3-sphere

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/3-sphere

Stereographic projection of the hypersphere's parallels (red), meridians (blue) and hypermeridians (green). Because this projection is conformal, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles: the curves that intersect ⟨0,0,0,1⟩ have infinite radius (= straight line). In this picture, the whole 3D space maps the surface of the hypersphere, whereas in the next picture the 3D space contained the shadow of the bulk hypersphere.

Direct projection of 3-sphere into 3D space and covered with surface grid, showing structure as stack of 3D spheres (2-spheres)

In mathematics, a 3-sphere, glome or hypersphere is a higher-dimensional analogue of a sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensions is an ordinary sphere (or 2-sphere, a two-dimensional surface), the boundary of a ball in four dimensions (a gongyl) is a 3-sphere (a 3-dimensional "surface"). Topologically, a 3-sphere is an example of a 3-manifold, and it is also an n-sphere.

Definition

In coordinates, a 3-sphere with center (C₀, C₁, C₂, C₃) and radius r is the set of all points (x₀, x₁, x₂, x₃) in real, 4-dimensional space (R⁴) such that

${\displaystyle \sum _{i=0}^3(x_i-C_i{)}^2=(x_0-C_0{)}^2+(x_1-C_1{)}^2+(x_2-C_2{)}^2+(x_3-C_3{)}^2=r^2.}$

The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere and is usually denoted S³:

${\displaystyle S^3=\left\{(x_0,x_1,x_2,x_3)\in {\mathbb{R}}^4:x_0^2+x_1^2+x_2^2+x_3^2=1\right\}.}$

It is often convenient to regard R⁴ as the space with 2 complex dimensions (C²) or the quaternions (H). The unit 3-sphere is then given by

${\displaystyle S^3=\left\{(z_1,z_2)\in {\mathbb{C}}^2:|z_1{|}^2+|z_2{|}^2=1\right\}}$

or

${\displaystyle S^3=\left\{q\in \mathbb{H}:\Vert q\Vert =1\right\}.}$

This description as the quaternions of norm one identifies the 3-sphere with the versors in the quaternion division ring. Just as the unit circle is important for planar polar coordinates, so the 3-sphere is important in the polar view of 4-space involved in quaternion multiplication. See polar decomposition of a quaternion for details of this development of the three-sphere. This view of the 3-sphere is the basis for the study of elliptic space as developed by Georges Lemaître.

Properties

Elementary properties

The 3-dimensional surface volume of a 3-sphere of radius r is

${\displaystyle SV=2{\pi }^2{r}^3}$

while the 4-dimensional hypervolume (the content of the 4-dimensional region bounded by the 3-sphere) is

${\displaystyle H=\frac{1}{2}{\pi }^2{r}^4.}$

Every non-empty intersection of a 3-sphere with a three-dimensional hyperplane is a 2-sphere (unless the hyperplane is tangent to the 3-sphere, in which case the intersection is a single point). As a 3-sphere moves through a given three-dimensional hyperplane, the intersection starts out as a point, then becomes a growing 2-sphere that reaches its maximal size when the hyperplane cuts right through the "equator" of the 3-sphere. Then the 2-sphere shrinks again down to a single point as the 3-sphere leaves the hyperplane.

In a given three-dimensional hyperplane, a 3-sphere can rotate about an "equatorial plane" (analogous to a 2-sphere rotating about a central axis), in which case it appears to be a 2-sphere whose size is constant.

Topological properties

A 3-sphere is a compact, connected, 3-dimensional manifold without boundary. It is also simply connected. What this means, in the broad sense, is that any loop, or circular path, on the 3-sphere can be continuously shrunk to a point without leaving the 3-sphere. The Poincaré conjecture, proved in 2003 by Grigori Perelman, provides that the 3-sphere is the only three-dimensional manifold (up to homeomorphism) with these properties.

The 3-sphere is homeomorphic to the one-point compactification of R³. In general, any topological space that is homeomorphic to the 3-sphere is called a topological 3-sphere.

The homology groups of the 3-sphere are as follows: H₀(S³, Z) and H₃(S³, Z) are both infinite cyclic, while H_i(S³, Z) = {} for all other indices i. Any topological space with these homology groups is known as a homology 3-sphere. Initially Poincaré conjectured that all homology 3-spheres are homeomorphic to S³, but then he himself constructed a non-homeomorphic one, now known as the Poincaré homology sphere. Infinitely many homology spheres are now known to exist. For example, a Dehn filling with slope 1/n on any knot in the 3-sphere gives a homology sphere; typically these are not homeomorphic to the 3-sphere.

As to the homotopy groups, we have π₁(S³) = π₂(S³) = {} and π₃(S³) is infinite cyclic. The higher-homotopy groups (k ≥ 4) are all finite abelian but otherwise follow no discernible pattern. For more discussion see homotopy groups of spheres.

Homotopy groups of S³

k	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
πk(S3)	0	0	0	Z	Z2	Z2	Z12	Z2	Z2	Z3	Z15	Z2	Z2⊕Z2	Z12⊕Z2	Z84⊕Z2⊕Z2	Z2⊕Z2	Z6

Geometric properties

The 3-sphere is naturally a smooth manifold, in fact, a closed embedded submanifold of R⁴. The Euclidean metric on R⁴ induces a metric on the 3-sphere giving it the structure of a Riemannian manifold. As with all spheres, the 3-sphere has constant positive sectional curvature equal to 1/r² where r is the radius.

Much of the interesting geometry of the 3-sphere stems from the fact that the 3-sphere has a natural Lie group structure given by quaternion multiplication (see the section below on group structure). The only other spheres with such a structure are the 0-sphere and the 1-sphere (see circle group).

Unlike the 2-sphere, the 3-sphere admits nonvanishing vector fields (sections of its tangent bundle). One can even find three linearly independent and nonvanishing vector fields. These may be taken to be any left-invariant vector fields forming a basis for the Lie algebra of the 3-sphere. This implies that the 3-sphere is parallelizable. It follows that the tangent bundle of the 3-sphere is trivial. For a general discussion of the number of linear independent vector fields on a n-sphere, see the article vector fields on spheres.

There is an interesting action of the circle group T on S³ giving the 3-sphere the structure of a principal circle bundle known as the Hopf bundle. If one thinks of S³ as a subset of C², the action is given by

${\displaystyle (z_1,z_2)\cdot \lambda =(z_1\lambda ,z_2\lambda )\mathrm{\forall }\lambda \in \mathbb{T}}$.

The orbit space of this action is homeomorphic to the two-sphere S². Since S³ is not homeomorphic to S² × S¹, the Hopf bundle is nontrivial.

Topological construction

There are several well-known constructions of the three-sphere. Here we describe gluing a pair of three-balls and then the one-point compactification.

Gluing

A 3-sphere can be constructed topologically by "gluing" together the boundaries of a pair of 3-balls. The boundary of a 3-ball is a 2-sphere, and these two 2-spheres are to be identified. That is, imagine a pair of 3-balls of the same size, then superpose them so that their 2-spherical boundaries match, and let matching pairs of points on the pair of 2-spheres be identically equivalent to each other. In analogy with the case of the 2-sphere (see below), the gluing surface is called an equatorial sphere.

Note that the interiors of the 3-balls are not glued to each other. One way to think of the fourth dimension is as a continuous real-valued function of the 3-dimensional coordinates of the 3-ball, perhaps considered to be "temperature". We take the "temperature" to be zero along the gluing 2-sphere and let one of the 3-balls be "hot" and let the other 3-ball be "cold". The "hot" 3-ball could be thought of as the "upper hemisphere" and the "cold" 3-ball could be thought of as the "lower hemisphere". The temperature is highest/lowest at the centers of the two 3-balls.

This construction is analogous to a construction of a 2-sphere, performed by gluing the boundaries of a pair of disks. A disk is a 2-ball, and the boundary of a disk is a circle (a 1-sphere). Let a pair of disks be of the same diameter. Superpose them and glue corresponding points on their boundaries. Again one may think of the third dimension as temperature. Likewise, we may inflate the 2-sphere, moving the pair of disks to become the northern and southern hemispheres.

One-point compactification

After removing a single point from the 2-sphere, what remains is homeomorphic to the Euclidean plane. In the same way, removing a single point from the 3-sphere yields three-dimensional space. An extremely useful way to see this is via stereographic projection. We first describe the lower-dimensional version.

Rest the south pole of a unit 2-sphere on the xy-plane in three-space. We map a point P of the sphere (minus the north pole N) to the plane by sending P to the intersection of the line NP with the plane. Stereographic projection of a 3-sphere (again removing the north pole) maps to three-space in the same manner. (Notice that, since stereographic projection is conformal, round spheres are sent to round spheres or to planes.)

A somewhat different way to think of the one-point compactification is via the exponential map. Returning to our picture of the unit two-sphere sitting on the Euclidean plane: Consider a geodesic in the plane, based at the origin, and map this to a geodesic in the two-sphere of the same length, based at the south pole. Under this map all points of the circle of radius π are sent to the north pole. Since the open unit disk is homeomorphic to the Euclidean plane, this is again a one-point compactification.

The exponential map for 3-sphere is similarly constructed; it may also be discussed using the fact that the 3-sphere is the Lie group of unit quaternions.

Coordinate systems on the 3-sphere

The four Euclidean coordinates for S³ are redundant since they are subject to the condition that x₀² + x₁² + x₂² + x₃² = 1. As a 3-dimensional manifold one should be able to parameterize S³ by three coordinates, just as one can parameterize the 2-sphere using two coordinates (such as latitude and longitude). Due to the nontrivial topology of S³ it is impossible to find a single set of coordinates that cover the entire space. Just as on the 2-sphere, one must use at least two coordinate charts. Some different choices of coordinates are given below.

Hyperspherical coordinates

It is convenient to have some sort of hyperspherical coordinates on S³ in analogy to the usual spherical coordinates on S². One such choice — by no means unique — is to use (ψ, θ, φ), where

${\displaystyle \begin{array}{rl}{x}_0& =r\cos \psi \\ x_1& =r\sin \psi \cos \theta \\ x_2& =r\sin \psi \sin \theta \cos \phi \\ x_3& =r\sin \psi \sin \theta \sin \phi \end{array}}$

where ψ and θ run over the range 0 to π, and φ runs over 0 to 2π. Note that, for any fixed value of ψ, θ and φ parameterize a 2-sphere of radius ${\displaystyle r\sin \psi }$, except for the degenerate cases, when ψ equals 0 or π, in which case they describe a point.

The round metric on the 3-sphere in these coordinates is given by

${\displaystyle d{s}^2=r^2\left[d{\psi }^2+{\sin }^2\psi \left(d{\theta }^2+{\sin }^2\theta d{\phi }^2\right)\right]}$

and the volume form by

${\displaystyle dV=r^3\left({\sin }^2\psi \sin \theta \right)d\psi \wedge d\theta \wedge d\phi .}$

These coordinates have an elegant description in terms of quaternions. Any unit quaternion q can be written as a versor:

${\displaystyle q=e^{\tau \psi }=\cos \psi +\tau \sin \psi }$

where τ is a unit imaginary quaternion; that is, a quaternion that satisfies τ² = −1. This is the quaternionic analogue of Euler's formula. Now the unit imaginary quaternions all lie on the unit 2-sphere in Im H so any such τ can be written:

${\displaystyle \tau =(\cos \theta )i+(\sin \theta \cos \phi )j+(\sin \theta \sin \phi )k}$

With τ in this form, the unit quaternion q is given by

${\displaystyle q=e^{\tau \psi }=x_0+x_{1}i+x_{2}j+x_{3}k}$

where x_0,1,2,3 are as above.

When q is used to describe spatial rotations (cf. quaternions and spatial rotations), it describes a rotation about τ through an angle of 2ψ.

Hopf coordinates

The Hopf fibration can be visualized using a stereographic projection of S³ to R³ and then compressing R³ to a ball. This image shows points on S² and their corresponding fibers with the same color.

For unit radius another choice of hyperspherical coordinates, (η, ξ₁, ξ₂), makes use of the embedding of S³ in C². In complex coordinates (z₁, z₂) ∈ C² we write

${\displaystyle \begin{array}{rl}{z}_1& =e^{i{\xi }_1}\sin \eta \\ z_2& =e^{i{\xi }_2}\cos \eta .\end{array}}$

This could also be expressed in R⁴ as

${\displaystyle \begin{array}{rl}{x}_0& =\cos {\xi }_1\sin \eta \\ x_1& =\sin {\xi }_1\sin \eta \\ x_2& =\cos {\xi }_2\cos \eta \\ x_3& =\sin {\xi }_2\cos \eta .\end{array}}$

Here η runs over the range 0 to π/2, and ξ₁ and ξ₂ can take any values between 0 and 2π. These coordinates are useful in the description of the 3-sphere as the Hopf bundle

${\displaystyle S^1\to S^3\to S^2.}$

A diagram depicting the poloidal (ξ₁) direction, represented by the red arrow, and the toroidal (ξ₂) direction, represented by the blue arrow, although the terms poloidal and toroidal are arbitrary in this flat torus case.

For any fixed value of η between 0 and π/2, the coordinates (ξ₁, ξ₂) parameterize a 2-dimensional torus. Rings of constant ξ₁ and ξ₂ above form simple orthogonal grids on the tori. See image to right. In the degenerate cases, when η equals 0 or π/2, these coordinates describe a circle.

The round metric on the 3-sphere in these coordinates is given by

${\displaystyle d{s}^2=d{\eta }^2+{\sin }^2\eta d{\xi }_1^2+{\cos }^2\eta d{\xi }_2^2}$

and the volume form by

${\displaystyle dV=\sin \eta \cos \eta d\eta \wedge d{\xi }_1\wedge d{\xi }_2.}$

To get the interlocking circles of the Hopf fibration, make a simple substitution in the equations above^[2]

${\displaystyle \begin{array}{rl}{z}_1& =e^{i({\xi }_1+{\xi }_2)}\sin \eta \\ z_2& =e^{i({\xi }_2-{\xi }_1)}\cos \eta .\end{array}}$

In this case η, and ξ₁ specify which circle, and ξ₂ specifies the position along each circle. One round trip (0 to 2π) of ξ₁ or ξ₂ equates to a round trip of the torus in the 2 respective directions.

Stereographic coordinates

Another convenient set of coordinates can be obtained via stereographic projection of S³ from a pole onto the corresponding equatorial R³ hyperplane. For example, if we project from the point (−1, 0, 0, 0) we can write a point p in S³ as

${\displaystyle p=\left(\frac{1-\Vert u{\Vert }^2}{1+\Vert u{\Vert }^2},\frac{2\mathbf{u}}{1+\Vert u{\Vert }^2}\right)=\frac{1+\mathbf{u}}{1-\mathbf{u}}}$

where u = (u₁, u₂, u₃) is a vector in R³ and ‖u‖² = u₁² + u₂² + u₃². In the second equality above, we have identified p with a unit quaternion and u = u₁i + u₂j + u₃k with a pure quaternion. (Note that the numerator and denominator commute here even though quaternionic multiplication is generally noncommutative). The inverse of this map takes p = (x₀, x₁, x₂, x₃) in S³ to

${\displaystyle \mathbf{u}=\frac{1}{1+x_0}\left(x_1,x_2,x_3\right).}$

We could just as well have projected from the point (1, 0, 0, 0), in which case the point p is given by

${\displaystyle p=\left(\frac{-1+\Vert v{\Vert }^2}{1+\Vert v{\Vert }^2},\frac{2\mathbf{v}}{1+\Vert v{\Vert }^2}\right)=\frac{-1+\mathbf{v}}{1+\mathbf{v}}}$

where v = (v₁, v₂, v₃) is another vector in R³. The inverse of this map takes p to

${\displaystyle \mathbf{v}=\frac{1}{1-x_0}\left(x_1,x_2,x_3\right).}$

Note that the u coordinates are defined everywhere but (−1, 0, 0, 0) and the v coordinates everywhere but (1, 0, 0, 0). This defines an atlas on S³ consisting of two coordinate charts or "patches", which together cover all of S³. Note that the transition function between these two charts on their overlap is given by

${\displaystyle \mathbf{v}=\frac{1}{\Vert u{\Vert }^2}\mathbf{u}}$

and vice versa.

Group structure

When considered as the set of unit quaternions, S³ inherits an important structure, namely that of quaternionic multiplication. Because the set of unit quaternions is closed under multiplication, S³ takes on the structure of a group. Moreover, since quaternionic multiplication is smooth, S³ can be regarded as a real Lie group. It is a nonabelian, compact Lie group of dimension 3. When thought of as a Lie group S³ is often denoted Sp(1) or U(1, H).

It turns out that the only spheres that admit a Lie group structure are S¹, thought of as the set of unit complex numbers, and S³, the set of unit quaternions (The degenerate case S⁰ which consists of the real numbers 1 and −1 is also a Lie group, albeit a 0-dimensional one). One might think that S⁷, the set of unit octonions, would form a Lie group, but this fails since octonion multiplication is nonassociative. The octonionic structure does give S⁷ one important property: parallelizability. It turns out that the only spheres that are parallelizable are S¹, S³, and S⁷.

By using a matrix representation of the quaternions, H, one obtains a matrix representation of S³. One convenient choice is given by the Pauli matrices:

${\displaystyle x_1+x_{2}i+x_{3}j+x_{4}k\mapsto \left(\begin{array}{cc}{x}_1+i{x}_2& x_3+i{x}_4\\ -x_3+i{x}_4& x_1-i{x}_2\end{array}\right).}$

This map gives an injective algebra homomorphism from H to the set of 2 × 2 complex matrices. It has the property that the absolute value of a quaternion q is equal to the square root of the determinant of the matrix image of q.

The set of unit quaternions is then given by matrices of the above form with unit determinant. This matrix subgroup is precisely the special unitary group SU(2). Thus, S³ as a Lie group is isomorphic to SU(2).

Using our Hopf coordinates (η, ξ₁, ξ₂) we can then write any element of SU(2) in the form

${\displaystyle \left(\begin{array}{cc}{e}^{i{\xi }_1}\sin \eta & e^{i{\xi }_2}\cos \eta \\ -e^{-i{\xi }_2}\cos \eta & e^{-i{\xi }_1}\sin \eta \end{array}\right).}$

Another way to state this result is if we express the matrix representation of an element of SU(2) as a exponential of a linear combination of the Pauli matrices. It is seen that an arbitrary element U ∈ SU(2) can be written as

${\displaystyle U=\exp \left(\sum _{i=1}^3{\alpha }_i{J}_i\right).}$

The condition that the determinant of U is +1 implies that the coefficients α₁ are constrained to lie on a 3-sphere.

In literature

In Edwin Abbott Abbott's Flatland, published in 1884, and in Sphereland, a 1965 sequel to Flatland by Dionys Burger, the 3-sphere is referred to as an oversphere, and a 4-sphere is referred to as a hypersphere.

Writing in the American Journal of Physics, Mark A. Peterson describes three different ways of visualizing 3-spheres and points out language in The Divine Comedy that suggests Dante viewed the Universe in the same way; Carlo Rovelli supports the same idea.

In Art Meets Mathematics in the Fourth Dimension, Stephen L. Lipscomb develops the concept of the hypersphere dimensions as it relates to art, architecture, and mathematics.

n-sphere

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/N-sphere

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 ${\displaystyle n}$-dimensional generalization of the ${\displaystyle 1}$-dimensional circle and ${\displaystyle 2}$-dimensional sphere to any non-negative integer ${\displaystyle n}$. The ${\displaystyle n}$-sphere is the setting for ${\displaystyle n}$-dimensional spherical geometry.

Considered extrinsically, as a hypersurface embedded in ${\displaystyle (n+1)}$-dimensional Euclidean space, an ${\displaystyle n}$-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 ${\displaystyle (n+1)}$-dimensional ball. In particular:

• The ${\displaystyle 0}$-sphere is the pair of points at the ends of a line segment (${\displaystyle 1}$-ball).
• The ${\displaystyle 1}$-sphere is a circle, the circumference of a disk (${\displaystyle 2}$-ball) in the two-dimensional plane.
• The ${\displaystyle 2}$-sphere, often simply called a sphere, is the boundary of a ${\displaystyle 3}$-ball in three-dimensional space.
• The 3-sphere is the boundary of a ${\displaystyle 4}$-ball in four-dimensional space.
• The ${\displaystyle (n-1)}$-sphere is the boundary of an ${\displaystyle n}$-ball.

Given a Cartesian coordinate system, the unit ${\displaystyle n}$-sphere of radius ${\displaystyle 1}$ can be defined as:

${\displaystyle S^n=\left\{x\in {\mathbb{R}}^{n+1}:\Vert x\Vert =1\right\}.}$

Considered intrinsically, when ${\displaystyle n\ge 1}$, the ${\displaystyle n}$-sphere is a Riemannian manifold of positive constant curvature, and is orientable. The geodesics of the ${\displaystyle n}$-sphere are called great circles.

The stereographic projection maps the ${\displaystyle n}$-sphere onto ${\displaystyle n}$-space with a single adjoined point at infinity; under the metric thereby defined, ${\displaystyle {\mathbb{R}}^n\cup \{\mathrm{\infty }\}}$ is a model for the ${\displaystyle n}$-sphere.

In the more general setting of topology, any topological space that is homeomorphic to the unit ${\displaystyle n}$-sphere is called an ${\displaystyle n}$-sphere. Under inverse stereographic projection, the ${\displaystyle n}$-sphere is the one-point compactification of ${\displaystyle n}$-space. The ${\displaystyle n}$-spheres admit several other topological descriptions: for example, they can be constructed by gluing two ${\displaystyle n}$-dimensional spaces together, by identifying the boundary of an ${\displaystyle n}$-cube with a point, or (inductively) by forming the suspension of an ${\displaystyle (n-1)}$-sphere. When ${\displaystyle n\ge 2}$ it is simply connected; the ${\displaystyle 1}$-sphere (circle) is not simply connected; the ${\displaystyle 0}$-sphere is not even connected, consisting of two discrete points.

Description

For any natural number ${\displaystyle n}$, an ${\displaystyle n}$-sphere of radius ${\displaystyle r}$ is defined as the set of points in ${\displaystyle (n+1)}$-dimensional Euclidean space that are at distance ${\displaystyle r}$ from some fixed point ${\displaystyle \mathbf{c}}$, where ${\displaystyle r}$ may be any positive real number and where ${\displaystyle \mathbf{c}}$ may be any point in ${\displaystyle (n+1)}$-dimensional space. In particular:

• a 0-sphere is a pair of points ${\displaystyle \{c-r,c+r\}}$, and is the boundary of a line segment (${\displaystyle 1}$-ball).
• a 1-sphere is a circle of radius ${\displaystyle r}$ centered at ${\displaystyle \mathbf{c}}$, and is the boundary of a disk (${\displaystyle 2}$-ball).
• a 2-sphere is an ordinary ${\displaystyle 2}$-dimensional sphere in ${\displaystyle 3}$-dimensional Euclidean space, and is the boundary of an ordinary ball (${\displaystyle 3}$-ball).
• a 3-sphere is a ${\displaystyle 3}$-dimensional sphere in ${\displaystyle 4}$-dimensional Euclidean space.

Cartesian coordinates

The set of points in ${\displaystyle (n+1)}$-space, ${\displaystyle (x_1,x_2,\dots ,x_n+1)}$, that define an ${\displaystyle n}$-sphere, ${\displaystyle S^n(r)}$, is represented by the equation:

${\displaystyle r^2=\sum _{i=1}^{n+1}(x_i-c_i{)}^2,}$

where ${\displaystyle \mathbf{c}=(c_1,c_2,\dots ,c_{n+1})}$ is a center point, and ${\displaystyle r}$ is the radius.

The above ${\displaystyle n}$-sphere exists in ${\displaystyle (n+1)}$-dimensional Euclidean space and is an example of an ${\displaystyle n}$-manifold. The volume form ${\displaystyle \omega }$ of an ${\displaystyle n}$-sphere of radius ${\displaystyle r}$ is given by

${\displaystyle \omega =\frac{1}{r}\sum _{j=1}^{n+1}(-1{)}^{j-1}{x}_{j}d{x}_1\wedge \cdots \wedge d{x}_{j-1}\wedge d{x}_{j+1}\wedge \cdots \wedge d{x}_{n+1}=\star dr}$

where ${\displaystyle \star }$ is the Hodge star operator; see Flanders (1989, §6.1) for a discussion and proof of this formula in the case ${\displaystyle r=1}$. As a result,

${\displaystyle dr\wedge \omega =d{x}_1\wedge \cdots \wedge d{x}_{n+1}.}$

${\displaystyle n}$-ball

Main article: Ball (mathematics)

The space enclosed by an ${\displaystyle n}$-sphere is called an ${\displaystyle (n+1)}$-ball. An ${\displaystyle (n+1)}$-ball is closed if it includes the ${\displaystyle n}$-sphere, and it is open if it does not include the ${\displaystyle n}$-sphere.

Specifically:

• A ${\displaystyle 1}$-ball, a line segment, is the interior of a 0-sphere.
• A ${\displaystyle 2}$-ball, a disk, is the interior of a circle (${\displaystyle 1}$-sphere).
• A ${\displaystyle 3}$-ball, an ordinary ball, is the interior of a sphere (${\displaystyle 2}$-sphere).
• A ${\displaystyle 4}$-ball is the interior of a 3-sphere, etc.

Topological description

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

Volume and area

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

Let ${\displaystyle S_{n-1}}$ be the surface area of the unit ${\displaystyle (n-1)}$-sphere of radius ${\displaystyle 1}$ embedded in ${\displaystyle n}$-dimensional Euclidean space, and let ${\displaystyle V_n}$ be the volume of its interior, the unit ${\displaystyle n}$-ball. The surface area of an arbitrary ${\displaystyle (n-1)}$-sphere is proportional to the ${\displaystyle (n-1)}$st power of the radius, and the volume of an arbitrary ${\displaystyle n}$-ball is proportional to the ${\displaystyle n}$th power of the radius.

S_{n &minus 1}) of n-balls of radius 1.

The ${\displaystyle 0}$-ball is sometimes defined as a single point. The ${\displaystyle 0}$-dimensional Hausdorff measure is the number of points in a set. So

${\displaystyle V_0=1.}$

A unit ${\displaystyle 1}$-ball is a line segment whose points have a single coordinate in the interval ${\displaystyle [-1,1]}$ of length ${\displaystyle 2}$, and the ${\displaystyle 0}$-sphere consists of its two end-points, with coordinate ${\displaystyle \{-1,1\}}$.

${\displaystyle S_0=2,V_1=2.}$

A unit ${\displaystyle 1}$-sphere is the unit circle in the Euclidean plane, and its interior is the unit disk (${\displaystyle 2}$-ball).

${\displaystyle S_1=2\pi ,V_2=\pi .}$

The interior of a 2-sphere in three-dimensional space is the unit ${\displaystyle 3}$-ball.

${\displaystyle S_2=4\pi ,V_3=\frac{4}{3}\pi .}$

In general, ${\displaystyle S_{n-1}}$ and ${\displaystyle V_n}$ are given in closed form by the expressions

${\displaystyle S_{n-1}=\frac{2{\pi }^{n/2}}{\mathrm{\Gamma }(\frac{n}{2})},V_n=\frac{{\pi }^{n/2}}{\mathrm{\Gamma }(\frac{n}{2}+1)}}$

where ${\displaystyle \mathrm{\Gamma }}$ is the gamma function.

As ${\displaystyle n}$ tends to infinity, the volume of the unit ${\displaystyle n}$-ball (ratio between the volume of an ${\displaystyle n}$-ball of radius ${\displaystyle 1}$ and an ${\displaystyle n}$-cube of side length ${\displaystyle 1}$) tends to zero.

Recurrences

The surface area, or properly the ${\displaystyle n}$-dimensional volume, of the ${\displaystyle n}$-sphere at the boundary of the ${\displaystyle (n+1)}$-ball of radius ${\displaystyle R}$ is related to the volume of the ball by the differential equation

${\displaystyle S_n{R}^n=\frac{d{V}_{n+1}{R}^{n+1}}{dR}=(n+1)V_{n+1}{R}^n.}$

Equivalently, representing the unit ${\displaystyle n}$-ball as a union of concentric ${\displaystyle (n-1)}$-sphere shells,

${\displaystyle V_{n+1}={\int }_0^1{S}_n{r}^{n}dr=\frac{1}{n+1}{S}_n.}$

We can also represent the unit ${\displaystyle (n+2)}$-sphere as a union of products of a circle (${\displaystyle 1}$-sphere) with an ${\displaystyle n}$-sphere. Then ${\displaystyle S_{n+2}=2\pi V_{n+1}}$. Since ${\displaystyle S_1=2\pi V_0}$, the equation

${\displaystyle S_{n+1}=2\pi V_n}$

holds for all ${\displaystyle n}$. Along with the base cases ${\displaystyle S_0=2}$, ${\displaystyle V_1=2}$ 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 ${\displaystyle n}$-dimensional Euclidean space which is analogous to the spherical coordinate system defined for ${\displaystyle 3}$-dimensional Euclidean space, in which the coordinates consist of a radial coordinate ${\displaystyle r}$, and ${\displaystyle n-1}$ angular coordinates ${\displaystyle {\phi }_1,{\phi }_2,\dots ,{\phi }_{n-1}}$, where the angles ${\displaystyle {\phi }_1,{\phi }_2,\dots ,{\phi }_{n-2}}$ range over ${\displaystyle [0,\pi ]}$ radians (or ${\displaystyle [0,180]}$ degrees) and ${\displaystyle {\phi }_{n-1}}$ ranges over ${\displaystyle [0,2\pi )}$ radians (or ${\displaystyle [0,360)}$ degrees). If ${\displaystyle x_i}$ are the Cartesian coordinates, then we may compute ${\displaystyle x_1,\dots ,x_n}$ from ${\displaystyle r,{\phi }_1,\dots ,{\phi }_{n-1}}$ with:

${\displaystyle \begin{array}{rl}{x}_1& =r\cos ({\phi }_1),\\ x_2& =r\sin ({\phi }_1)\cos ({\phi }_2),\\ x_3& =r\sin ({\phi }_1)\sin ({\phi }_2)\cos ({\phi }_3),\\ & ⋮\\ x_{n-1}& =r\sin ({\phi }_1)\cdots \sin ({\phi }_{n-2})\cos ({\phi }_{n-1}),\\ x_n& =r\sin ({\phi }_1)\cdots \sin ({\phi }_{n-2})\sin ({\phi }_{n-1}).\end{array}}$

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

${\displaystyle \begin{array}{rl}r& =\sqrt{{x_n}^2+{x_{n-1}}^2+\cdots +{x_2}^2+{x_1}^2},\\ {\phi }_1& =\mathrm{atan2}\left(\sqrt{{x_n}^2+{x_{n-1}}^2+\cdots +{x_2}^2},x_1\right),\\ {\phi }_2& =\mathrm{atan2}\left(\sqrt{{x_n}^2+{x_{n-1}}^2+\cdots +{x_3}^2},x_2\right),\\ & ⋮\\ {\phi }_{n-2}& =\mathrm{atan2}\left(\sqrt{{x_n}^2+{x_{n-1}}^2},x_{n-2}\right),\\ {\phi }_{n-1}& =\mathrm{atan2}\left(x_n,x_{n-1}\right).\end{array}}$

where atan2 is the two-argument arctangent function.

There are some special cases where the inverse transform is not unique; ${\displaystyle {\phi }_k}$ for any ${\displaystyle k}$ will be ambiguous whenever all of ${\displaystyle x_k,x_{k+1},\dots x_n}$ are zero; in this case ${\displaystyle {\phi }_k}$ may be chosen to be zero. (For example, for the ${\displaystyle 2}$-sphere, when the polar angle is ${\displaystyle 0}$ or ${\displaystyle \pi }$ then the point is one of the poles, zenith or nadir, and the choice of azimuthal angle is arbitrary.)

Spherical volume and area elements

To express the volume element of ${\displaystyle n}$-dimensional Euclidean space in terms of spherical coordinates, let ${\displaystyle s_k=\sin {\phi }_k}$ and ${\displaystyle c_k=\cos {\phi }_k}$ for concision, then observe that the Jacobian matrix of the transformation is:

${\displaystyle J_n=\left(\begin{array}{cccccc}{c}_1& -r{s}_1& 0& 0& \cdots & 0\\ s_1{c}_2& r{c}_1{c}_2& -r{s}_1{s}_2& 0& \cdots & 0\\ ⋮& ⋮& ⋮& & \ddots & ⋮\\ & & & & & 0\\ s_1\cdots s_{n-2}{c}_{n-1}& \cdots & \cdots & & & -r{s}_1\cdots s_{n-2}{s}_{n-1}\\ s_1\cdots s_{n-2}{s}_{n-1}& r{c}_1\cdots s_{n-1}& \cdots & & & \phantom{}r{s}_1\cdots s_{n-2}{c}_{n-1}\end{array}\right).}$

The determinant of this matrix can be calculated by induction. When ${\displaystyle n=2}$, a straightforward computation shows that the determinant is ${\displaystyle r}$. For larger ${\displaystyle n}$, observe that ${\displaystyle J_n}$ can be constructed from ${\displaystyle J_{n-1}}$ as follows. Except in column ${\displaystyle n}$, rows ${\displaystyle n-1}$ and ${\displaystyle n}$ of ${\displaystyle J_n}$ are the same as row ${\displaystyle n-1}$ of ${\displaystyle J_{n-1}}$, but multiplied by an extra factor of ${\displaystyle \cos {\phi }_{n-1}}$ in row ${\displaystyle n-1}$ and an extra factor of ${\displaystyle \sin {\phi }_{n-1}}$ in row ${\displaystyle n}$. In column ${\displaystyle n}$, rows ${\displaystyle n-1}$ and ${\displaystyle n}$ of ${\displaystyle J_n}$ are the same as column ${\displaystyle n-1}$ of row ${\displaystyle n-1}$ of ${\displaystyle J_{n-1}}$, but multiplied by extra factors of ${\displaystyle \sin {\phi }_{n-1}}$ in row ${\displaystyle n-1}$ and ${\displaystyle \cos {\phi }_{n-1}}$ in row ${\displaystyle n}$, respectively. The determinant of ${\displaystyle J_n}$ can be calculated by Laplace expansion in the final column. By the recursive description of ${\displaystyle J_n}$, the submatrix formed by deleting the entry at ${\displaystyle (n-1,n)}$ and its row and column almost equals ${\displaystyle J_{n-1}}$, except that its last row is multiplied by ${\displaystyle \sin {\phi }_{n-1}}$. Similarly, the submatrix formed by deleting the entry at ${\displaystyle (n,n)}$ and its row and column almost equals ${\displaystyle J_{n-1}}$, except that its last row is multiplied by ${\displaystyle \cos {\phi }_{n-1}}$. Therefore the determinant of ${\displaystyle J_n}$ is

${\displaystyle \begin{array}{rl}|J_n|& =(-1{)}^{(n-1)+n}(-r{s}_1\cdots s_{n-2}{s}_{n-1})(s_{n-1}|J_{n-1}|)\\ & +(-1{)}^{n+n}(r{s}_1\cdots s_{n-2}{c}_{n-1})(c_{n-1}|J_{n-1}|)\\ & =(r{s}_1\cdots s_{n-2}|J_{n-1}|(s_{n-1}^2+c_{n-1}^2)\\ & =(r{s}_1\cdots s_{n-2})|J_{n-1}|.\end{array}}$

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

${\displaystyle \begin{array}{rl}{d}^{n}V& =\left|det\frac{\mathrm{\partial }(x_i)}{\mathrm{\partial }\left(r,{\phi }_j\right)}\right|drd{\phi }_{1}d{\phi }_2\cdots d{\phi }_{n-1}\\ & =r^{n-1}{\sin }^{n-2}({\phi }_1){\sin }^{n-3}({\phi }_2)\cdots \sin ({\phi }_{n-2})drd{\phi }_{1}d{\phi }_2\cdots d{\phi }_{n-1}.\end{array}}$

The formula for the volume of the ${\displaystyle n}$-ball can be derived from this by integration.

Similarly the surface area element of the ${\displaystyle (n-1)}$-sphere of radius ${\displaystyle r}$, which generalizes the area element of the ${\displaystyle 2}$-sphere, is given by

${\displaystyle d_{S^{n-1}}V=R^{n-1}{\sin }^{n-2}({\phi }_1){\sin }^{n-3}({\phi }_2)\cdots \sin ({\phi }_{n-2})d{\phi }_{1}d{\phi }_2\cdots d{\phi }_{n-1}.}$

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

${\displaystyle \begin{array}{rl}& {\int }_0^{\pi }{\sin }^{n-j-1}\left({\phi }_j\right)C_s^{\left(\frac{n-j-1}{2}\right)}\cos \left({\phi }_j\right)C_{s^{\prime }}^{\left(\frac{n-j-1}{2}\right)}\cos \left({\phi }_j\right)d{\phi }_j\\ & =\frac{2^{3-n+j}\pi \mathrm{\Gamma }(s+n-j-1)}{s!(2s+n-j-1){\mathrm{\Gamma }}^2\left(\frac{n-j-1}{2}\right)}{\delta }_{s,s^{\prime }}\end{array}}$

for ${\displaystyle j=1,2,\dots ,n-2}$, and the ${\displaystyle e^{is{\phi }_j}}$ for the angle ${\displaystyle j=n-1}$ in concordance with the spherical harmonics.

Polyspherical coordinates

The standard spherical coordinate system arises from writing ${\displaystyle {\mathbb{R}}^n}$ as the product ${\displaystyle \mathbb{R}\times {\mathbb{R}}^{n-1}}$. These two factors may be related using polar coordinates. For each point ${\displaystyle \mathbf{x}}$ of ${\displaystyle {\mathbb{R}}^n}$, the standard Cartesian coordinates

${\displaystyle \mathbf{x}=(x_1,\dots ,x_n)=(y_1,z_1,\dots ,z_{n-1})=(y_1,\mathbf{z})}$

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

${\displaystyle \mathbf{x}=(r\sin \theta ,(r\cos \theta )\hat{\mathbf{z}}).}$

This says that points in ${\displaystyle {\mathbb{R}}^n}$ may be expressed by taking the ray starting at the origin and passing through ${\displaystyle \hat{\mathbf{z}}=\mathbf{z}/\Vert \mathbf{z}\Vert \in S^{n-2}}$, rotating it towards ${\displaystyle (1,0,\dots ,0)}$ by ${\displaystyle \theta =\arcsin y_1/r}$, and traveling a distance ${\displaystyle r=\Vert \mathbf{x}\Vert }$ 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 ${\displaystyle {\mathbb{R}}^n}$ is split as the product of two Euclidean spaces of smaller dimension, but neither space is required to be a line. Specifically, suppose that ${\displaystyle p}$ and ${\displaystyle q}$ are positive integers such that ${\displaystyle n=p+q}$. Then ${\displaystyle {\mathbb{R}}^n={\mathbb{R}}^p\times {\mathbb{R}}^q}$. Using this decomposition, a point ${\displaystyle x\in {\mathbb{R}}^n}$ may be written as

${\displaystyle \mathbf{x}=(x_1,\dots ,x_n)=(y_1,\dots ,y_p,z_1,\dots ,z_q)=(\mathbf{y},\mathbf{z}).}$

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

${\displaystyle \mathbf{x}=((r\sin \theta )\hat{\mathbf{y}},(r\cos \theta )\hat{\mathbf{z}}).}$

Here ${\displaystyle \hat{\mathbf{y}}}$ and ${\displaystyle \hat{\mathbf{z}}}$ are the unit vectors associated to ${\displaystyle \mathbf{y}}$ and ${\displaystyle \mathbf{z}}$. This expresses ${\displaystyle \mathbf{x}}$ in terms of ${\displaystyle \hat{\mathbf{y}}\in S^{p-1}}$, ${\displaystyle \hat{\mathbf{z}}\in S^{q-1}}$, ${\displaystyle r\ge 0}$, and an angle ${\displaystyle \theta }$. It can be shown that the domain of ${\displaystyle \theta }$ is ${\displaystyle [0,2\pi )}$ if ${\displaystyle p=q=1}$, ${\displaystyle [0,\pi ]}$ if exactly one of ${\displaystyle p}$ and ${\displaystyle q}$ is ${\displaystyle 1}$, and ${\displaystyle [0,\pi /2]}$ if neither ${\displaystyle p}$ nor ${\displaystyle q}$ are ${\displaystyle 1}$. The inverse transformation is

${\displaystyle \begin{array}{rl}r& =\Vert \mathbf{x}\Vert ,\\ \theta & =\arcsin \frac{\Vert \mathbf{y}\Vert }{\Vert \mathbf{x}\Vert }=\arccos \frac{\Vert \mathbf{z}\Vert }{\Vert \mathbf{x}\Vert }=\arctan \frac{\Vert \mathbf{y}\Vert }{\Vert \mathbf{z}\Vert }.\end{array}}$

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 ${\displaystyle \hat{\mathbf{y}}}$ and ${\displaystyle \hat{\mathbf{z}}}$ are spheres, so the coordinates of a polyspherical coordinate system are a non-negative radius and ${\displaystyle n-1}$ angles. The possible polyspherical coordinate systems correspond to binary trees with ${\displaystyle n}$ 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 ${\displaystyle {\mathbb{R}}^n}$, and its immediate children represent the first splitting into ${\displaystyle {\mathbb{R}}^p}$ and ${\displaystyle {\mathbb{R}}^q}$. Leaf nodes correspond to Cartesian coordinates for ${\displaystyle S^{n-1}}$. 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 ${\displaystyle {\theta }_i}$, taking the left branch introduces a factor of ${\displaystyle \sin {\theta }_i}$ and taking the right branch introduces a factor of ${\displaystyle \cos {\theta }_i}$. 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 ${\displaystyle {\mathbb{R}}^n={\mathbb{R}}^p\times {\mathbb{R}}^q}$ determines a subgroup

${\displaystyle {\mathrm{SO}}_p(\mathbb{R})\times {\mathrm{SO}}_q(\mathbb{R})\subseteq {\mathrm{SO}}_n(\mathbb{R}).}$

This is the subgroup that leaves each of the two factors ${\displaystyle S^{p-1}\times S^{q-1}\subseteq S^{n-1}}$ 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 ${\displaystyle {\mathbb{R}}^n}$ and the area measure on ${\displaystyle S^{n-1}}$ are products. There is one factor for each angle, and the volume measure on ${\displaystyle {\mathbb{R}}^n}$ also has a factor for the radial coordinate. The area measure has the form:

${\displaystyle d{A}_{n-1}=\prod _{i=1}^{n-1}{F}_i({\theta }_i)d{\theta }_i,}$

where the factors ${\displaystyle F_i}$ are determined by the tree. Similarly, the volume measure is

${\displaystyle d{V}_n=r^{n-1}dr\prod _{i=1}^{n-1}{F}_i({\theta }_i)d{\theta }_i.}$

Suppose we have a node of the tree that corresponds to the decomposition ${\displaystyle {\mathbb{R}}^{n_1+n_2}={\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2}}$ and that has angular coordinate ${\displaystyle \theta }$. The corresponding factor ${\displaystyle F}$ depends on the values of ${\displaystyle n_1}$ and ${\displaystyle n_2}$. When the area measure is normalized so that the area of the sphere is ${\displaystyle 1}$, these factors are as follows. If ${\displaystyle n_1=n_2=1}$, then

${\displaystyle F(\theta )=\frac{d\theta }{2\pi }.}$

If ${\displaystyle n_1>1}$ and ${\displaystyle n_2=1}$, and if ${\displaystyle \mathrm{B}}$ denotes the beta function, then

${\displaystyle F(\theta )=\frac{{\sin }^{n_1-1}\theta }{\mathrm{B}(\frac{n_1}{2},\frac{1}{2})}d\theta .}$

If ${\displaystyle n_1=1}$ and ${\displaystyle n_2>1}$, then

${\displaystyle F(\theta )=\frac{{\cos }^{n_2-1}\theta }{\mathrm{B}(\frac{1}{2},\frac{n_2}{2})}d\theta .}$

Finally, if both ${\displaystyle n_1}$ and ${\displaystyle n_2}$ are greater than one, then

${\displaystyle F(\theta )=\frac{({\sin }^{n_1-1}\theta )({\cos }^{n_2-1}\theta )}{\frac{1}{2}\mathrm{B}(\frac{n_1}{2},\frac{n_2}{2})}d\theta .}$

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 ${\displaystyle n}$-sphere can be mapped onto an ${\displaystyle n}$-dimensional hyperplane by the ${\displaystyle n}$-dimensional version of the stereographic projection. For example, the point ${\displaystyle [x,y,z]}$ on a two-dimensional sphere of radius ${\displaystyle 1}$ maps to the point ${\displaystyle [\frac{x}{1-z},\frac{y}{1-z}]}$ on the ${\displaystyle xy}$-plane. In other words,

${\displaystyle [x,y,z]\mapsto \left[\frac{x}{1-z},\frac{y}{1-z}\right].}$

Likewise, the stereographic projection of an ${\displaystyle n}$-sphere ${\displaystyle S^n}$ of radius ${\displaystyle 1}$ will map to the ${\displaystyle (n-1)}$-dimensional hyperplane ${\displaystyle {\mathbb{R}}^{n-1}}$ perpendicular to the ${\displaystyle x_n}$-axis as

${\displaystyle [x_1,x_2,\dots ,x_n]\mapsto \left[\frac{x_1}{1-x_n},\frac{x_2}{1-x_n},\dots ,\frac{x_{n-1}}{1-x_n}\right].}$

Probability distributions

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

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 ${\displaystyle (n-1)}$-sphere (that is, the surface of the unit ${\displaystyle n}$-ball), Marsaglia (1972) gives the following algorithm.

Generate an ${\displaystyle n}$-dimensional vector of normal deviates (it suffices to use ${\displaystyle N(0,1)}$, although in fact the choice of the variance is arbitrary), ${\displaystyle \mathbf{x}=(x_1,x_2,\dots ,x_n)}$. Now calculate the "radius" of this point:

${\displaystyle r=\sqrt{x_1^2+x_2^2+\cdots +x_n^2}.}$

The vector ${\displaystyle \frac{1}{r}\mathbf{x}}$ is uniformly distributed over the surface of the unit ${\displaystyle n}$-ball.

An alternative given by Marsaglia is to uniformly randomly select a point ${\displaystyle \mathbf{x}=(x_1,x_2,\dots ,x_n)}$ in the unit n-cube by sampling each ${\displaystyle x_i}$ independently from the uniform distribution over ${\displaystyle (-1,1)}$, computing ${\displaystyle r}$ as above, and rejecting the point and resampling if ${\displaystyle r\ge 1}$ (i.e., if the point is not in the ${\displaystyle n}$-ball), and when a point in the ball is obtained scaling it up to the spherical surface by the factor ${\displaystyle \frac{1}{r}}$; then again ${\displaystyle \frac{1}{r}\mathbf{x}}$ is uniformly distributed over the surface of the unit ${\displaystyle n}$-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 ${\displaystyle {10}^{-24}}$ 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 ${\displaystyle (n-1)}$-sphere (e.g., by using Marsaglia's algorithm), one needs only a radius to obtain a point uniformly at random from within the unit ${\displaystyle n}$-ball. If ${\displaystyle u}$ is a number generated uniformly at random from the interval ${\displaystyle [0,1]}$ and ${\displaystyle \mathbf{x}}$ is a point selected uniformly at random from the unit ${\displaystyle (n-1)}$-sphere, then ${\displaystyle u^{1/n}\mathbf{x}}$ is uniformly distributed within the unit ${\displaystyle n}$-ball.

Alternatively, points may be sampled uniformly from within the unit ${\displaystyle n}$-ball by a reduction from the unit ${\displaystyle (n+1)}$-sphere. In particular, if ${\displaystyle (x_1,x_2,\dots ,x_{n+2})}$ is a point selected uniformly from the unit ${\displaystyle (n+1)}$-sphere, then ${\displaystyle (x_1,x_2,\dots ,x_n)}$ is uniformly distributed within the unit ${\displaystyle n}$-ball (i.e., by simply discarding two coordinates).

If ${\displaystyle n}$ is sufficiently large, most of the volume of the ${\displaystyle n}$-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.

Distribution of the first coordinate

Let ${\displaystyle y=x_1^2}$ be the square of the first coordinate of a point sampled uniformly at random from the ${\displaystyle (n-1)}$-sphere, then its probability density function, for ${\displaystyle y\in [0,1]}$, is

$${\displaystyle \rho (y)=\frac{\mathrm{\Gamma }(\frac{n}{2})}{\sqrt{\pi }\mathrm{\Gamma }(\frac{n-1}{2})}(1-y{)}^{(n-3)/2}{y}^{-1/2}.}$$

Let ${\displaystyle z=y/N}$ be the appropriately scaled version, then at the ${\displaystyle N\to \mathrm{\infty }}$ limit, the probability density function of ${\displaystyle z}$ converges to ${\displaystyle (2\pi z{e}^z{)}^{-1/2}}$. This is sometimes called the Porter–Thomas distribution.

Specific spheres

0-sphere

The pair of points ${\displaystyle \{\pm R\}}$ with the discrete topology for some ${\displaystyle R>0}$. 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.

2-sphere

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

3-sphere

Parallelizable, principal U(1)-bundle over the ${\displaystyle 2}$-sphere, Lie group structure Sp(1).

4-sphere

Homeomorphic to the quaternionic projective line, ${\displaystyle {\mathbf{H}\mathbf{P}}^1}$. ${\displaystyle \mathrm{SO}(5)/\mathrm{SO}(4)}$.

5-sphere

Principal U(1)-bundle over the complex projective space ${\displaystyle {\mathbf{C}\mathbf{P}}^2}$. ${\displaystyle \mathrm{SO}(6)/\mathrm{SO}(5)=\mathrm{SU}(3)/\mathrm{SU}(2)}$. It is undecidable whether a given ${\displaystyle n}$-dimensional manifold is homeomorphic to ${\displaystyle S^n}$ for ${\displaystyle n\ge 5}$.

6-sphere

Possesses an almost complex structure coming from the set of pure unit octonions. ${\displaystyle \mathrm{SO}(7)/\mathrm{SO}(6)=G_2/\mathrm{SU}(3)}$. 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 ${\displaystyle \mathrm{Sp}(1)}$-bundle over S^4. Parallelizable. ${\displaystyle \mathrm{SO}(8)/\mathrm{SO}(7)=\mathrm{SU}(4)/\mathrm{SU}(3)=\mathrm{Sp}(2)/\mathrm{Sp}(1)=\mathrm{Spin}(7)/G_2=\mathrm{Spin}(6)/\mathrm{SU}(3)}$. The ${\displaystyle 7}$-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 ${\displaystyle {\mathbf{O}\mathbf{P}}^1}$.

23-sphere

A highly dense sphere-packing is possible in ${\displaystyle 24}$-dimensional space, which is related to the unique qualities of the Leech lattice.

Octahedral sphere

The octahedral ${\displaystyle n}$-sphere is defined similarly to the ${\displaystyle n}$-sphere but using the 1-norm

${\displaystyle S^n=\left\{x\in {\mathbb{R}}^{n+1}:{\Vert x\Vert }_1=1\right\}}$

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

The octahedral ${\displaystyle 1}$-sphere is a square (without its interior). The octahedral ${\displaystyle 2}$-sphere is a regular octahedron; hence the name. The octahedral ${\displaystyle n}$-sphere is the topological join of ${\displaystyle n+1}$ 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.