Since the edge is rough (i.e., there is no normal for any given point on the edge) a whole lot of things stop working. It becomes a much more interesting problem if you have to work at infinite iteration, since, even if you find a point that is on the edge (i.e., a points such that (x,y) is inside and (x+epsilon*cos(theta),y+epsilon*sin(theta)), I'm not convinced there is even a way to find the next point (assuming that theta is the direction your hand is in, and your 'facing' theta + pi / 4).

That's not exactly true - the Mandelbrot set has infinite circumference (and so does every branch of it), so while that strategy would work in theory, in practice, not so much. However, you can connect the two points with the 2 connected green lines. Most other paths would just take one line, but the top (or bottom) to the left is the most difficult 'route'). At iteration 10, the 'hardest' path that you can choose which maximizes the number of connected straight lines is 7, which is from the bottom to the top.Īt iteration 1 or iteration 2: Max = 1 line (since it's a circle, any part can be reached from a single line max to any other part)Īt iteration 3: Max = 2 lines (slightly warped circle, so to get from the top to the left, you need to connect two straight lines. Some paths only require two straight lines, some only one.

The idea is to reach any part of the Mandelbrot from any other part.

I don't understand your example why would 7 be the maximum number of lines at iteration 10? In fact, don't you mean minimum number of lines? Wouldn't the maximum number always be infinity?

The hardest maze ever