Inmathematics, ann-sphere orhypersphere is an⁠${\displaystyle n}$⁠-dimensional generalization of the⁠${\displaystyle 1}$⁠-dimensionalcircle and⁠${\displaystyle 2}$⁠-dimensionalsphere to any non-negativeinteger⁠${\displaystyle n}$⁠.

The circle is considered 1-dimensional, and the sphere 2-dimensional, because the surfaces themselves are 1- and 2-dimensional respectively,not because they exist as shapes in 1- and 2-dimensional space. 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${\displaystyle n}$⁠-sphere is embedded in an⁠${\displaystyle n+1}$⁠-dimensional space. The termhypersphere is commonly used to distinguish spheres of dimension⁠${\displaystyle n\ge 3}$⁠ which are thus embedded in a space of dimension⁠${\displaystyle n+1\ge 4}$⁠, which means that they cannot be easily visualized. The⁠${\displaystyle n}$⁠-sphere is the setting for⁠${\displaystyle n}$⁠-dimensionalspherical geometry.

Considered extrinsically, as ahypersurface embedded in⁠${\displaystyle (n+1)}$⁠-dimensionalEuclidean space, an⁠${\displaystyle n}$⁠-sphere is thelocus ofpoints at equaldistance (theradius) from a givencenter point. Itsinterior, consisting of all points closer to the center than the radius, is an⁠${\displaystyle (n+1)}$⁠-dimensionalball. In particular:

• The⁠${\displaystyle 0}$⁠-sphere is the pair of points at the ends of aline segment (⁠${\displaystyle 1}$⁠-ball).
• The⁠${\displaystyle 1}$⁠-sphere is acircle, thecircumference of adisk (⁠${\displaystyle 2}$⁠-ball) in the two-dimensionalplane.
• The⁠${\displaystyle 2}$⁠-sphere, often simply called a sphere, is theboundary of a⁠${\displaystyle 3}$⁠-ball inthree-dimensional space.
• The3-sphere is the boundary of a⁠${\displaystyle 4}$⁠-ball infour-dimensional space.
• The⁠${\displaystyle (n-1)}$⁠-sphere is the boundary of an⁠${\displaystyle n}$⁠-ball.

Given aCartesian coordinate system, theunit⁠${\displaystyle n}$⁠-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 aRiemannian manifold of positiveconstant curvature, and isorientable. Thegeodesics of the⁠${\displaystyle n}$⁠-sphere are calledgreat circles.

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

In the more general setting oftopology, anytopological space that ishomeomorphic to the unit⁠${\displaystyle n}$⁠-sphere is called an⁠${\displaystyle n}$⁠-sphere. Under inverse stereographic projection, the⁠${\displaystyle n}$⁠-sphere is theone-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 thesuspension of an⁠${\displaystyle (n-1)}$⁠-sphere. When⁠${\displaystyle n\ge 2}$⁠ it issimply connected; the⁠${\displaystyle 1}$⁠-sphere (circle) is not simply connected; the⁠${\displaystyle 0}$⁠-sphere is not even connected, consisting of two discrete points.

For anynatural number⁠${\displaystyle n}$⁠, an⁠${\displaystyle n}$⁠-sphere of radius⁠${\displaystyle r}$⁠ is defined as the set of points in⁠${\displaystyle (n+1)}$⁠-dimensionalEuclidean space that are at distance⁠${\displaystyle r}$⁠ from some fixed point⁠${\displaystyle \mathbf{c}}$⁠, where⁠${\displaystyle r}$⁠ may be anypositivereal 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).
• a1-sphere is acircle of radius⁠${\displaystyle r}$⁠ centered at⁠${\displaystyle \mathbf{c}}$⁠, and is the boundary of a disk (⁠${\displaystyle 2}$⁠-ball).
• a2-sphere is an ordinary⁠${\displaystyle 2}$⁠-dimensionalsphere in⁠${\displaystyle 3}$⁠-dimensional Euclidean space, and is the boundary of an ordinary ball (⁠${\displaystyle 3}$⁠-ball).
• a3-sphere is a⁠${\displaystyle 3}$⁠-dimensional sphere in⁠${\displaystyle 4}$⁠-dimensional Euclidean space.

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. Thevolume 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 theHodge star operator; seeFlanders (1989, §6.1) for a discussion and proof of this formula in the case⁠${\displaystyle r=1}$⁠. As a result,

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

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

Specifically:

• A⁠${\displaystyle 1}$⁠-ball, aline segment, is the interior of a 0-sphere.
• A⁠${\displaystyle 2}$⁠-ball, adisk, is the interior of acircle (⁠${\displaystyle 1}$⁠-sphere).
• A⁠${\displaystyle 3}$⁠-ball, an ordinaryball, is the interior of asphere (⁠${\displaystyle 2}$⁠-sphere).
• A⁠${\displaystyle 4}$⁠-ball is the interior of a3-sphere, etc.

Topologically, an⁠${\displaystyle n}$⁠-sphere can be constructed as aone-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 becomeshomeomorphic to${\displaystyle {\mathbb{R}}^n}$. This forms the basis forstereographic projection.^[1]

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.

The⁠${\displaystyle 0}$⁠-ball is sometimes defined as a single point. The⁠${\displaystyle 0}$⁠-dimensionalHausdorff 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 theunit circle in the Euclidean plane, and its interior is theunit disk (⁠${\displaystyle 2}$⁠-ball).

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

The interior of a2-sphere inthree-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 thegamma 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.^[2]

Thesurface 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${\displaystyle (n-1)}$⁠-sphereshells,

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

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.

We may define a coordinate system in an⁠${\displaystyle n}$⁠-dimensional Euclidean space which is analogous to thespherical 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:^[3][a]

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

whereatan2 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.)

The arc length element is$${\displaystyle d{s}^2=d{r}^2+\sum _{k=1}^{n-1}{r}^2\left(\prod _{m=1}^{k-1}{\sin }^2\left({\phi }_m\right)\right)d{\phi }_k^2}$$To express thevolume 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 theJacobian matrix of the transformation is:

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 byLaplace 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

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

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 thearea 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 ofultraspherical polynomials,

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 thespherical harmonics.

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.^[4] 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

These splittings may be repeated as long as one of the factors involved has dimension two or greater. Apolyspherical 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 thespecial 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 thebeta 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

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

Just as a two-dimensional sphere embedded in three dimensions can be mapped onto a two-dimensional plane by astereographic projection, an⁠${\displaystyle n}$⁠-sphere can be mapped onto an⁠${\displaystyle n}$⁠-dimensionalhyperplane 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

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

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 ofnormal 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 unitn-cube by sampling each⁠${\displaystyle x_i}$⁠ independently from theuniform 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.

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).^[5]

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-calledcurse of dimensionality that arises in some numerical and other applications.

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.^[6]

0-sphere

The pair of points⁠${\displaystyle \{\pm R\}}$⁠ with thediscrete topology for some⁠${\displaystyle R>0}$⁠. The only sphere that is notpath-connected.Parallelizable.

1-sphere

Commonly called acircle. Has a nontrivial fundamental group. Abelian Lie group structureU(1); thecircle group.Homeomorphic to thereal projective line.Parallelizable

2-sphere

Commonly simply called asphere. For its complex structure, seeRiemann sphere. Homeomorphic to thecomplex projective line

3-sphere

Parallelizable,principalU(1)-bundleover the⁠${\displaystyle 2}$⁠-sphere,Lie group structureSp(1) =SU(2).

4-sphere

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

5-sphere

PrincipalU(1)-bundle over thecomplex projective space⁠${\displaystyle {\mathbf{C}\mathbf{P}}^2}$⁠.⁠${\displaystyle \mathrm{SO}(6)/\mathrm{SO}(5)=\mathrm{SU}(3)/\mathrm{SU}(2)}$⁠. It isundecidable whether a given⁠${\displaystyle n}$⁠-dimensional manifold is homeomorphic to⁠${\displaystyle S^n}$⁠ for⁠${\displaystyle n\ge 5}$⁠.^[7]

6-sphere

Possesses analmost complex structure coming from the set of pure unitoctonions.⁠${\displaystyle \mathrm{SO}(7)/\mathrm{SO}(6)=G_2/\mathrm{SU}(3)}$⁠. The question of whether it has acomplex structure is known as theHopf problem, afterHeinz Hopf.^[8]

7-sphere

Topologicalquasigroup structure as the set of unitoctonions. Principal⁠${\displaystyle \mathrm{Sp}(1)}$⁠-bundle over⁠${\displaystyle S^4}$⁠.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 firstexotic spheres were discovered.

8-sphere

Homeomorphic to the octonionic projective line⁠${\displaystyle {\mathbf{O}\mathbf{P}}^1}$⁠.

23-sphere

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

Theoctahedral⁠${\displaystyle n}$⁠-sphere is defined similarly to the⁠${\displaystyle n}$⁠-sphere but using the1-norm

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

In general, it takes the shape of across-polytope.

The octahedral⁠${\displaystyle 1}$⁠-sphere is a square (without its interior). The octahedral⁠${\displaystyle 2}$⁠-sphere is a regularoctahedron; hence the name. The octahedral⁠${\displaystyle n}$⁠-sphere is thetopological join of⁠${\displaystyle n+1}$⁠ pairs of isolated points.^[9] 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.

• Conformal geometry – Study of angle-preserving transformations of a geometric space
• Exotic sphere – Smooth manifold that is homeomorphic but not diffeomorphic to a sphere
• Homology sphere – Topological manifold whose homology coincides with that of a sphere
• Homotopy groups of spheres – How spheres of various dimensions can wrap around each other
• Inversive geometry – Study of angle-preserving transformations
• Möbius transformation – Rational function of the form (az + b)/(cz + d)