n-sphere

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2-sphere wireframe as an orthogonal projection
2-sphere wireframe as an orthogonal projection
Just as a stereographic projection can project a sphere's surface to a plane, it can also project a 3-sphere's surface into 3-space. This image shows three coordinate directions projected to 3-space: parallels (red), meridians (blue) and hypermeridians (green). Due to the conformal property of the stereographic projection, the curves intersect each other orthogonally (in the yellow points) as in 4D. All of the curves are circles: the curves that intersect <0,0,0,1> have an infinite radius (= straight line).
Just as a stereographic projection can project a sphere's surface to a plane, it can also project a 3-sphere's surface into 3-space. This image shows three coordinate directions projected to 3-space: parallels (red), meridians (blue) and hypermeridians (green). Due to the conformal property of the stereographic projection, the curves intersect each other orthogonally (in the yellow points) as in 4D. All of the curves are circles: the curves that intersect <0,0,0,1> have an infinite radius (= straight line).

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:

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

An n-sphere is the surface or boundary of an (n + 1)-dimensional ball, 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.

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[edit] 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 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} containing a line segment.
  • a 1-sphere is a circle of radius r. These contain disks.
  • a 2-sphere is an ordinary sphere in 3-dimensional Euclidean space that contains a ball.
  • a 3-sphere is a sphere in 4-dimensional Euclidean space.

[edit] Euclidean coordinates in (n + 1)-space

The set of points in (n + 1)-space: (x_1,x_2,x_3,\dots,x_{n+1}) that define an n-sphere, (\mathbf S^n) is represented by the equation:

r^2=\sum_{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 volume element ω of n-sphere of radius r is given by

\omega = {1 \over r} \sum_{j=1}^{n+1} (-1)^{j-1} x_j \,dx_1 \wedge \cdots \wedge dx_{j-1} \wedge dx_j \cdots \wedge dx_{n+1}

In fact, dr \wedge \omega = dx_1 \wedge \cdots \wedge dx_{n+1}

[edit] 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.

[edit] 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).

[edit] Volume of the n-ball

The hyperdimensional volume of the space which a (n − 1)-sphere encloses (the n-ball) is given by

V_n={\pi^\frac{n}{2}R^n\over\Gamma(\frac{n}{2} + 1)}={C_n R^n},

where Γ is the gamma function. (For even n, \Gamma\left(\frac{n}{2}+1\right)= \left(\frac{n}{2}\right)!; for odd n, \Gamma\left(\frac{n}{2}+1\right)= \sqrt{\pi} \frac{n!!}{2^{(n+1)/2}}, where n!! denotes the double factorial.)

From this, it follows that the value of the constant Cn for a given n is:

C_n={\frac{\pi^k}{k!}}, for even n=2k, and
C_n=C_{2k+1}=\frac{2^{2k+1} k!\, \pi^{k}}{(2k+1)!} for odd n=2k+1.

The "surface area" of this (n-1)-sphere is

S_{n-1}=\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}}

The following relationships hold between the n-spherical surface area and volume:

V_n/S_{n-1} = R/n\,
S_{n+1}/V_n = 2\pi R\,

This leads to the recurrence relation:

V_n = \frac{2 \pi R^2}{n} V_{n-2}\,

The interior of an n-sphere, the set of all points whose distance from the center is less than R, 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 R), a closed hyperball.

[edit] Examples

For small values of n, the volumes, Vn , of the n-ball of radius R are:

V_0\, (point) = 1\,    
V_1\, (line segment) = 2\,R    
V_2\, (disk) = \pi\,R^2 = 3.14159\ldots\,R^2
V_3\, (ball) = \frac{4 \pi}{3}\,R^3 = 4.18879\ldots\,R^3
V_4\, = \frac{\pi^2}{2}\,R^4 = 4.93480\ldots\,R^4
V_5\, = \frac{8 \pi^2}{15}\,R^5 = 5.26379\ldots\,R^5
V_6\, = \frac{\pi^3}{6}\,R^6 = 5.16771\ldots\,R^6
V_7\, = \frac{16 \pi^3}{105}\,R^7 = 4.72477\ldots\,R^7
V_8\, = \frac{\pi^4}{24}\,R^8 = 4.05871\ldots\,R^8
\lim_{n\rightarrow\infty} \frac{V_n}{R^n}\, = 0\,

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.

[edit] 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 \ r, and \ n-1 angular coordinates \ \phi _1 , \phi _2 , ... , \phi _{n-1}. If \ x_i are the Cartesian coordinates, then we may define

x_1=r\cos(\phi_1)\,
x_2=r\sin(\phi_1)\cos(\phi_2)\,
x_3=r\sin(\phi_1)\sin(\phi_2)\cos(\phi_3)\,
\cdots\,
x_{n-1}=r\sin(\phi_1)\cdots\sin(\phi_{n-2})\cos(\phi_{n-1})\,
x_n~~\,=r\sin(\phi_1)\cdots\sin(\phi_{n-2})\sin(\phi_{n-1})\,

While the inverse transformations can be derived from those above:

\tan(\phi_{n-1})=\frac{x_n}{x_{n-1}}
\tan(\phi_{n-2})=\frac{\sqrt{{x_n}^2+{x_{n-1}}^2}}{x_{n-2}}
\cdots\,
\tan(\phi_{1})=\frac{\sqrt{{x_n}^2+{x_{n-1}}^2+\cdots+{x_2}^2}}{x_{1}}

Note that last angle φn − 1 has a range of 2π while the other angles have a range of π. This range covers the whole sphere.

The volume element in n-dimensional Euclidean space will be found from the Jacobian of the transformation:

d_{\mathbb{R}^n}V = \left|\det\frac{\partial (x_i)}{\partial(r,\phi_j)}\right| dr\,d\phi_1 \, d\phi_2\ldots d\phi_{n-1}
=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}

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

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_{\mathbb{R}^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}(\phi_1)\sin^{n-3}(\phi_2)\cdots \sin(\phi_{n-2})\, d\phi_1 \, d\phi_2\ldots d\phi_{n-1}

[edit] 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,y,z] \mapsto \left[\frac{x}{1-z},\frac{y}{1-z}\right] on the \ xy plane. In other words:

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

Likewise, the stereographic projection of an n-sphere \mathbf{S}^{n-1} of radius 1 will map to the n-1 dimensional hyperplane \mathbf{R}^{n-1} perpendicular to the \ x_n axis as:

[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].

[edit] Generating points on the surface of the n-ball

To generate points on 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), \mathbf{x}=(x_1,x_2,\ldots,x_n).

Now calculate the "radius" of this point, r=\sqrt{x_1^2+x_2^2+\ldots+x_n^2}.

The vector \frac1r \mathbf{x} is uniformly distributed over the surface of the n-ball.

[edit] See also

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