, an -sphere is a topological space
that is homeomorphic
to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space
that are situated at a constant distance from a fixed point, called the ''center''. It is the generalization of an ordinary sphere
in the ordinary three-dimensional space
. The "radius" of a sphere is the constant distance of its points to the center. When the sphere has unit radius, it is usual to call it the unit -sphere or simply the -sphere for brevity. In terms of the standard norm, the -sphere is defined as
and an -sphere of radius can be defined as
The dimension of -sphere is , and must not be confused with the dimension of the Euclidean space in which it is naturally embedded
. An -sphere is the surface or boundary of an -dimensional ball
*the pair of points at the ends of a (one-dimensional) line segment
is a 0-sphere,
, which is the one-dimensional circumference
of a (two-dimensional) disk
, is a 1-sphere,
*the two-dimensional surface of a three-dimensional ball is a 2-sphere, often simply called a sphere,
*the three-dimensional boundary
of a (four-dimensional) 4-ball is a 3-sphere
*the dimensional boundary of a (-dimensional) -ball is an -sphere.
For , the -spheres that are differential manifold
s can be characterized (up to
) as the simply connected
s of constant, positive curvature
. The -spheres admit several other topological descriptions: for example, they can be constructed by gluing two -dimensional Euclidean spaces together, by identifying the boundary of an -cube
with a point, or (inductively) by forming the suspension
of an -sphere. The 1-sphere is the 1-manifold that is a circle, which is not simply connected. The 0-sphere is the 0-manifold consisting of two points, which is not even connected.
For any natural number
, an -sphere of radius is defined as the set of points in -dimensional Euclidean space
that are at distance from some fixed point , where may be any positive real number
and where may be any point in -dimensional space. In particular:
* a 0-sphere is a pair of points , and is the boundary of a line segment (1-ball).
* a 1-sphere
is a circle
of radius centered at , and is the boundary of a disk (2-ball).
* a 2-sphere
is an ordinary 2-dimensional sphere
in 3-dimensional Euclidean space, and is the boundary of an ordinary ball (3-ball).
* a 3-sphere
is a 3-dimensional sphere in 4-dimensional Euclidean space.
Euclidean coordinates in -space
The set of points in -space, , that define an -sphere,
, is represented by the equation:
where is a center point, and is the radius.
The above -sphere exists in -dimensional Euclidean space and is an example of an -manifold
. The volume form
of an -sphere of radius is given by
where is the Hodge star operator
; see for a discussion and proof of this formula in the case . As a result,
The space enclosed by an -sphere is called an -ball
. An -ball is closed
if it includes the -sphere, and it is open
if it does not include the -sphere.
* A 1-''ball'', a line segment
, is the interior of a 0-sphere.
* A 2-''ball'', a disk
, is the interior of a circle
* A 3-''ball'', an ordinary ball
, is the interior of a sphere
* A 4-''ball'' is the interior of a 3-sphere
, an -sphere can be constructed as a one-point compactification
of -dimensional Euclidean space. Briefly, the -sphere can be described as , which is -dimensional Euclidean space plus a single point representing infinity in all directions.
In particular, if a single point is removed from an -sphere, it becomes homeomorphic
to . This forms the basis for stereographic projection
Volume and surface area
and are the -dimensional volume of the -ball
and the surface area of the -sphere embedded in dimension , respectively, of radius .
The constants and (for , the unit ball and sphere) are related by the recurrences:
The surfaces and volumes can also be given in closed form:
where is the gamma function
. Derivations of these equations are given in this section.
In general, the volume of the -ball in -dimensional Euclidean space, and the surface area of the -sphere in -dimensional Euclidean space, of radius , are proportional to the th power of the radius, (with different constants of proportionality that vary with ). We write for the volume of the -ball and for the surface area of the -sphere, both of radius , where and are the values for the unit-radius case.
In theory, one could compare the values of and for . However, this is not well-defined. For example, if and then the comparison is like comparing a number of square meters to a different number of cubic meters. The same applies to a comparison of and for .
The 0-ball consists of a single point. The 0-dimensional Hausdorff measure
is the number of points in a set. So,
The 0-sphere consists of its two end-points, . So,
The unit 1-ball is the interval of length 2. So,
The unit 1-sphere is the unit circle in the Euclidean plane, and this has circumference (1-dimensional measure)
The region enclosed by the unit 1-sphere is the 2-ball, or unit disc, and this has area (2-dimensional measure)
Analogously, in 3-dimensional Euclidean space, the surface area (2-dimensional measure) of the unit 2-sphere is given by
and the volume enclosed is the volume (3-dimensional measure) of the unit 3-ball, given by
The ''surface area'', or properly the -dimensional volume, of the -sphere at the boundary of the -ball of radius is related to the volume of the ball by the differential equation
or, equivalently, representing the unit -ball as a union of concentric -sphere ''shells
We can also represent the unit -sphere as a union of products of a circle (1-sphere) with an -sphere. Let and , so that and . Then,
Since , the equation
holds for all .
This completes the derivation of the recurrences:
Combining the recurrences, we see that
So it is simple to show by induction on ''k'' that,
where denotes the double factorial
, defined for odd natural numbers by and similarly for even numbers .
In general, the volume, in -dimensional Euclidean space, of the unit -ball, is given by
where is the gamma function
, which satisfies , , and , and so , and where we conversely define x! = for any x.
By multiplying by , differentiating with respect to , and then setting , we get the closed form
for the (n-1)-dimensional volume of the sphere ''S''n-1
The recurrences can be combined to give a "reverse-direction" recurrence relation for surface area, as depicted in the diagram:
Index-shifting to then yields the recurrence relations:
where , , and .
The recurrence relation for can also be proved via integration
with 2-dimensional polar coordinates
We may define a coordinate system in an -dimensional Euclidean space which is analogous to the spherical coordinate system
defined for 3-dimensional Euclidean space, in which the coordinates consist of a radial coordinate , and angular coordinates , where the angles range over radians (or over degrees) and ranges over radians (or over degrees). If are the Cartesian coordinates, then we may compute from with:
Except in the special cases described below, the inverse transformation is unique:
where if for some but all of are zero then when , and (180 degrees) when .
There are some special cases where the inverse transform is not unique; for any will be ambiguous whenever all of are zero; in this case may be chosen to be zero.
Spherical volume and area elements
To express the volume element
of -dimensional Euclidean space in terms of spherical coordinates, first observe that the Jacobian matrix
of the transformation is:
The determinant of this matrix can be calculated by induction. When , a straightforward computation shows that the determinant is . For larger , observe that can be constructed from as follows. Except in column , rows and of are the same as row of , but multiplied by an extra factor of in row and an extra factor of in row . In column , rows and of are the same as column of row of , but multiplied by extra factors of in row and in row , respectively. The determinant of can be calculated by Laplace expansion
in the final column. By the recursive description of , the submatrix formed by deleting the entry at and its row and column almost equals , except that its last row is multiplied by . Similarly, the submatrix formed by deleting the entry at and its row and column almost equals , except that its last row is multiplied by . Therefore the determinant of is
Induction then gives a closed-form expression for the volume element in spherical coordinates
The formula for the volume of the -ball can be derived from this by integration.
Similarly the surface area element of the -sphere of radius , which generalizes the area element
of the 2-sphere, is given by
The natural choice of an orthogonal basis over the angular coordinates is a product of ultraspherical polynomials
for , and the for the angle in concordance with the spherical harmonics
The standard spherical coordinate system arises from writing as the product . These two factors may be related using polar coordinates. For each point of , the standard Cartesian coordinates
can be transformed into a mixed polar–Cartesian coordinate system:
This says that points in may be expressed by taking the ray starting at the origin and passing through , rotating it towards the first basis vector by , and traveling a distance 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 is split as the product of two Euclidean spaces of smaller dimension, but neither space is required to be a line. Specifically, suppose that and are positive integers such that . Then . Using this decomposition, a point may be written as
This can be transformed into a mixed polar–Cartesian coordinate system by writing:
are the unit vectors associated to and . This expresses in terms of
, , and an angle . It can be shown that the domain of is if , if exactly one of and is 1, and if neither nor are 1. The inverse transformation is
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
are spheres, so the coordinates of a polyspherical coordinate system are a non-negative radius and angles. The possible polyspherical coordinate systems correspond to binary trees with 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 , and its immediate children represent the first splitting into and . Leaf nodes correspond to Cartesian coordinates for . 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 , taking the left branch introduces a factor of and taking the right branch introduces a factor of . 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 determines a subgroup
This is the subgroup that leaves each of the two factors
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 and the area measure on are products. There is one factor for each angle, and the volume measure on also has a factor for the radial coordinate. The area measure has the form:
where the factors are determined by the tree. Similarly, the volume measure is
Suppose we have a node of the tree that corresponds to the decomposition and that has angular coordinate . The corresponding factor depends on the values of and . When the area measure is normalized so that the area of the sphere is 1, these factors are as follows. If , then
If and , and if denotes the beta function
If and , then
Finally, if both and are greater than one, then
Just as a two-dimensional sphere embedded in three dimensions can be mapped onto a two-dimensional plane by a stereographic projection
, an -sphere can be mapped onto an -dimensional hyperplane by the -dimensional version of the stereographic projection. For example, the point on a two-dimensional sphere of radius 1 maps to the point on the -plane. In other words,
Likewise, the stereographic projection of an -sphere of radius 1 will map to the -dimensional hyperplane perpendicular to the -axis as
Generating random points
Uniformly at random on the -sphere
To generate uniformly distributed random points on the unit -sphere (that is, the surface of the unit -ball), gives the following algorithm.
Generate an -dimensional vector of normal deviates
(it suffices to use , although in fact the choice of the variance is arbitrary), . Now calculate the "radius" of this point:
The vector is uniformly distributed over the surface of the unit -ball.
An alternative given by Marsaglia is to uniformly randomly select a point in the unit -cube
by sampling each independently from the uniform distribution
over , computing as above, and rejecting the point and resampling if (i.e., if the point is not in the -ball), and when a point in the ball is obtained scaling it up to the spherical surface by the factor ; then again is uniformly distributed over the surface of the unit -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
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 -ball
With a point selected uniformly at random from the surface of the unit -sphere (e.g., by using Marsaglia's algorithm), one needs only a radius to obtain a point uniformly at random from within the unit -ball. If is a number generated uniformly at random from the interval and is a point selected uniformly at random from the unit -sphere, then is uniformly distributed within the unit -ball.
Alternatively, points may be sampled uniformly from within the unit -ball by a reduction from the unit -sphere. In particular, if is a point selected uniformly from the unit -sphere, then is uniformly distributed within the unit -ball (i.e., by simply discarding two coordinates).
If is sufficiently large, most of the volume of the -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.
; 0-sphere : The pair of points with the discrete topology for some . The only sphere that is not path-connected
. Has a natural Lie group structure; isomorphic to O(1). Parallelizable.
: Also known as the circle. Has a nontrivial fundamental group. Abelian Lie group structure U(1)
; the circle group
. Topologically equivalent to the real projective line
. Parallelizable. SO(2) = U(1).
: Also known as the sphere. Complex structure; see Riemann sphere
. Equivalent to the complex projective line
: Also known as the glome
. Parallelizable, principal U(1)-bundle over
the 2-sphere, Lie group structure Sp(1)
, where also
; 4-sphere : Equivalent to the quaternionic projective line
; 5-sphere : Principal U(1)-bundle
. SO(6)/SO(5) = SU(3)/SU(2).
; 6-sphere : Possesses an almost complex structure
coming from the set of pure unit octonion
s. SO(7)/SO(6) = ''G''2
/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 octonion
s. Principal Sp(1)-bundle over ''S''4
. Parallelizable. SO(8)/SO(7) = SU(4)/SU(3) = Sp(2)/Sp(1) = Spin(7)/''G''2
= Spin(6)/SU(3). The 7-sphere is of particular interest since it was in this dimension that the first exotic sphere
s were discovered.
; 8-sphere : Equivalent to the octonionic projective line OP1
; 23-sphere : A highly dense sphere-packing
is possible in 24-dimensional space, which is related to the unique qualities of the Leech lattice
The octahedral ''n''-sphere is defined similarly to the ''n''-sphere but using the 1-norm
The octahedral 1-sphere is a square (without its interior). The octahedral 2-sphere is a regular octahedron
; hence the name. The octahedral ''n''-sphere is the topological join
of ''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.
*Homotopy groups of spheres
*Volume of an -ball
*Wigner semicircle distribution