n-sphere

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  (Redirected from Hypersphere) 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.

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 which are at distance r from a fixed point, where r may be any positive real number. In particular:

Euclidean coordinates in (n + 1)-space

The set of points in (n + 1)-space: (x1,x1,x2,…,xn+1) that define an n-sphere, (Sn) 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 form 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+1}\wedge \cdots \wedge dx_{n+1} = * dr
\scriptstyle{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 included the equality, and open otherwise.

Specifically:


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}
\Gamma\left(\frac{n}{2}+1\right)= \left(\frac{n}{2}\right)!

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

C_n={\frac{\pi^k}{k!}}
C_n=C_{2k+1}=\frac{2^{2k+1} k!\, \pi^{k}}{(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}\,

Conventionally, these formulas can also be proven directly by integration in n-dimensional spherical coordinates (Stewart 2006, p. 881).

Examples

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

V_0\,1\,V_1\,2\,RV_2\,\pi\,R^23.14159\ldots\,R^2V_3\,\frac{4 \pi}{3}\,R^34.18879\ldots\,R^3V_4\,\frac{\pi^2}{2}\,R^44.93480\ldots\,R^4V_5\,\frac{8 \pi^2}{15}\,R^55.26379\ldots\,R^5V_6\,\frac{\pi^3}{6}\,R^65.16771\ldots\,R^6V_7\,\frac{16 \pi^3}{105}\,R^74.72477\ldots\,R^7V_8\,\frac{\pi^4}{24}\,R^84.05871\ldots\,R^8\lim_{n\rightarrow\infty} \frac{V_n}{R^n}\,0\,
=    
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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

\ r
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}

Stereographic projection

\ [x,y,z]
\ [x,y,z] \mapsto \left[\frac{x}{1-z},\frac{y}{1-z}\right].
\mathbf{S}^{n-1}
[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 points on the surface of the n-ball

To generate points on the surface of the n ball, Marsaglia (1972) gives the following algorithm.

\mathbf{x}=(x_1,x_2,\ldots,x_n)r=\sqrt{x_1^2+x_2^2+\ldots+x_n^2}\frac1r \mathbf{x}

The normal distribution e-x^2 when expanded over another axis e-y^2 after multiplication takes the form e-(x^2+y^2) or e-(r^2) and so is only dependent on distance from the origin.

Another way to generate a random distribution on a hypersphere is to make a uniform one over a hypercube that includes the unit hypersphere, exclude those points that are outside the hypersphere, then project the remaining interior points outward from the origin onto the surface. This will give a uniform distribution, but it is necessary to remove the exterior points. As the relative volume of the hypersphere to the hypercube decreases very rapidly with dimension it will only work for fairly small numbers of dimensions.

Specific spheres

0-sphere 
The pair of points {±1} with the discrete topology. The only sphere which is disconnected. Has a natural Lie group structure; isomorphic to O(1). Parallelizable.
1-sphere 
Also known as the unit 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 
Complex structure; see Riemann sphere. Equivalent to the complex projective line, CP1. SO(3)/SO(2).
3-sphere 
Lie group structure Sp(1). Principal U(1)-bundle over the 2-sphere. Parallelizable. SO(4)/SO(3) = SU(2) = Sp(1) = 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 
Almost complex structure coming from the set of pure unit octonions. SO(7)/SO(6) = G2/SU(3).
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).