Here are a number of assumptions I have reached about the Mandlebrot set. I have no mathematical training and have not attempted to verify any of these, I'm going entirely on intuition. I'm going to post them here and I'd greatly appreciate it if those with more knowledge than me could tell me if any number of them are correct or incorrect (or unproven/poorly stated). I'm fully prepared to be completely wrong on all of these

1. The curve its boundary traces is comprised of regions of recursive self-similarity, where there will be a series of iterations—visually, "shapes" (trying to be delicate with that term since I know it's not rigorously defined here)—such that as one relates to the one prior to it, so will the next iteration relate to it (e.g. concentric circles; circle B is to circle A as circle C is to B, and so on)

2. While each recursive iteration is ostensibly isomorphic to those before and after it, no two sequential iterations in the entire boundary can ever be identical in both shape and size, as that would imply an infinite sequence of duplicates in an endless straight line, which is impossible given that no element in the set has an absolute value >2

3. No segment of the boundary is differentiable

4. Except for any patterns/iterations falling horizontally along the real number line (the set's only axis of symmetry) no identifiable recursive sequence within the boundary iterates across a straight line, even if decreasing iteration sizes leave the sequence's limit within the 2-unit disc. In other words, every repeating shape "curves" one way or the other as it iterates, even if only slightly so

5. Any function describing the curve along which some identifiable recursive sequence iterates is smooth, i.e. infinitely differentiable

Thanks for satisfying my curiosity! If anyone else has uneducated assumptions they want evaluated, feel free to post them in this thread