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In mathematics, a 'hypersphere' is a higher dimensional analog of the usual 2-sphere. It can be constructed by gluing two Euclidean spaces together and can be embedded in Euclidean space as the locus of points a fixed distance from the origin.
A hypersphere of dimension ''n'' is called an $n$-sphere and denoted $mathbf\{S\}^n$. It is an ''n''-dimensional manifold and can be embedded in Euclidean (''n''+1)-space.

Contents

Euclidean coordinates in (''n''+1)-space

''n''-ball

Notation

Hyperspherical volume

Hyperspherical volume - some examples

Hyperspherical coordinates

Stereographic projection

References

Euclidean coordinates in (''n''+1)-space

The set of points in (''n''+1)-space: $(x\_1,x\_2,x\_3,...,x\_\{n+1\})$ 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 hypersphere exists in $n+1$-dimensional Euclidean space and is an example of an $n$-manifold.

''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 hyperspheres 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 hyperspheres.[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).

Hyperspherical volume

The hyperdimensional volume of the space which a $(n-1)$-sphere encloses (the '$n$-ball') is:
:
where $Gamma$ is the gamma function. (For even $n$,
ight)= left(rac{n}{2}
ight)!; for odd $n$,
ight)= sqrt{pi} rac{n!!}{2^{(n+1)/2}}, where $n!!$ denotes the double factorial.)
The "surface area" of this (n-1)-sphere is
:
The following relationships hold between the hyperspherical surface area and volume:
:$V\_n/S\_n\; =\; R/n,$
:$S\_\{n+2\}/V\_n\; =\; 2pi\; R,$
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.

Hyperspherical volume - some examples

For small values of $n$, the volumes, $V\_n$ , of the unit $n$-ball ($R=1$) are:
:{|
|-
|$V\_0,$
|=
|$1,$
| 
| 
|-
|$V\_1,$ (''line segment'')
|=
|$2,$
| 
| 
|-
|$V\_2,$ (''disk'')
|=
|$pi,$
|=
|$3.14159ldots,$
|-
|$V\_3,$ (''ball'')
|=
|
|=
|$4.18879ldots,$
|-
|$V\_4,$
|=
|
|=
|$4.93480ldots,$
|-
|$V\_5,$
|=
|
|=
|$5.26379ldots,$
|-
|$V\_6,$
|=
|
|=
|$5.16771ldots,$
|-
|$V\_7,$
|=
|
|=
|$4.72477ldots,$
|-
|$V\_8,$
|=
|
|=
|$4.05871ldots,$
|-
|$lim\_\{n$
ightarrowinfty} 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... 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 2^''n''; the ratio of the volume of the hypersphere to its circumscribed hypercube decreases monotonically as the dimension increases.
This apparently strange behavior of the unit n-sphere volume can be understood by assigning units of length to each dimension. It then becomes clearly meaningless to compare the unit-sphere volumes in different n's, just as it is meaningless to compare a length to an area. A meaningful comparison is obtained by using a dimensionless measure of the volume, such as the ratio of the hypersphere 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 $r$, and $n-1$ angular coordinates $phi\; \_1\; ,\; phi\; \_2\; ,\; ...\; ,\; phi\; \_\{n-1\}$. If $x\_i$ are the
Cartesian coordinates, then we may define
:$x\_1=rcos(phi\_1),$
:$x\_2=rsin(phi\_1)cos(phi\_2),$
:$x\_3=rsin(phi\_1)sin(phi\_2)cos(phi\_3),$
:$cdots,$
:$x\_\{n-1\}=rsin(phi\_1)cdotssin(phi\_\{n-2\})cos(phi\_\{n-1\}),$
:$x\_n~~,=rsin(phi\_1)cdotssin(phi\_\{n-2\})sin(phi\_\{n-1\}),$
While the inverse transformations can be derived from those above:
:
:
:$cdots,$
:
Note that last angle $phi\; \_\{n-1\}$ has a range of $2pi$ while the other angles have a range of $pi$. This range covers the whole sphere.
The hyperspherical volume element will be found from the Jacobian of the transformation:
:$d^nr\; =$
left|detrac{partial (x_i)}{partial(r,phi_i)}
ight|
dr,dphi_1 , dphi_2ldots dphi_{n-1}
:$=r^\{n-1\}sin^\{n-2\}(phi\_1)sin^\{n-3\}(phi\_2)cdots\; sin(phi\_\{n-2\}),$
dr,dphi_1 , dphi_2cdots dphi_{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}^piint_{phi_{n-1}=0}^{2pi}d^nr. ,

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
ight] on the $xy$ plane. In other words:
:
ight]
Likewise, the stereographic projection of a hypersphere $mathbf\{S\}^\{n-1\}$ of radius 1 will map to the n-1 dimensional hyperplane $mathbf\{R\}^\{n-1\}$ perpendicular to the $x\_n$ axis as:
:
ight]
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hhhhhghf
hfgdhgfdhgfhgfhgfhgf

fghgfdhgdhgfhgfhggh

blockqugote>''
hg

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