Studying Mathematics Thread

[polymath257]

Other than Mandelbrot himself? Oh yes. At least one difficulty is that there is no agreed upon definition of what it means to be a fractal. yes, self-similarity is one key, but the notion of Hausdorff (fractal) dimension is also widely recognized. The two lead to different notions, however.

Here's an actual mathematics book:


https://www.amazon.com/Geometry-Fractal-...s+falconer

There are many unanswered mathematical questions even concerning the Mandlebrot set: for example, is it locally connected?

Here are some recent papers:
https://scholar.google.com/scholar?hl=en...nsion&btnG=

In any case, fractals seem to be less 'in vogue' than they were 20 years ago, but still being actively studied in various ways.

Problem is that this stuff is really, really hard, not only to learn but to actually improve upon the work of others.  My problem was that I had a love of physics and math; just was not particularly good at either subject.  Still, got through 3-semesters of calculus and taught myself ordinary and partial differential equations with a little tensor calculus over the years.  They are beautiful subjects, and I see no reason to believe that what modern science observes in the heavens and on earth is just the product of invisible angelic beings who are the true cause of particle motion, whether it be atomic, microscopic or macroscopic.

The Universe can be modeled just fine with higher math, although, just because something is correctly mathematically does not mean that such corresponds to physical reality.

Yes, research in math is hard.

The concept of a fractal dimension goes back to Hausdorff and is around a century old. Using various forms of iteration to produce rather complicated sets is even a bit older (the Cantor ternary set is, perhaps, the first example of a set now recognizable as a fractal).

All I can say is that a lot depends on your definitions. And there are several competing definitions for what constitutes a 'fractal'. But iterative methods involving self-similarity are widespread in math.