n-sphere

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2-sphere wireframe as an orthogonal projection

Just as a stereographic projection can project a sphere's surface to a plane, it can also project a 3-sphere 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 or hypersphere is an Template:Tmath-dimensional generalization of the Template:Tmath-dimensional circle and Template:Tmath-dimensional sphere to any non-negative integer Template:Tmath.

The circle is considered 1-dimensional and the sphere 2-dimensional because a point within them has one and two degrees of freedom respectively. However, the typical embedding of the 1-dimensional circle is in 2-dimensional space, the 2-dimensional sphere is usually depicted embedded in 3-dimensional space, and a general Template:Tmath-sphere is embedded in an Template:Tmath-dimensional space. The term hypersphere is commonly used to distinguish spheres of dimension Template:Tmath which are thus embedded in a space of dimension Template:Tmath, which means that they cannot be easily visualized. The Template:Tmath-sphere is the setting for Template:Tmath-dimensional spherical geometry.

Considered extrinsically, as a hypersurface embedded in Template:Tmath-dimensional Euclidean space, an Template:Tmath-sphere is the locus of points at equal distance (the radius) from a given center point. Its interior, consisting of all points closer to the center than the radius, is an Template:Tmath-dimensional ball. In particular:

• The Template:Tmath-sphere is the pair of points at the ends of a line segment (Template:Tmath-ball).
• The Template:Tmath-sphere is a circle, the circumference of a disk (Template:Tmath-ball) in the two-dimensional plane.
• The Template:Tmath-sphere, often simply called a sphere, is the boundary of a Template:Tmath-ball in three-dimensional space.
• The 3-sphere is the boundary of a Template:Tmath-ball in four-dimensional space.
• The Template:Tmath-sphere is the boundary of an Template:Tmath-ball.

Given a Cartesian coordinate system, the [[unit n-sphere|unit Template:Tmath-sphere]] of radius Template:Tmath can be defined as:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle S^{n}=\left\{x\in \mathbb {R} ^{n+1}:\left\|x\right\|=1\right\}.}

Considered intrinsically, when Template:Tmath, the Template:Tmath-sphere is a Riemannian manifold of positive constant curvature, and is orientable. The geodesics of the Template:Tmath-sphere are called great circles.

The stereographic projection maps the Template:Tmath-sphere onto Template:Tmath-space with a single adjoined point at infinity; under the metric thereby defined, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {R} ^{n}\cup \{\infty \}} is a model for the Template:Tmath-sphere.

In the more general setting of topology, any topological space that is homeomorphic to the unit Template:Tmath-sphere is called an Template:Tmath-sphere. Under inverse stereographic projection, the Template:Tmath-sphere is the one-point compactification of Template:Tmath-space. The Template:Tmath-spheres admit several other topological descriptions: for example, they can be constructed by gluing two Template:Tmath-dimensional spaces together, by identifying the boundary of an [[hypercube|Template:Tmath-cube]] with a point, or (inductively) by forming the suspension of an Template:Tmath-sphere. When Template:Tmath it is simply connected; the Template:Tmath-sphere (circle) is not simply connected; the Template:Tmath-sphere is not even connected, consisting of two discrete points.

Description

For any natural number Template:Tmath, an Template:Tmath-sphere of radius Template:Tmath is defined as the set of points in Template:Tmath-dimensional Euclidean space that are at distance Template:Tmath from some fixed point Template:Tmath, where Template:Tmath may be any positive real number and where Template:Tmath may be any point in Template:Tmath-dimensional space. In particular:

• a 0-sphere is a pair of points Template:Tmath, and is the boundary of a line segment (Template:Tmath-ball).
• a 1-sphere is a circle of radius Template:Tmath centered at Template:Tmath, and is the boundary of a disk (Template:Tmath-ball).
• a 2-sphere is an ordinary Template:Tmath-dimensional sphere in Template:Tmath-dimensional Euclidean space, and is the boundary of an ordinary ball (Template:Tmath-ball).
• a 3-sphere is a Template:Tmath-dimensional sphere in Template:Tmath-dimensional Euclidean space.

Cartesian coordinates

The set of points in Template:Tmath-space, Template:Tmath, that define an Template:Tmath-sphere, Template:Tmath, is represented by the equation:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r^{2}=\sum _{i=1}^{n+1}(x_{i}-c_{i})^{2},}

where Template:Tmath is a center point, and Template:Tmath is the radius.

The above Template:Tmath-sphere exists in Template:Tmath-dimensional Euclidean space and is an example of an Template:Tmath-manifold. The volume form Template:Tmath of an Template:Tmath-sphere of radius Template:Tmath is given by

$\omega ={\frac {1}{r}}\sum _{j=1}^{n+1}(-1)^{j-1}x_{j}\,dx_{1}\wedge \cdots \wedge dx_{j-1}\wedge dx_{j+1}\wedge \cdots \wedge dx_{n+1}={\star }dr$

where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\star }} is the Hodge star operator; see Flanders (1989, §6.1) for a discussion and proof of this formula in the case Template:Tmath. As a result,

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle dr\wedge \omega =dx_{1}\wedge \cdots \wedge dx_{n+1}.}

n-ball

Main article: Ball (mathematics)

The space enclosed by an Template:Tmath-sphere is called an Template:Tmath-ball. An Template:Tmath-ball is closed if it includes the Template:Tmath-sphere, and it is open if it does not include the Template:Tmath-sphere.

Specifically:

• A Template:Tmath-ball, a line segment, is the interior of a 0-sphere.
• A Template:Tmath-ball, a disk, is the interior of a circle (Template:Tmath-sphere).
• A Template:Tmath-ball, an ordinary ball, is the interior of a sphere (Template:Tmath-sphere).
• A Template:Tmath-ball is the interior of a 3-sphere, etc.

Topological description

Topologically, an Template:Tmath-sphere can be constructed as a one-point compactification of Template:Tmath-dimensional Euclidean space. Briefly, the Template:Tmath-sphere can be described as Template:Tmath, which is Template:Tmath-dimensional Euclidean space plus a single point representing infinity in all directions. In particular, if a single point is removed from an Template:Tmath-sphere, it becomes homeomorphic to $\mathbb {R} ^{n}$. This forms the basis for stereographic projection.

Volume and area

See also: Volume of an n-ball and Unit sphere § Volume and area

Let Template:Tmath be the surface area of the unit Template:Tmath-sphere of radius Template:Tmath embedded in Template:Tmath-dimensional Euclidean space, and let Template:Tmath be the volume of its interior, the unit Template:Tmath-ball. The surface area of an arbitrary Template:Tmath-sphere is proportional to the Template:Tmathst power of the radius, and the volume of an arbitrary Template:Tmath-ball is proportional to the Template:Tmathth power of the radius.

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Graphs of volumes (Template:Tmath) and surface areas (Template:Tmath) of n-balls of radius 1.

The Template:Tmath-ball is sometimes defined as a single point. The Template:Tmath-dimensional Hausdorff measure is the number of points in a set. So

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V_{0}=1.}

A unit Template:Tmath-ball is a line segment whose points have a single coordinate in the interval Template:Tmath of length Template:Tmath, and the Template:Tmath-sphere consists of its two end-points, with coordinate Template:Tmath.

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle S_{0}=2,\quad V_{1}=2.}

A unit Template:Tmath-sphere is the unit circle in the Euclidean plane, and its interior is the unit disk (Template:Tmath-ball).

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle S_{1}=2\pi ,\quad V_{2}=\pi .}

The interior of a 2-sphere in three-dimensional space is the unit Template:Tmath-ball.

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle S_{2}=4\pi ,\quad V_{3}={\tfrac {4}{3}}\pi .}

In general, Template:Tmath and Template:Tmath are given in closed form by the expressions

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle S_{n-1}={\frac {2\pi ^{n/2}}{\Gamma {\bigl (}{\frac {n}{2}}{\bigr )}}},\quad V_{n}={\frac {\pi ^{n/2}}{\Gamma {\bigl (}{\frac {n}{2}}+1{\bigr )}}}}

where Template:Tmath is the gamma function. Note that Template:Tmath's values at half-integers contain a factor of Template:Tmath that cancels out the factor in the numerator.

As Template:Tmath tends to infinity, the volume of the unit Template:Tmath-ball (ratio between the volume of an Template:Tmath-ball of radius Template:Tmath and an [[hypercube| Template:Tmath-cube]] of side length Template:Tmath) tends to zero.

Recurrences

The surface area, or properly the Template:Tmath-dimensional volume, of the Template:Tmath-sphere at the boundary of the Template:Tmath-ball of radius Template:Tmath is related to the volume of the ball by the differential equation

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle S_{n}R^{n}={\frac {dV_{n+1}R^{n+1}}{dR}}={(n+1)V_{n+1}R^{n}}.}

Equivalently, representing the unit Template:Tmath-ball as a union of concentric Template:Tmath-sphere shells,

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V_{n+1}=\int _{0}^{1}S_{n}r^{n}\,dr={\frac {1}{n+1}}S_{n}.}

We can also represent the unit Template:Tmath-sphere as a union of products of a circle (Template:Tmath-sphere) with an Template:Tmath-sphere. Then Template:Tmath. Since Template:Tmath, the equation

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle S_{n+1}=2\pi V_{n}}

holds for all Template:Tmath. Along with the base cases Template:Tmath, Template:Tmath from above, these recurrences can be used to compute the surface area of any sphere or volume of any ball.

Spherical coordinates

We may define a coordinate system in an Template:Tmath-dimensional Euclidean space which is analogous to the spherical coordinate system defined for Template:Tmath-dimensional Euclidean space, in which the coordinates consist of a radial coordinate Template:Tmath, and Template:Tmath angular coordinates Template:Tmath, where the angles Template:Tmath range over Template:Tmath radians (or Template:Tmath degrees) and Template:Tmath ranges over Template:Tmath radians (or Template:Tmath degrees). If Template:Tmath are the Cartesian coordinates, then we may compute Template:Tmath from Template:Tmath with:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}x_{1}&=r\cos(\varphi _{1}),\\[5mu]x_{2}&=r\sin(\varphi _{1})\cos(\varphi _{2}),\\[5mu]x_{3}&=r\sin(\varphi _{1})\sin(\varphi _{2})\cos(\varphi _{3}),\\&\qquad \vdots \\x_{n-1}&=r\sin(\varphi _{1})\cdots \sin(\varphi _{n-2})\cos(\varphi _{n-1}),\\[5mu]x_{n}&=r\sin(\varphi _{1})\cdots \sin(\varphi _{n-2})\sin(\varphi _{n-1}).\end{aligned}}}

Except in the special cases described below, the inverse transformation is unique:

${\begin{aligned}r&={\textstyle {\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}+\cdots +{x_{2}}^{2}+{x_{1}}^{2}}}},\\[5mu]\varphi _{1}&=\operatorname {atan2} \left({\textstyle {\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}+\cdots +{x_{2}}^{2}}}},x_{1}\right),\\[5mu]\varphi _{2}&=\operatorname {atan2} \left({\textstyle {\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}+\cdots +{x_{3}}^{2}}}},x_{2}\right),\\&\qquad \vdots \\\varphi _{n-2}&=\operatorname {atan2} \left({\textstyle {\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}}}},x_{n-2}\right),\\[5mu]\varphi _{n-1}&=\operatorname {atan2} \left(x_{n},x_{n-1}\right).\end{aligned}}$

where atan2 is the two-argument arctangent function.

There are some special cases where the inverse transform is not unique; Template:Tmath for any Template:Tmath will be ambiguous whenever all of Template:Tmath are zero; in this case Template:Tmath may be chosen to be zero. (For example, for the Template:Tmath-sphere, when the polar angle is Template:Tmath or Template:Tmath then the point is one of the poles, zenith or nadir, and the choice of azimuthal angle is arbitrary.)

Spherical volume and area elements

The arc length element isFailed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle ds^{2}=dr^{2}+\sum _{k=1}^{n-1}r^{2}\left(\prod _{m=1}^{k-1}\sin ^{2}\left(\varphi _{m}\right)\right)d\varphi _{k}^{2}} To express the volume element of Template:Tmath-dimensional Euclidean space in terms of spherical coordinates, let Template:Tmath and Template:Tmath for concision, then observe that the Jacobian matrix of the transformation is:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle J_{n}={\begin{pmatrix}c_{1}&-rs_{1}&0&0&\cdots &0\\s_{1}c_{2}&rc_{1}c_{2}&-rs_{1}s_{2}&0&\cdots &0\\\vdots &\vdots &\vdots &&\ddots &\vdots \\&&&&&0\\s_{1}\cdots s_{n-2}c_{n-1}&\cdots &\cdots &&&-rs_{1}\cdots s_{n-2}s_{n-1}\\s_{1}\cdots s_{n-2}s_{n-1}&rc_{1}\cdots s_{n-1}&\cdots &&&{\phantom {-}}rs_{1}\cdots s_{n-2}c_{n-1}\end{pmatrix}}.}

The determinant of this matrix can be calculated by induction. When Template:Tmath, a straightforward computation shows that the determinant is Template:Tmath. For larger Template:Tmath, observe that Template:Tmath can be constructed from Template:Tmath as follows. Except in column Template:Tmath, rows Template:Tmath and Template:Tmath of Template:Tmath are the same as row Template:Tmath of Template:Tmath, but multiplied by an extra factor of Template:Tmath in row Template:Tmath and an extra factor of Template:Tmath in row Template:Tmath. In column Template:Tmath, rows Template:Tmath and Template:Tmath of Template:Tmath are the same as column Template:Tmath of row Template:Tmath of Template:Tmath, but multiplied by extra factors of Template:Tmath in row Template:Tmath and Template:Tmath in row Template:Tmath, respectively. The determinant of Template:Tmath can be calculated by Laplace expansion in the final column. By the recursive description of Template:Tmath, the submatrix formed by deleting the entry at Template:Tmath and its row and column almost equals Template:Tmath, except that its last row is multiplied by Template:Tmath. Similarly, the submatrix formed by deleting the entry at Template:Tmath and its row and column almost equals Template:Tmath, except that its last row is multiplied by Template:Tmath. Therefore the determinant of Template:Tmath is

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}|J_{n}|&=(-1)^{(n-1)+n}(-rs_{1}\dotsm s_{n-2}s_{n-1})(s_{n-1}|J_{n-1}|)\\&\qquad {}+(-1)^{n+n}(rs_{1}\dotsm s_{n-2}c_{n-1})(c_{n-1}|J_{n-1}|)\\&=(rs_{1}\dotsm s_{n-2}|J_{n-1}|(s_{n-1}^{2}+c_{n-1}^{2})\\&=(rs_{1}\dotsm s_{n-2})|J_{n-1}|.\end{aligned}}}

Induction then gives a closed-form expression for the volume element in spherical coordinates

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}d^{n}V&=\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}\\&=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{aligned}}}

The formula for the volume of the Template:Tmath-ball can be derived from this by integration.

Similarly the surface area element of the Template:Tmath-sphere of radius Template:Tmath, which generalizes the area element of the Template:Tmath-sphere, is given by

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle d_{S^{n-1}}V=R^{n-1}\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}.}

The natural choice of an orthogonal basis over the angular coordinates is a product of ultraspherical polynomials,

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}&{}\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{aligned}}}

for Template:Tmath, and the Template:Tmath for the angle Template:Tmath in concordance with the spherical harmonics.

Polyspherical coordinates

The standard spherical coordinate system arises from writing Template:Tmath as the product Template:Tmath. These two factors may be related using polar coordinates. For each point Template:Tmath of $\mathbb {R} ^{n}$, the standard Cartesian coordinates

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbf {x} =(x_{1},\dots ,x_{n})=(y_{1},z_{1},\dots ,z_{n-1})=(y_{1},\mathbf {z} )}

can be transformed into a mixed polar–Cartesian coordinate system:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbf {x} =(r\sin \theta ,(r\cos \theta ){\hat {\mathbf {z} }}).}

This says that points in Template:Tmath may be expressed by taking the ray starting at the origin and passing through ${\hat {\mathbf {z} }}=\mathbf {z} /\lVert \mathbf {z} \rVert \in S^{n-2}$, rotating it towards Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (1,0,\dots ,0)} by Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \theta =\arcsin y_{1}/r} , and traveling a distance Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r=\lVert \mathbf {x} \rVert } along the ray. Repeating this decomposition eventually leads to the standard spherical coordinate system.

Polyspherical coordinate systems arise from a generalization of this construction. The space Template:Tmath is split as the product of two Euclidean spaces of smaller dimension, but neither space is required to be a line. Specifically, suppose that Template:Tmath and Template:Tmath are positive integers such that Template:Tmath. Then Template:Tmath. Using this decomposition, a point Template:Tmath may be written as

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbf {x} =(x_{1},\dots ,x_{n})=(y_{1},\dots ,y_{p},z_{1},\dots ,z_{q})=(\mathbf {y} ,\mathbf {z} ).}

This can be transformed into a mixed polar–Cartesian coordinate system by writing:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbf {x} =((r\sin \theta ){\hat {\mathbf {y} }},(r\cos \theta ){\hat {\mathbf {z} }}).}

Here Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\hat {\mathbf {y} }}} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\hat {\mathbf {z} }}} are the unit vectors associated to Template:Tmath and Template:Tmath. This expresses Template:Tmath in terms of Template:Tmath, Template:Tmath, Template:Tmath, and an angle Template:Tmath. It can be shown that the domain of Template:Tmath is Template:Tmath if Template:Tmath, Template:Tmath if exactly one of Template:Tmath and Template:Tmath is Template:Tmath, and Template:Tmath if neither Template:Tmath nor Template:Tmath are Template:Tmath. The inverse transformation is

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}r&=\lVert \mathbf {x} \rVert ,\\\theta &=\arcsin {\frac {\lVert \mathbf {y} \rVert }{\lVert \mathbf {x} \rVert }}=\arccos {\frac {\lVert \mathbf {z} \rVert }{\lVert \mathbf {x} \rVert }}=\arctan {\frac {\lVert \mathbf {y} \rVert }{\lVert \mathbf {z} \rVert }}.\end{aligned}}}

These splittings may be repeated as long as one of the factors involved has dimension two or greater. A polyspherical coordinate system is the result of repeating these splittings until there are no Cartesian coordinates left. Splittings after the first do not require a radial coordinate because the domains of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\hat {\mathbf {y} }}} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\hat {\mathbf {z} }}} are spheres, so the coordinates of a polyspherical coordinate system are a non-negative radius and Template:Tmath angles. The possible polyspherical coordinate systems correspond to binary trees with Template:Tmath leaves. Each non-leaf node in the tree corresponds to a splitting and determines an angular coordinate. For instance, the root of the tree represents Template:Tmath, and its immediate children represent the first splitting into Template:Tmath and Template:Tmath. Leaf nodes correspond to Cartesian coordinates for Template:Tmath. The formulas for converting from polyspherical coordinates to Cartesian coordinates may be determined by finding the paths from the root to the leaf nodes. These formulas are products with one factor for each branch taken by the path. For a node whose corresponding angular coordinate is Template:Tmath, taking the left branch introduces a factor of Template:Tmath and taking the right branch introduces a factor of Template:Tmath. The inverse transformation, from polyspherical coordinates to Cartesian coordinates, is determined by grouping nodes. Every pair of nodes having a common parent can be converted from a mixed polar–Cartesian coordinate system to a Cartesian coordinate system using the above formulas for a splitting.

Polyspherical coordinates also have an interpretation in terms of the special orthogonal group. A splitting Template:Tmath determines a subgroup

$\operatorname {SO} _{p}(\mathbb {R} )\times \operatorname {SO} _{q}(\mathbb {R} )\subseteq \operatorname {SO} _{n}(\mathbb {R} ).$

This is the subgroup that leaves each of the two factors Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle S^{p-1}\times S^{q-1}\subseteq S^{n-1}} fixed. Choosing a set of coset representatives for the quotient is the same as choosing representative angles for this step of the polyspherical coordinate decomposition.

In polyspherical coordinates, the volume measure on Template:Tmath and the area measure on Template:Tmath are products. There is one factor for each angle, and the volume measure on Template:Tmath also has a factor for the radial coordinate. The area measure has the form:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle dA_{n-1}=\prod _{i=1}^{n-1}F_{i}(\theta _{i})\,d\theta _{i},}

where the factors Template:Tmath are determined by the tree. Similarly, the volume measure is

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle dV_{n}=r^{n-1}\,dr\,\prod _{i=1}^{n-1}F_{i}(\theta _{i})\,d\theta _{i}.}

Suppose we have a node of the tree that corresponds to the decomposition Template:Tmath and that has angular coordinate Template:Tmath. The corresponding factor Template:Tmath depends on the values of Template:Tmath and Template:Tmath. When the area measure is normalized so that the area of the sphere is Template:Tmath, these factors are as follows. If Template:Tmath, then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(\theta) = \frac{d\theta}{2\pi}.}

If Template:Tmath and Template:Tmath, and if Template:Tmath denotes the beta function, then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(\theta) = \frac{\sin^{n_1 - 1}\theta}{\Beta(\frac{n_1}{2}, \frac{1}{2})}\,d\theta.}

If Template:Tmath and Template:Tmath, then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(\theta) = \frac{\cos^{n_2 - 1}\theta}{\Beta(\frac{1}{2}, \frac{n_2}{2})}\,d\theta.}

Finally, if both Template:Tmath and Template:Tmath are greater than one, then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(\theta) = \frac{(\sin^{n_1 - 1}\theta)(\cos^{n_2 - 1}\theta)}{\frac{1}{2}\Beta(\frac{n_1}{2}, \frac{n_2}{2})}\,d\theta.}

Stereographic projection

Main article: 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 Template:Tmath-sphere can be mapped onto an Template:Tmath-dimensional hyperplane by the Template:Tmath-dimensional version of the stereographic projection. For example, the point Template:Tmath on a two-dimensional sphere of radius Template:Tmath maps to the point Template:Tmath on the Template:Tmath-plane. In other words,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [x,y,z] \mapsto \left[\frac{x}{1-z},\frac{y}{1-z}\right].}

Likewise, the stereographic projection of an Template:Tmath-sphere Template:Tmath of radius Template:Tmath will map to the Template:Tmath-dimensional hyperplane Template:Tmath perpendicular to the Template:Tmath-axis as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [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].}

Probability distributions

Uniformly at random on the (n − 1)-sphere

See also: Von Mises–Fisher distribution § The uniform hypersphere distribution.

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A set of points drawn from a uniform distribution on the surface of a unit 2-sphere, generated using Marsaglia's algorithm.

To generate uniformly distributed random points on the unit Template:Tmath-sphere (that is, the surface of the unit Template:Tmath-ball), Marsaglia (1972) gives the following algorithm.

Generate an Template:Tmath-dimensional vector of normal deviates (it suffices to use Template:Tmath, although in fact the choice of the variance is arbitrary), Template:Tmath. Now calculate the "radius" of this point:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=\sqrt{x_1^2+x_2^2+\cdots+x_n^2}.}

The vector Template:Tmath is uniformly distributed over the surface of the unit Template:Tmath-ball.

An alternative given by Marsaglia is to uniformly randomly select a point Template:Tmath in the unit n-cube by sampling each Template:Tmath independently from the uniform distribution over Template:Tmath, computing Template:Tmath as above, and rejecting the point and resampling if Template:Tmath (i.e., if the point is not in the Template:Tmath-ball), and when a point in the ball is obtained scaling it up to the spherical surface by the factor Template:Tmath; then again Template:Tmath is uniformly distributed over the surface of the unit Template:Tmath-ball. This method becomes very inefficient for higher dimensions, as a vanishingly small fraction of the unit cube is contained in the sphere. In ten dimensions, less than 2% of the cube is filled by the sphere, so that typically more than 50 attempts will be needed. In seventy dimensions, less than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^{-24}} of the cube is filled, meaning typically a trillion quadrillion trials will be needed, far more than a computer could ever carry out.

Uniformly at random within the n-ball

With a point selected uniformly at random from the surface of the unit Template:Tmath-sphere (e.g., by using Marsaglia's algorithm), one needs only a radius to obtain a point uniformly at random from within the unit Template:Tmath-ball. If Template:Tmath is a number generated uniformly at random from the interval Template:Tmath and Template:Tmath is a point selected uniformly at random from the unit Template:Tmath-sphere, then Template:Tmath is uniformly distributed within the unit Template:Tmath-ball.

Alternatively, points may be sampled uniformly from within the unit Template:Tmath-ball by a reduction from the unit Template:Tmath-sphere. In particular, if Template:Tmath is a point selected uniformly from the unit Template:Tmath-sphere, then Template:Tmath is uniformly distributed within the unit Template:Tmath-ball (i.e., by simply discarding two coordinates).

If Template:Tmath is sufficiently large, most of the volume of the Template:Tmath-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.

Distribution of the first coordinate

Let Template:Tmath be the square of the first coordinate of a point sampled uniformly at random from the Template:Tmath-sphere, then its probability density function, for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y\in [0, 1]} , is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho(y) = \frac{\Gamma\bigl(\frac{n}{2} \bigr)}{\sqrt\pi \; \Gamma\bigl(\frac{n-1}{2}\bigr)} (1-y)^{(n-3)/2}y^{-1/2}. }

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = y/N} be the appropriately scaled version, then at the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N\to \infty} limit, the probability density function of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} converges to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2\pi ze^z)^{-1/2}} . This is sometimes called the Porter–Thomas distribution.

Specific spheres

0-sphere

The pair of points Template:Tmath with the discrete topology for some Template:Tmath. The only sphere that is not path-connected. Parallelizable.

1-sphere

Commonly called a circle. Has a nontrivial fundamental group. Abelian Lie group structure U(1); the circle group. Homeomorphic to the real projective line. Parallelizable

2-sphere

Commonly simply called a sphere. For its complex structure, see Riemann sphere. Homeomorphic to the complex projective line

3-sphere

Parallelizable, principal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{U}(1)} -bundle over the Template:Tmath-sphere, Lie group structure Sp(1) = SU(2).

4-sphere

Homeomorphic to the quaternionic projective line, Template:Tmath. Template:Tmath.

5-sphere

Principal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{U}(1)} -bundle over the complex projective space Template:Tmath. Template:Tmath. It is undecidable whether a given Template:Tmath-dimensional manifold is homeomorphic to Template:Tmath for Template:Tmath.

6-sphere

Possesses an almost complex structure coming from the set of pure unit octonions. Template:Tmath. The question of whether it has a complex structure is known as the Hopf problem, after Heinz Hopf.

7-sphere

Topological quasigroup structure as the set of unit octonions. Principal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{SU}(2)} -bundle over Template:Tmath. Parallelizable. Template:Tmath. The Template:Tmath-sphere is of particular interest since it was in this dimension that the first exotic spheres were discovered.

8-sphere

Homeomorphic to the octonionic projective line Template:Tmath.

23-sphere

A highly dense sphere-packing is possible in Template:Tmath-dimensional space, which is related to the unique qualities of the Leech lattice.

Octahedral sphere

The octahedral Template:Tmath-sphere is defined similarly to the Template:Tmath-sphere but using the 1-norm

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S^n = \left\{ x \in \R^{n+1} : \left\| x \right\|_1 = 1 \right\}}

In general, it takes the shape of a cross-polytope.

The octahedral Template:Tmath-sphere is a square (without its interior). The octahedral Template:Tmath-sphere is a regular octahedron; hence the name. The octahedral Template:Tmath-sphere is the topological join of Template:Tmath pairs of isolated points. Intuitively, the topological join of two pairs is generated by drawing a segment between each point in one pair and each point in the other pair; this yields a square. To join this with a third pair, draw a segment between each point on the square and each point in the third pair; this gives a octahedron.

See also

• Conformal geometry
• Exotic sphere
• Homology sphere
• Homotopy groups of spheres
• Inversive geometry
• Möbius transformation

References

• Marsaglia, G. (1972). "Choosing a Point from the Surface of a Sphere". Annals of Mathematical Statistics. 43 (2): 645–646. doi:10.1214/aoms/1177692644.
• Huber, Greg (1982). "Gamma function derivation of n-sphere volumes". Amer. Math. Monthly. 89 (5): 301–302. doi:10.2307/2321716. JSTOR 2321716. MR 1539933.
• 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).CS1 maint: postscript (link)
• Kalnins, E. G.; Miller, W. (1986). "Separation of variables on n-dimensionsional Riemannian manifolds. I. the n-sphere S_n and Euclidean n-sparce R_n". J. Math. Phys. 27: 1721–1746. doi:10.1063/1.527088. hdl:10289/1219.
• Flanders, Harley (1989). Differential forms with applications to the physical sciences. New York: Dover Publications. ISBN 978-0-486-66169-8.
• Moura, Eduarda; Henderson, David G. (1996). Experiencing geometry: on plane and sphere. Prentice Hall. ISBN 978-0-13-373770-7 (Chapter 20: 3-spheres and hyperbolic 3-spaces).CS1 maint: postscript (link)
• Barnea, Nir (1999). "Hyperspherical functions with arbitrary permutational symmetry: Reverse construction". Phys. Rev. A. 59 (2): 1135–1146. Bibcode:1999PhRvA..59.1135B. doi:10.1103/PhysRevA.59.1135.

External links

• Weisstein, Eric W. "Hypersphere". MathWorld.