</math> (ball) = <math>\frac{4 \pi}{3}\,</math> = <math>4.18879\ldots\,</math> <math>V_4\,</math> = <math>\frac{\pi^2}{2}\,</math> = <math>4.93480\ldots\,</math> <math>V_5\,</math> = <math>\frac{8 \pi^2}{15}\,</math> = <math>5.26379\ldots\,</math> <math>V_6\,</math> = <math>\frac{\pi^3}{6}\,</math> = <math>5.16771\ldots\,</math> <math>V_7\,</math> = <math>\frac{16 \pi^3}{105}\,</math> = <math>4.72477\ldots\,</math> <math>V_8\,</math> = <math>\frac{\pi^4}{24}\,</math> = <math>4.05871\ldots\,</math> <math>\lim_{n\rightarrow\infty} V_n\,</math> = <math>0\,</math>
If the dimension n, is not limited to integral values, the n-sphere volume is a continuous function of n with a global maximum for the unit sphere in "dimension" n = 5.2569464... where the "volume" is 5.277768... It has a hypervolume of 1 when n = 0 or when n = 12.76405... The hypercube circumscribed around the unit n-sphere has an edge length of 2 and hence a volume of 2n; the ratio of the volume of the n-sphere to its circumscribed hypercube decreases monotonically as the dimension increases. The non-monotonic behaviour of the numerical value of n-spheres as a function of n may seem strange at first glance. However, by assigning units of length to each dimension one can see it is meaningless to compare the unit-sphere volumes in different n's, just as it is meaningless to compare a length to an area in other contexts. A meaningful comparison is obtained by using a dimensionless measure of the volume, such as the ratio of the n-sphere and its circumscribed hypercube volumes. Using this measure restores the intuitively normal behavior of a monotonic decline in the volume as the dimension increases.
Hyperspherical 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 <math>\ r</math>, and <math>\ n-1</math> angular coordinates <math>\ \phi _1 , \phi _2 , ... , \phi _{n-1}</math>. If <math>\ x_i</math> are the Cartesian coordinates, then we may define
- <math>x_1=r\cos(\phi_1)\,</math>
- <math>x_2=r\sin(\phi_1)\cos(\phi_2)\,</math>
- <math>x_3=r\sin(\phi_1)\sin(\phi_2)\cos(\phi_3)\,</math>
- <math>\cdots\,</math>
- <math>x_{n-1}=r\sin(\phi_1)\cdots\sin(\phi_{n-2})\cos(\phi_{n-1})\,</math>
- <math>x_n~~\,=r\sin(\phi_1)\cdots\sin(\phi_{n-2})\sin(\phi_{n-1})\,</math>
While the inverse transformations can be derived from those above:
- <math>\tan(\phi_{n-1})=\frac{x_n}{x_{n-1}}</math>
- <math>\tan(\phi_{n-2})=\frac{\sqrt{{x_n}^2+{x_{n-1}}^2}}{x_{n-2}}</math>
- <math>\cdots\,</math>
- <math>\tan(\phi_{1})=\frac{\sqrt{{x_n}^2+{x_{n-1}}^2+\cdots+{x_2}^2}}{x_{1}}</math>
Note that last angle <math>\phi _{n-1}</math> has a range of <math>2\pi</math> while the other angles have a range of <math>\pi</math>. This range covers the whole sphere. The n-spherical volume element will be found from the Jacobian of the transformation:
- <math>d^nr =
\left|\det\frac{\partial (x_i)}{\partial(r,\phi_j)}\right| dr\,d\phi_1 \, d\phi_2\ldots d\phi_{n-1}</math>
- <math>=r^{n-1}\sin^{n-2}(\phi_1)\sin^{n-3}(\phi_2)\cdots \sin(\phi_{n-2})\,
dr\,d\phi_1 \, d\phi_2\cdots d\phi_{n-1}</math> and the above equation for the volume of the n-sphere can be recovered by integrating:
- <math>V_n=\int_{r=0}^R \int_{\phi_1=0}^\pi
\cdots \int_{\phi_{n-2}=0}^\pi\int_{\phi_{n-1}=0}^{2\pi}d^nr. \,</math>
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 <math>\ [x,y,z]</math> on a two-dimensional sphere of radius 1 maps to the point <math>\ [x,y,z] \mapsto \left[\frac{x}{1-z},\frac{y}{1-z}\right]</math> on the <math>\ xy</math> plane. In other words:
- <math>\ [x,y,z] \mapsto \left[\frac{x}{1-z},\frac{y}{1-z}\right].</math>
Likewise, the stereographic projection of an n-sphere <math>\mathbf{S}^{n-1}</math> of radius 1 will map to the n-1 dimensional hyperplane <math>\mathbf{R}^{n-1}</math> perpendicular to the <math>\ x_n</math> axis as:
- <math>[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].</math>
See also
References
- David W. Henderson, Experiencing Geometry: In Euclidean, Spherical, and Hyperbolic Spaces, second edition, 2001, [2] (Chapter 20: 3-spheres and hyperbolic 3-spaces.)
- Jeffrey R. Weeks, The Shape of Space: How to Visualize Surfaces and Three-dimensional Manifolds, 1985, (Chapter 14: The Hypersphere)
- Exploring Hyperspace with the Geometric Product
