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 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).