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Re: The Density of Foam
Posted:
Jul 31, 1996 6:31 PM
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In article <31FD037B.12A7@math.okstate.edu> David Ullrich <ullrich@math.okstate.edu> writes:
>R E Sheskey wrote:
>>
>> Hi. I have an infinite collection of identical disks arranged
>> randomly on the plane subject to two conditions:
>>
>> 1. No two of the disks overlap.
>> 2. No disk can be added without violating #1.
>>
>> Question: what fraction of the plane is covered?
>
> By which we mean presumably the limit as R tends to
>infinity of A(R), where A(R) is the fraction of the disc
>about 0 of radius R that's covered by your collection of
>discs. Or something like that...
> There's no reason that A(R) should actually have
>a limit as R tends to infinity.
> [...]
>
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Sure, it would be easy to create a configuration of non-overlapping unit disks
in the plane for which A(R) does not approach a limit as R -> oo.
But there is good reason to expect that there will be (with probability 1)
a definite numerical limit that is approached, statistically.
Let D_R denote the disk of radius R about 0 in the plane.
Define B(R) as the average fraction of D_R covered, averaged over all
configurations of unit disks within D_R satisfying 1. and 2.
(Intuitively, this could be made precise by considering one "try" to consist of
the following: Pick points c_1, c_2,... at random from D_R, thinking of them
as centers of unit disks -- rejecting any c_(n+1) which violates rule 1. Stop
when there is no room for another disk. Repeating tries K times should, in the
limit as K -> oo, lead to an average fractional coverage B(R). This limiting
average should exist, I think, with probability 1.)
(A cleaner method might be to consider sequences of points c_1,...,c_n of D_R
that are maximal with respect to the condition that all pairwise distances
||c_i - c_j|| < 1. Such sequences form a subset S (open) of D_R x ... x D_R
(n factors), with a natural uniform measure on it inherited from R^(2n).
Then it is straightforward to average the fraction of D_R associated to disks
centered at c_1,...,c_n over the subset S, obtaining a number F(R,n). The rub
is that the number n can vary. The correct definition for B(R) ought to be
the weighted average w_1*F(R,1) + w_2*F(R,2) + ... (this infinite sum is
actually finite), where w_n is the probability that n is the size of the
maximal sequence c_1,...,c_n. But the w_i may be difficult to determine.)
Then I believe that B(R) will approach a limit as R -> oo.
The reason is that the "edge effects" -- due to the boundary of D_R -- should
approach zero influence on B(R) as R -> oo.
Dr. Daniel Asimov
Senior Research Scientist
Mail Stop T27A-1
NASA Ames Research Center
Moffett Field, CA 94035-1000
asimov@nas.nasa.gov
(415) 604-4799 w
(415) 604-3957 fax
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