The Double Pendulum Fractal 

0:06 It's actually fascinating how the texts morph, must be AI's work

Why is this comment so high up with only 6 Likes

@@bendank9762 verified

@@dlrss1v274 I want the text transitions now lol

Wow it’s carykh

l i t e r a l l y

You've got cockroaches in your bathroom? That's pretty gross...

don't slip!

Reddy what country are you from?

​@@axelrios276 he probably lives in the same place where i live

Basicly: "A tiny change cause an entirely different outcome" has been made visual. A tiny step (left, right, up or down) has a different color (a different outcome). It also shows that small angles (center) tend to stay ordered. As they move slow and stable.

@@suddeneevee9441 thank you! I understand now. I was looking for a comment like this 😅

@@suddeneevee9441 Yes. I understand that and also much interested a lot in the chaos theory stuff. But what is the source of the color image? And what exactly is happening to it?

​@Hritik Vaishnav the color image is the angles mapped to an image (I think) And the angles changing is what's happening to it

💫synth toilet💫

Best comment

r/brandnewsentence

synth toilet is my new favorite phrase

✨synth toilet✨

Assuming x and y correspond to increasing angle starting from completely upwards the "spikes" would be roughly along the line y=-cx, where c is slightly less than 1. To my intuition this is where the bottom mass is almost directly below the static point of the first pendulum, forming a sideways v. Idk the mathmatical implications but that may help visualize.

the relatively stable region also contains those branches

I actually more interested in the lines in the top left and bottom right just in middle of the chaos.

@@828burke yes, so basically the entire double pendulum oscillates like "> < >

@@APozzi Yup they look like a small region of persevering order for some time

Exactly! And I think I could render it to much higher precision if I had a GPU. To simulate a pendulum I used a time step of 0.0005, and I know it could definitely get smaller. I think part of the reason why the random noise appears in the fractal is due to the inaccuracy of the simulation. This is an unavoidable feature of the simulation, but I think the random noise would be delayed with higher precision.

@@sammaksimovich2654 all those scalpers are ruining so much stuff... I hate em

@@sammaksimovich2654 It would be cool to see how different time steps affects the fractal. I am going to try and simulate this in unity with compute shaders :)

@@sammaksimovich2654 couldn't you theoretically adaptively choose the time step based on the state of the simulation to cut down on computation where the expected error is the lowest (or even smaller than the representation of the numbers you use can represent)? Also, this looks to me like something that could benefit *a lot* from using fixed point numbers for accuracy.

Maybe this could be more efficient too if it was implemented as a texture shader?

and they get so small that they’re smaller than a pixel, making it less opaque and then add a ton of other semi opaque colors onto it and they mix

Of course, the destabilizer for the methamphetamine ununoctide is quintesentially conclusive to the equilibrium that instigates the proprietarianist conjugation that had been discovered

our eyes are so overpowered and our vision is so easily screwed with lol

Is it You that doesnt understand english?

Atomic shrimp I see you everywhere man. Love you videos man!

oh awesome to seeu here

It looks very similar to 3b1b

Hello, glad to see you here!

This observation is incredibly beautiful to me. I imagine this chaotic geometry as a tiny sliver of some global geometry of a category of symplectic maps.

I don't think this is actually true... I think the bottom one rotating would make the top rotate too

@@EmersonPeters that's why Kattenelvis threw the word "roughly" in there. So indeed, not exactly true, but not exactly false either :'-)

@@EmersonPeters Well the only way to find out who's right would be to actually try it in a simulation

The center oval corresponds to the maximum energy state at which you can guarantee the lower pendulum will not flip. So if you start the pendulum from stationary at a given choice of angles, the states in the oval correspond to too little potential energy to ever let the lower pendulum flip - it bounds off the states which have the lower one as flipped over. The motion is still chaotic though in some regions of that oval. A better way to see this is to draw a fractal of the time taken for the second pendulum to flip, and then the same oval forms as a region of infinite time taken.

you can do anything with I̵̼͖̪͌͂M̵̲͖̺̳̯̀̂͆̾̈́̐̚͝Ă̵̪̙͛̚͜Ģ̴̧̲̭̳͑͛͗́̏̏̓̅̽̆I̴̯̽̄N̵̡̨̳̣̑Ą̷̡̮̱͍̹̻̈́Ţ̸͕̖̫͚̃̃͗̄̉̉̏̍͐Ï̵̻̬̱̬͒̄̎̈O̶̗̼̱͌N̴̹̂́

@@highonlife2323 Why not portals, too? Everything is possible when you got explosive lemons, math, science and Portals!

My thoughts exactly

took a screenshot of it for this reason

@@PeptoAbismol that's plagiarism bud

@@geetargato never said i’d use it for profit lol i just like how it looks like an album cover

@@geetargato it's math.. anyone could pop this into a program render it and then use it cause they made it. It's math.

Me too

Same here! I wish to make stuff as beautiful as this someday

I understand the coding is its own part of the process, but what are they using to make the visuals? iirc, didn't 3b1b write his own software? This looks like the same visual style as his videos, especially the text morphs.

@@dreamer097 This uses a community maintained version of 3b1b’s animation engine. Both it and the original engine are open source: github.com/ManimCommunity/manim/

Is that worth the risk, though? You can't predict how those axes will swing. Dangerous!

That's just pendulum's phase space. Which is interesting is to find some surface in the phase space and see how trajectories go through it, thus see it as moving point on that surface. That's Poincare section (map) which greatly helps to analyze such chaotic dynamic systems. Using that we can track those trajectories, identify periodic orbits, see how those orbits split in two or otherwize change on bifurcations and so on.

Can't wait to see it on my 4d screen

I misread that and thought you were talking about an axe on each double pendulum. Which would be effective at cutting trees, I should add

There is thus a smooth continuum from each initial state, via chaos, to the final state.

I think, there is no friction in this simulation, so it will never stop changing.

I thought this was a computer crashing joke, I’m not smart enough for this

Thats because the center of the picture is the origin of the cordinate system, and in this the closer a pendulum is to the center the less it moves, with the center pendulum not moving at all, I think.

@@BlastinRope You shed some light : it does make sense indeed, Kattenelvis' comment was along this line too. Thanks !

Symmetry across the diagonal y = x is expected because it's equivalent to mirroring each pendulum across its vertical axis (based on the current position of its pivot).

My high ass watched this in between calculus homework

The human brain likes familiarity. Which also, the human brain is chaotic.

@@GeekBrony tame impala?

Fractals have a unique property that just make them easily recognizable

If tame impala made watch the throne

Seek medical attention.

It uses his animation engine, Manim. Its open source and there's also one kept up by the community, github.com/ManimCommunity/manim/

@@maxdavison914 oh it's open source software too? Wow even better.

You see a graphics simulation and automatically assume it has to do something with the man that became famous for using it in his mathematics video format...

@@syngos98 He is also the person that created it, so the assumption that 3b1b has something to do with it would be correct.

I chose that specific coloring because the colors on either edge are the same. This way, if the pendulum flips on its side, (meaning the it wraps around to the other side of the plot e.g 0:55) the change in color remains continuous. The way I derived this color mapping was by embedding a sphere in the center of the RGB cube and projecting that to the 2d plane. My intuition tells me if I changed the color mapping the end behavior would be the same, but with a different coloring scheme

@@sammaksimovich2654 ooh cool that's a useful property to have! i think you might be right! the calm area I think is related only to the raw coordinates and which ones don't move "too" much on successive iterations. so each point could be colored whatever as long as the color mapping was continuous and you'd still have the same calm area

@@sammaksimovich2654 Doing it on the sphere itself, we would have no edges and it might give a more clear view of what's happening.

Me too that really surprised me when I first rendered the fractal! I still have no clue, but I'll definitely explore this area when I revisit this project in the future

From 0:48 you can see the orange pendulum passing through that zone several times. Although it has some left over momentum from the previous swings, it seemed to be somewhat stable for a few iterations, before veering away. That should give you an idea of what motion that zone corresponds to.

Hi, I'm a zoomer! Old people are dumb. **waits around for fellow zoomer validation**

@@cubixthree3495 lord I hope people know i was joking about being a boomer. I’m only 20 don’t come for me

@@liliththefirehawk796 lmao i wondered if that was the case. its all good =)

i'm a millenial, so when I see "its behavior is chaotic" I just say "same"

You know, it's funny, when his rendering started to take shape, I thought the general form looked tampon-like. Perhaps that could explain something?