Ball
المؤلف:
Apostol, T. M
المصدر:
Mathematical Analysis. Reading, MA: Addison-Wesley, 1974.
الجزء والصفحة:
...
6-4-2020
1117
Ball
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 ball of radius 
centered at point ![<span style=]()
{x,y,z}" src="https://mathworld.wolfram.com/images/equations/Ball/Inline6.gif" style="height:15px; width:46px" /> is implemented in the Wolfram Language as Ball[![<span style=]()
{" src="https://mathworld.wolfram.com/images/equations/Ball/Inline7.gif" style="height:15px; width:5px" />x, y, z![<span style=]()
}" src="https://mathworld.wolfram.com/images/equations/Ball/Inline8.gif" style="height:15px; width:5px" />, r].
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
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 
(OEIS A074455) (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 (OEIS A072345 and A072346), ratio of the volume of the 
-ball to that of a circumscribed hypercube (OEIS A087299), and surface area of the 
-ball (OEIS 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
so
![sum_(n=0)^inftyV_n=e^(piR^2)[1+erf(Rsqrt(pi))],](https://mathworld.wolfram.com/images/equations/Ball/NumberedEquation4.gif) |
(9)
|
where 
is the erf function (Freden 1993).
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.
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