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| {{DISPLAYTITLE:''n''-sphere}}
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| [[Image:Sphere wireframe.svg|thumb|2-sphere wireframe as an [[orthogonal projection]]]]
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| [[Image: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 3-sphere into 3-space. This image shows three coordinate directions projected to 3-space:
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| parallels (red), [[Meridian (perimetry, visual field)|meridians]] (blue) and hypermeridians (green).
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| Due to the [[conformal map|conformal]] property of the stereographic projection,
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| the curves intersect each other orthogonally (in the yellow points) as in 4D.
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| All of the curves are circles: the curves that intersect <0,0,0,1> have an infinite radius (= straight line).]]
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| In [[mathematics]], an '''''n''-sphere''' is a generalization of the surface of an ordinary [[sphere]] to a [[n-dimensional space]]. 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 central point, where the radius ''r'' may be any [[Positive number|positive]] [[real number]]. Thus, the n-sphere centred at the origin is defined by:
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| :<math>S^n = \left\{ x \in \mathbb{R}^{n+1} : \|x\| = r\right\}.</math>
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| It is an ''n''-dimensional [[manifold]] in Euclidean (''n'' + 1)-space.
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| In particular:
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| :a 0-sphere is the pair of points at the ends of a (one-dimensional) [[line segment]],
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| :a 1-sphere is the [[circle]], which is the one-dimensional circumference of a (two-dimensional) disk in the plane,
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| :a 2-sphere is the two-dimensional surface of a (three-dimensional) ball in three-dimensional space.
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| Spheres of dimension ''n'' > 2 are sometimes called '''hyperspheres''', with [[3-sphere]]s sometimes known as '''glomes'''. The ''n''-sphere of unit radius centered at the origin is called the '''unit ''n''-sphere''', denoted ''S''<sup>''n''</sup>. The unit ''n''-sphere is often referred to as ''the'' ''n''-sphere.
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| An ''n''-sphere is the surface or boundary of an (''n'' + 1)-dimensional [[ball (mathematics)|ball]], and is an ''n''-dimensional [[manifold]]. For ''n'' ≥ 2, the ''n''-spheres are the [[simply connected]] ''n''-dimensional [[manifold]]s of constant, positive curvature. The ''n''-spheres admit several other topological descriptions: for example, they can be constructed by gluing two ''n''-dimensional [[Euclidean space]]s together, by identifying the boundary of an [[hypercube|''n''-cube]] with a point, or (inductively) by forming the [[suspension (topology)|suspension]] of an (''n'' − 1)-sphere.
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| ==Description==
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| For any [[natural number]] ''n'', an ''n''-sphere of radius ''r'' is defined as the set of points in (''n'' + 1)-dimensional [[Euclidean space]] that are at distance ''r'' from some fixed point '''c''', where ''r'' may be any [[Positive number|positive]] [[real number]] and where '''c''' may be any point in (''n'' + 1)-dimensional space. In particular:
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| * a 0-sphere is a pair of points {''c'' − ''r'', ''c'' + ''r''}, and is the boundary of a line segment (1-ball).
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| * a [[1-sphere]] is a [[circle]] of radius ''r'' centered at '''c''', and is the boundary of a disk (2-ball).
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| * a [[2-sphere]] is an ordinary 2-dimensional [[sphere]] in 3-dimensional Euclidean space, and is the boundary of an ordinary ball (3-ball).
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| * a [[3-sphere]] is a sphere in 4-dimensional Euclidean space.
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| ===Euclidean coordinates in (''n'' + 1)-space ===
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| The set of points in (''n'' + 1)-space: (''x''<sub>1</sub>,''x''<sub>2</sub>,…,''x''<sub>''n''+1</sub>) that define an ''n''-sphere, (''S''<sup>''n''</sub>) is represented by the equation:
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| :<math>r^2=\sum_{i=1}^{n+1} (x_i - c_i)^2.\,</math>
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| where ''c'' is a center point, and ''r'' is the radius.
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| The above ''n''-sphere exists in (''n'' + 1)-dimensional Euclidean space and is an example of an ''n''-[[manifold]]. The [[volume form]] ω of an ''n''-sphere of radius ''r'' is given by
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| :<math>\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</math>
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| where * is the [[Hodge star operator]]; see {{harvtxt|Flanders|1989|loc=§6.1}} for a discussion and proof of this formula in the case ''r'' = 1. As a result, <math>\scriptstyle{dr \wedge \omega = dx_1 \wedge \cdots \wedge dx_{n+1}}.</math>
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| === ''n''-ball ===<!-- This section is linked from [[n-ball]], a redirect page. -->
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| : {{main|ball (mathematics)}}
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| The space enclosed by an ''n''-sphere is called an (''n'' + 1)-[[Ball (mathematics)|ball]]. An (''n'' + 1)-ball is [[Closed set|closed]] if it includes the ''n''-sphere, and it is [[Open set|open]] if it does not include the ''n''-sphere.
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| Specifically:
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| * A 1-''ball'', a [[line segment]], is the interior of a (0-sphere).
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| * A 2-''ball'', a [[Disk (mathematics)|disk]], is the interior of a [[circle]] (1-sphere).
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| * A 3-''ball'', an ordinary [[Ball (mathematics)|ball]], is the interior of a [[sphere]] (2-sphere).
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| * A 4-''ball'' is the interior of a [[3-sphere]], etc.
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| ===Topological description===
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| [[Topology|Topologically]], an ''n''-sphere can be constructed as a [[Alexandroff extension|one-point compactification]] of ''n''-dimensional Euclidean space. Briefly, the ''n''-sphere can be described as <math>S^n = \mathbb{R}^n \cup \{ \infty \}</math>, which is ''n''-dimensional Euclidean space plus a single point representing infinity in all directions.
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| In particular, if a single point is removed from an ''n''-sphere, it becomes [[Homeomorphism|homeomorphic]] to <math>\mathbb{R}^n</math>. This forms the basis for [[stereographic projection]].<ref>James W. Vick (1994). ''Homology theory'', p. 60. Springer</ref>
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| ==Volume and surface area==
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| {{seealso|Volume of an n-ball}}
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| <div style="display:inline; width:300px; float:right; background-color:#eeeeff; border: solid 1px #cccccc; padding: 10px">
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| <math>V_n (R)</math> and <math>S_n (R)</math> are the ''n''-dimensional volumes of the [[Ball (mathematics)|''n''-ball]] and ''n''-sphere of radius <math>R</math>, respectively.
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| The constants <math>V_n</math> and <math>S_n</math> (for the unit ball and sphere) are related by the recurrences:
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| :<math>V_0=1\qquad V_{n+1}=S_n/(n+1)</math>
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| :<math>S_0=2\qquad S_{n+1}=2\pi V_n</math>
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| The surfaces and volumes can also be given in closed form:
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| :<math>\begin{array}{ll}
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| S_{n-1}(R) &= \displaystyle{\frac{n\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}R^{n-1}} \\[1 em]
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| V_n(R) &= \displaystyle{\frac{\pi^{n/2}}{\Gamma(\frac{n}{2} + 1)}}R^n
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| \end{array}</math>
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| where <math>\Gamma \,</math> is the [[gamma function]]. Derivations of these equations are given in this section.
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| </div>
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| In general, the volumes of the ''n''-ball in ''n''-dimensional Euclidean space, and the ''n''-sphere in (''n'' + 1)-dimensional Euclidean, of radius ''R'', are proportional to the ''n''th power of the radius, ''R''. We write <math>V_n(R) = V_n R^n</math> for the volume of the ''n''-ball and <math>S_n(R) = S_n R^n</math> for the surface of the ''n''-sphere, both of radius <math>R</math>.
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| ===Examples===
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| The 0-ball consists of a single point. The 0-dimensional [[Hausdorff measure]] is the number of points in a set, so
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| :<math>V_0=1</math>.
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| The unit 1-ball is the interval <math>[-1,1]</math> of length 2. So,
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| :<math>V_1 = 2.</math>
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| The 0-sphere consists of its two end-points, <math>\{-1,1\}</math>. So
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| :<math>S_0 = 2</math>.
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| The unit 1-sphere is the unit circle in the Euclidean plane, and this has circumference (1-dimensional measure)
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| :<math>S_1 = 2\pi. \,</math>
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| The region enclosed by the unit 1-sphere is the 2-ball, or unit disc, and this has area (2-dimensional measure)
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| :<math>V_2 = \pi. \,</math>
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| Analogously, in 3-dimensional Euclidean space, the surface area (2-dimensional measure) of the unit 2-sphere is given by
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| :<math>S_2 = 4\pi \,</math>
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| and the volume enclosed is the volume (3-dimensional measure) of the unit 3-ball, given by
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| :<math>V_3 = \frac{4}{3} \pi. \,</math>
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| ===Recurrences===
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| The ''surface area'', or properly the ''n''-dimensional volume, of the ''n''-sphere at the boundary of the (''n'' + 1)-ball of radius <math>R</math> is related to the volume of the ball by the differential equation
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| :<math>S_{n}R^{n}=\frac{dV_{n+1}R^{n+1}}{dR}={(n+1)V_{n+1}R^{n}}</math>,
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| or, equivalently, representing the unit ''n''-ball as a union of concentric (''n'' − 1)-sphere ''shells'',
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| :<math>V_{n+1} = \int_0^1 S_{n}r^{n}\,dr</math>
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| So,
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| :<math>V_{n+1} = \frac{S_n}{n+1}</math>.
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| We can also represent the unit (''n'' + 2)-sphere as a union of tori, each the product of a circle (1-sphere) with an ''n''-sphere. Let <math>r = \cos\theta</math> and <math>r^2 + R^2 = 1</math>, so that <math>R = \sin\theta</math> and <math>dR = \cos\theta\,d\theta</math>. Then,
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| :<math>
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| \begin{align}
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| S_{n+2} &= \int_0^{\pi/2}S_1 r . S_n R^n\, d\theta =\int_0^{\pi/2}S_1 . S_n R^n\cos\theta\,d\theta\\
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| &=\int_0^1 S_1 . S_n R^n \,dR= S_1 \int_0^1 S_n R^n \,dR\\
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| &= 2\pi V_{n+1}
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| \end{align}
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| </math>
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| Since <math>S_1 = 2\pi V_0</math>, the equation
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| <math>S_{n+1} = 2\pi V_{n}</math>
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| holds for all ''n''.
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| This completes our derivation of the recurrences:
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| :<math>V_0=1\qquad V_{n+1}=S_n/(n+1)</math>
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| :<math>S_0=2\qquad S_{n+1}=2\pi V_n</math>
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| ===Closed forms===
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| Combining the recurrences, we see that <math>V_{n+2}=2\pi V_n/(n+2)</math>. So it is simple to show by induction on ''k'' that,
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| :<math>V_{2k} = \frac{\pi^k}{k!}</math>
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| :<math>V_{2k+1} = \frac{2(2\pi)^k}{(2k+1)!!} = \frac{2 k! (4\pi)^k}{(2k+1)!}</math>
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| where <math>!!</math> denotes the [[Factorial#Double_factorial|double factorial]], defined for odd integers {{nowrap|2''k'' + 1}} by {{nowrap|1=(2''k'' + 1)!! = 1 · 3 · 5 ··· (2''k'' − 1) · (2''k'' + 1)}}.
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| In general, the volume, in ''n''-dimensional Euclidean space, of the unit ''n''-ball, is given by
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| :<math>V_n = \frac{\pi^\frac{n}{2}}{\Gamma(\frac{n}{2} + 1)}</math>
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| where <math>\Gamma \,</math> is the [[gamma function]], which satisfies <math>\Gamma(1/2) = \sqrt{\pi}; \Gamma(1) = 1; \Gamma(x + 1) = x\Gamma(x)</math>.
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| By multiplying <math>V_n</math> by <math>R^n</math>, differentiating with respect to <math>R</math>, and then setting <math>R = 1</math>, we get the closed form
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| :<math>S_{n-1} = \frac{2\pi^\frac{n}{2}}{\Gamma(\frac{n}{2} )}</math>.
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| ===Other relations===
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| The recurrences can be combined to give a "reverse-direction" recurrence relation for surface area, as depicted in the diagram:
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| :<math>S_{n-1} = \frac{n}{2 \pi} S_{n-1+2}</math>
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| [[File:N SpheresVolumeAndSurfaceArea.png|thumb|580px|The curved red arrows show the relationship between formulas for different n. The formula coefficient at each arrow's tip equals the formula coefficient at that arrow's tail times the factor in the arrowhead. If the direction of the bottom arrows were reversed, their arrowheads would say to multiply by <big>{{frac2|2{{pi}}|n − 2}}</big>]]
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| Index-shifting n to n − 2 then yields the recurrence relations:
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| :<math>V_n = \frac{2 \pi}{n} V_{n-2}</math>
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| :<math>S_{n-1} = \frac{2 \pi}{n-2} S_{n-1-2}</math>
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| where S<sub>0</sub> = 2, V<sub>1</sub> = 2, S<sub>1</sub> = 2{{pi}} and V<sub>2</sub> = {{pi}}.
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| The recurrence relation for <math>V_n</math> can also be proved via [[integral|integration]] with 2-dimensional [[Polar coordinate system|polar coordinates]]:
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| :<math>\begin{align}
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| V_n
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| & = \int_0^1 \int_0^{2\pi} V_{n-2}(\sqrt{1-r^2})^{n-2} \, r \, d\theta \, dr \\[6pt]
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| & = \int_0^1 \int_0^{2\pi} V_{n-2} (1-r^2)^{n/2-1}\, r \, d\theta \, dr \\[6pt]
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| & = 2 \pi V_{n-2} \int_{0}^{1} (1-r^2)^{n/2-1}\, r \, dr \\[6pt]
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| & = 2 \pi V_{n-2} \left[ -\frac{1}{n}(1-r^2)^{n/2} \right]^{r=1}_{r=0} \\[6pt]
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| & = 2 \pi V_{n-2} \frac{1}{n} = \frac{2 \pi}{n} V_{n-2}.
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| \end{align}</math>
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| ==Spherical coordinates==<!-- This section is linked from [[Spherical coordinate system]] -->
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| We may define a coordinate system in an ''n''-dimensional Euclidean space which is analogous
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| to the [[Spherical coordinates|spherical coordinate system]] defined for 3-dimensional Euclidean space, in which the coordinates consist of a radial coordinate, <math>r \,,</math> and ''n'' − 1 angular coordinates <math>\phi _1 , \phi _2 , \dots , \phi _{n-1} \,</math> where <math>\phi_{n-1} \,</math> ranges over <math>[0, 2\pi) \,</math> radians (or over [0, 360) degrees) and the other angles range over <math>[0, \pi] \,</math> radians (or over [0, 180] degrees). If <math>\ x_i</math> are the Cartesian coordinates, then we may compute <math>x_1,\ldots,x_n</math> from <math>r, \phi_1,\ldots,\phi_{n-1}</math> with:
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| :<math>
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| \begin{align}
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| x_1 &= r \cos(\phi_1) \\
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| x_2 &= r \sin(\phi_1) \cos(\phi_2) \\
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| x_3 &= r \sin(\phi_1) \sin(\phi_2) \cos(\phi_3) \\
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| &\vdots\\
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| x_{n-1} &= r \sin(\phi_1) \cdots \sin(\phi_{n-2}) \cos(\phi_{n-1}) \\
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| x_n &= r \sin(\phi_1) \cdots \sin(\phi_{n-2}) \sin(\phi_{n-1}) \,.
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| \end{align}
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| </math>
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| Except in the special cases described below, the inverse transformation is unique:
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| :<math>
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| \begin{align}
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| r &= \sqrt{{x_n}^2 + {x_{n-1}}^2 + \cdots + {x_2}^2 + {x_1}^2} \\
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| \phi_1 &= \arccos \frac{x_{1}}{\sqrt{{x_n}^2+{x_{n-1}}^2+\cdots+{x_1}^2}} \\
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| \phi_2 &= \arccos \frac{x_{2}}{\sqrt{{x_n}^2+{x_{n-1}}^2+\cdots+{x_2}^2}} \\
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| &\vdots\\
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| \phi_{n-2} &= \arccos \frac{x_{n-2}}{\sqrt{{x_n}^2+{x_{n-1}}^2+{x_{n-2}}^2}} \\
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| \phi_{n-1} &= \begin{cases}
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| \arccos \frac{x_{n-1}}{\sqrt{{x_n}^2+{x_{n-1}}^2}} & x_n\geq 0 \\
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| 2 \pi - \arccos \frac{x_{n-1}}{\sqrt{{x_n}^2+{x_{n-1}}^2}} & x_n < 0
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| \end{cases} \,.
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| \end{align}
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| </math>
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| where if <math>x_k \ne 0</math> for some <math>k</math> but all of <math>x_{k+1},\ldots,x_n</math> are zero then <math>\phi_k = 0</math> when <math>x_k > 0</math>, and <math>\phi_k = \pi</math> radians (180 degrees) when <math>x_k < 0</math>.
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| There are some special cases where the inverse transform is not unique; <math>\phi_k</math> for any <math>k</math> will be ambiguous whenever all of <math>x_k,x_{k+1},\ldots,x_n</math> are zero; in this case <math>\phi_k</math> may be chosen to be zero.
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| ===Spherical volume element===
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| Expressing the angular measures in radians, the [[volume element]] in ''n''-dimensional Euclidean space will be found from the [[Jacobian matrix and determinant|Jacobian]] of the transformation:
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| :<math>
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| \begin{align}
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| d^nV & =
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| \left|\det\frac{\partial (x_i)}{\partial(r,\phi_j)}\right|
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| dr\,d\phi_1 \, d\phi_2\cdots d\phi_{n-1} \\[6pt]
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| & = r^{n-1}\sin^{n-2}(\phi_1)\sin^{n-3}(\phi_2)\cdots \sin(\phi_{n-2})\,
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| dr\,d\phi_1 \, d\phi_2\cdots d\phi_{n-1}
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| \end{align}
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| </math>
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| and the above equation for the volume of the ''n''-ball can be recovered by integrating:
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| :<math>V_n=\int_{\phi_{n-1}=0}^{2\pi} \int_{\phi_{n-2}=0}^\pi
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| \cdots \int_{\phi_1=0}^\pi\int_{r=0}^R d^nV. \,</math>
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| The volume element of the (''n''-1)–sphere, which generalizes the [[area element]] of the 2-sphere, is given by
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| :<math>d_{S^{n-1}}V =
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| \sin^{n-2}(\phi_1)\sin^{n-3}(\phi_2)\cdots \sin(\phi_{n-2})\, d\phi_1 \, d\phi_2\cdots d\phi_{n-1}.</math>
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| The natural choice of an orthogonal basis over the angular coordinates is a product of [[Gegenbauer polynomial|ultraspherical polynomials]],
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| :<math>
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| \begin{align}
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| & {} \quad \int_0^\pi \sin^{n-j-1}(\phi_j) C_s^{((n-j-1)/2)}(\cos \phi_j)C_{s'}^{((n-j-1)/2)}(\cos\phi_j) \, d\phi_j \\[6pt]
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| & = \frac{\pi 2^{3-n+j}\Gamma(s+n-j-1)}{s!(2s+n-j-1)\Gamma^2((n-j-1)/2)}\delta_{s,s'}
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| \end{align}
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| </math>
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| for ''j'' = 1, 2, ..., ''n'' − 2, and the ''e''<sup> ''isφ''<sub>''j''</sup></sup>
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| for the angle ''j'' = ''n'' − 1 in concordance with the [[spherical harmonics]].
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| == Stereographic projection ==
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| : {{main|Stereographic projection}}
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| 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 <math>\ [x,y,z]</math> on a two-dimensional sphere of radius 1 maps to the point <math>\left[\frac{x}{1-z},\frac{y}{1-z}\right]</math> on the <math>\ xy</math> plane. In other words,
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| :<math>\ [x,y,z] \mapsto \left[\frac{x}{1-z},\frac{y}{1-z}\right].</math>
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| Likewise, the stereographic projection of an ''n''-sphere <math>\mathbf{S}^{n-1}</math> of radius 1 will map to the <math>n-1</math> dimensional hyperplane <math>\mathbf{R}^{n-1}</math> perpendicular to the <math>\ x_n</math> axis as
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| :<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>
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| == Generating random points ==
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| === Uniformly at random from the (''n'' − 1)-sphere ===
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| To generate [[continuous uniform distribution|uniformly distributed]] random points on the (''n'' − 1)-sphere (''i.e.'', the surface of the ''n''-ball), {{harvtxt|Marsaglia|1972}} gives the following algorithm.
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| Generate an ''n''-dimensional vector of [[normal distribution|normal deviates]] (it suffices to use N(0, 1), although in fact the choice of the variance is arbitrary), <math>\mathbf{x}=(x_1,x_2,\ldots,x_n)</math>.
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| Now calculate the "radius" of this point, <math>r=\sqrt{x_1^2+x_2^2+\cdots+x_n^2}.</math>
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| The vector <math>\frac{1}{r} \mathbf{x}</math> is uniformly distributed over the surface of the unit ''n''-ball.
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| ==== Examples ====
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| For example, when ''n'' = 2 the normal distribution exp(−''x''<sub>1</sub><sup>2</sup>) when expanded over another axis exp(−''x''<sub>2</sub><sup>2</sup>) after multiplication takes the form exp(−''x''<sub>1</sub><sup>2</sup>−''x''<sub>2</sub><sup>2</sup>) or exp(−''r''<sup>2</sup>) and so is only
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| dependent on distance from the origin.
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| ==== Alternatives ====
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| Another way to generate a random distribution on a hypersphere is to make a uniform distribution
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| over a hypercube that includes the unit hyperball, exclude those points that are outside the hyperball, 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 hyperball to the hypercube decreases very rapidly with dimension, this procedure will succeed with high probability only for fairly small numbers of dimensions.
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| [[Wendel's theorem]] gives the probability that all of the points generated will lie in the same half of the hypersphere.
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| === Uniformly at random from the ''n''-ball ===
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| With a point selected from the surface of the ''n''-ball uniformly at random, one needs only a radius to obtain a point uniformly at random within the ''n''-ball. If ''u'' is a number generated uniformly at random from the interval [0, 1] and '''x''' is a point selected uniformly at random from the surface of the ''n''-ball then u<sup>1/n</sup>'''x''' is uniformly distributed over the entire unit ''n''-ball.
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| ==Specific spheres==
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| ; 0-sphere : The pair of points {±''R''} with the discrete topology for some ''R'' > 0. The only sphere that is disconnected. Has a natural Lie group structure; isomorphic to O(1). Parallelizable.
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| ; 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).
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| ; 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).
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| ; 3-sphere : Parallelizable, Principal U(1)-bundle [[Hopf fibration|over]] the 2-sphere, Lie group structure [[Sp(1)]], where also
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| :<math>\mathrm{Sp}(1) \cong \mathrm{SO}(4)/\mathrm{SO}(3) \cong \mathrm{SU}(2) \cong \mathrm{Spin}(3)</math>.
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| ; 4-sphere : Equivalent to the [[quaternionic projective line]], '''H'''P<sup>1</sup>. SO(5)/SO(4).
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| ; 5-sphere : Principal U(1)-bundle over '''C'''P<sup>2</sup>. SO(6)/SO(5) = SU(3)/SU(2).
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| ; 6-sphere : Almost complex structure coming from the set of pure unit [[octonion]]s. SO(7)/SO(6) = ''G''<sub>2</sub>/SU(3).
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| ; 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.
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| ; 8-sphere : Equivalent to the octonionic projective line '''O'''P<sup>1</sup>.
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| ; 23-sphere : A highly dense [[sphere-packing]] is possible in 24 dimensional space, which is related to the unique qualities of the [[Leech lattice]].
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| ==See also==
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| *[[Affine sphere]]
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| *[[Conformal geometry]]
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| *[[Homology sphere]]
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| *[[Homotopy groups of spheres]]
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| *[[Homotopy sphere]]
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| *[[Hyperbolic group]]
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| *[[Hypercube]]
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| *[[Inversive geometry]]
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| *[[Loop (topology)]]
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| *[[Manifold]]
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| *[[Möbius transformation]]
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| *[[Orthogonal group]]
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| *[[Spherical cap]]
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| *[[Volume of an n-ball|Volume of an ''n''-ball]]
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| == Notes ==
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| <references />
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| == References ==
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| * {{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}}.
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| * {{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=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}} (Chapter 20: 3-spheres and hyperbolic 3-spaces.)
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| * {{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=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}} (Chapter 14: The Hypersphere)
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| * {{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}}
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| * {{cite journal|first=Greg|last=Huber|title=Gamma function derivation of n-sphere volumes|journal=Am. Math. Monthly|volume=89|year=1982|pages=301–302|mr=1539933 |jstor=2321716|issue=5|doi=10.2307/2321716|ref=harv
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| }}
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| * {{cite journal|first1=Nir | last1=Barnea | title=Hyperspherical functions with arbitrary permutational symmetry: Reverse construction
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| |year=1999 | journal = Phys. Rev. A | doi=10.1103/PhysRevA.59.1135
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| |volume=59 | number=2|pages=1135–1146}}
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| ==External links==
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| *[http://www.bayarea.net/~kins/thomas_briggs/ Exploring Hyperspace with the Geometric Product]
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| * {{MathWorld|title=Hypersphere|urlname=Hypersphere}}
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| {{Dimension topics}}
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| [[Category:Multi-dimensional geometry]]
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| [[Category:Spheres]]
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