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Ball


The n-ball, denoted B^n, is the interior of a sphere S^(n-1), and sometimes also called the n-disk. (Although physicists often use the term "sphere" to mean the solid ball, mathematicians definitely do not!)

The ball of radius r centered at point {x,y,z} is implemented in the Wolfram Language as Ball[{x, y, z}, r].

BallVolume

The equation for the surface area of the n-dimensional unit hypersphere S^n gives the recurrence relation

 S_(n+2)=(2piS_n)/n.
(1)

Using Gamma(n+1)=nGamma(n) then gives the hypercontent of the n-ball B^n of radius R as

 V_n=(S_nR^n)/n=(pi^(n/2)R^n)/((1/2n)Gamma(1/2n))=(pi^(n/2)R^n)/(Gamma(1+1/2n))
(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 n increases. The point of maximal content of a unit n-ball satisfies

(dV_n)/(dn)=(pi^(n/2)[lnpi-psi_0(1+1/2n)])/(2Gamma(1+1/2n))
(3)
=(pi^(n/2)[gamma+lnpi-H_(n/2)])/(nGamma(1/2n))
(4)
=0,
(5)

where psi_0(x) is the digamma function, Gamma(z) is the gamma function, gamma is the Euler-Mascheroni constant, and H_n is a harmonic number. This equation cannot be solved analytically for n, but the numerical solution to

 gamma+lnpi-H_(n/2)=0
(6)

is n=5.25694... (OEIS A074455) (Wells 1986, p. 67). As a result, the five-dimensional unit ball B^5 has maximal content (Le Lionnais 1983; Wells 1986, p. 60).

The following table gives the content for the unit radius n-ball (OEIS A072345 and A072346), ratio of the volume of the n-ball to that of a circumscribed hypercube (OEIS A087299), and surface area of the n-ball (OEIS A072478 and A072479).

nV_nV_(ball)/V_(cube)S_n
0110
1212
2pi1/4pi2pi
34/3pi1/6pi4pi
41/2pi^21/(32)pi^22pi^2
58/(15)pi^21/(60)pi^28/3pi^2
61/6pi^31/(384)pi^3pi^3
7(16)/(105)pi^31/(840)pi^3(16)/(15)pi^3
81/(24)pi^41/(6144)pi^41/3pi^4
9(32)/(945)pi^41/(15120)pi^4(32)/(105)pi^4
101/(120)pi^51/(122880)pi^51/(12)pi^5

Let V_n denote the volume of an n-dimensional ball of radius R. Then

sum_(n=0,2,4,...)^(infty)V_n=e^(piR^2)
(7)
sum_(n=1,3,5,...)^(infty)V_n=e^(piR^2)erf(sqrt(pi)R),
(8)

so

 sum_(n=0)^inftyV_n=e^(piR^2)[1+erf(Rsqrt(pi))],
(9)

where erf(x) 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, Hairy Ball Theorem, Hypersphere, Open Ball, Sphere, Tennis Ball Theorem, Unit Ball, Wild Point

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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, A074455, and A087299 in "The On-Line Encyclopedia of Integer Sequences."Sommerville, D. M. Y. An Introduction to the Geometry of n 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:

Weisstein, Eric W. "Ball." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Ball.html

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