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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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 (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.
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.
Euclidean coordinates in (n + 1)-space
The set of points in (n + 1)-space: Failed to parse (Missing texvc executable; please see math/README to configure.): (x_1,x_2,x_3,\dots,x_{n+1})
that define an n-sphere, (Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf S^n
) is represented by the equation:
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \omega
of n-sphere of radius Failed to parse (Missing texvc executable; please see math/README to configure.): r
is given by:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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}
In fact, Failed to parse (Missing texvc executable; please see math/README to configure.): dr \wedge \omega = dx_1 \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:
- 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).
Volume of the n-ball
The hyperdimensional volume of the space which a Failed to parse (Missing texvc executable; please see math/README to configure.): (n-1) -sphere encloses (the Failed to parse (Missing texvc executable; please see math/README to configure.): n -ball) is given by
- Failed to parse (Missing texvc executable; please see math/README to configure.): V_n={\pi^\frac{n}{2}R^n\over\Gamma(\frac{n}{2} + 1)}={C_n R^n}
,
where Failed to parse (Missing texvc executable; please see math/README to configure.): \Gamma
is the gamma function. (For even Failed to parse (Missing texvc executable; please see math/README to configure.): n
, Failed to parse (Missing texvc executable; please see math/README to configure.): \Gamma\left(\frac{n}{2}+1\right)= \left(\frac{n}{2}\right)!
- for odd Failed to parse (Missing texvc executable; please see math/README to configure.): n
, Failed to parse (Missing texvc executable; please see math/README to configure.): \Gamma\left(\frac{n}{2}+1\right)= \sqrt{\pi} \frac{n!!}{2^{(n+1)/2}} , where Failed to parse (Missing texvc executable; please see math/README to configure.): n!!
denotes the double factorial.)
From this, it follows that the value of the constant Failed to parse (Missing texvc executable; please see math/README to configure.): C_n
for a given Failed to parse (Missing texvc executable; please see math/README to configure.): n
is:
- Failed to parse (Missing texvc executable; please see math/README to configure.): C_n={\frac{\pi^r}{r!}}
, for even Failed to parse (Missing texvc executable; please see math/README to configure.): n
such that Failed to parse (Missing texvc executable; please see math/README to configure.): n={2r}
, and
- Failed to parse (Missing texvc executable; please see math/README to configure.): C_n={\frac{2^{(n+1)/2}\pi^{(n-1)/2}}{n!!}}
, for odd Failed to parse (Missing texvc executable; please see math/README to configure.): n .
The "surface area" of this (n-1)-sphere is
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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}}
The following relationships hold between the n-spherical surface area and volume:
- Failed to parse (Missing texvc executable; please see math/README to configure.): V_n/S_n = R/n\,
- Failed to parse (Missing texvc executable; please see math/README to configure.): S_{n+2}/V_n = 2\pi R\,
The interior of an n-sphere, the set of all points whose distance from the center is less than Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): R ), a closed hyperball.
Examples
For small values of Failed to parse (Missing texvc executable; please see math/README to configure.): n , the volumes, Failed to parse (Missing texvc executable; please see math/README to configure.): V_n
, of the Failed to parse (Missing texvc executable; please see math/README to configure.): n
-ball of radius Failed to parse (Missing texvc executable; please see math/README to configure.): R
are:
-
Failed to parse (Missing texvc executable; please see math/README to configure.): V_0\,
(point)
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Failed to parse (Missing texvc executable; please see math/README to configure.): 1\, |
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Failed to parse (Missing texvc executable; please see math/README to configure.): V_1\,
(line segment)
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Failed to parse (Missing texvc executable; please see math/README to configure.): 2\,R |
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Failed to parse (Missing texvc executable; please see math/README to configure.): V_2\,
(disk)
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Failed to parse (Missing texvc executable; please see math/README to configure.): \pi\,R^2 |
= |
Failed to parse (Missing texvc executable; please see math/README to configure.): 3.14159\ldots\,R^2 |
Failed to parse (Missing texvc executable; please see math/README to configure.): V_3\,
(ball)
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= |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{4 \pi}{3}\,R^3 |
= |
Failed to parse (Missing texvc executable; please see math/README to configure.): 4.18879\ldots\,R^3 |
| Failed to parse (Missing texvc executable; please see math/README to configure.): V_4\, |
= |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\pi^2}{2}\,R^4 |
= |
Failed to parse (Missing texvc executable; please see math/README to configure.): 4.93480\ldots\,R^4 |
| Failed to parse (Missing texvc executable; please see math/README to configure.): V_5\, |
= |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{8 \pi^2}{15}\,R^5 |
= |
Failed to parse (Missing texvc executable; please see math/README to configure.): 5.26379\ldots\,R^5 |
| Failed to parse (Missing texvc executable; please see math/README to configure.): V_6\, |
= |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\pi^3}{6}\,R^6 |
= |
Failed to parse (Missing texvc executable; please see math/README to configure.): 5.16771\ldots\,R^6 |
| Failed to parse (Missing texvc executable; please see math/README to configure.): V_7\, |
= |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{16 \pi^3}{105}\,R^7 |
= |
Failed to parse (Missing texvc executable; please see math/README to configure.): 4.72477\ldots\,R^7 |
| Failed to parse (Missing texvc executable; please see math/README to configure.): V_8\, |
= |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\pi^4}{24}\,R^8 |
= |
Failed to parse (Missing texvc executable; please see math/README to configure.): 4.05871\ldots\,R^8 |
| Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{n\rightarrow\infty} \frac{V_n}{R^n}\, |
= |
Failed to parse (Missing texvc executable; please see math/README to configure.): 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.
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 Failed to parse (Missing texvc executable; please see math/README to configure.): \ r , and Failed to parse (Missing texvc executable; please see math/README to configure.): \ n-1
angular coordinates Failed to parse (Missing texvc executable; please see math/README to configure.): \ \phi _1 , \phi _2 , ... , \phi _{n-1}
. If Failed to parse (Missing texvc executable; please see math/README to configure.): \ x_i
are the
Cartesian coordinates, then we may define
- Failed to parse (Missing texvc executable; please see math/README to configure.): x_1=r\cos(\phi_1)\,
- Failed to parse (Missing texvc executable; please see math/README to configure.): x_2=r\sin(\phi_1)\cos(\phi_2)\,
- Failed to parse (Missing texvc executable; please see math/README to configure.): x_3=r\sin(\phi_1)\sin(\phi_2)\cos(\phi_3)\,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \cdots\,
- Failed to parse (Missing texvc executable; please see math/README to configure.): x_{n-1}=r\sin(\phi_1)\cdots\sin(\phi_{n-2})\cos(\phi_{n-1})\,
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \tan(\phi_{n-1})=\frac{x_n}{x_{n-1}}
- Failed to parse (Missing texvc executable; please see math/README to configure.): \tan(\phi_{n-2})=\frac{\sqrt{{x_n}^2+{x_{n-1}}^2}}{x_{n-2}}
- Failed to parse (Missing texvc executable; please see math/README to configure.): \cdots\,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \tan(\phi_{1})=\frac{\sqrt{{x_n}^2+{x_{n-1}}^2+\cdots+{x_2}^2}}{x_{1}}
Note that last angle Failed to parse (Missing texvc executable; please see math/README to configure.): \phi _{n-1}
has a range of Failed to parse (Missing texvc executable; please see math/README to configure.): 2\pi
while the other angles have a range of Failed to parse (Missing texvc executable; please see math/README to configure.): \pi
. This range covers the whole sphere.
The volume element in n-dimensional Euclidean space will be found from the Jacobian of the transformation:
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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}
- Failed to parse (Missing texvc executable; please see math/README to configure.): =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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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
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 Failed to parse (Missing texvc executable; please see math/README to configure.): \ [x,y,z]
on a two-dimensional sphere of radius 1 maps to the point Failed to parse (Missing texvc executable; please see math/README to configure.): \ [x,y,z] \mapsto \left[\frac{x}{1-z},\frac{y}{1-z}\right]
on the Failed to parse (Missing texvc executable; please see math/README to configure.): \ xy
plane. In other words:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \ [x,y,z] \mapsto \left[\frac{x}{1-z},\frac{y}{1-z}\right].
Likewise, the stereographic projection of an n-sphere Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{S}^{n-1}
of radius 1 will map to the n-1 dimensional hyperplane Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{R}^{n-1}
perpendicular to the Failed to parse (Missing texvc executable; please see math/README to configure.): \ x_n
axis as:
- Failed to parse (Missing texvc executable; please see math/README to configure.): [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].
See also
References
- Moura, Eduarda & Henderson, David G. (1996), Experiencing geometry: on plane and sphere, Prentice Hall, ISBN 978-0-13-373770-7, <http://www.math.cornell.edu/~henderson/books/eg00> (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)
External links
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