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.
(In reply to
Solution - last paragraph - and irregular polygons by brianjn)
Yes, looks like the configuration you describe generates the same number of edge intersections and thus (by Euler) the same number of regions. Maybe I would add a fourth condition to make sure no three edges intersect in one single point, which would waste a region. But that can be easily achieved by slightly nudging the polygons.
BTW, if we pose the more general problem of convex n-sided polygons (your first condition only), no more than the n*m*(m-1)+2 regions (for the regular case) are possible. Again this follows from Euler and from the fact that no polygon edge can intersect more than two other edges of another polygon, altogether giving no more than n*m*(m-1) intersection points.
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Posted by JLo
on 2006-09-21 13:16:23 |
re: Solution - last paragraph - and irregular polygons
