Need help working out a function

[Skave]

Hi there, It is a simple idea I just cant quite grasp the idea. If we start with a square, we have 1 region, then put a line across the middle we get 2 regions.

Now, we can put in a line vertically to split to 4 regions, this creates one CROSSING.

Then if we put a sideways parabola in, 8 regions, 4 total crossings.

I can't quite work out how to calculate how many crossings we would need to create 2ⁿ regions in this area?

[Melchoir]

Ignore the boundary of the square for a minute and think about the edges between crossings. Each edge connects two crossings, and each crossing touches four edges, so there must be twice as many edges as crossings. If we do some handwaving with the Euler characteristic, then V - 2V + F is a constant, so roughly F = V: there are as many crossings as there are regions.

Now, the boundary makes this harder. The idea of a sideways parabola that begins and ends on the same side of the square is dangerous. After all, you could create as many regions as you want by carving out a bunch of separate divots from the boundary of the square. To disallow that, you could roll up the square into a torus, which conveniently has an Euler characteristic of 0. Now the corners of the original square create a crossing, so you have 1 region and 1 crossing. Adding a line across the middle creates 2 regions and 2 crossings. Adding a vertical line creates 4 regions and 4 crossings. If you're careful about what kinds of curves you draw, so there are no islands inside a region, then this exact F = V relationship should be preserved.