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.

This problem stated regular polygons and that is how we have addressed it.

I believe that we can still get the same result with m n-sided irregular polygons, provided that:
1. The exterior angle at each vertex is greater than 180º
2. The vertices of each newly overlaid polygon are similarly displaced (ie anti/clockwise) from its 'neighbour'.
and
3. All vertices each have a vertex in common with an m*n 'virtual' polygon which forms the 'circumference' of the m polygonal array.  The only restriction on the vertex angles of this polygon is that it forms a 'perimeter'; its sides do not cross any other lines.

With all of that in mind, the polygons don't need to have a shared point of rotation as was used in the way that we have approached the problem; actually, this scenario can be applied to the regular case.

Yes?

Posted by brianjn on 2006-09-21 02:15:13