Chaos Equations - Simple Mathematical Art
ฝัง
- เผยแพร่เมื่อ 28 ก.พ. 2019
- This is based on a very old project I made originally in Game Maker, but I updated it to a new polished program.
Download (Windows 64-bit): github.com/HackerPoet/Chaos-E...
Source Code: github.com/HackerPoet/Chaos-E...
Music:
Arabesque No.1 - Claude Debussy
Performed by: Luc Laporte Sr
/ luc-laporte-sr
www.orangefreesounds.com/
Whoa, these do look like strange attractors! Especially the one at 2:37, the spirals there almost remind me of ones you see in Mandelbrot sets. Also... Debussy's First Arabesque is one of the few pieces I can still play on the piano, so i recognized it right away heehee
of course id see you here lol.
Debussy is my favorite classical composer, I also learned to play First Arabesque and Clair De Lune.
Hey Cary! Fancy seeing you here! Lel
What is this, a crossover episode?
They are indeed related to Mandelbrot sets. Ok, closer to Julia sets actually. Parametrically, the Julia set describes all points c=(x,y) such that the set generated by the recursion x’=x^2-y^2+a, y’=2xy+b and c is closed, where a and b are reals. That said, a special case of the video described by x’=x^2-y^2+f(t), y’=2xy+g(t) would generate subsets of the Julia set at point f+g*i. Doing a transformation of this equation would give a ‘warped’, somewhat empty, Julia like object. Which is why a lot of the video’s equations here have those spirals, which btw are misiurewicz points of the recurrence. Now, the reason why it looks like the Mandelbrot set as well is because the Mandelbrot set behaves like a Julia fractal locally, which can easily be seen from the eqns that generate them.
"CODE: HELPME"
I think the math is holding him hostage
FIND X OR YOUR FAMILY GETS IT
try lackno
This is quiet beautiful and dramatic.
PRQSTS
OGIUOC
OHSHIT
OH SHIT
The first one is giving me interdimensional-dickbutt-vibes. I know the meme is long dead but I can't help it.
@@taxxon_ XD
You just break my brain
MAXIME is also beautiful
Things like these is why I decided to study math again from the start on my own, but this time, I have no tests to worry about, and I'm actually gonna put that knowledge to use now that I'm learning to code.
It seems like math in school was purposefully boring, I'm an F average but Im obsessed these videos and documentaries, it would be a slap in the face for me to start doing math after failing it every year of my life,
@@kennygaming208 My grades were awful as well, but there's something so rewarding about learning and applying math in a project, no matter how simple the concept is, don't let the education system make you hate something you're passionate about.
Quality content has been officially computer generated.
Quality content has been logically computer generated.
@@f.jideament I don't see the _logic_ behind that statement
*ba dum diiiiing*
This is the basis for a fun philosophical debate: obviously there is nothing wrong with your statement that this is computer generated. But whom would be the author of the art work? Do we say that the coder is the one who was the architect of the code and thus the author of it, or do we recognize the people that have worked towards building the machines that made it possible to do this. And what about the pure maths. Does that receive any praise?
@@drvanon I would say that the author of the code is the author of the work, if a painter creates a work of art, the credit goes to the painter, not the person who made the paintbrush or paint itself.
drvanon I can do one better, *if the machine is somewhat intelligent, is it the author?*
3:00 Are you okay, CodeParade?
Hehe, I was trying some funny codes as a joke, but I liked how it looked so much, I had to keep it.
@@CodeParade That's just what someone in distress would say to throw off the scent! =P
@@TheBigLou13 Wooooosh
@@sussybaka3603you're*
PixelTheif "Haha, you're correcting me in a youtube comment section. Fucking Loser." "Oh... I'm also going to correct you." how desperate for attention do you have to be lol
Teacher: today we are going to start with algebra
My brain: chaos equations
Would you like to upsize for $1?
*chaos equations*
My brain : a black hole?
Algebra? That is a calculus thing. Those are literally differential equations.
The colors, the shapes, the music, these are legit just functions, but theyre giving me serious goosebumps
When math is better at art than you
What do we call it as? Mathist?
@@f.jideament Martist.... Artician... Arthimatician?
@@f.jideament Generative artist
en.wikipedia.org/wiki/Generative_art
math is not, it's not artist, but a tool
Artmathist
Cool! My profile picture came from doing the same with a function called an LSTM in machine learning, just recursively applying it to a point cloud. It’s surprising how complex these patterns in your video can be coming from such simple equations! Thanks for sharing, love your work.
That's really neat, I never though to do it with neural networks.
CodeParade Yeah! Thought it’d reveal something about their inner structure or modeling power but in the end I just got some pretty pictures :-) I think the patterns did change a bit with different activation functions or bigger nets, will have to revisit it sometime.
Interesting. Well the input would be a vector V=(x,y,x^2,y^2,t,1). And then you'd apply M^n V where M is a matrix of weights. But in a neural net you'd also apply like a sigmoid function. Yes, I can see the connection. So it's like these are patterns represent the complex things an recurrent neural network can replicate. Maybe even be Turing complete with very precise input values. Fascinating. Maybe you'd have to allow quadratic neural networks instead of just linear ones even though. hmm...
@@giantneuralnetwork hey! I Have a question where did you applied the functions (sofwares)? Do you if it can be done on proccesing?
@@mauroacosta5115 I did it in python in a jupyter notebook! I used opencv to plot the points in an image after. You could do it in processing but you'd have to implement your neural net there.. while many neural net libraries exist for python.
This is one of the most beautiful things I’ve seen on the internet in a long time. I’m amazed by how simple it is and am so glad I was able to see it. Great choice of music too!
it was beautiful to watch, the music fits well, good job dude, thank you for making this
Next project; 4d game engine
I'm legit thinking about it. It's been 10 years, and Miegakure still isn't out!
@@CodeParade I once made a 4d ray tracer, you can have the code
CodeParade There’s more than one way to do a 4D engine!
@@CodeParade I made a 4d minecraft clone
@@ShroomLab can you please share the github link? I would love to experiment with it!
Thanks for uploading this. I'm doing research for my own amusement but also for the purpose of putting together a video for a scholarship contest, so I can go to school to study physics. I'm thinking of making my topic Chaos theory, and so a search brought up your video. It's so lovely. The way some of them got increasingly clomplex and then seemed to dissolve or evaporate, leaving a shell behind which gradually became a simpler and simpler outline until they seemed to spiral down a drain and poof from existence is just breathtaking when paired with the music. I know others have said it, so I'll just confirm; the music was a great choice.
chaos equation's: *chaotic yet beautiful*
What’s really cool is that you could see these as cross-sections of 3D fractals.
That is not 'Simple Mathematical Art', it is awesomeeeee
Simple in the sense that it is simple to make, like the Mandelbrot set.
This channel is such a hidden gem! Awesome stuff.
Yeah
Amazing to see your code and eqations visualized. Great work!
Why is this channel still slept on? Keep it up man and you will be rewarded 100%
Magnificent.
I do love your creative activity.
Studying this for 3rd year engineering at university, we call them "Nonlinear Dynamical Systems", with attractors being a subset of that categorization I believe
for me, this is one of the best definitions of beauty
Good choice of music! Man, didn't knew Debussy's Arabesque 1 would really fit the theme of mathematical art. Just so damn amazing
This is so beautiful O:
If you try the code : CHAOS_ you got a perfect circle at t=0. What a coincidence
I thought you were wilburgur
Hey, I recognize you from GM48. The internet suddenly feels smaller.
@@ivarangquist9184 Oh, that's sweet, nice to see other fellow gm48 people over here! :D
I could be totally wrong here, but the use of "t" as a third, outside variable kind reminded me of parametric equations. Maybe this is a just a really cool way to visualize them. There might be something more to it, though, judging by how you use those same equations to generate the particles (like nested functions).
A parametric equation is where you define the x and y with a third parameter so they kind of are parametric equations
I'm so glad your channel is finally getting the recognition it deserves xD been a Sub since 6k which wasn't that long go
Just beautiful stuff as always. I love your videos!
As someone studying MIS with no idea what I want to do in life, I'm really drawn to these videos and wish I would have taken a path like this for life.
The coefficient for each term is either -1, 0 or 1, and each iterative equation would have variables x^2, y^2, t^2, xy, xt, yt, x, y and t. That gives us 3^18 permutations, and you used one letter of alphabet along with an underscore to contain 27 combinations, in order to use 6 letters to represent all possible combinations. Neat.
Wild guesses below:
x^2, y^2, t^2, xy, xt, yt, x, y, t coefficients for x' correlates with the first three letter, while the nine coefficients for y' is contained by the latter three.
_ is (-1,-1,-1) for the three terms it corresponds, and A is (-1,-1,0), D is (-1,0,0), so it increments by balanced ternary system.
I don't usually post comments, but I just wanted to mention I love your work and you're an amazing and talented person! It's a lot of fun listening to your explanations and looking at your creations. Keep up the amazing work! =D
This is amazing! Thank you for sharing the code/program
Amazing. I used to try to make these back in 1989. My computer was just too weak and slow to get this kind of look. These are amazing.
This reminds me of Chaoscope, some program I used to toy with attractors years ago. Really fun to play with
I could cry, it's so beautiful... The forms, they just make sense
Awesome music choice :) and a very cool video to boot!
Some of the ones I have found I like recently are: ABSURD, HIGHER, and BOTTLE
Some of these look like the lifespan of some sort of deep sea creature made of bioluminescent nerves, very weird how even random equations can form these beautiful organic patterns ! Thanks for the upload this was very well displayed👌🎥
Beautiful! I love the music. It's a different perspective for me. Thank you for sharing.
This is so cool and crazy to me. This is literally how I always pictured the band Orchid’s music, and their debut album was called Chaos Is Me. I think you picked a great name !
The universe is just a complex chaos equation and i love it
What I find amazing is how many cosmic things look like many of the demos. I saw spiral galaxies and solar flares a lot in this
I really love this channel, keep it up man
Beautiful animations! And excellent choice of music :)
I don’t know what it is but something about this makes me feel incredibly touched
Probably the music
Just learned about these 2 hours ago, (well i heard of them). And now you have a video out.
this was one of the most beautiful videos I have ever seen
Equation is a true magic and then solving the equation and realizing everything is related is mind blowing.
I guess the technical term for these equations in nonlinear dynamics would be "a family of discrete chaotic maps". If you want to treat them as a dynamical system your t is not really time. Instead the time is characterized by the number of iterations n of the equations. You start with a point (x_0, y_0) = (t,t) (initial condition) and each iteration of the equations (the map) gives you the next point on the trajectory. In other words you have (x_n+1, y_n+1) = f(x_n, y_n) which is a discrete map because the "time" n is not continous.
If you look at the result for some fixed t you are looking at one trajectory of the dynamical system up to the time n which equals the number of points you draw. Because your map also depends on t it is actually not a single map but a family of parameterized maps. You could say if you change t you are changing the "laws of nature" by which your dynamical system operates in a controlled manner.
A well-known one-dimensional example of what you are doing would be the logistic map en.wikipedia.org/wiki/Logistic_map where the control parameter r corresponds to your t.
All of this also means that you are indeed looking at strange attractors in many of these images.
So What it's going to be called if it is a 2D version?
We currently want to parallelize this algorithm using Parareal en.wikipedia.org/wiki/Parareal and wonder if it is possible to do that.
It would be great help if you can answer this. Thank you!
Cringe
It's beautiful...
Hey, thanks for sharing this! Super cool and super simple! I've got a lot of ideas for how to use it already :D
Somewhere inside all of this is a complete understanding of the entire universe.
In the future, entertainment will be randomly generated
WEED EATER
I love how this is suddenly everywhere while it's from 2001 or so
Yeah
They’re like the awesome old screensavers that became the only redeeming quality of elementary school computers!
(Just better!)
Hey CodeParade. I'd like to say first of all that what you made is absolutely amazing. I've come across one equation in particular- to me- resembles works by Piet Mondrian; the code is OMMDMM. Again, wonderful work.
This is one of the best channel ever.
The music makes it even better.
This is very beautiful.. Btw, if you're interested, I think these are modili spaces for the orbit under a certain polynomial dynamical system.
Beautiful - from Galoise to Lorenz to Feigenbaum.... it never gets old.... thank you....
This makes the universe make sence
They might not be strange attractors, but some of them are attractors nontheless. If we consider the x-y-plane as the phase space with x = x and y = d/dt x you just modelled non linear systems. This is pretty common in numerical physics.
“Simple things middle schoolers can understand”...... suppose I was left behind.
I dont think it was a very good explanation
That’s because it’s not true. Chaos theory and dynamics are very rich and fascinating fields of math and it’s very unhelpful to present them as something trivial. You shouldn’t feel dumb for feeling left behind. It’s difficult stuff, not middle school math
@@gillesdeleuze1900 thank you
I was starting to think I was dumb as hell
Honestly, that was a lie. They are things that one can explain to a middle schooler and have them understand it, but certainly not something a middle schooler should be able to look at and understand. These are called 3-variable differential equations.
The first idea is to have a line where the slope is constantly changing. It would bend, since the slope never stays the same. You can even write an equation for how the slope would change. This is what happens when you do that.
Adam Freed If I had a better memory and more patience, I feel as if I would enjoy math much more. But considering it’s been 3 full semesters that I haven’t covered slopes, I’ve kinda forget all about them. Perhaps if it was the field I was going into and felt passionate about the field, I could retain a bit more... but I really cannot. Also three semesters ago in college, not high school lmao. So sometimes I do feel dumb, but I understand I could be more dumb than I am currently, as I am attempting to work on my self-esteem and look at things as they are rather than my low self esteem outlook on myself. Perhaps if I dedicated a study session or two to this subject, maybe I could fully comprehend what the subject matter of this video fully meant.
My dude, you are an inspiration for so many of us viewers.
The foundation of beauty is knowledge.
Nice! Love to add a z axis and see what cool stuff happens in vr
Dont you mean 3D?
These are awesome, they look like fractals but they also look like a sliced pieces of a 3d structure as the time progresses , im pretty sure that if you put time in a vertical axis it will be a structure! Trust me i'm a sculptor :P
@meme I didnt know what these functions where until you said it. Thanks for the info! Parametric area is beautiful!
Phenomenal my guy, well done. Thanks for sharing.
this what the people needa see. much love man 🙏🙏
3:37 looks like the visual representation of the golden ratio as seeds in a flower that Numberphile did a while back. 🤔
It looks like a portal
Or just infinite density before infinite expansion of the universe
Imagine calling fluxions "time dependent equations"
THIS POST WAS MADE BY NEWTON GANG
@@tf2excession I think talking down to people on a platform from which you have access to Google's search engine is just silly and very demeaning to the reader.
You sir, you created something magnificant!
man you have an amazing channel
Btw. The t - time, has nothing to do with this being a 'dynamical system'. Even for a single t, it is a dynamical system. The dynamics relates to a single point trajectory at single parameter. It is dynamic because point is jumping at each 'step' of applying equations.
I would rather say t is a parameter, not a 'time' in a sense of dynamical system.
Also not every dynamical system is chaotic and/or interesting, so I wouldn't call them 'chaos equations' either.
Nevertheless I do love chaos theory, and dynamic systems (discrete and continues) and pretty pictures generated by simple rules. Simple to learn and code, yet creating complex things. In fact these are the things that were the first I ever coded in my life (i.e. logistic equation evolution, bifurcation diagrams, Lorentz equations, etc)
DMY_GD is one of my favorites. The reason I say "one of my favorites" is because there are still like thousands of more to see!
New favorite, ------ LIIOLX ------ 0 trail persistance, 2 speed, and point size 1!
omg it's fricking awesome
@@akramaka1111 haha thanks! Just revisiting some old liked videos and I'm glad to see you enjoyed it 👍
this is truly beautiful. Thank you.
Terrence McKenna lectures brought me here lol... It's funny how I never liked math, but watching stuff like this makes me want to understand it more... Really wish mathematics wasn't taught so boringly in school.
I'd like to recommend making a program that automatically creates scrollwork within a drawn shape. Scrollwork is all based on formula and mathematics, but takes talent to actually draw it out. I feel it could be done via computer if someone decided to make it software for it. That'd be amazing!
I think the mathematical concept closest to this is Differential Equations. How the new coordinates are calculated isn't exactly clear from the explanation in the video, so I can't tell for sure, but it looks very much like the solutions to some (ordinary two-dimensional) differential equations (of first order). Interesting stuff, there's a lot of literature about it.
I unterstand it like this: at the beginnig of each frame, x and y are set to t. After that, the two equations are executed several (thousand?) times and after each iteration, the new point gets plotted onto the screen. After that (for a new frame), the screen gets cleared, t gets incremented by a very small amount and everything starts again.
I hope that my explanation is correct and clears this topic up...
breathtaking. thank you for sharing!
criminally underrated channel
Brother, you should make a 1080p or 4k version for a Live wallpaper.
oh yeah and Good choice of music, Debussy always fits with things like immersions.
Cool! I used to make parametric equations on my Apple ][e. Good times! Lately, I've been experimenting with primordial particle systems. I'll post more video soon!
The most beautiful thing I've ever seen 🔥🔥🔥
IDK why but when you put the music over the visualized equations it looks like an anime intro and I absolutely adore it
Could you use "t" as a Z dimension and show those shapes as 3D entities? Because right now they look like cross sections of some interesting things :)
you need to live stream this looping
Just found your channel. All your videos are awesome keep it up!!!
Lovely. Thank you for posting this.
1:57 (paused)
cardioid's back at it again :0!
3:00 "Code: HELPME"
Also these would make really cool loading/menu screens for a game
I have no idea how surprising a simple equation can be! I always work with them and they always surprises me! I love math!!!
This is some real behind the curtain stuff
Respect brother
Very nice work!
I read a book in 1993 "Strange Attractors: Creating Patterns in Chaos" by Julien Sprott that created 2D works based on higher dimensional projections that looked a lot like these, and they also had been given letter codes to generate with the software that came on diskette with 3D-Glasses! Check it out!
so the characters can be devided in 2 groups
group a and group b
group a controlls x and group b controlls y
setting one group to MMM will result in 0 for that axis
when group a and group b is identical a 45deg line will be drawn
dude. this is beautiful
i don't know how to express it... while i am watching this, i am on the verge of tears... so beautiful.... i mean, if school teach me first about the beauty of math before blocks of formula, maybe i can endure the hardness of learning it when i was in scholl
If someone asks for the background music: Claude Debussy - Arabesque No. 1
You're welcome
Would be great if you make a full tutorial on one of these!
This look amazing on an OLED display