Into how many regions can you partition the plane with m n-sided
regular polygons?
For example, with two squares you can achieve up to 10 regions by choosing the right size and position of your squares.
I believe that I can correctly make the following observations
with the limit to which infinite polygons could said to approach.
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Case1.
Take ‘∞’
triangles and rotate them, equi-angularly around a central point.
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This defines 3 regions.
There will be an annulus of width 1 unit which contains a circle of
diameter 2 units. These are enclosed in
an infinite plane.
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Case2.
Now take ‘∞’ ‘∞’-gons and treat them similarly. As the limit of a regular polygon with
infinite sides is the circumference of a circle, there are two regions, an
outer and an inner.
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In between these extremes, the width of the annulus can be
given by:
Radius * (1 – Cos[180/n]º ), n is the number of sides of the
polygon in question.
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Posted by brianjn
on 2006-09-07 20:05:27 |
Observations
