Working on it reminded me of the little "maths for fun" exercises I used to do while learning programming in early days. Just trying things out, getting fascinated and geeky, and being surprised by the results. I spent way too much time on this, but it was extreme fun.
My favorite part: someone pointed me to the Simone Attractor on Threads. It is a 2D attractor and I asked GPT to extrapolate it to 3D, not sure if it’s mathematically correct, but it’s the coolest by far. I have left all the params configurable, so give it a try. I called it Simone (Maybe).
If you like math-art experiments, check it out. Would love feedback, especially from folks who know more about the math side.
Still they've had a strong impact in how I see systems - orbits, instability, etc.
It actually shaped my post doc work quite a bit and shifted my focus from individual classroom education to strategic systems analysis of entire university and k-12 institutions. Somewhere along the way, a switch flipped and allowed me to view complicated hierarchies like college systems as 2-d fractal geometry in my mind. I can't really explain it, but now that I consult, I can feel when a department is broken before I can prove it with data. It's like they don't fit or reflect the main structure of the institution.
I would not suggest taking this route though. Maybe just take some graduate courses or something.
Fun fact, though, defending your dissertation to a room of around 200 people while still feeling the effects of dmt is a really good way to induce a panic attack. Source: it's me. I'm source material.
e.g. https://i.imgur.com/ZjiBF8f.png
just a coincidence?
Side note: Did anyone else know it was AI before reading the post? Mathematicians would be argent enough to assume the name was enough, displaying the algo when clicking the name was the give away.
(I wonder if there are slick ways to visualise the >3D case. Like, we can view 3D cross sections surely.
Or maybe could we follow a Lagrangian particle and have it change colour according to the D (or combination of D) it is traversing? And do this for lots of particles? And plot their distributions to get a feeling for how much of phase space is being traversed?)
This visualization also reminds me of the early debates in the history of statistical mechanics: How Boltzmann, Gibbs, Ehrenfest, Loschmidt and that entire conference of Geniuses must have all grappled with phase space and how macroscopic systems reach equilibrium.
Great work Shashank!
There isn't always "a" correct extension into higher dimensions. There may be many, there may be none, and either way something "close enough" may well be interesting in its own right.
If you'd like something concrete to poke at you can try searching around for people's adventures in trying to make a 3D Mandelbrot. I've seen a couple of good write-ups on those adventures. I don't know if anyone has ever landed on a "correct" solution, it's been years since I last looked, but certainly some very interesting possibilities have been found.
It reminded me of one of my (cranky) musings from back in college about galaxy formation and whether they were more like tossed pizzas (i.e. spreading out) than like whirlpools getting sucked in.
- Hypster by Nonlinear Circuits (https://modulargrid.net/e/nonlinearcircuits-ian-fritz-s-hyps...)
- Orbit 3 by Joranalogue (https://modulargrid.net/e/joranalogue-audio-design-orbit-3)
https://github.com/gradientwolf/fractals_SFML
Your post gives me so much joy. These tiny little things take me back to teenage years, simpler times & when interests were different. (I put a little note as "why" in my GH repo readme)
I made a similar experiment a while ago and randomized the parameters. Given it's difficult to stumble on a stable arrangement, I turned it into a small game to find pretty ones: (big disclaimer: this involves NFT tech, please skip if you're against that sort of stuff) https://karimjedda.com/symmetry-in-chaos-my-first-generative...
This might be inspiration to try to grasp these ideas again.
Rotating the Lorenz makes me think otherwise though because given the amount of time I put into this, I should understand that much more than I do.
Chance and Chaos by David Ruelle is a wonderful little book.
People seem to have surprisingly different internal experiences. I don't know how common 4d visualization is, and I suspect even those capable require exposure to the concepts and practice. However I do think it possible.
It's such a typical object of its time. Garishly colored cover, comes with a floppy disk (!) and there are even 3D glasses to view some of the stereoscopic color plates (unfortunately these were missing from the used copy I got). I was surprised to find that most of the programs are in BASIC (maybe easier to do graphics on Windows back then?), though a small number of them are in C.
It's a nice book, and the author seems to have a lot of publications about chaotic systems. Anyone know him? He seems to still be teaching at the University of Wisconsin - Madison.
[1] https://sprott.physics.wisc.edu/fractals/booktext/SABOOK.PDF
I think this is slightly inaccurate. The butterfly effect is about the evolution of two nearby states in phase space into well-separated states. But the parameter a is not a state. To see the butterfly effect by changing a we would need to let the system settle down, give the parameter a small change, and then change it back. The evolution during the changed time acts as a perturbation on states.
Instead, showing that the attractor changes qualitatively as a function of the parameter is more akin to a phase transition.
The arrangement of these neurons physically corresponds to reality, and so things are pretty hardwired.
Repurposing these neurons might be possible with advanced training and nootropics, but I'm not sure. You might have better luck engaging other parts of your brain, for example using metaphor or abstraction such as mathematics.
