# N-sphere

An **n-sphere** is the set of points in an (n + 1)-dimensional [Euclidean space](https://www.edgepedia.org/euclidean-space) that lie at a fixed distance, the radius, from a central point; in symbols, \( S^n = \{x \in E^{n+1} : \rho(x, x_0) = R\} \).<sup>[1](https://encyclopediaofmath.org/wiki/Sphere)</sup> The 0-sphere is a pair of points, the 1-sphere is the circle, the 2-sphere is the familiar surface of a ball, and for n greater than 2 the object is sometimes called a hypersphere.<sup>[1](https://encyclopediaofmath.org/wiki/Sphere)</sup> The sphere is the surface only; the solid interior is properly called a ball.<sup>[2](https://mathworld.wolfram.com/Sphere.html)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( S^n = \{x \in E^{n+1} : \rho(x, x_0) = R\} \), the set of points at distance R from a center in (n + 1)-dimensional Euclidean space<sup>[1](https://encyclopediaofmath.org/wiki/Sphere)</sup> |
| Dimension conventions | Geometers count coordinates in the ambient space (a two-dimensional sphere is a circle); topologists count the dimension of the surface itself, so the usual sphere is S²<sup>[2](https://mathworld.wolfram.com/Sphere.html)</sup> |
| "Volume" (content) of the surface | \( v(S^n) = \frac{2\pi^{(n+1)/2}}{\Gamma((n+1)/2)} R^n \); in particular \( v(S^1) = 2\pi R \), \( v(S^2) = 4\pi R^2 \), \( v(S^3) = 2\pi^2 R^3 \), \( v(S^4) = \frac{8}{3}\pi^2 R^4 \)<sup>[1](https://encyclopediaofmath.org/wiki/Sphere)</sup> |
| Curvature | A space form of constant positive curvature \( 1/r^2 \) for radius r<sup>[3](https://encyclopediaofmath.org/wiki/Space_forms)</sup> |
| Geodesics | All geodesics are closed great circles of length \( 2\pi R \)<sup>[1](https://encyclopediaofmath.org/wiki/Sphere)</sup> |
| Symmetry | The isometry group is O(n + 1), acting 2-transitively; the sphere is the homogeneous space \( S^n = O(n+1)/O(n) \)<sup>[1](https://encyclopediaofmath.org/wiki/Sphere)</sup> |
| Manifold structure | Two stereographic projection charts cover S^n; S^n is the one-point compactification of \( \mathbb{R}^n \)<sup>[4](https://ncatlab.org/nlab/show/stereographic%2Bprojection)</sup><sup> • </sup><sup>[5](https://www.math.ucla.edu/~sharifi/notes/topology-ch03.html)</sup> |

## Definition and conventions

The Cartesian equation of a sphere of radius R centered at \( x_0 \) in \( E^{n+1} \) is \( \sum_{i=1}^{n+1} (x^i - x_0^i)^2 = R^2 \), a quadric of special form.<sup>[1](https://encyclopediaofmath.org/wiki/Sphere)</sup> For the unit sphere in \( \mathbb{R}^{n+1} \) this reads \( x_1^2 + \cdots + x_{n+1}^2 = 1 \).<sup>[2](https://mathworld.wolfram.com/Sphere.html)</sup>

The term "n-sphere" carries a genuine ambiguity between fields. Geometers refer to the number of coordinates in the underlying space, so a two-dimensional sphere is a circle; topologists refer to the dimension of the surface itself, so the n-dimensional sphere \( S^n \) is the set of points \( (x_1, \dots, x_{n+1}) \) in \( E^{n+1} \) satisfying \( x_1^2 + \cdots + x_{n+1}^2 = 1 \).<sup>[2](https://mathworld.wolfram.com/Sphere.html)</sup> Under the topological convention, geometers call the surface of the usual ball a 3-sphere while topologists call it the 2-sphere and write \( S^2 \).<sup>[2](https://mathworld.wolfram.com/Sphere.html)</sup> In higher-dimensional work the common practice is simply to say "sphere" or "d-sphere", with a circle a 1-sphere and the standard three-dimensional sphere a 2-sphere.<sup>[6](https://people.eecs.berkeley.edu/~jrs/highd/)</sup>

## Geometry

As a Riemannian space the sphere has constant positive curvature; the sphere \( S^n \) of radius r in \( E^{n+1} \) is a space form of curvature \( 1/r^2 \), the positive-curvature member of the family completed by flat Euclidean space and hyperbolic space of negative curvature.<sup>[3](https://encyclopediaofmath.org/wiki/Space_forms)</sup> Its geodesics are the great circles, the intersections of \( S^n \) with two-dimensional planes through the center, and every one of them is closed with length \( 2\pi R \).<sup>[1](https://encyclopediaofmath.org/wiki/Sphere)</sup> All normals meet at one point, every point is an umbilical point, and the sphere has constant mean curvature.<sup>[1](https://encyclopediaofmath.org/wiki/Sphere)</sup>

The measure of the surface itself, which the Encyclopedia of Mathematics calls the volume of \( S^n \), is \( v(S^n) = \frac{2\pi^{(n+1)/2}}{\Gamma((n+1)/2)} R^n \), where \( \Gamma \) is the gamma function.<sup>[1](https://encyclopediaofmath.org/wiki/Sphere)</sup> The first values are \( 2\pi R \) for the circle, \( 4\pi R^2 \) for the ordinary sphere, \( 2\pi^2 R^3 \) for the 3-sphere, and \( \frac{8}{3}\pi^2 R^4 \) for the 4-sphere.<sup>[1](https://encyclopediaofmath.org/wiki/Sphere)</sup> The same sequence appears as the surface-area formula \( S_n = \frac{2\pi^{n/2} R^{n-1}}{\Gamma(n/2)} \) for a hypersphere in n dimensions, with coefficients 2, 2, 4, 2, 8/3, 1, 16/15 for successive dimensions.<sup>[7](https://mathworld.wolfram.com/Four-DimensionalGeometry.html)</sup>

## Stereographic projection and manifold structure

[Stereographic projection](https://www.edgepedia.org/stereographic-projection) is a homeomorphism, for any \( n \in \mathbb{N} \), from the n-sphere with one point \( p \) removed to Euclidean space \( \mathbb{R}^n \).<sup>[4](https://ncatlab.org/nlab/show/stereographic%2Bprojection)</sup> Geometrically, a line is drawn through the projection point (for example the north pole \( N = (0, \dots, 0, 1) \)) and a point \( P \) of the unit sphere, and the map records where that line meets a fixed hyperplane.<sup>[8](https://people.math.wisc.edu/~angenent/519.2014s/coursenotes/stereographic-projection.html)</sup> The projection is one-to-one onto the plane except that the image of the projection center is undefined, or may be set to infinity.<sup>[9](https://sites.math.washington.edu/~king/coursedir/m445w03/class/01-29-Stereographic1.html)</sup>

Two such charts, projecting from the north and south poles, cover the whole sphere: \( S^n = U_N \cup U_S \) with \( U_N = S^n \setminus \{N\} \) and \( U_S = S^n \setminus \{S\} \), each mapped into \( \mathbb{R}^n \).<sup>[10](https://www.math.utoronto.ca/mgualt/courses/MAT1300F-2016/docs/1300-2016-notes.pdf)</sup> The projection is conformal: the round metric on the sphere pulls back to a scalar multiple of the Euclidean metric on the hyperplane.<sup>[11](https://www.math.ucla.edu/~mattkowalski/documents/khan.pdf)</sup> The same picture identifies the sphere with the one-point compactification of \( \mathbb{R}^n \): viewing \( S^n \) as the unit sphere centered at \( \infty = (0, \dots, 0, 1) \in \mathbb{R}^{n+1} \), the one-point compactification of \( \mathbb{R}^n \) is homeomorphic to \( S^n \).<sup>[5](https://www.math.ucla.edu/~sharifi/notes/topology-ch03.html)</sup> In complex analysis the construction gives the [Riemann sphere](https://www.edgepedia.org/riemann-sphere), where stereographic projection is an isometry between the sphere with its Euclidean metric and the extended plane with the chordal metric.<sup>[12](https://www.math.ucla.edu/~mbonk/complana.pdf)</sup>

## Symmetry

The orthogonal group \( O(n+1) \) of \( E^{n+1} \) acts 2-transitively on \( S^n \): for any two pairs of points with equal distances between them, a rotation maps one pair onto the other. This group is the complete group of isometries of the sphere, and the sphere is the homogeneous space \( S^n = O(n+1)/O(n) \).<sup>[1](https://encyclopediaofmath.org/wiki/Sphere)</sup>

## Role in topology

Spheres are central objects in topology partly because they serve as the domains that define homotopy groups: the nth homotopy group \( \pi_n(X) \) of a space X is the set of homotopy classes of maps from the n-sphere to X, with a group structure.<sup>[13](https://mathworld.wolfram.com/HomotopyGroup.html)</sup> They also anchor the most famous problem about manifolds. In 1904 [Henri Poincaré](https://www.edgepedia.org/henri-poincare) asked whether the three-dimensional sphere is characterized as the unique simply connected three-manifold; the question turned out to be extraordinarily difficult, and nearly a century passed before Grigoriy Perelman announced its solution in preprints posted in 2002 and 2003, building on Richard Hamilton's theory of [Ricci flow](https://www.edgepedia.org/ricci-flow).<sup>[14](https://www.claymath.org/millennium/poincare-conjecture/)</sup> The generalized statement, that a compact n-manifold homotopy equivalent to the n-sphere is homeomorphic to it, was proved by [Stephen Smale](https://www.edgepedia.org/stephen-smale) for dimensions greater than 4 in 1961 and by Michael Freedman for dimension 4 in 1982.<sup>[15](https://web.stanford.edu/class/cspmath07/Lecture_4.pdf)</sup> Freedman's topological methods also gave a complete classification of closed simply connected topological 4-manifolds.<sup>[16](https://www.claymath.org/library/monographs/MPPc.pdf)</sup>

The sphere's smooth structure is a subtler matter. In 1956 [John Milnor](https://www.edgepedia.org/john-milnor) constructed the first exotic spheres, smooth manifolds homeomorphic but not diffeomorphic to \( S^7 \).<sup>[17](https://www.ams.org/bookstore/pspdf/gsm-111-prev.pdf)</sup> The set of equivalence classes of smooth structures on \( S^n \) forms a monoid under connected sum, a group for \( n \neq 4 \) isomorphic to the finite abelian group \( \Theta_n \) of h-cobordism classes of oriented homotopy n-spheres.<sup>[18](https://www.ams.org/bull/2019-56-02/S0273-0979-2018-01654-6/S0273-0979-2018-01654-6.pdf)</sup> [Dimension](https://www.edgepedia.org/dimension) 4 remains genuinely delicate: it is not known whether every smooth homotopy 4-sphere is diffeomorphic to \( S^4 \), the smooth four-dimensional [Poincaré conjecture](https://www.edgepedia.org/poincare-conjecture).<sup>[16](https://www.claymath.org/library/monographs/MPPc.pdf)</sup>

## Analysis and high dimensions

On the sphere, the natural differential operator is the [Laplace–Beltrami operator](https://www.edgepedia.org/laplace-beltrami-operator), the curved-space analog of the Laplacian. Its eigenfunctions are the spherical harmonics, which form a complete orthonormal basis of the square-integrable functions on the unit sphere; the eigenvalues are \( -l(l + N - 2) \) for \( l = 0, 1, \dots \), each with multiplicity \( h(l, N) = \frac{(2l + N - 2)(l + N - 3)!}{(N-2)!\, l!} \).<sup>[19](https://ir.cwi.nl/pub/10902/10902D.pdf)</sup> For the ordinary 2-sphere the eigenvalue takes the familiar form \( -l(l+1) \).<sup>[20](https://www.cis.upenn.edu/~cis6100/sharmonics.pdf)</sup> These functions underlie expansions of data on spherical maps, and spherical convolutional networks built on graph representations of the sphere approximate the spherical harmonic basis while avoiding its computational cost.<sup>[21](https://arxiv.org/abs/1810.12186)</sup>

High-dimensional spheres behave in ways that low-dimensional intuition does not suggest. The volume of the unit d-sphere goes to 0 as d grows: it increases from dimensions one to five, then decreases rapidly toward 0 after dimension six.<sup>[6](https://people.eecs.berkeley.edu/~jrs/highd/)</sup> At the same time, nearly all of the surface area of a high-dimensional sphere lies a very small distance from any chosen equator, and any choice of equator works equally well because the sphere is spherically symmetric.<sup>[6](https://people.eecs.berkeley.edu/~jrs/highd/)</sup> These concentration phenomena are one face of the curse of dimensionality that shapes data analysis in high dimensions.<sup>[22](https://arxiv.org/abs/cs/9901004)</sup>

## References

1. [Sphere, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Sphere)
2. [Sphere, Wolfram MathWorld](https://mathworld.wolfram.com/Sphere.html)
3. [Space forms, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Space_forms)
4. [Stereographic projection, nLab](https://ncatlab.org/nlab/show/stereographic%2Bprojection)
5. [Point-Set Topology, Chapter 3: Connected and compact spaces, Romyar Sharifi, UCLA](https://www.math.ucla.edu/~sharifi/notes/topology-ch03.html)
6. [High Dimensional Space, John R. Sack, UC Berkeley (course notes)](https://people.eecs.berkeley.edu/~jrs/highd/)
7. [Four-Dimensional Geometry, Wolfram MathWorld](https://mathworld.wolfram.com/Four-DimensionalGeometry.html)
8. [Math 519: Stereographic projection, University of Wisconsin](https://people.math.wisc.edu/~angenent/519.2014s/coursenotes/stereographic-projection.html)
9. [Stereographic Projection, University of Washington course notes](https://sites.math.washington.edu/~king/coursedir/m445w03/class/01-29-Stereographic1.html)
10. [MAT1300 course notes, University of Toronto](https://www.math.utoronto.ca/mgualt/courses/MAT1300F-2016/docs/1300-2016-notes.pdf)
11. [Problem 6: stereographic projection is conformal, UCLA](https://www.math.ucla.edu/~mattkowalski/documents/khan.pdf)
12. [Complex Analysis notes (Riemann sphere), UCLA](https://www.math.ucla.edu/~mbonk/complana.pdf)
13. [Homotopy Group, Wolfram MathWorld](https://mathworld.wolfram.com/HomotopyGroup.html)
14. [Poincaré Conjecture, Clay Mathematics Institute](https://www.claymath.org/millennium/poincare-conjecture/)
15. [The Generalized Poincaré Conjecture, Stanford lecture notes](https://web.stanford.edu/class/cspmath07/Lecture_4.pdf)
16. [The Poincaré Conjecture, Clay Mathematics Institute monograph](https://www.claymath.org/library/monographs/MPPc.pdf)
17. [Theorems in geometry (exotic spheres and the Differentiable Sphere Theorem), AMS Graduate Studies in Mathematics 111 preview](https://www.ams.org/bookstore/pspdf/gsm-111-prev.pdf)
18. [Bulletin of the American Mathematical Society 56 (2019), on Milnor and exotic spheres](https://www.ams.org/bull/2019-56-02/S0273-0979-2018-01654-6/S0273-0979-2018-01654-6.pdf)
19. [Spherical harmonics on the N-sphere, CWI report](https://ir.cwi.nl/pub/10902/10902D.pdf)
20. [Notes on Spherical Harmonics on the 2-Sphere, University of Pennsylvania](https://www.cis.upenn.edu/~cis6100/sharmonics.pdf)
21. [DeepSphere: Efficient spherical Convolutional Neural Network with HEALPix sampling, arXiv:1810.12186](https://arxiv.org/abs/1810.12186)
22. [On the geometry of similarity search: dimensionality curse and concentration of measure, arXiv:cs/9901004](https://arxiv.org/abs/cs/9901004)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Geometry and topology › General and set-theoretic topology*

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