Re: Mike Throll 5

In article <9oachv$795$1...@news.fas.harvard.edu>, uncle monty wrote:
>Ilmari Karonen <ilt...@sci.invalid> wrote:
>: In article <3ba...@News.CIS.DFN.DE>, Timofei Shatrov wrote:
>:>On 17 Sep 2001 17:38:49 GMT, uncle monty <unc...@babyfishmouth.org> tried to
>:>confuse everyone with this message:
>:>
>:>>Given an initial separation of the plane RxR into a subset A and not-A,
>:>>and an iterative process which at each step assigns membership in A to
>:>>each point outside A whose circular neighbourhood of radius 1 has a
>:>>greater intersection with A than with not-A, and removes each point
>:>>previously within A whose circular neighbourhood of radius 1 has a greater
>:>>intersection with not-A than with A, is it possible to choose initial A
>:>>such that its area will increase by a factor of 1000 or more during the
>:>>process?
>
>:>>I imagine it is some 
>:>>impossible fractal shape to begin with.
>:>
>:>No way. It's not very complicated. Just try to invent blot that can increase and
>:>then magic word "asymptotics".

My original idea was simply a ring with inner radius R and outer radius
R+0.5+epsilon, for some humongous value of R and the smallest value of
epsilon you can get away with.  It's obvious that such a ring will get
thicker as it shrinks, but unfortunately it's not at all obvious if its
area will increase or not.

And just to add to the frustration, after struggling with the trig and
computing the hard parts by brute force, it seems I still don't know.
Yes, the area appears to increase, but very slightly, and I'm not sure
if the possible increase has an upper bound or not.


>Well, I found a blot that can increase, but not a thousand-fold... take a 
>very large "sheet" of ink and punch holes in it, with diameter slightly 
>less than 1/sqrt(2pi)... unless they are spaced too close together they 
>will all be filled after one generation, while only the outer edge of the 
>whole sheet will have started degrading... am I on the right track?

Hmm.. you just gave me an idea.  Take one big blot.  Fill it with N > 1
large circular holes, just far enough apart not to disturb each other's
shrinking.  Now fill the *remaining* blot with smaller circular holes,
and repeat until the holes get too small.

Now your blot is a fractal foam.  If you make it big enough, I see no
reason why you couldn't get an arbitrarily high area increase.  Not that
I could prove this either, but I'm a lot more confident in making that
conjecture than with any of my previous ideas.

Of course, if there's a non-fractal answer, as Timofei seems to imply
above, I still have no idea what it could be.

ObAscii: foam
      _..._                _     __
   ./'     `\,---.-'"".--.(_)  /"  "\
  ,' .'"`. ."   / .--:    : __: _    ;
  |  `._.'-(  _: (    )__.7'   (_)_./
 -'-._  ;  `.(_)-"`--'-.  ;  ,--.|-.
   .--".--. /_    .---(_)/  ;_   |  :
  ;    `;  ;(_)--`-.    `;-.(_)-'--'

-- 
 .    _, .. j u s t.  h o w l.i n g   i n  .t h e   n.i g h t .. ._,   .
,    )''        .       /\_    .     '     ,/\   .          ,    ``(  ,
_\__/ |__.'\._______,--;_'_`-.___,.______,/_,_`.__,-.__'__,/`-._,_| \____
 ,_f_)\. Ilmari Karonen  ilt...@sci.fi  http://www.sci.fi/~iltzu/  /(_|_,.

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