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I was wondering if it was possible to sort of enumerate all points on the border of the Mandelbrot set. Similar to the idea of external angles, which map (at least the rational points) of an infinitely large surrounding circle to points on the border.

But external angles are related to the idea of (electrical) field lines, which leads to a somewhat uneven distribution. Simply speaking, approximately all field lines starting infinitely far away will land on antennas and spikes of the Mandelbrot border, and approximately zero field lines will land deep in spirals.

Ever since I first read about external angles, their uneven distribution has irked me. If there existed a more evenly distributed mapping, we could do the following: generate a random number in [0 .. 1], evenly distributed (possibly with very high precision), and map it to the Mandelbrot border. Zoom in, and be surprised!

This algorithm does not work all that well with external angles, because those tend to cover only the same boring few regions over and over.


I guess maybe one could use the level sets to construct some sort of "equidistant" mapping. Basically the borders of all level sets are smooth curves, and their arc length can be determined (albeit not necessarily in closed form). So an equidistant mapping from [0 .. 1] to the corresponding fraction of total arc length can be defined for each and every level set.

The question is: would the series of mappings at all converge as the number of iterations approaches infinity? Would the resulting mapping indeed be somewhat equidistant (in the sense that it could enable a random selection of targets to cover all of the border with equal probability)?

I don't really know where to start looking for an answer.

Strictly speaking, no, it's not possible.  The boundary contains an uncountable infinity of points, so it's impossible to enumerate them all, even with infinite time and memory.

One simple alternative is to tick off key points of the boundary of some components.  For example, the main cardioid and the large disk centered at (-1,0) both have simple equations parameterized by an angle (around (0,0) for the cardioid and around (-1,0) for the disk).  Then, one could create lists of {angle, x, y, interesting note} for each piece.

For other components, the equations are substantially messier, but I think one can find points numerically based on the period of the component and the derivative of the composition of the iterates.  But, I haven't tried it.

For the level set thing, you could try following the path given by directional estimation... I posted about that in another thread. It's like distance estimation, but instead to estimates the direction to the nearest point on the set. By following these lines - which would, admittedly, be a computationally intensive process - you're traveling directly at the Mandelbrot. Just by choosing an initial point anywhere on the circle (radius=2). The one problem I can think of is how to get "into" some parts of the set - the points where two protruding stalks produce a saddle node, if the path you take is viewed as a vector field. Because the distance estimation does not have a continuous derivative, it is not very possible to make useful decision making about navigating between two spikes. Using the derivative of the iteration count in conjunction, though, could help guide the path an potentially lead to useful navigating. First one would need to work out what the equations for directional estimation are, because as far as I know, they don't exist anywhere yet.