This might be interesting for us. A paper about @Scott_Draves Fractal Flame Algorithm. Contains great general hints how to improve picture and animation quality when doing art based on math.
If you don´t know his work yet please have a look at https://electricsheep.org/ (Don´t miss the videos.)
I wonder how much from his catalog of variations (page 16+) could be applied in real time on led animations…
http://flam3.com/flame_draves.pdf
I read that a while ago, its a DEEP dive into the matetial. A lot of these things are difficult to translate into the low reaolution space we all work with. As an example,younlast few posts you use a 16x16 grid, there limited resolution there to get anything very complex to look good. Id love to be corected! I personlay keep trying to push the envelope in this small space,but i fins it chalenging
I agree that many of the techniques simply make no sense with a pretty limited space and color resolution. At the other hand some techniques are the very same - no matter if stuff is rendered in icon size or in 4k: it is all about generating, filtering, modifying and combining data in order to get an angreeable visual effect. If this is done internally with high resolution the effect looks cool even scaled down onto a small matrix. I got the first glimpse of that when I wrote the noise smearing where I used fractional shifting the first time. I think the idea of dynamic “directed tails” still has potential. For example I didn´t got a universal and good looking swirl or fisheye filter coded yet because I didn´t knew enough about the math behind. In the paper I found already some informations how this could be achieved. Also the part about color and gamma mapping I found inspiring because it touches exactely the same challenges we face when displaying data on leds. So I have no results yet to show, but my imagination was expanded by the paper and that´s for me always the first step before (maybe) manifesting something “new”.
I love fractals, however the challenge I see with them at the Arduino AVR level is their iterative nature. It can take many passes at a routine before you arrive at a decent static image. Then there’s the animation of said image, which is usually done by rendering each image and then combining them into a movie or gif. Creating animated fractals on this hardware would be pretty incredible. Stefan is well ahead of the game on this one, and I hope he comes up with something great out of this.
I’m also wondering if Perlin Noise, which is built into FastLED, might be able to product decent flames using a LOT less CPU.
Just to keep the expectations on a realistic level: right now I´m exited about bitmap manipulation based on Scotts “Catalog of Variations” because this covers some questions I have since a long time. So I don´t see animated fractals coming soon. Also the advanced bitmap manipulation will be impossible at decent speed unless a FPU and a fast processor like a Teensy 3.6 is present. If the RAM allows it I could imagine to kind of render single fractals “in the background” while playing on a bitmap level with previously rendered ones and then fade or blend somehow to the new bitmap. Also I´m just a hobbyist with limited time and skills. Basically I´m still using the math I learned in school and try to utilize this in creative ways. Some experienced coder might later optimize the formulas to hell and back, but I´m not good at this myself. Back to fractals again: This name covers a wide variety of formulas and not all are expensive. I hope to find some attractors which are easy to calculate. Even if I get just some few values calculated they could be fed as control parameters into other algos. And isn´t the gardient noise a fractal function as well? So again, I see no realtime Mandelbrot zoom coming, but I can imagine some interesting and maybe even “new” effects based on math. I think I have to move over to Processing for a while where trying, debugging and visualizing formulas is way easier until I understand the math enough to do it on ARM. Sideeffect: I can do this everywhere without any need to have a led setup present. Exciting. 
+Andrew Tuline The FastLED noise implementation has some minor flaws - we had an interesting discussion about different gardient noise functions here: https://github.com/FastLED/FastLED/issues/416