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- Hacker News
- It would be so much fun to use these as a setting for some kind of gaming experience. Like, I wanna hide behind parts of these and pop around a corner and blast my friend with a laser. Or to race gocarts along the surface of one, or I dunno, something frogger-esque to get a feel for the directionality of the flows... I love how they look, but I need more interaction to get a feel for the thing.
- Thank you for sharing. This really inspired me to check out three.js. This website of yours might be one of the more beautiful things on the internet.
- Really visually wonderful. I tried to self learn about nonlinear dynamics after reading about Takens's theorem last year but I have to admit, I have no idea what an attractor is actually showing like this.
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.
- Coincidentally enough, I dug out my 11th grade CS project on generating fractals from 2002 & modernized it using SFML graphics lib just this week.
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)
by srvmshr - Thanks a lot, it was clearly worth the effort.
- Hobbyists hacking around and sharing their art, best part of the Internet!by orzig
- I don't care about the math, the computation, the physics. This is just by far the most beautiful thing(s) I have ever seen.by aniijbod
- > A small change in the parameter a can lead to vastly different particle trajectories and the overall shape of the attractor. Change this value in the control panel and observe the butterfly effect in action.
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.
by evanb - "not sure if it’s mathematically correct,"
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.
by jerf - This is so cool. Back in highschool during the Jurassic age I used ti play with attractors a lot. Unfortunately on a 486 it took 20-30 minutes to draw one even at low resolution. This renders in realtime and in 3D. Great work!
Still they've had a strong impact in how I see systems - orbits, instability, etc.
by Grosvenor - If that was in the Jurassic, I guess I went to high school in the Cambrian. But hey, I like trilobites!by mcswell
- Fractint4life https://fractint.org/by anjel
- I bought "Strange Attractors: Creating Patterns in Chaos" (1993) by J. C. Sprott recently, which is a fun book about these kinds of attractors. The whole book can be downloaded online [1] from the author's web site [2].
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
by atombender - I have a pdf of this book and was using LLM to translate the old code into modern, idiomatic python and it is very cool. I wonder if somebody will re-release it with modern code and tooling? In fact , google Gemini was able to do it on the fly using the posted links.
- The book software https://sprott.physics.wisc.edu/fractals/bookdisk/by AlexeyBrin
- Neat :) When I was a teenager, some 25+ years ago, I wrote a chaotic attractor visualiser like this — but only in 2D — and it occurred to me, “What if instead of visualising it, I rendered it to audio?” I don’t remember the details: I think frequency was correlated with polar angle and amplitude to magnitude. It forced me to learn how to write WAV format — which was my first introduction to endianness — but the result wasn’t completely inaudible! A bit like the sound effects for computers in old sci-fi movies; random(ish) but not discordant beeps and boops!by Xophmeister
- Along these lines there are at least two modules that I know of in Eurorack focused on strange attractors, and they're both a LOT of fun adding this kind of unpredictable-but-cyclical movement to your sounds:
- 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)
by gausswho - Visualizations like this truly highlight how much there is to be gained from viewing the 3D phase space, but also how much richness we miss in >3D!
(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!
by cableclasper - The conclusion I’ve come to from works like Flatland, 4D toys, etc., is that we simply don’t have the neural circuitry to grasp anything beyond three dimensions. We can reason about them, we can make inferences about the whole from partial understanding, but we cannot truly grasp more than three, or perhaps only for an instant of forced conceptualization using heuristics like you mentioned. Even three is a stretch, our minds have adapted to build a three dimensional realm from something like a 2.5 dimensional field of combined visual, tactile, and auditory stimuli. I suspect 3D reasoning itself is a huge adaptive trait compared to most other animals.by flatline