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.

Questions:

(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?

(In reply to tabulating F for the triangle, square and regular pentagon by Charlie)

In the program, by changing the increment of angles from the center to the vertices of the original n-gon, as follows:

   incr = 270 / sides

(with the 270 instead of 360), one large side is created.  In the case of a quadrilateral, this results in a trapezoid, so it's not really a completely irregular figure, which might explain the below result of the 4-sided figure matching the F function for the square.  The irregular pentagon does not agree with the regular one:

 n     r    area ratio for reg  area ratio for irregular   
 3   0.01    0.999694002448065  0.999694008815178 
 3   0.03    0.997138618058553  0.997138626420883 
 3   0.05    0.991758241758228  0.991758244975405 
 3   0.07    0.983287858117326  0.983287858075113 
 3   0.09    0.971492257156317  0.971492257185587 
 3   0.11    0.956180589088248  0.956180588798681 
 3   0.13    0.937221396737034  0.937221396672242 
 3   0.15    0.914556962053026  0.914556962043385 
 3   0.17    0.88821557513045   0.888215575129495 
 3   0.19    0.858320251407409  0.858320251407505 
 3   0.21    0.825092544080909  0.825092544080891 
 3   0.23    0.788850453087247  0.788850453087276 
 3   0.25    0.750000000931304  0.750000000931342 
 3   0.27    0.709020756711921  0.709020756711909 
 3   0.29    0.666446325560179  0.666446325560207 
 3   0.31    0.622841444882343  0.622841444882339 
 3   0.33    0.578777720820273  0.578777720820272 
 3   0.35    0.534810126733177  0.534810126733175 
 3   0.37    0.49145616646746   0.49145616646746 
 3   0.39    0.449179140522402  0.449179140522415 
 3   0.41    0.408376349133236  0.408376349133272 
 3   0.43    0.369372442019184  0.369372442019171 
 3   0.45    0.332417582417581  0.332417582417563 
 3   0.47    0.297689699025004  0.297689699024282 
 3   0.49    0.265299877600972  0.265299877532814 
 
 4   0.01    0.999796001631998  0.99979600163196 
 4   0.03    0.998092412038989  0.998092412038998 
 4   0.05    0.994505494505502  0.994505494505476 
 4   0.07    0.988858572078206  0.988858572078226 
 4   0.09    0.980994838104205  0.980994838104228 
 4   0.11    0.970787059392139  0.970787059392178 
 4   0.13    0.95814759782472   0.958147597824691 
 4   0.15    0.943037974702018  0.943037974702006 
 4   0.17    0.925477050086977  0.925477050086964 
 4   0.19    0.905546834271634  0.905546834271592 
 4   0.21    0.883395029387269  0.883395029387291 
 4   0.23    0.859233635391501  0.859233635391501 
 4   0.25    0.833333333954223  0.833333333954213 
 4   0.27    0.806013837807946  0.806013837807956 
 4   0.29    0.777630883706801  0.777630883706793 
 4   0.31    0.748560963254881  0.748560963254884 
 4   0.33    0.719185147213503  0.719185147213516 
 4   0.35    0.689873417822104  0.689873417822131 
 4   0.37    0.660970777644983  0.660970777644973 
 4   0.39    0.632786093681606  0.632786093681614 
 4   0.41    0.605584232755492  0.605584232755487 
 4   0.43    0.579581628012783  0.579581628012805 
 4   0.45    0.554945054945063  0.554945054945053 
 4   0.47    0.531793132683342  0.53179313268333 
 4   0.49    0.510199918400665  0.510199918400672 
 
 5   0.01    0.999859040594551  0.999855622645019 
 5   0.03    0.998681889137223  0.998649927904468 
 5   0.05    0.996203390079002  0.996111330801772 
 5   0.07    0.992301462647655  0.992114790379764 
 5   0.09    0.986867756110856  0.986549328643809 
 5   0.11    0.979814354495635  0.979324897855736 
 5   0.13    0.971080701352272  0.97037947320765 
 5   0.15    0.960640208553128  0.959685821869484 
 5   0.17    0.94850590808105   0.94725729182049 
 5   0.19    0.934734467654193  0.933151924843888 
 5   0.21    0.919427946935218  0.917474255320241 
 5   0.23    0.902732834291905  0.90037432363185 
 5   0.25    0.88483616615817   0.882043701426706 
 5   0.27    0.865958858598862  0.862708661488885 
 5   0.29    0.846346719665549  0.842620972163964 
 5   0.31    0.826259898658383  0.822047090789371 
 5   0.33    0.805961708997442  0.80125671554504 
 5   0.35    0.7857078021225    0.780511696845035 
 5   0.37    0.765736568942403  0.760056206043067 
 5   0.39    0.7462614313048    0.740108839988099 
 5   0.41    0.727465407683486  0.720857054941485 
 5   0.43    0.709498049704287  0.702454028820639 
 5   0.45    0.692474596397636  0.685017794944093 
 5   0.47    0.676477011567258  0.668632304553971 
 5   0.49    0.661556467461076  0.653349970710514 

Note that in all these, there are differences in the low order portions, but for the pentagons, they differ in significant portions.

Posted by Charlie on 2007-02-28 22:09:39