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| Spheres (mathematics) | |
|---|---|
| Name | Sphere |
| Caption | A 2-sphere embedded in 3-space |
| Curvature | constant positive |
| Related | Ball, Hypersphere, Manifold, Geodesic |
Spheres (mathematics) are the sets of points at a fixed distance from a central point in Euclidean or more general metric spaces. They serve as fundamental examples in Euclidean space, Riemannian geometry, topology, algebraic geometry, and mathematical physics, connecting classical constructions studied by Euclid and modern theories developed by figures such as Bernhard Riemann, Henri Poincaré, and Sophus Lie.
A sphere of radius r centered at a point c in n-dimensional Euclidean space R^n is the locus {x in R^n : ||x - c|| = r}, introduced in works by Archimedes and formalized in René Descartes' analytic geometry; common examples include the 2-sphere studied by Carl Friedrich Gauss and the 3-sphere appearing in research by William Rowan Hamilton. Spheres are homogeneous under the action of the orthogonal group O(n) and the special orthogonal group SO(n), and their symmetry leads to transitive isometry groups such as Lie group actions studied by Élie Cartan. Boundary relations with balls and embeddings into spaces like Minkowski space and Hilbert space appear in works by David Hilbert and John von Neumann.
Geometric properties of spheres include geodesics, great circles, and intrinsic metrics studied by Bernhard Riemann in the context of curved spaces; great circles on S^2 are intersections with planes through the center, a fact exploited in navigation by explorers like Ferdinand Magellan and referenced in Ptolemy's astronomical models. Distance on the sphere uses the arc length from the induced metric from embedding in R^{n+1}, a setup central to studies by Hermann Minkowski and in applications by Isaac Newton in gravitational theories. Spherical geometry contrasts with Euclidean geometry and hyperbolic geometry as explored by Nikolai Lobachevsky and János Bolyai.
Algebraically, an n-sphere S^n is defined by the quadratic equation x_0^2 + x_1^2 + ... + x_n^2 = r^2 in projective or affine space, a formulation used in Algebraic geometry by scholars like Alexander Grothendieck and André Weil. Parametrizations include spherical coordinates linked to Johannes Kepler's work on planetary motion and stereographic projection from a pole, classical since Ptolemy and exploited by Augustin-Jean Fresnel; stereographic maps relate S^n\{point} to R^n and are conformal, a property central to George Gabriel Stokes's and James Clerk Maxwell's analytical techniques. Harmonic analysis on spheres uses spherical harmonics developed by Pierre-Simon Laplace and Adrien-Marie Legendre, applied in methods by Vito Volterra and Norbert Wiener.
Topologically, spheres are compact, simply connected for n ≥ 2, and serve as basic examples in classification theorems by Henri Poincaré and later work by William Thurston and Michael Freedman; the Poincaré conjecture for S^3 was resolved by Grigori Perelman. Homotopy groups of spheres form deep problems investigated by Edwin E. Moise, J. H. C. Whitehead, and in stable homotopy by J. Frank Adams; phenomena such as exotic spheres were discovered by John Milnor and furthered in research by Simon Donaldson. Spheres also underlie constructions in K-theory and connect to the Bott periodicity theorem by Raoul Bott.
As Riemannian manifolds with constant positive sectional curvature, spheres are canonical models in differential geometry; the Gauss map for surfaces in S^2 was studied by Carl Friedrich Gauss and links to the Gauss-Bonnet theorem proven by Pierre Ossian Bonnet and Élie Cartan. The shape operator, mean curvature, and Ricci curvature on spheres provide examples for comparison theorems developed by Jeff Cheeger and Mikhail Gromov. Geodesic completeness and eigenvalue spectra of the Laplace–Beltrami operator on spheres were analyzed by Yau and A. Lichnerowicz in relation to isoperimetric inequalities by Joseph-Louis Lagrange and Lord Rayleigh.
Volume and surface measure on spheres derive from integration in spherical coordinates, classical since Archimedes computed the area and volume relations for the sphere and circumscribed cylinder; modern measure-theoretic treatments use Haar measure on SO(n) studied by Alfréd Haar and Fourier analysis on spheres by Norbert Wiener. Explicit formulas for n-ball volume involve the gamma function from Leonhard Euler, and concentration of measure phenomena on high-dimensional spheres were explored by Paul Lévy and applied in probability by Wassily Kolmogorov and Andrey Kolmogorov's schools. Techniques of spherical integration appear in physical theories developed by Albert Einstein and computational methods by John von Neumann.
Spheres appear in astrophysics via models by Johannes Kepler and Galileo Galilei, in signal processing with spherical harmonics used by J. H. Conway and engineers, and in computer graphics through parametrizations employed by studios like Pixar and institutions like NASA. Generalizations include hyperspheres in functional analysis contexts studied by Stefan Banach, unit spheres in normed spaces central to Banach space theory, orbifolds and spherical space forms classified by Seifert and Thurston, and spherical buildings in algebraic groups examined by Jacques Tits. Spheres also intersect number theory in the study of integer representations by sums of squares as in results by Srinivasa Ramanujan and Carl Ludwig Siegel.