Intersections of Polygon DiagonalsDate: 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/ |
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