Chaos Equations - Simple Mathematical Art

[@carykh]

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

[@tygermarez2556]

of course id see you here lol.

[@CodeParade]

Debussy is my favorite classical composer, I also learned to play First Arabesque and Clair De Lune.

[@Magnogen]

Hey Cary! Fancy seeing you here! Lel

[@SonicPman]

What is this, a crossover episode?

[@julespoon2884]

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.

[@crunchiesjl]

"CODE: HELPME" I think the math is holding him hostage

[@portal6347]

FIND X OR YOUR FAMILY GETS IT

[@DTorto]

try lackno

[@QuakDucc]

Quality content has been officially computer generated.

[@f.jideament]

Quality content has been logically computer generated.

[@QuakDucc]

@@f.jideament I don't see the _logic_ behind that statement *ba dum diiiiing*

[@drvanon]

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?

[@dudejoe8705]

@@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.

[@vbgvbg1133]

drvanon I can do one better, *if the machine is somewhat intelligent, is it the author?*

[@THExRISER]

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.

[@kennygaming208]

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,

[@THExRISER]

​@@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.

[@oscill8ocelot]

3:00 Are you okay, CodeParade?

[@CodeParade]

Hehe, I was trying some funny codes as a joke, but I liked how it looked so much, I had to keep it.

[@oscill8ocelot]

@@CodeParade That's just what someone in distress would say to throw off the scent! =P

[@Woetson]

@@TheBigLou13 Wooooosh

[@Woetson]

@@sussybaka3603you're*

[@isabelle5547]

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

[@cajunbeats909]

The colors, the shapes, the music, these are legit just functions, but theyre giving me serious goosebumps

[@giantneuralnetwork]

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.

[@CodeParade]

That's really neat, I never though to do it with neural networks.

[@giantneuralnetwork]

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.

[@WildAnimalChannel]

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...

[@mauroacosta5115]

@@giantneuralnetwork hey! I Have a question where did you applied the functions (sofwares)? Do you if it can be done on proccesing?

[@giantneuralnetwork]

@@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.

[@prec1sion548]

Teacher: today we are going to start with algebra My brain: chaos equations

[@swine13]

Would you like to upsize for $1? *chaos equations*

[@Open6a-fx4qf]

My brain : a black hole?

[@joda7697]

Algebra? That is a calculus thing. Those are literally differential equations.

[@alexanderliang2437]

This is quiet beautiful and dramatic. PRQSTS OGIUOC OHSHIT

[@roxxane019]

OH SHIT

[@StainlessHelena]

The first one is giving me interdimensional-dickbutt-vibes. I know the meme is long dead but I can't help it.

[@alexanderliang2437]

@@roxxane019 XD

[@danny.belanger]

You just break my brain

[@lachainedudeveloppeur2556]

MAXIME is also beautiful

[@thecobra8508]

chaos equation's: *chaotic yet beautiful*

[@lucbrisebois767]

This channel is such a hidden gem! Awesome stuff.

[@anon89461]

Yeah

[@leonardosoaresfuks294]

for me, this is one of the best definitions of beauty

[@floatingturtle2512]

Next project; 4d game engine

[@CodeParade]

I'm legit thinking about it. It's been 10 years, and Miegakure still isn't out!

[@ShroomLab]

@@CodeParade I once made a 4d ray tracer, you can have the code

[@henryseg]

CodeParade There’s more than one way to do a 4D engine!

[@codynelson424]

@@CodeParade I made a 4d minecraft clone

[@aldobernaltvbernal8745]

@@ShroomLab can you please share the github link? I would love to experiment with it!

[@xenontesla122]

What’s really cool is that you could see these as cross-sections of 3D fractals.

[@pmqtpqbtmz3956]

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.

[@singularity-]

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.

[@Xeon2112]

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!

[@santoriomaker69]

Good choice of music! Man, didn't knew Debussy's Arabesque 1 would really fit the theme of mathematical art. Just so damn amazing

[@argile5]

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.

[@triston_03]

The closing of JJPVDN (2:46) was so satisfying--it game me legitimate chills.

[@WangleLine]

This is so beautiful O:

[@tgfjrfjfgjgfj]

If you try the code : CHAOS_ you got a perfect circle at t=0. What a coincidence

[@dudeawsomeness1]

I thought you were wilburgur

[@ivarangquist9184]

Hey, I recognize you from GM48. The internet suddenly feels smaller.

[@WangleLine]

@@ivarangquist9184 Oh, that's sweet, nice to see other fellow gm48 people over here! :D

[@landsman420]

I could cry, it's so beautiful... The forms, they just make sense

[@kira_15_R3D]

it was beautiful to watch, the music fits well, good job dude, thank you for making this

[@galihdwikaputraragayatsu892]

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

[@ps8883]

That is not 'Simple Mathematical Art', it is awesomeeeee

[@CodeParade]

Simple in the sense that it is simple to make, like the Mandelbrot set.

[@olivergurney9622]

Why is this channel still slept on? Keep it up man and you will be rewarded 100%

[@traderslick9763]

Magnificent. I do love your creative activity.

[@EJlol]

This reminds me of Chaoscope, some program I used to toy with attractors years ago. Really fun to play with

[@SolarizedWasTaken]

The universe is just a complex chaos equation and i love it

[@MDud]

Amazing to see your code and eqations visualized. Great work!

[@dylansedits8047]

I don’t know what it is but something about this makes me feel incredibly touched

[@Kycilak]

Probably the music

[@lunaeclipse5768]

Equation is a true magic and then solving the equation and realizing everything is related is mind blowing.

[@wesleythomas6858]

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

[@the-selfish-meme7585]

Beautiful - from Galoise to Lorenz to Feigenbaum.... it never gets old.... thank you....

[@aguiscard2452]

Studying this for 3rd year engineering at university, we call them "Nonlinear Dynamical Systems", with attractors being a subset of that categorization I believe

[@SebastianVik]

2:31 wow, 2 black holes with spiral galaxies colliding in the middle. this is amazing man

[@taylorkinsey7737]

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).

[@josecontreras3396]

A parametric equation is where you define the x and y with a third parameter so they kind of are parametric equations

[@anindyabiswas1551]

I have no idea how surprising a simple equation can be! I always work with them and they always surprises me! I love math!!!

[@only2ndplace]

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.

[@topchrischang471]

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!

[@nictibbetts]

Cringe

[@safir2241]

My dude, you are an inspiration for so many of us viewers.

[@halfnwhole751]

When math is better at art than you

[@f.jideament]

What do we call it as? Mathist?

[@votemefordictator4910]

@@f.jideament Martist.... Artician... Arthimatician?

[@TernaryM01]

@@f.jideament Generative artist en.wikipedia.org/wiki/Generative_art

[@DiamondSane]

math is not, it's not artist, but a tool

[@nivlescollection3509]

Artmathist

[@Ludifant]

It´s new to me, reminiscent of thousand point sculptures, but I like that it is discreet, has no random element and still fills the field at some values. Very interesting and useful!

[@EBTS-3]

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👌🎥

[@koalatycontent1305]

I really enjoy the simplicity yet shear beauty of ssnnnj. It progresses through many different "phases".

[@delta_yd]

In the future, entertainment will be randomly generated

[@igorchistyakov8876]

WEED EATER

[@jinjunliu2401]

I love how this is suddenly everywhere while it's from 2001 or so

[@Hello-qg4yk]

Yeah

[@steve-usmcvet8934]

Somewhere inside all of this is a complete understanding of the entire universe.

[@marbleswan6664]

Just learned about these 2 hours ago, (well i heard of them). And now you have a video out.

[@karrensusan4825]

POO_ _ _ is so chill and relaxing, looks like an animated logo as a background

[@Tundra-ec3ii]

They’re like the awesome old screensavers that became the only redeeming quality of elementary school computers! (Just better!)

[@diaanlouw5039]

this was one of the most beautiful videos I have ever seen

[@vishnuvs6121]

The music makes it even better.

[@gastrointestinalgaming7629]

IDK why but when you put the music over the visualized equations it looks like an anime intro and I absolutely adore it

[@edwardlau892]

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.

[@lucascyrne]

As an artist, and don't get me wrong, I truly guess we all are; this is some of the most beautiful things I ever see. Fuck academics and shit, this seems to take a deep trip inside me and the way I see life. Thanks for the found and the share! I certainly gonna play with this

[@AbsurdAsparagus]

Math is so much more than the dumbass equations hw that they shove down our throats in school. Its tragic that math class has so little art despite math being so artistic in nature.

[@monochr0m]

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.

[@thedanieljamesjeff]

This is some real behind the curtain stuff Respect brother

[@morganstoodley8591]

It's beautiful...

[@shardgunner4815]

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 !

[@istream222]

I just love how maths and computation work in a symbiotic manner.

[@raydencreed1524]

Was that Desmos in there? At around 1:10

[@CodeParade]

Haha, yes it is! I was too lazy to animate it myself :)

[@kangar1797]

Desmos will conquer the world mouahshaha !!!!

[@telecorpse1957]

@@kangar1797 It's funny how I have just added Desmos to my bookmarks for helping me check whether or not I solve equations/inequalities correctly.

[@ittotaq]

@@telecorpse1957 There's an app too.

[@user-ro1cc8tz6d]

You can do the same with geogebra but i should check desmos too

[@maya-cc2sx]

Looking at this makes me feel like physics is actually a tool for mathematics so we can see all these beautiful graphs

[@BuildEver]

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 :)

[@zxGHOSTr]

You sir, you created something magnificant!

[@Assault_Butter_Knife]

Imagine calling fluxions "time dependent equations" THIS POST WAS MADE BY NEWTON GANG

[@joalampela8612]

@@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.

[@liquiddonkey6530]

[@DeDeNoM]

Reminds me of lissajous patterns. Have you tried adding trigonometric functions to the mix?

[@user-lt2rw5nr9s]

Maybe someone will add them to the source code

[@jackbrand4507]

God I want to see that so badly

[@charleslambert3368]

You don't need trig for lissajous. Just some extra hidden variables behind the scenes. y'=y+dy x'=x+dx dy'=(1-C)dy-Ay dx'=(1-D)dx-By A and B are close to a whole number ratio (e.g. 3 and 2.01). C and D are small or zero. Or get some broomsticks and weights and build a harmonograph.

[@ineednochannelyoutube5384]

Trygonometry all just boils down to the cosine and Pythagoras laws. They can be expressed algebraically.

[@benedictus5657]

im writing my own version of this in julia right now, if you're still interested i can show you the results with trig functions :)

[@ggz7651]

This is one of the best channel ever.

[@AnastasisKr]

This is very beautiful.. Btw, if you're interested, I think these are modili spaces for the orbit under a certain polynomial dynamical system.

[@sunkingarthur]

The last one really takes form in my life as smoke. When someone lets out a small billowing cloud these forms take hold.

[@zangarkhan]

Nice! Love to add a z axis and see what cool stuff happens in vr

[@IsraeliXdude]

Dont you mean 3D?

[@Blueyzachary]

The music makes this amazing to watch!!

[@Exaspatial]

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!

[@akramaka1111]

omg it's fricking awesome

[@Exaspatial]

@@akramaka1111 haha thanks! Just revisiting some old liked videos and I'm glad to see you enjoyed it 👍

[@nodustollens9183]

criminally underrated channel

[@trevordendel4257]

3:37 looks like the visual representation of the golden ratio as seeds in a flower that Numberphile did a while back. 🤔

[@poraodoojimo]

It looks like a portal

[@thomasbauden198]

Or just infinite density before infinite expansion of the universe

[@N3G4T3]

This look amazing on an OLED display

[@GhostStealth590]

“Simple things middle schoolers can understand”...... suppose I was left behind.

[@jetison333]

I dont think it was a very good explanation

[@gillesdeleuze1900]

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

[@hdjfhhjbfjdhjk4256]

@@gillesdeleuze1900 thank you I was starting to think I was dumb as hell

[@adamfreed2291]

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.

[@GhostStealth590]

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.

[@donutdude4174]

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

[@Mlksgf]

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

[@Mlksgf]

@meme I didnt know what these functions where until you said it. Thanks for the info! Parametric area is beautiful!

[@chase_h.01]

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!

[@hughjazz4936]

That's the exact opposite of chaos but it's beautiful and mesmerising anyway. I particulary enjoyed the last one, don't know why. Very nicely done!

[@zeekowolf]

it would be nice if you could pan around some 3d versions of this

[@CodeParade]

Yeah, the algorithm generalizes very naturally to 3D. Maybe I'll explore this in the future.

[@SpektralJo]

@@CodeParade couldnt you create the actual Lorenz atractor then?

[@xenontesla122]

SpektralJo (Edit: You could do it because the Lorenz attractor is 1st order.) That’s based on differential equations, so a point in space won’t always go to the same next point (depending on speed). But here it always goes the same point.

[@zoltankurti]

@@xenontesla122 you are wrong. In case of the lorentz attractor, it only depends on the point since it's a first order equation. You are talking about second order differential equations, those need position AND speed as initial conditions. Not first order ones.

[@zoltankurti]

@@xenontesla122 and this also is based on differential equations...

[@BiskviLover]

Doesn't look too chaotic to me. Just beautiful.

[@secretfurry6346]

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

[@PaintedSkies248]

Nice music choice, very relevant. Chopin is organized chaos

[@kadmani]

Would be great if you make a full tutorial on one of these!

[@Jan-ex3wj]

please never stop making videos

[@TheRainHarvester]

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!

[@isaacdonnan7316]

The foundation of beauty is knowledge.

[@sbilldmilk]

3:00 "Code: HELPME" Also these would make really cool loading/menu screens for a game

[@Dezomm]

Awesome music choice :) and a very cool video to boot!

[@anfiwa]

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.

[@floriandaler5327]

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...

[@JhonatanNeneh]

Beautiful! I love the music. It's a different perspective for me. Thank you for sharing.

[@iamasalad9080]

1:57 (paused) cardioid's back at it again :0!

[@TyronTention]

Nice addition of the Debussy Arabesque.

[@eggyherman]

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!

[@robbiekavanagh2802]

I really love this channel, keep it up man

[@KomoliRihyoh]

Its interesting that a lot of the images have seven points/parts. Not something i’d expect from random maths

[@QuakDucc]

OH I SEE IT

[@agcummings11]

I think he's just applying the transformation 7 times

[@CodeParade]

Yeah I see a lot of 3's, 4's, 5's, and 7's. Also surprising to me. And, no, I apply 800 iterations so almost everything blends together into a white anyway, it looks the same no matter how may iterations you use.

[@Rubikorigami]

Komoli Rihyoh It seems like he mostly used quadratic equations for the images, that may have something to do with it.

[@agcummings11]

@@CodeParade Right after I commented that it was 7 times I looked at the source code and realized I was wrong rip

[@sciencoking]

Tha's absolutely amazing. I didn't think most equations would produce attractors

[@Klex816]

If someone asks for the background music: Claude Debussy - Arabesque No. 1 You're welcome