In mathematics, an nsphere or hypersphere is an dimensional generalization of the dimensional circle and dimensional sphere to any nonnegative integer . The sphere is the setting for dimensional spherical geometry.
Considered extrinsically, as a hypersurface embedded in dimensional Euclidean space, an 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 dimensional ball. In particular:
 The sphere is the pair of points at the ends of a line segment (ball).
 The sphere is a circle, the circumference of a disk (ball) in the twodimensional plane.
 The sphere, often simply called a sphere, is the boundary of a ball in threedimensional space.
 The 3sphere is the boundary of a ball in fourdimensional space.
 The sphere is the boundary of an ball.
Given a Cartesian coordinate system, the unit sphere of radius can be defined as:
Considered intrinsically, when , the sphere is a Riemannian manifold of positive constant curvature, and is orientable. The geodesics of the sphere are called great circles.
The stereographic projection maps the sphere onto space with a single adjoined point at infinity; under the metric thereby defined, is a model for the sphere.
In the more general setting of topology, any topological space that is homeomorphic to the unit sphere is called an sphere. Under inverse stereographic projection, the sphere is the onepoint compactification of space. The spheres admit several other topological descriptions: for example, they can be constructed by gluing two dimensional spaces together, by identifying the boundary of an cube with a point, or (inductively) by forming the suspension of an sphere. When it is simply connected; the sphere (circle) is not simply connected; the sphere is not even connected, consisting of two discrete points.
YouTube Encyclopedic

1/5Views:45 335353 035233 76419 734163 302

How to derive the volume of an ndimensional hypersphere (the long version)

Volume of a Hypersphere

How to see a sphere in 4D

How to derive the volume of an ndimensional hypersphere in 3 minutes.

So why do the volumes of all even dimensional unit spheres sum to e^π?
Transcription
Description
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 0sphere is a pair of points , and is the boundary of a line segment (ball).
 a 1sphere is a circle of radius centered at , and is the boundary of a disk (ball).
 a 2sphere is an ordinary dimensional sphere in dimensional Euclidean space, and is the boundary of an ordinary ball (ball).
 a 3sphere is a dimensional sphere in dimensional Euclidean space.
Cartesian coordinates
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 Flanders (1989, §6.1) for a discussion and proof of this formula in the case . As a result,
nball
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.
Specifically:
 A ball, a line segment, is the interior of a 0sphere.
 A ball, a disk, is the interior of a circle (sphere).
 A ball, an ordinary ball, is the interior of a sphere (sphere).
 A ball is the interior of a 3sphere, etc.
Topological description
Topologically, an sphere can be constructed as a onepoint 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.^{[1]}
Volume and area
Let be the surface area of the unit sphere of radius embedded in dimensional Euclidean space, and let be the volume of its interior, the unit ball. The surface area of an arbitrary sphere is proportional to the st power of the radius, and the volume of an arbitrary ball is proportional to the th power of the radius.
The ball is sometimes defined as a single point. The dimensional Hausdorff measure is the number of points in a set. So
A unit ball is a line segment whose points have a single coordinate in the interval of length , and the sphere consists of its two endpoints, with coordinate .
A unit sphere is the unit circle in the Euclidean plane, and its interior is the unit disk (ball).
The interior of a 2sphere in threedimensional space is the unit ball.
In general, and are given in closed form by the expressions
where is the gamma function.
As tends to infinity, the volume of the unit ball (ratio between the volume of an ball of radius and an cube of side length ) tends to zero.^{[2]}
Recurrences
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
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 (sphere) with an sphere. Then . Since , the equation
holds for all . Along with the base cases , 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 dimensional Euclidean space which is analogous to the spherical coordinate system defined for dimensional Euclidean space, in which the coordinates consist of a radial coordinate , and angular coordinates , where the angles range over radians (or degrees) and ranges over radians (or degrees). If are the Cartesian coordinates, then we may compute from with:^{[3]}^{[a]}
Except in the special cases described below, the inverse transformation is unique:
where atan2 is the twoargument arctangent function.
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. (For example, for the sphere, when the polar angle is or 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 dimensional Euclidean space in terms of spherical coordinates, let and for concision, then 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 closedform 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 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.
Polyspherical coordinates
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 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.^{[4]} 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:
Here and 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 , and if neither nor are . 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 and are spheres, so the coordinates of a polyspherical coordinate system are a nonnegative radius and angles. The possible polyspherical coordinate systems correspond to binary trees with leaves. Each nonleaf 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 , these factors are as follows. If , then
If and , and if denotes the beta function, then
If and , then
Finally, if both and are greater than one, then
Stereographic projection
Just as a twodimensional sphere embedded in three dimensions can be mapped onto a twodimensional 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 twodimensional sphere of radius maps to the point on the plane. In other words,
Likewise, the stereographic projection of an sphere of radius will map to the dimensional hyperplane perpendicular to the axis as
Probability distributions
Uniformly at random on the (n − 1)sphere
To generate uniformly distributed random points on the unit sphere (that is, the surface of the unit ball), Marsaglia (1972) 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 ncube 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 nball
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).^{[5]}
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 socalled curse of dimensionality that arises in some numerical and other applications.
Distribution of the first coordinate
Let be the square of the first coordinate of a point sampled uniformly at random from the sphere, then its probability density function, for , is
Let be the appropriately scaled version, then at the limit, the probability density function of converges to . This is sometimes called the Porter–Thomas distribution.^{[6]}
Specific spheres
 0sphere
 The pair of points with the discrete topology for some . The only sphere that is not pathconnected. Parallelizable.
 1sphere
 Commonly called a circle. Has a nontrivial fundamental group. Abelian Lie group structure U(1); the circle group. Homeomorphic to the real projective line.
 2sphere
 Commonly simply called a sphere. For its complex structure, see Riemann sphere. Homeomorphic to the complex projective line
 3sphere
 Parallelizable, principal U(1)bundle over the sphere, Lie group structure Sp(1).
 4sphere
 Homeomorphic to the quaternionic projective line, . .
 5sphere
 Principal U(1)bundle over the complex projective space . . It is undecidable whether a given dimensional manifold is homeomorphic to for .^{[7]}
 6sphere
 Possesses an almost complex structure coming from the set of pure unit octonions. . The question of whether it has a complex structure is known as the Hopf problem, after Heinz Hopf.^{[8]}
 7sphere
 Topological quasigroup structure as the set of unit octonions. Principal bundle over S^4. Parallelizable. . The sphere is of particular interest since it was in this dimension that the first exotic spheres were discovered.
 8sphere
 Homeomorphic to the octonionic projective line .
 23sphere
 A highly dense spherepacking is possible in dimensional space, which is related to the unique qualities of the Leech lattice.
Octahedral sphere
The octahedral sphere is defined similarly to the sphere but using the 1norm
In general, it takes the shape of a crosspolytope.
The octahedral sphere is a square (without its interior). The octahedral sphere is a regular octahedron; hence the name. The octahedral sphere is the topological join of 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.
See also
 Conformal geometry – Study of anglepreserving 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 anglepreserving transformations
 Möbius transformation – Rational function of the form (az + b)/(cz + d)
Notes
 ^ Formally, this formula is only correct for . For , the line beginning with must be omitted, and for , the formula for polar coordinates must be used. The case reduces to . Using capitalpi notation and the usual convention for the empty product, a formula valid for is given by and for .
 ^ James W. Vick (1994). Homology theory, p. 60. Springer
 ^ Smith, David J.; Vamanamurthy, Mavina K. (1989). "How Small Is a Unit Ball?". Mathematics Magazine. 62 (2): 101–107. doi:10.1080/0025570X.1989.11977419. JSTOR 2690391.
 ^ Blumenson, L. E. (1960). "A Derivation of nDimensional Spherical Coordinates". The American Mathematical Monthly. 67 (1): 63–66. doi:10.2307/2308932. JSTOR 2308932.
 ^ N. Ja. Vilenkin and A. U. Klimyk, Representation of Lie groups and special functions, Vol. 2: Class I representations, special functions, and integral transforms, translated from the Russian by V. A. Groza and A. A. Groza, Math. Appl., vol. 74, Kluwer Acad. Publ., Dordrecht, 1992, ISBN 0792314921, pp. 223–226.
 ^ Voelker, Aaron R.; Gosmann, Jan; Stewart, Terrence C. (2017). Efficiently sampling vectors and coordinates from the nsphere and nball (Report). Centre for Theoretical Neuroscience. doi:10.13140/RG.2.2.15829.01767/1.
 ^ Livan, Giacomo; Novaes, Marcel; Vivo, Pierpaolo (2018), Livan, Giacomo; Novaes, Marcel; Vivo, Pierpaolo (eds.), "One Pager on Eigenvectors", Introduction to Random Matrices: Theory and Practice, SpringerBriefs in Mathematical Physics, Cham: Springer International Publishing, pp. 65–66, doi:10.1007/9783319708850_9, ISBN 9783319708850, retrieved 20230519
 ^ Stillwell, John (1993), Classical Topology and Combinatorial Group Theory, Graduate Texts in Mathematics, vol. 72, Springer, p. 247, ISBN 9780387979700.
 ^ Agricola, Ilka; Bazzoni, Giovanni; Goertsches, Oliver; Konstantis, Panagiotis; Rollenske, Sönke (2018). "On the history of the Hopf problem". Differential Geometry and Its Applications. 57: 1–9. arXiv:1708.01068. doi:10.1016/j.difgeo.2017.10.014. S2CID 119297359.
 ^ Meshulam, Roy (20010101). "The Clique Complex and Hypergraph Matching". Combinatorica. 21 (1): 89–94. doi:10.1007/s004930170006. ISSN 14396912. S2CID 207006642.
References
 Marsaglia, G. (1972). "Choosing a Point from the Surface of a Sphere". Annals of Mathematical Statistics. 43 (2): 645–646. doi:10.1214/aoms/1177692644.
 Huber, Greg (1982). "Gamma function derivation of nsphere volumes". Amer. Math. Monthly. 89 (5): 301–302. doi:10.2307/2321716. JSTOR 2321716. MR 1539933.
 Weeks, Jeffrey R. (1985). The Shape of Space: how to visualize surfaces and threedimensional manifolds. Marcel Dekker. ISBN 9780824774370(Chapter 14: The Hypersphere).
{{cite book}}
: CS1 maint: postscript (link)  Kalnins, E. G.; Miller, W. (1986). "Separation of variables on ndimensionsional Riemannian manifolds. I. the nsphere S_n and Euclidean nsparce R_n". J. Math. Phys. 27: 1721–1746. doi:10.1063/1.527088. hdl:10289/1219.
 Flanders, Harley (1989). Differential forms with applications to the physical sciences. New York: Dover Publications. ISBN 9780486661698.
 Moura, Eduarda; Henderson, David G. (1996). Experiencing geometry: on plane and sphere. Prentice Hall. ISBN 9780133737707(Chapter 20: 3spheres and hyperbolic 3spaces).
{{cite book}}
: CS1 maint: postscript (link)  Barnea, Nir (1999). "Hyperspherical functions with arbitrary permutational symmetry: Reverse construction". Phys. Rev. A. 59 (2): 1135–1146. Bibcode:1999PhRvA..59.1135B. doi:10.1103/PhysRevA.59.1135.