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
interesting comparison: regular vs irregular by Charlie)
With a quadrilateral that's more irregular than a trapezoid, we do get different numbers:
r area ratio
4 0.01 0.992585184560644
4 0.03 0.976711455250238
4 0.05 0.959530975162394
4 0.07 0.941145303807876
4 0.09 0.921747612750326
4 0.11 0.901469613592343
4 0.13 0.881071727952944
4 0.15 0.860101205722898
4 0.17 0.838616102659335
4 0.19 0.816661344413935
4 0.21 0.794428980033211
4 0.23 0.772013461755896
4 0.25 0.749211518985067
4 0.27 0.726055772435752
4 0.29 0.702675594782874
4 0.31 0.679331112355033
4 0.33 0.656368891988905
4 0.35 0.633939944936173
4 0.37 0.61399106434176
4 0.39 0.596606076300455
4 0.41 0.584430752007476
4 0.43 0.580753909783496
4 0.45 0.585390263606205
4 0.47 0.589820614770201
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Posted by Charlie
on 2007-02-28 22:39:05 |
re: interesting comparison: regular vs irregular
