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.

Surely the plane was meant as being 2D space.

However, I have just found a plane which allows some interesting possibilities.

The curved surface of a cylinder allows me to take a triangle and overlay it on this surface. If I choose my cylinder correctly one vertex of my triangle will overlay the opposite side of the triangle and incringe upon the interior space of the triangle. Thus I now have 3 regions instead of two in 2D space.

[Let's take our nm(m-1)+2 regions, put them onto a rubber sheet and stretch them around the cyclinder so that distortion is only in the direction of the circular stretching.  My mind is being boggled by the imagery.]

Posted by brianjn on 2006-09-07 20:58:16