unit sphere. Unlike a ball, even a large sphere may be an empty set. For example, in with Euclidean metric, a sphere of radius is nonempty only if can be written as sum of squares of integers. An octahedron is a sphere in taxicab geometry, and a cube is a sphere in geometry using the Chebyshev distance. History The geometry of the sphere was studied by the Greeks. Euclid's Elements defines the sphere in book XI, discusses various properties of the sphere in book XII, and shows how to inscribe the five regular polyhedra within a sphere in book XIII. Euclid does not include the area and volume of a sphere, only a theorem that the volume of a sphere varies as the third power of its diameter, probably due to Eudoxus of Cnidus. The volume and area formulas were first determined in Archimedes's On the Sphere and Cylinder by the method of exhaustion. Zenodorus was the first to state that, for a given surface area, the sphere is the solid of maximum volume. Archimedes wrote about the problem of dividing a sphere into segments whose volumes are in a given ratio, but did not solve it. A solution by means of the parabola and hyperbola was given by Dionysodorus. A similar problem — to construct a segment equal in volume to a given segment, and in surface to another segment — was solved later by al-Quhi. Gallery Regions Hemisphere Spherical cap Spherical lune Spherical polygon Spherical sector Spherical segment Spherical wedge Spherical zone See also 3-sphere Affine sphere Alexander horned sphere Celestial spheres Curvature Directional statistics Dyson sphere Gauss map Hand with Reflecting Sphere, M.C. Escher self-portrait drawing illustrating reflection and the optical properties of a mirror sphere Hoberman sphere Homology sphere Homotopy groups of spheres Homotopy sphere Lenart Sphere Napkin ring problem Orb (optics) Pseudosphere Riemann sphere Solid angle Sphere packing Spherical coordinates Spherical cow Spherical helix, tangent indicatrix of a curve of constant precession Spherical polyhedron Sphericity Tennis ball theorem Zoll sphere Notes and references Notes References Further reading . . . . External links Mathematica/Uniform Spherical Distribution Surface area of sphere proof Differential geometry Differential topology Elementary geometry Elementary shapes Homogeneous spaces Surfaces Topology