Take a polygon with area S1 and pick a number r in [0,1/2]. Take vertex A that connects sides AB and AC and add points M and N on these sides so that AM/AB=AN/AC=r. Cut corner A along MN. Cut all other corners the same way.
After repeating these steps infinite times we will get a figure with an area S2. Let's F(r)=S2/S1. It's clear that F(0)=1 and F(½)=0.
(a) What is this function for square?
(b) What is this function for equilateral triangle?
(c) Is it possible to get a circle from a square or from an equilateral triangle this way?
(d) Is it possible that this function is universal for all triangles, or for all rectangles, or for all polygons?
If r<1/2, then one interesting observation is part of each original side remains, although that part goes to 0 as the number of cuts goes to infinity. So r=1/2 and r=0 are really just special cases -- the other cuts double the number of corners.
As far as going to a circle, it seems after each cycle, the angles would stay the same if r=1/4 for squares and r=1/3 for triangles.
It seems like if the curvature stays constant in the limit (ie the angles stay the same) then the limit could be a circle, even though the sides don't converge at the same rate. (ie 1/3k = 1/k = 0 as x->infinity)
Posted by Gamer
on 2007-02-27 22:24:32