Hypersphere

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Stereographic projection of the hypersphere's parallels (red), meridians (blue) and hypermeridians (green). Due to conformal property of Stereographic projection, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles: the curves that intersects <0,0,0,1> have infinite radius (= straight line).
Stereographic projection of the hypersphere's parallels (red), meridians (blue) and hypermeridians (green). Due to conformal property of Stereographic projection, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles: the curves that intersects <0,0,0,1> have infinite radius (= straight line).

In mathematics, a hypersphere is a higher dimensional sphere.

The term n-sphere, or \mathbb S^n, is a higher dimensional sphere, with n surface dimensions and embedded in (n+1)-space.

Specifically:

  1. A 0-sphere represents two points on a line.
  2. A 1-sphere is a circle on a plane.
  3. A 2-sphere is an ordinary sphere in 3 dimensional space.
  4. And higher, a 3-sphere exists in 4 dimensional space, etc.

This notation (used throughout this article) is using the convention common in geometrical use. Potentially confusingly, topologists routinely use the dimensionality of the surface to label hyperspheres.[1] Thus a circle in a plane is a 1-sphere, a ball in 3D is a 2-sphere, etc.

Contents

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

The set of points in (n+1)-space: (x1,x2,x3,...,xn + 1) that define an n-sphere, (\mathbb 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 hypersphere exists in n + 1-dimensional Euclidean space and is an example of an n-manifold.

[edit] n-ball

The interior of 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] Hyperspherical volume

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

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

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.)

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

S_n=\frac{dV_n}{dR}=\frac{nV_n}{R}={2\pi^\frac{n}{2}R^{n-1}\over\Gamma(\frac{n}{2})}

The following relationships hold between the hyperspherical surface area and volume:

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

The interior of a hypersphere, that is the set of all points whose distance from the centre is less than R, is called a hyperball, or if the hypersphere itself is included, a closed hyperball.

[edit] Hyperspherical volume - some examples

For small values of n, the volumes, Vn , of the unit n-ball (R = 1) are:

V_0\, = 1\,    
V_1\, = 2\,    
V_2\, = \pi\, = 3.14159\ldots\,
V_3\, = \frac{4 \pi}{3}\, = 4.18879\ldots\,
V_4\, = \frac{\pi^2}{2}\, = 4.93480\ldots\,
V_5\, = \frac{8 \pi^2}{15}\, = 5.26379\ldots\,
V_6\, = \frac{\pi^3}{6}\, = 5.16771\ldots\,
V_7\, = \frac{16 \pi^3}{105}\, = 4.72478\ldots\,
V_8\, = \frac{\pi^4}{24}\, = 4.05871\ldots\,
\lim_{n\rightarrow\infty} V_n\, = 0\,

If the dimension \ n , is not limited to integral values, the hypersphere 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...

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 hypersphere to its circumscribed hypercube decreases monotonically 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 hyperspherical volume element will be found from the Jacobian of the transformation:

d^nr = \left|\det\frac{\partial (x_i)}{\partial(r,\phi_i)}\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 hypersphere 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^nr. \,

[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-dimensional hypersphere 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 a hypersphere \mathbb{S}^{n-1} of radius 1 will map to the n-1 dimensional hyperplane \mathbb{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] See also

[edit] 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
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