Why fractals are interesting

Fractals are often simple parametric equations that end up intersecting multiple planes. The mandlebrot set only consists of the scary mathematical concept of recursion, that is, that the function ends up calling itself in order to substitute variables. As a result, this set grows to massive complexity very quickly, but it is not chaotic at all. Rather than having any kind of chaos, if you zoom in on it, it will resolve into parts that go together to make the whole, and then somehow transition to make a part that, other than scale, seems to mirror the whole very closely.
Fractals are not interesting because of their repetition, rather they are interesting because they are a surprisingly close analogue to structures formed in biology. A favorite one of these is the tree, which has been generalized into an algorithm that produces a decently randomized tree(called speedtree). The internal structure of the branches is similar to certain divisions of the trunk, and as a result, the entire tree is far more robust than any organism that is as specialized as humans are. Fractals are found more in plants, this might be due to the unusually efficient method of cramming a nearly infinite surface area into a finite volume, such a compressed reaction pathway is not convenient for most animal cells, except perhaps for the mitochondria(which itself is probably more like plants than our cells are).

Fractals were never meant to be a piece of math that was interesting. In all honesty, unless used subtly, even the most interesting deep zoom fractal quickly becomes boring to watch. There was however a strong tendency in the infancy of computers to devise the fastest method of estimating a fractal so that it could be rendered as quickly as possible. These algorithms have likely gone on to become legitimately useful in another field of math that deserves attention. But Fractals themselves are not that interesting, and my math teacher who called me out on it was right; the only reason I found it interesting was because I did not understand the math well enough to see how boring it was.