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{{DISPLAYTITLE:''n''-sphere}}
[[File:Sphere wireframe.svg|thumb|2-sphere wireframe as an [[orthogonal projection]]]]
[[File:Hypersphere coord.PNG|right|thumb|Just as a [[stereographic projection]] can project a sphere's surface to a plane, it can also project the surface of a 4-sphere into 3-space. This image shows three coordinate directions projected to 3-space:
parallels (red), [[Meridian (geography)|meridians]] (blue) and hypermeridians (green).
Due to the [[conformal map|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 {{angbr|0,0,0,1}} have an infinite radius (= straight line).]]
In [[mathematics]], the '''{{math|''n''}}-sphere''' is the generalization of the ordinary [[sphere]] to [[n-dimensional space|spaces of arbitrary dimension]].  It is an {{math|''n''}}-dimensional [[manifold]] that can be embedded in Euclidean {{math|(''n'' + 1)}}-space.

The 0-sphere is a pair of points, the 1-sphere is a circle, and the 2-sphere is an ordinary sphere.  Generally, when embedded in an {{math|(''n'' + 1)}}-dimensional Euclidean space, an {{math|''n''}}-sphere is the surface or boundary of an {{math|(''n'' + 1)}}-dimensional [[Ball (mathematics)|ball]].  That is, for any [[natural number]] {{math|''n''}}, an {{math|''n''}}-sphere of radius {{math|''r''}} may be defined in terms of an [[embedding]] in {{math|(''n'' + 1)}}-dimensional [[Euclidean space]] as the set of points that are at distance {{math|''r''}} from a central point, where the radius {{math|''r''}} may be any [[Positive number|positive]] [[real number]]. Thus, the {{math|''n''}}-sphere would be defined by:
:<math> S^n = \left\{ x \in \mathbb{R}^{n+1} : \left\| x \right\| = r \right\} .</math>

In particular:
*the pair of points at the ends of a (one-dimensional) [[line segment]] is a 0-sphere,
*the [[circle]], which is the one-dimensional circumference of a (two-dimensional) disk in the plane is a 1-sphere,
*the two-dimensional surface of a (three-dimensional) ball in three-dimensional space is a 2-sphere, often simply called a [[sphere]],
*the three-dimensional boundary of a (four-dimensional) 4-ball in four-dimensional Euclidean is a [[3-sphere]], also known as a '''[[glome]]'''.
*the n-1 dimensional boundary of a (n-dimensional) n-ball is a general n-sphere which can be denoted as a "glone". 

An ''n''-sphere embedded in an {{math|(''n'' + 1)}}-dimensional Euclidean space is called a '''[[hypersphere]]'''.  The {{math|''n''}}-sphere of unit radius is called the '''unit {{math|''n''}}-sphere''', denoted {{math|''S''<sup>''n''</sup>}}, often referred to as ''the'' {{math|''n''}}-sphere.

When embedded as described, an {{math|''n''}}-sphere is the surface or boundary of an {{math|(''n'' + 1)}}-dimensional [[Ball (mathematics)|ball]]. For {{math|''n'' ≥ 2}}, the {{math|''n''}}-spheres are the [[simply connected]] {{math|''n''}}-dimensional [[manifold]]s of constant, positive [[Constant curvature|curvature]]. The {{math|''n''}}-spheres admit several other topological descriptions: for example, they can be constructed by gluing two {{math|''n''}}-dimensional Euclidean spaces together, by identifying the boundary of an [[hypercube|{{math|''n''}}-cube]] with a point, or (inductively) by forming the [[suspension (topology)|suspension]] of an {{math|(''n'' − 1)}}-sphere.

==Description==

For any [[natural number]] {{math|''n''}}, an {{math|''n''}}-sphere of radius {{math|''r''}} is defined as the set of points in {{math|(''n'' + 1)}}-dimensional [[Euclidean space]] that are at distance {{math|''r''}} from some fixed point {{math|'''c'''}}, where {{math|''r''}} may be any [[Positive number|positive]] [[real number]] and where {{math|'''c'''}} may be any point in {{math|(''n'' + 1)}}-dimensional space. In particular:
* a 0-sphere is a pair of points {{math|{''c'' − ''r'', ''c'' + ''r''<nowiki>}</nowiki>}}, and is the boundary of a line segment (1-ball).
* a [[1-sphere]] is a [[circle]] of radius {{math|''r''}} centered at {{math|'''c'''}}, and is the boundary of a disk (2-ball).
* a [[2-sphere]] is an ordinary 3-dimensional [[sphere]] in 3-dimensional Euclidean space, and is the boundary of an ordinary ball (3-ball).
* a [[3-sphere]] is a sphere in 4-dimensional Euclidean space.

===Euclidean coordinates in {{math|(''n'' + 1)}}-space ===

The set of points in {{math|(''n'' + 1)}}-space, {{math|(''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''+1</sub>)}}, that define an {{math|''n''}}-sphere, {{math|''S''<sup>''n''</sup>}}, is represented by the equation:

:<math>r^2=\sum_{i=1}^{n+1} \left(x_i - c_i\right)^2</math>

where {{math|'''c'''}}={{math|(''c''<sub>1</sub>, ''c''<sub>2</sub>, ..., ''c''<sub>''n''+1</sub>)}} is a center point, and {{math|''r''}} is the radius.

The above {{math|''n''}}-sphere exists in {{math|(''n'' + 1)}}-dimensional Euclidean space and is an example of an {{math|''n''}}-[[manifold]].  The [[volume form]] {{math|''ω''}} of an {{math|''n''}}-sphere of radius {{math|''r''}} is given by

:<math>\omega = \frac{1}{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</math>

where {{math|∗}} is the [[Hodge star operator]]; see {{harvtxt|Flanders|1989|loc=§6.1}} for a discussion and proof of this formula in the case {{math|''r'' {{=}} 1}}.  As a result,
:<math>dr \wedge \omega = dx_1 \wedge \cdots \wedge dx_{n+1}.</math>

=== {{math|''n''}}-ball ===<!-- This section is linked from [[n-ball]], a redirect page. -->

{{main|Ball (mathematics)}}

The space enclosed by an {{math|''n''}}-sphere is called an {{math|(''n'' + 1)}}-[[Ball (mathematics)|ball]]. An {{math|(''n'' + 1)}}-ball is [[Closed set|closed]] if it includes the {{math|''n''}}-sphere, and it is [[Open set|open]] if it does not include the {{math|''n''}}-sphere.

Specifically:
* A 1-''ball'', a [[line segment]], is the interior of a 0-sphere.
* A 2-''ball'', a [[Disk (mathematics)|disk]], is the interior of a [[circle]] (1-sphere).
* A 3-''ball'', an ordinary [[Ball (mathematics)|ball]], is the interior of a [[sphere]] (2-sphere).
* A 4-''ball'' is the interior of a [[3-sphere]], etc.

===Topological description===

[[Topology|Topologically]], an {{math|''n''}}-sphere can be constructed as a [[Alexandroff extension|one-point compactification]] of {{math|''n''}}-dimensional Euclidean space.  Briefly, the {{math|''n''}}-sphere can be described as {{math|'''S'''<sup>''n''</sup> {{=}} '''R'''<sup>''n''</sup> ∪ {∞<nowiki>}</nowiki>}}, which is {{math|''n''}}-dimensional Euclidean space plus a single point representing infinity in all directions.
In particular, if a single point is removed from an {{math|''n''}}-sphere, it becomes [[Homeomorphism|homeomorphic]] to {{math|'''R'''<sup>''n''</sup>}}. This forms the basis for [[stereographic projection]].<ref>James W. Vick (1994). ''Homology theory'', p. 60. Springer</ref>

==Volume and surface area==
{{see also|Volume of an n-ball}}
<div style="display:block; width:300px; float:right; background-color:#eeeeff; border: solid 1px #cccccc; padding: 10px">
{{math|''V<sub>n</sub>''(''R'')}} and {{math|''S<sub>n</sub>''(''R'')}} are the {{math|''n''}}-dimensional volume of the [[Ball (mathematics)|{{math|''n''}}-ball]] and the surface area of the {{math|''n''}}-sphere embedded in dimension {{math|''n'' + 1}}, respectively, of radius {{math|''R''}}.

The constants {{math|''V<sub>n</sub>''}} and {{math|''S<sub>n</sub>''}} (for {{math|''R'' {{=}} 1}}, the unit ball and sphere) are related by the recurrences:
:<math>\begin{align}
V_0&=1 & V_{n+1}&=\frac{S_n}{n+1} \\[6pt]
S_0&=2 & S_{n+1}&=2\pi V_n
\end{align}</math>

The surfaces and volumes can also be given in closed form:
:<math>\begin{align}
S_{n}(R) &= \frac{2\,\pi^\frac{n+1}{2}}{\Gamma\left(\frac{n+1}{2}\right)}R^{n} \\[6pt]
V_n(R) &= \frac{\pi^\frac{n}{2}}{\Gamma\left(\frac{n}{2} + 1\right)}R^n
\end{align}</math>

where {{math|''Γ''}} is the [[gamma function]]. Derivations of these equations are given in this section. 
</div>
{{hypersphere_volume_and_surface_area_graphs.svg}}
<div style="overflow-wrap: normal"><!-- Prevent words from breaking in the middle due to the column being very narrow (e.g. on mobile). -->In general, the volume of the {{math|''n''}}-ball in {{math|''n''}}-dimensional Euclidean space, and the surface area of the {{math|''n''}}-sphere in {{math|(''n'' + 1)}}-dimensional Euclidean space, of radius {{math|''R''}}, are proportional to the {{math|''n''}}th power of the radius, {{math|''R''}} (with different constants of proportionality that vary with {{math|''n''}}). We write {{math|''V<sub>n</sub>''(''R'') {{=}} ''V<sub>n</sub>R<sup>n</sup>''}} for the volume of the {{math|''n''}}-ball and {{math|''S<sub>n</sub>''(''R'') {{=}} ''S<sub>n</sub>R<sup>n</sup>''}} for the surface area of the {{math|''n''}}-sphere, both of radius {{math|''R''}}, where {{math|''V<sub>n</sub>'' {{=}} ''V<sub>n</sub>''(1)}} and {{math|''S<sub>n</sub>'' {{=}} ''S<sub>n</sub>''(1)}} are the values for the unit-radius case.<div>

In theory, one could compare the values of {{math|''S<sub>n</sub>''(''R'')}} and  {{math|''S<sub>m</sub>''(''R'')}} for {{math|''n'' ≠ ''m''}}.  However, this is somewhat nonsensical.  For example, if {{math|1=''n'' = 2}} and {{math|1=''m'' = 3}} then the comparison is like comparing a number of square meters to a different number of cubic meters.  The same lack of sense applies to a comparison of {{math|''V<sub>n</sub>''(''R'')}} and  {{math|''V<sub>m</sub>''(''R'')}} for {{math|''n'' ≠ ''m''}}.

===Examples===
The 0-ball consists of a single point. The 0-dimensional [[Hausdorff measure]] is the number of points in a set. So, 
:<math>V_0=1.</math>
The unit 1-ball is the interval {{math|[−1,1]}} of length 2. So,
:<math>V_1 = 2.</math>
The 0-sphere consists of its two end-points, {{math|{−1,1<nowiki>}</nowiki>}}. So,
:<math>S_0 = 2.</math>
The unit 1-sphere is the unit circle  in the Euclidean plane, and this has circumference (1-dimensional measure)
:<math>S_1 = 2\pi.</math>
The region enclosed by the unit 1-sphere is the 2-ball, or unit disc, and this has area (2-dimensional measure)
:<math>V_2 = \pi.</math>
Analogously, in 3-dimensional Euclidean space, the surface area (2-dimensional measure) of the unit 2-sphere is given by
:<math>S_2 = 4\pi.</math>
and the volume enclosed is the volume (3-dimensional measure) of the unit 3-ball, given by
:<math>V_3 = \tfrac{4}{3} \pi.</math>

===Recurrences===
The ''surface area'', or properly the {{math|''n''}}-dimensional volume, of the {{math|''n''}}-sphere at the boundary of the {{math|(''n'' + 1)}}-ball of radius {{math|''R''}} is related to the volume of the ball by the differential equation
:<math>S_{n}R^{n}=\frac{dV_{n+1}R^{n+1}}{dR}={(n+1)V_{n+1}R^{n}},</math>
or, equivalently, representing the unit {{math|''n''}}-ball as a union of concentric {{math|(''n'' − 1)}}-sphere ''[[spherical shell|shells]]'',
:<math>V_{n+1} = \int_0^1 S_{n}r^{n}\,dr.</math>
So, 
:<math>V_{n+1} = \frac{S_n}{n+1}.</math>

We can also represent the unit {{math|(''n'' + 2)}}-sphere as a union of [[Torus|tori]], each the product of a circle (1-sphere) with an {{math|''n''}}-sphere. Let {{math|''r'' {{=}} cos ''θ''}} and {{math|''r''<sup>2</sup> + ''R''<sup>2</sup> {{=}} 1}}, so that {{math|''R'' {{=}} sin ''θ''}} and {{math|''dR'' {{=}} cos ''θ'' ''dθ''}}. Then,
:<math>
\begin{align}
S_{n+2} &= \int_0^\frac{\pi}{2}S_1 r \cdot S_n R^n\, d\theta \\[6pt]
&=\int_0^\frac{\pi}{2}S_1 \cdot S_n R^n\cos\theta\,d\theta\\[6pt]
&=\int_0^1 S_1 \cdot S_n R^n \,dR\\[6pt]
&= S_1 \int_0^1 S_n R^n \,dR\\[6pt]
&= 2\pi V_{n+1}.
\end{align}
</math>
Since {{math|''S''<sub>1</sub> {{=}} 2π ''V''<sub>0</sub>}}, the equation
:<math>S_{n+1} =  2\pi V_{n}</math>
holds for all {{math|''n''}}.

This completes the derivation of the recurrences:
:<math>\begin{align}
V_0&=1 & V_{n+1}&=\frac{S_n}{n+1} \\[6pt]
S_0&=2 & S_{n+1}&=2\pi V_n
\end{align}</math>

===Closed forms===
Combining the recurrences, we see that
:<math>V_{n+2}=2\pi \frac{V_n}{n+2}.</math>
So it is simple to show by induction on ''k'' that,
:<math>\begin{align}
V_{2k} &= \frac{\pi^k}{k!} \\[6pt]
V_{2k+1} &= \frac{2\left(2\pi\right)^k}{(2k+1)!!} = \frac{2 k! \left(4\pi\right)^k}{(2k+1)!}
\end{align}</math>
where {{math|!!}} denotes the [[Factorial#Double factorial|double factorial]], defined for odd integers {{math|2''k'' + 1}} by {{math|1=(2''k'' + 1)!! = 1 × 3 × 5 ... (2''k'' − 1) × (2''k'' + 1)}}.

In general, the volume, in {{math|''n''}}-dimensional Euclidean space, of the unit {{math|''n''}}-ball, is given by

:<math>V_n = \frac{\pi^\frac{n}{2}}{\Gamma\left(\frac{n}{2} + 1\right)}</math>

where {{math|''Γ''}} is the [[gamma function]], which satisfies {{math|''Γ''({{sfrac|1|2}}) {{=}} {{sqrt|π}}}}, {{math|''Γ''(1) {{=}} 1}}, and {{math|''Γ''(''x'' + 1) {{=}} ''xΓ''(''x'')}}.

By multiplying {{math|''V<sub>n</sub>''}} by {{math|''R<sup>n</sup>''}}, differentiating with respect to {{math|''R''}}, and then setting {{math|R {{=}} 1}}, we get the closed form

:<math>S_{n-1} = \frac{n\pi^\frac{n}{2}}{\Gamma\left(\frac{n}{2}+1 \right)}.</math>

===Other relations===
The recurrences can be combined to give a "reverse-direction" recurrence relation for surface area, as depicted in the diagram:

:<math>S_{n-1} = \frac{n}{2 \pi} S_{n+1}</math>

[[File:N SpheresVolumeAndSurfaceArea.png|thumb|580px|{{math|''n''}} refers to the dimension of the ambient Euclidean space, which is also the intrinsic dimension of the solid whose volume is listed here, but which is 1 more than the intrinsic dimension of the sphere whose surface area is listed here. The curved red arrows show the relationship between formulas for different {{math|''n''}}. The formula coefficient at each arrow's tip equals the formula coefficient at that arrow's tail times the factor in the arrowhead (where the {{math|''n''}} in the arrowhead refers to the {{math|''n''}} value that the arrowhead points to). If the direction of the bottom arrows were reversed, their arrowheads would say to multiply by {{math|{{sfrac|2π|''n'' − 2}}}}. Alternatively said, the surface area {{math|''S''<sub>''n''+1</sub>}} of the sphere in {{math|''n'' + 2}} dimensions is exactly {{math|2π''R''}} times the volume {{math|''V''<sub>''n''</sub>}} enclosed by the sphere in {{math|''n''}} dimensions.]]

Index-shifting {{math|''n''}} to {{math|''n'' − 2}} then yields the recurrence relations:

:<math>\begin{align}
V_n &= \frac{2 \pi}{n} V_{n-2} \\[6pt]
S_{n-1} &= \frac{2 \pi}{n-2} S_{n-3}
\end{align}</math>

where {{math|''S''<sub>0</sub> {{=}} 2}}, {{math|''V''<sub>1</sub> {{=}} 2}}, {{math|''S''<sub>1</sub> {{=}} 2π}} and {{math|''V''<sub>2</sub> {{=}} π}}.

The recurrence relation for {{math|''V''<sub>n</sub>}} can also be proved via [[integral|integration]] with 2-dimensional [[Polar coordinate system|polar coordinates]]:

:<math>\begin{align}
V_n 
& = \int_0^1 \int_0^{2\pi} V_{n-2}\left(\sqrt{1-r^2}\right)^{n-2} \, r \, d\theta \, dr \\[6pt]
& = \int_0^1 \int_0^{2\pi} V_{n-2} \left(1-r^2\right)^{\frac{n}{2}-1}\, r \, d\theta \, dr \\[6pt]
& = 2 \pi V_{n-2} \int_{0}^{1} \left(1-r^2\right)^{\frac{n}{2}-1}\, r \, dr \\[6pt]
& = 2 \pi V_{n-2} \left[ -\frac{1}{n}\left(1-r^2\right)^\frac{n}{2} \right]^{r=1}_{r=0} \\[6pt]
& = 2 \pi V_{n-2} \frac{1}{n} = \frac{2 \pi}{n} V_{n-2}.
\end{align}</math>

==Spherical coordinates==<!-- This section is linked from [[Spherical coordinate system]] -->

We may define a coordinate system in an {{math|''n''}}-dimensional Euclidean space which is analogous to the [[Spherical coordinates|spherical coordinate system]] defined for 3-dimensional Euclidean space, in which the coordinates consist of a radial coordinate {{math|''r''}}, and {{math|''n'' − 1}} angular coordinates {{math|''φ''<sub>1</sub>, ''φ''<sub>2</sub>, ... ''φ''<sub>''n''−1</sub>}}, where the angles {{math|''φ''<sub>1</sub>, ''φ''<sub>2</sub>, ... ''φ''<sub>''n''−2</sub>}} range over {{math|[0,π]}} radians (or over {{math|[0,180]}} degrees) and {{math|''φ''<sub>''n''−1</sub>}} ranges over {{math|[0,2π)}} radians (or over {{math|[0,360)}} degrees). If {{math|''x<sub>i</sub>''}} are the Cartesian coordinates, then we may compute {{math|''x''<sub>1</sub>, ... ''x<sub>n</sub>''}} from {{math|''r'', ''φ''<sub>1</sub>, ... ''φ''<sub>''n''−1</sub>}} with: <ref>{{cite journal |last1=Blumenson |first1=L. E. |title=A Derivation of n-Dimensional Spherical Coordinates |journal=The American Mathematical Monthly |date=1960 |volume=67 |issue=1 |pages=63–66 |jstor=2308932 |doi=10.2307/2308932 }}</ref> 

:<math>\begin{align}
x_1 &= r \cos(\varphi_1) \\
x_2 &= r \sin(\varphi_1) \cos(\varphi_2) \\
x_3 &= r \sin(\varphi_1) \sin(\varphi_2) \cos(\varphi_3) \\
    &\vdots\\
x_{n-1} &= r \sin(\varphi_1) \cdots \sin(\varphi_{n-2}) \cos(\varphi_{n-1}) \\
x_n     &= r \sin(\varphi_1) \cdots \sin(\varphi_{n-2}) \sin(\varphi_{n-1}) .
\end{align}</math> 
Except in the special cases described below, the inverse transformation is unique:

:<math>
\begin{align}
r      &= \sqrt{{x_n}^2 + {x_{n-1}}^2 + \cdots + {x_2}^2 + {x_1}^2} \\[6pt]
\varphi_1 &= \arccot \frac{x_{1}}{\sqrt{{x_n}^2+{x_{n-1}}^2+\cdots+{x_2}^2}} &&= \arccos \frac{x_{1}}{\sqrt{{x_n}^2+{x_{n-1}}^2+\cdots+{x_1}^2}} \\[6pt]
\varphi_2 &= \arccot \frac{x_{2}}{\sqrt{{x_n}^2+{x_{n-1}}^2+\cdots+{x_3}^2}} &&= \arccos \frac{x_{2}}{\sqrt{{x_n}^2+{x_{n-1}}^2+\cdots+{x_2}^2}} \\[6pt]
       &\vdots &&\vdots \\[6pt]
\varphi_{n-2} &= \arccot \frac{x_{n-2}}{\sqrt{{x_n}^2+{x_{n-1}}^2}} &&= \arccos \frac{x_{n-2}}{\sqrt{{x_n}^2+{x_{n-1}}^2+{x_{n-2}}^2}} \\[6pt]
\varphi_{n-1} &= 2\arccot \frac{x_{n-1}+\sqrt{x_n^2+x_{n-1}^2}}{x_n} &&= \begin{cases}
    \arccos \frac{x_{n-1}}{\sqrt{{x_n}^2+{x_{n-1}}^2}} & x_n\geq 0 \\[6pt]
     2\pi - \arccos \frac{x_{n-1}}{\sqrt{{x_n}^2+{x_{n-1}}^2}} & x_n < 0
\end{cases} \,.
\end{align}
</math>

where if {{math|''x<sub>k</sub>'' ≠ 0}} for some {{math|''k''}} but all of {{math|''x''<sub>''k''+1</sub>, ... ''x<sub>n</sub>''}} are zero then {{math|''φ<sub>k</sub>'' {{=}} 0}} when {{math|''x<sub>k</sub>'' > 0}}, and {{math|''φ<sub>k</sub>'' {{=}} π}} (180 degrees) when {{math|''x<sub>k</sub>'' < 0}}.

There are some special cases where the inverse transform is not unique; {{math|''φ<sub>k</sub>''}} for any {{math|''k''}} will be ambiguous whenever all of {{math|''x<sub>k</sub>'', ''x''<sub>''k''+1</sub>, ... ''x<sub>n</sub>''}} are zero; in this case {{math|''φ<sub>k</sub>''}} may be chosen to be zero.

===Spherical volume element===
Expressing the angular measures in radians, the [[volume element]] in {{math|''n''}}-dimensional Euclidean space will be found from the [[Jacobian matrix and determinant|Jacobian]] of the transformation:

:<math>
\begin{pmatrix} 
\cos(\varphi_{1})                                       &-r\sin(\varphi_{1})                      &0                                 &0&\cdots    &0 \\
\sin(\varphi_{1})\cos(\varphi_{2})                       &r\cos(\varphi_{1})\cos(\varphi_{2})       &-r\sin(\varphi_{1})\sin(\varphi_{2})&0&\cdots   &0 \\
\vdots                                                 &        \vdots                          & \vdots                           &\ddots &  &\vdots\\
                                                       &                                        &                                  & &       &0    \\
\sin(\varphi_{1})\cdots\sin(\varphi_{n-2})\cos(\varphi_{n-1})&   \cdots                                     &\cdots                                  & &       &-r\sin(\varphi_{1})\cdots\sin(\varphi_{n-2})\sin(\varphi_{n-1})    \\
\sin(\varphi_{1})\cdots\sin(\varphi_{n-2})\sin(\varphi_{n-1})& r\cos(\varphi_{1})\cdots\sin(\varphi_{n-1})& \cdots                           & &      &r\sin(\varphi_{1})\cdots\sin(\varphi_{n-2})\cos(\varphi_{n-1})
\end{pmatrix}
</math>

: <math>\begin{align}
d^nV & = 
\left|\det\frac{\partial (x_i)}{\partial\left(r,\varphi_j\right)}\right|
dr\,d\varphi_1 \, d\varphi_2\cdots d\varphi_{n-1} \\[6pt]
& = r^{n-1}\sin^{n-2}(\varphi_1)\sin^{n-3}(\varphi_2)\cdots \sin(\varphi_{n-2})\,
dr\,d\varphi_1 \, d\varphi_2\cdots d\varphi_{n-1}
\end{align}</math>

and the above equation for the volume of the {{math|''n''}}-ball can be recovered by integrating:

: <math>V_n=\int_{\varphi_{n-1}=0}^{2\pi} \int_{\varphi_{n-2}=0}^\pi \cdots \int_{\varphi_1=0}^\pi\int_{r=0}^R d^nV.</math>

The volume element of the {{math|(''n'' − 1)}}-sphere, which generalizes the [[area element]] of the 2-sphere, is given by

: <math>d_{S^{n-1}}V = \sin^{n-2}(\varphi_1)\sin^{n-3}(\varphi_2)\cdots \sin(\varphi_{n-2})\, d\varphi_1 \, d\varphi_2\cdots d\varphi_{n-1}.</math>

The natural choice of an orthogonal basis over the angular coordinates is a product of [[Gegenbauer polynomial|ultraspherical polynomials]],

: <math>\begin{align}
& {} \quad \int_0^\pi \sin^{n-j-1}\left(\varphi_j\right) C_s^{\left(\frac{n-j-1}{2}\right)}\cos \left(\varphi_j \right)C_{s'}^{\left(\frac{n-j-1}{2}\right)}\cos \left(\varphi_j\right) \, d\varphi_j \\[6pt]
& = \frac{2^{3-n+j}\pi \Gamma(s+n-j-1)}{s!(2s+n-j-1)\Gamma^2\left(\frac{n-j-1}{2}\right)}\delta_{s,s'}
\end{align}</math>

for {{math|''j'' {{=}} 1, 2,... ''n'' − 2}}, and the {{math|''e''<sup>''isφ''<sub>''j''</sub></sup>}} for the angle {{math|''j'' {{=}} ''n'' − 1}} in concordance with the [[spherical harmonics]].

== Stereographic projection ==

{{main|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 {{math|''n''}}-sphere can be mapped onto an {{math|''n''}}-dimensional hyperplane by the {{math|''n''}}-dimensional version of the stereographic projection. For example, the point {{math|[''x'',''y'',''z'']}} on a two-dimensional sphere of radius 1 maps to the point {{math|[{{sfrac|''x''|1 − ''z''}},{{sfrac|''y''|1 − ''z''}}]}} on the {{math|''xy''}}-plane. In other words,
:<math>[x,y,z] \mapsto \left[\frac{x}{1-z},\frac{y}{1-z}\right].</math>

Likewise, the stereographic projection of an {{math|''n''}}-sphere {{math|'''S'''<sup>''n''−1</sup>}} of radius 1 will map to the {{math|(''n'' − 1)}}-dimensional hyperplane {{math|'''R'''<sup>''n''−1</sup>}} perpendicular to the {{math|''x<sub>n</sub>''}}-axis as
:<math>[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].</math>

== Generating random points ==

=== Uniformly at random on the {{math|(''n'' − 1)}}-sphere ===

[[File:2sphere-uniform.png|thumbnail|A set of uniformly distributed points on the surface of a unit 2-sphere generated using Marsaglia's algorithm.]]

To generate uniformly distributed random points on the unit {{math|(''n'' − 1)}}-sphere (that is, the surface of the unit {{math|''n''}}-ball), {{harvtxt|Marsaglia|1972}} gives the following algorithm.

Generate an {{math|''n''}}-dimensional vector of [[normal distribution|normal deviates]] (it suffices to use {{math|N(0, 1)}}, although in fact the choice of the variance is arbitrary), {{math|'''x''' {{=}} (''x''<sub>1</sub>, ''x''<sub>2</sub>,... ''x<sub>n</sub>'')}}. Now calculate the "radius" of this point:

:<math>r=\sqrt{x_1^2+x_2^2+\cdots+x_n^2}.</math>

The vector {{math|{{sfrac|1|''r''}}'''x'''}} is uniformly distributed over the surface of the unit {{math|''n''}}-ball.

An alternative given by Marsaglia is to uniformly randomly select a point {{math|'''x''' {{=}} (''x''<sub>1</sub>, ''x''<sub>2</sub>,... ''x<sub>n</sub>'')}} in the unit [[hypercube|{{math|''n''}}-cube]] by sampling each {{math|''x''<sub>''i''</sub>}} independently from the [[continuous uniform distribution|uniform distribution]] over {{math|(–1,1)}}, computing {{math|''r''}} as above, and rejecting the point and resampling if {{math|''r'' ≥ 1}} (i.e., if the point is not in the {{math|''n''}}-ball), and when a point in the ball is obtained scaling it up to the spherical surface by the factor {{math|{{sfrac|1|''r''}}}}; then again {{math|{{sfrac|1|''r''}}'''x'''}} is uniformly distributed over the surface of the unit {{math|''n''}}-ball.

=== Uniformly at random within the {{math|''n''}}-ball ===
With a point selected uniformly at random from the surface of the unit {{math|(''n'' - 1)}}-sphere (e.g., by using Marsaglia's algorithm), one needs only a radius to obtain a point uniformly at random from within the unit {{math|''n''}}-ball.  If {{math|''u''}} is a number generated uniformly at random from the interval {{math|[0, 1]}} and {{math|'''x'''}} is a point selected uniformly at random from the unit {{math|(''n'' - 1)}}-sphere, then {{math|''u''<sup>{{frac|1|''n''}}</sup>'''x'''}} is uniformly distributed within the unit {{math|''n''}}-ball.

Alternatively, points may be sampled uniformly from within the unit {{math|''n''}}-ball by a reduction from the unit {{math|(''n'' + 1)}}-sphere. In particular, if {{math|(''x''<sub>1</sub>,''x''<sub>2</sub>,...,''x''<sub>''n''+2</sub>)}} is a point selected uniformly from the unit {{math|(''n'' + 1)}}-sphere, then {{math|(''x''<sub>1</sub>,''x''<sub>2</sub>,...,''x''<sub>''n''</sub>)}} is uniformly distributed within the unit {{math|''n''}}-ball (i.e., by simply discarding two coordinates).<ref>{{cite report|first1=Aaron R. | last1=Voelker | first2=Jan | last2=Gosmann | first3=Terrence C. | last3=Stewart | title=Efficiently sampling vectors and coordinates from the n-sphere and n-ball | year=2017 | publisher=Centre for Theoretical Neuroscience | url=http://compneuro.uwaterloo.ca/publications/voelker2017.html | doi=10.13140/RG.2.2.15829.01767/1}}</ref>

If {{math|''n''}} is sufficiently large, most of the volume of the {{math|''n''}}-ball will be contained in the region very close to its surface, so a point selected from that volume will also probably be close to the surface.  This is one of the phenomena leading to the so-called [[curse of dimensionality]] that arises in some numerical and other applications.

==Specific spheres==

; 0-sphere : The pair of points {{math|{±''R''<nowiki>}</nowiki>}} with the discrete topology for some {{math|''R'' > 0}}. The only sphere that is not [[path-connected]]. Has a natural Lie group structure; isomorphic to O(1). Parallelizable.
; [[1-sphere]] : Also known as the circle. Has a nontrivial fundamental group. Abelian Lie group structure [[U(1)]]; the [[circle group]]. Topologically equivalent to the [[real projective line]], '''R'''P<sup>1</sup>. Parallelizable. SO(2) = U(1).
; [[2-sphere]] : Also known as the sphere. Complex structure; see [[Riemann sphere]]. Equivalent to the [[complex projective line]], '''C'''P<sup>1</sup>. SO(3)/SO(2).
; [[3-sphere]] : Also known as the [[glome]]. Parallelizable, [[principal bundle|principal]] [[circle bundle|U(1)-bundle]] [[Hopf fibration|over]] the 2-sphere, Lie group structure [[Sp(1)]], where also
:<math>\mathrm{Sp}(1) \cong \mathrm{SO}(4)/\mathrm{SO}(3) \cong \mathrm{SU}(2) \cong \mathrm{Spin}(3)</math>.
; 4-sphere : Equivalent to the [[quaternionic projective line]], '''H'''P<sup>1</sup>. SO(5)/SO(4).
; 5-sphere : [[principal bundle|Principal]] [[circle bundle|U(1)-bundle]] over [[complex projective space|'''C'''P<sup>2</sup>]]. SO(6)/SO(5) = SU(3)/SU(2).
; 6-sphere : Possesses an [[almost complex structure]] coming from the set of pure unit [[octonion]]s. SO(7)/SO(6) = ''G''<sub>2</sub>/SU(3). The question of whether it has a [[Complex manifold|complex structure]] is known as the ''Hopf problem,'' after [[Heinz Hopf]].<ref>{{cite journal|last1=Agricola |first1=Ilka |authorlink1=Ilka Agricola |first2=Giovanni |last2=Bazzoni |first3=Oliver |last3=Goertsches |first4=Panagiotis |last4=Konstantis |first5=Sönke |last5=Rollenske |title=On the history of the Hopf problem |arxiv=1708.01068 |journal=[[Differential Geometry and its Applications]] |year=2018 |volume=57 |pages=1–9}}</ref>
; 7-sphere : Topological [[quasigroup]] structure as the set of unit [[octonion]]s. Principal Sp(1)-bundle over ''S''<sup>4</sup>. Parallelizable. SO(8)/SO(7) = SU(4)/SU(3) = Sp(2)/Sp(1) = Spin(7)/''G''<sub>2</sub> = Spin(6)/SU(3). The 7-sphere is of particular interest since it was in this dimension that the first [[exotic sphere]]s were discovered.
; 8-sphere : Equivalent to the octonionic projective line '''O'''P<sup>1</sup>.
; 23-sphere : A highly dense [[sphere-packing]] is possible in 24-dimensional space, which is related to the unique qualities of the [[Leech lattice]].

==See also==
{{div col|colwidth=15em}}
*[[Affine sphere]]
*[[Conformal geometry]]
*[[Exotic sphere]]
*[[Homology sphere]]
*[[Homotopy groups of spheres]]
*[[Homotopy sphere]]
*[[Hyperbolic group]]
*[[Hypercube]]
*[[Inversive geometry]]
*[[Loop (topology)]]
*[[Manifold]]
*[[Möbius transformation]]
*[[Orthogonal group]]
*[[Spherical cap]]
*[[Volume of an n-ball|Volume of an {{math|''n''}}-ball]]
*[[Wigner semicircle distribution]]
{{div col end}}

== Notes ==
<references />

== References ==
* {{cite book | last1=Flanders | first1=Harley | title=Differential forms with applications to the physical sciences | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-66169-8 | year=1989|ref=harv}}
* {{Cite book | last1=Moura | first1=Eduarda | last2=Henderson | first2=David G. | title=Experiencing geometry: on plane and sphere | url=http://www.math.cornell.edu/~henderson/books/eg00 | publisher=[[Prentice Hall]] | isbn=978-0-13-373770-7 | year=1996 | ref=harv | postscript=&nbsp;(Chapter 20: 3-spheres and hyperbolic 3-spaces).}}
* {{Cite book | last1=Weeks | first1=Jeffrey R. | author1-link=Jeffrey Weeks (mathematician) | title=The Shape of Space: how to visualize surfaces and three-dimensional manifolds | publisher=Marcel Dekker | isbn=978-0-8247-7437-0 | year=1985 | ref=harv | postscript=&nbsp;(Chapter 14: The Hypersphere).}}
* {{cite journal|last=Marsaglia|first=G.|title=Choosing a Point from the Surface of a Sphere|journal=Annals of Mathematical Statistics|volume=43|pages=645–646|year=1972|doi=10.1214/aoms/1177692644|issue=2|ref=harv}}
* {{cite journal|first=Greg|last=Huber|title=Gamma function derivation of n-sphere volumes|journal=Amer. Math. Monthly|volume=89|year=1982|pages=301–302|mr=1539933 |jstor=2321716|issue=5|doi=10.2307/2321716|ref=harv
}}
* {{cite journal|first1=Nir | last1=Barnea | title=Hyperspherical functions with arbitrary permutational symmetry: Reverse construction
|year=1999 | journal = Phys. Rev. A | doi=10.1103/PhysRevA.59.1135
|volume=59 | number=2|pages=1135–1146}}

==External links==
*[https://web.archive.org/web/20141201023452/http://www.bayarea.net/~kins/thomas_briggs/ Exploring Hyperspace with the Geometric Product]
*[http://faculty.madisoncollege.edu/alehnen/sphere/Apendxa/Appendixa.htm Properties of Spherical Coordinates in n Dimensions]
* {{MathWorld|title=Hypersphere|urlname=Hypersphere}}

{{Dimension topics}}

[[Category:Multi-dimensional geometry]]
[[Category:Spheres]]