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Here is a test question:

If you draw a straight line across the Mandelbrot Set such that the line intersects the true "inside" of the Set and not just the boundary, then how many Mandelbrot Set boundary points lie on the line ?

I will unlock the topic when I think enough folks have considered the possible answer
(Please do not give the answer in other threads)

Anyone think they know the answer ?

I don't really understand your question. Is the only possibillity for this like line B or every line like line A as well



But actually I think there will be infinite cause of all the tiny cardioids on the outside regions of the Mandelbrot-set.


next question is of course how infinite is this?

The answer is infinity simply because (in the infinite limit) the dimension of the boundary is 2 (everywhere).

Please note that by "boundary" points I mean those points that are "inside" but adjacent to the "outside" to infinite resolution - the classic exampe being (-2,0).

As to the lines A and B, both would qualify - in fact any straight line drawn through the whole Set at any angle (from "outside" through "inside" back to "outside") would produce the same result.
The x (real) axis of course has an infinite number of boundary points in the spike.

Note that if a curved line were allowed then even restricting it such that it must start "outside" pass through to "inside" and back to "outside" the answer would be any number of points from 2 to infinity (assuming it isn't allowed to cross or touch itself).