Ball -- From MathWorld

The -ball, denoted , is the interior of a sphere , and sometimes also called the -disk. (Although physicists often use the term "sphere" to mean the solid ball, mathematicians definitely do not!) The equation for the surface area of the -dimensional unit hypersphere gives the recurrence relation (1) Using then gives the hypercontent of the -ball of radius as (2) (Sommerville 1958, p. 136; Apostol 1974, p. 430; Conway and Sloane 1993). Strangely enough, the content reaches a maximum and then decreases towards 0 as increases. The point of maximal content of a unit -ball satisfies (3) (4) (5) where is the digamma function, is the gamma function, is the Euler-Mascheroni constant, and is a harmonic number. This equation cannot be solved analytically for , but the numerical solution to (6) is (Sloane‘s A087300) (Wells 1986, p. 67). As a result, the five-dimensional unit ball has maximal content (Le Lionnais 1983; Wells 1986, p. 60). The following table gives the content for the unit radius -ball (Sloane‘s A072345 and A072346), ratio of the volume of the -ball to that of a circumscribed hypercube (Sloane‘s A087299), and surface area of the -ball (Sloane‘s A072478 and A072479). 0 1 1 0 1 2 1 2 2 3 4 5 6 7 8 9 10 Let denote the volume of an -dimensional ball of radius . Then (7) (8) so (9) where is the erf function (Freden 1993). SEE ALSO: Alexander‘s Horned Sphere, Ball Line Picking, Ball Point Picking, Ball Tetrahedron Picking, Ball Triangle Picking, Banach-Tarski Paradox, Bing‘s Theorem, Bishop‘s Inequality, Bounded Set, Closed Ball, Disk, Golf Ball, Hairy Ball Theorem, Hypersphere, Open Ball, Sphere, Tennis Ball Theorem, Unit Ball, Wild Point. [Pages Linking Here] REFERENCES: Apostol, T. M. Mathematical Analysis. Reading, MA: Addison-Wesley, 1974. Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, p. 9, 1993. Freden, E. Problem 10207. "Summing a Series of Volumes." Amer. Math. Monthly 100, 882, 1993. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 58, 1983. Sloane, N. J. A. Sequences A072345, A072346, A072478, A072479, A087299, and A087300 in "The On-Line Encyclopedia of Integer Sequences." Sommerville, D. M. Y. An Introduction to the Geometry of Dimensions. New York: Dover, p. 136, 1958. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986. CITE THIS AS: Eric W. Weisstein. "Ball." From MathWorld--A Wolfram Web Resource. http://mathworld./Ball.html 1999 CRC Press LLC, ?1999-2006 Wolfram Research, Inc. | Terms of Use