Intersections of Polygon Diagonals

Date: 8/26/96 at 9:32:56
From: Anonymous
Subject: Intersections of polygon diagonals

Given a regular polygon of v vertices write a formula f that gives the  
number of distinct zones z in which that polygon is divided by all its 
diagonals.  i.e.   z = f(v).

I have tried to find a solution by myself or to find the problem 
addressed in some text but with no luck.

By drawing the polygons and using a computer program to isolate the 
zones, I have found the following sequence:

    v      zo    ze

    3       1
    4             4
    5      11
    6            24
    7      50
    8            80
    9     154
   10           220
   11     375
   12           444
   13     781
   14           952
   15    1441
   16          1696
   17    2446           (I am not really sure about
   18          2446      these last two)


I have divided the sequence into even and odd numbers of vertices 
because, working on the problem, I noticed same regularities within a 
class (e.g. I found a formula for the number of layers of distinct 
intersection for the odd class).

After having tried several types of functions and series (obviously 
the sequence is not a polynomial of n < 17 as many other polygon 
properties are) I had to give up.

Even to know that the problem does not have a solution would be a 
relief!

Thanks very much for your help.

Sincerely,
   Franco Languasco

Date: 9/6/96 at 14:4:25
From: Doctor Ceeks
Subject: Re: Intersections of polygon diagonals

Hi,

This problem is actually quite complicated and has only recently
been solved by Bjorn Poonen and Mike Rubenstein.  It has a long
history; apparently it was posed several decades ago as a problem
with an award, and then the award was claimed by someone who thought
he had a solution.  Then Bjorn and Mike tackled the problem without
knowing of this other "solution" and got an answer, which showed, in 
fact, that the other "solution" (which got the award) was flawed!

I suggest you write e-mail to poonen@math.princeton.edu and ask him
for a preprint of his paper regarding this matter.  You can tell him
what you have below and say that you were refered to him by Dr. 
Math...he'll  know because he knows me (Doctor Ceeks).

Do you know about roots of unity and Euler characteristic? If not,
the paper isn't going to be so easy to read.  Still, the final answer
should be easy to get from the paper.

-Doctor Ceeks,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/