Posted by shashanktomar 4 days ago
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.
(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!
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.
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.
You can either sweep a cutting hyperplane through time or rotate a fixed projection or cut through time, but not both simultaneously.
- 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)
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
Still they've had a strong impact in how I see systems - orbits, instability, etc.
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.
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.
