For those curious, the equation of the relatively stable region is 3*cos(theta_1) + cos(theta_2) = 2, which corresponds to the pendulum not having enough energy to flip over, according to Wikipedia. The branches coming off of the top left and bottom right branches are probably of some interest though, seeing how resistant they are to embracing chaos.
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.
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 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?
It's cool how the center seems to radiate colors outwards and compress them on the edges until they become gray. Also, the gray means that adjacent pixels became so wildly different that they're pretty much random (different from all neighbors), and thus the average you take from a given area is gray - think of a pixel where you have individual RBG subpixels, but your vision combines them in a single color.
Of course, the destabilizer for the methamphetamine ununoctide is quintesentially conclusive to the equilibrium that instigates the proprietarianist conjugation that had been discovered
It’s currently 1:50 AM and I just got through cleaning for several hours. I pull up TH-cam and read this as “The Double Penetration Fractal” and it takes me nearly two minutes to figure out what is wrong with that assessment. I need sleep.
Makes sense that the stable region of the fractal looks like an oval shape, because if the initial conditions for the lower pendulum has a large angle while the top pendulum has angle around 0, then it basically becomes roughly a normal pendulum.
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.
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.
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 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.
If you added axes to track the momentum of the 2 weights you would have a 4d map of every possible state and be able to map each to the next iterative state.
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.
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/
fractals are crazy. How can such complex shapes arise from seemingly simple algorithms? it surely reveals something about our universe that we may never understand. something deep, woven into the fabric of reality itself.
Wow I was expecting every dots to behave like the pixels along the frame : ending up with noises and random colours coming together. Amazing this central structure that appears.
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.
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).
That's a nice one! Made something similar 5 years ago in Python, only had the mapping such that your corners were my center. To give some ideas: I remember that plotting the kinetic energy of a single mass was looking very cool. Encoding direction and modulus of speed in color and brightness is also spectacular. Maybe you give it a try.
very cool to see that on the edges after a while it turns into TV static looking stuff which perfectly shows how "random"/unpredictable the outcomes are! super cool visuals, thank you
History of histories, life's truth appears to me. A framework of opportunity ebbs and flows, Even a tiny shift will change where it goes. Time to live in each moment and let it be. But that itself is just one path of reality!
That moment when rainbow lines run from central point reminds me of seeing transparent stiff plastic piece being bent and deformations distorting the light
Everything in the Malkhut is a crystallization of what's, relatively, "above" itself and Ayn Soph is the "above/center" where the color stays the same since it like the eye of a hurricane 🌀 which is always completely still. I realized that you are jewish so I thought I'd clarify a interpretation of this in terms of Kabbalah (Apologies if I assumed wrongly regarding kabbalah).
Awesome nice work my friend!....beautiful chaos in motion like a fluidly dynamic fractal soap bubble. Crazy how it maintains a balanced symmetry like a mirrored reflection of itself. Wish I could up the resolution and zoom through the fine details!
Is this made from 3B1B's graphics library? Man it's amazing what one person can do. He literally implemented physics and graphics in his library all by himself, something that only pro coders do, as a mathematician. And as a coder, I can't do mathematics.
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...
This is very cool, interesting to see that there are a lot of adjacent initial states which result in similar, non-chaotic trajectories. There's a clear phase transition, inside is non-chaotic and outside is chaotic.
Congratulations! You’ve made the prettiest chaos portal Is there any connection between the double pendulum and entropy? The way the visualization of this system ripples into greater chaos as it further it gets from the center has me wondering. Not sure what the connection could be tho
This is genuinely incredible. I kept zooming in on it and reached a point where it can no longer calculate anymore precisely, so the pixels start getting bigger and bigger. Just goes to show the limits of our technology. Amazing.
how does this fractal change if you use a different method of assigning a color to a point? my intuition is the shape of the stable region depends on that mapping but curious if you had tried it out already
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
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.
I'm getting a water simulation vibe. Like the pendulum represents the water molecule and it's oscillating behavior sets up a motion fractal. Fascinating
i have no idea how pendulums work but these videos keep getting recommended to me and i’m starting to get hooked maybe the algorithm isn’t so bad after all
wow this is the perfect blend between math and art! Now we need a solid 3D colored block for triple pendulum
0:06 It's actually fascinating how the texts morph
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
It looks like an 70s or 80s album cover that has one song that defined a generation but all 9 of the other songs sound like synth toilet
💫synth toilet💫
Best comment
r/brandnewsentence
synth toilet is my new favorite phrase
✨synth toilet✨
For those curious, the equation of the relatively stable region is 3*cos(theta_1) + cos(theta_2) = 2, which corresponds to the pendulum not having enough energy to flip over, according to Wikipedia. The branches coming off of the top left and bottom right branches are probably of some interest though, seeing how resistant they are to embracing chaos.
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
Me trying to understand: "well at least it looks cool"
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
Really beautiful visualisations - thank you!
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!
It's cool how the center seems to radiate colors outwards and compress them on the edges until they become gray. Also, the gray means that adjacent pixels became so wildly different that they're pretty much random (different from all neighbors), and thus the average you take from a given area is gray - think of a pixel where you have individual RBG subpixels, but your vision combines them in a single color.
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
0:28 literally my legs when I see a cockroach on the floor while I'm taking a shower.
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
It’s currently 1:50 AM and I just got through cleaning for several hours. I pull up TH-cam and read this as “The Double Penetration Fractal” and it takes me nearly two minutes to figure out what is wrong with that assessment. I need sleep.
Makes sense that the stable region of the fractal looks like an oval shape, because if the initial conditions for the lower pendulum has a large angle while the top pendulum has angle around 0, then it basically becomes roughly a normal pendulum.
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.
Ive learned from the action lab that even in chaotic random behavior a trend can be stabilished and thats why statistics works
This is an amazing piece of chaotic beauty. Six hours of rendering? Worth it.
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?
never thought id see the day that someone would capture what it feels like to stand up too quickly in digital form
Buddy that IS what happens
I like thinking about the orange and blue pendulums as legs.
you can do anything with I̵̼͖̪͌͂M̵̲͖̺̳̯̀̂͆̾̈́̐̚͝Ă̵̪̙͛̚͜Ģ̴̧̲̭̳͑͛͗́̏̏̓̅̽̆I̴̯̽̄N̵̡̨̳̣̑Ą̷̡̮̱͍̹̻̈́Ţ̸͕̖̫͚̃̃͗̄̉̉̏̍͐Ï̵̻̬̱̬͒̄̎̈O̶̗̼̱͌N̴̹̂́
@@highonlife2323 Why not portals, too? Everything is possible when you got explosive lemons, math, science and Portals!
this is what the chosen one speaks of. it all makes sense now.
If you added axes to track the momentum of the 2 weights you would have a 4d map of every possible state and be able to map each to the next iterative state.
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
I suddenly realised that no matter the apparent color noise this color change is continuous and my brain started to melt.
dang, amazing, it can turn into an album cover
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.
I can't believe this is from a month ago, this looks like something that came out of 2008/2009 youtube
Subscribed.
I am here from 3b1b.
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/
fractals are crazy. How can such complex shapes arise from seemingly simple algorithms? it surely reveals something about our universe that we may never understand. something deep, woven into the fabric of reality itself.
Wow I was expecting every dots to behave like the pixels along the frame : ending up with noises and random colours coming together. Amazing this central structure that appears.
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).
Causally just made one of the hardest album covers of all time
What a cool and creative way to plot the behavior of a double pendulum. It is art
i don't understand a single thing but the visual is dope
it makes you realise that if this simulation went on long enough the whole screen would become the centre blue
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
Like someone desperately trying to walk
would be interesting to see this same effect being applied to other images for creative warping/distortion effects. really cool video!
It remembered me the colors that appear over a soap bubble before it pops, when it goes dark (gray in the video) it pops
This is EXACTLY what I needed to finish off my brief fascination with double pendulums. Thank you so much! This was awesome.
I fcking love the 3blue1brown art and animation. Amazing job on this.
I love how it seemingly creates an ellipsoid shape but pierced with a stick
2:22 Samsung's Smartphone wallpapers be like:
This is awesome!!! Please put more stuff like this on the internet.
1:46
an absolute GPU torture
1:53 *Enter the portal to The Land of Fractals*
“Alright, who tore the fabric of reality?” 2:03
excellent job
This is what I'd imagine acid to be like
Beautiful! This reminds me of the fractal created by the basins of attraction of 3 magnets!
damn i thought fulcrum was boutta hit a double penjamin blinker
That's a nice one! Made something similar 5 years ago in Python, only had the mapping such that your corners were my center.
To give some ideas: I remember that plotting the kinetic energy of a single mass was looking very cool. Encoding direction and modulus of speed in color and brightness is also spectacular. Maybe you give it a try.
"You cant run away from your problems"
Me: 0:29
the first time he shows the pathways it looks like a human heart
very cool to see that on the edges after a while it turns into TV static looking stuff which perfectly shows how "random"/unpredictable the outcomes are! super cool visuals, thank you
Why does the thumbnail look so familiar even though I'm sure I've never seen it before in my life-
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
History of histories, life's truth appears to me.
A framework of opportunity ebbs and flows,
Even a tiny shift will change where it goes.
Time to live in each moment and let it be.
But that itself is just one path of reality!
00:50 now I understand how god created the heart
That moment when rainbow lines run from central point reminds me of seeing transparent stiff plastic piece being bent and deformations distorting the light
woah. TH-cam just recommended me this thing. I have no idea what is this or what's happenning, but it looks amazing
as a someone in 9th grade geometry, i agree
i took that last year as well
im in 10th grade algebra 2
I don’t entirely understand what’s going on, nor is this a consistent interest of mine, but I get happy to be introduced to these kinds of things.
Everything in the Malkhut is a crystallization of what's, relatively, "above" itself and Ayn Soph is the "above/center" where the color stays the same since it like the eye of a hurricane 🌀 which is always completely still. I realized that you are jewish so I thought I'd clarify a interpretation of this in terms of Kabbalah (Apologies if I assumed wrongly regarding kabbalah).
@@yahya2925 ahh, lol everything you said went way over my head! I do not study the Kabbalah, But thank you anyway.
mom: **comes in my room**
me: **switches tab to video game**
Awesome nice work my friend!....beautiful chaos in motion like a fluidly dynamic fractal soap bubble.
Crazy how it maintains a balanced symmetry like a mirrored reflection of itself.
Wish I could up the resolution and zoom through the fine details!
I can't believe I am watching this to entertain myself
2:10
When Sauron gets a bit too high
This is honestly my fav video, nice calm music, and beautiful visuals. Thank u so much ❤️
2:11 kind of looks like a human heart.... bro....
Seek medical attention.
I wish I hadn't been so terrified of math at school. This is beautiful and fascinating. Thank you!
Very cool how it turns to noise as it strays from the center. Awesome video!
Is this made from 3B1B's graphics library? Man it's amazing what one person can do. He literally implemented physics and graphics in his library all by himself, something that only pro coders do, as a mathematician. And as a coder, I can't do mathematics.
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.
pov: you’re opening up an interdimensinal portal: 1:41
°Even in chaos, there is order。
Also thank you for your interest and skill。
This is very cool, interesting to see that there are a lot of adjacent initial states which result in similar, non-chaotic trajectories. There's a clear phase transition, inside is non-chaotic and outside is chaotic.
Which properties of the pendulum were mapped onto which components of the colour?
This is really interesting! Thanks 👍🏻
why is it when we try to make math into pictures its always like we’re on lsd
Congratulations! You’ve made the prettiest chaos portal
Is there any connection between the double pendulum and entropy? The way the visualization of this system ripples into greater chaos as it further it gets from the center has me wondering. Not sure what the connection could be tho
This is genuinely incredible. I kept zooming in on it and reached a point where it can no longer calculate anymore precisely, so the pixels start getting bigger and bigger. Just goes to show the limits of our technology. Amazing.
as someone who knows nothing about fractals...
the colours were pwettty :)
The way this flows is so psychedelic. Math is art !
“It’s behavior is chaotic”
HA! So is my wife’s. **waits around for fellow boomer validation**
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?
I really liked the way you presented the educational info. Thank you!
how does this fractal change if you use a different method of assigning a color to a point? my intuition is the shape of the stable region depends on that mapping but curious if you had tried it out already
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.
It's periodic
This is both very intuitive and something I never would have thought of without seeing it.
I'm curious what that little stable peninsula sticking out at each side is... What periodic motion does that correspond to?
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.
I wish you had more videos like this, I instantly thought of the other video when I saw the thumbnail!
"Small changes to the initial conditions results in wildly different behaviour"
Just like telling your girlfriend you need some space.
I Guess this is what you see when you do too many drugs
Man you can do a NFT with that, think about it
Three videos in and all the visuals are awsome, amazing work im definitely relaxing with some of these
0:30 how I run in my dreams
Stuff like this makes it easier to imagine an entire universe composed of waves of energy
Just amazing and beautiful. Thank you for sharing this with us!
I'm getting a water simulation vibe. Like the pendulum represents the water molecule and it's oscillating behavior sets up a motion fractal. Fascinating
the font and everything made it look like a 3 blue 1 brown video.
man that's some amazing stuff
This'd make for a pretty cool album cover
i have no idea how pendulums work but these videos keep getting recommended to me and i’m starting to get hooked
maybe the algorithm isn’t so bad after all
It almost looks like an eye, i think this would make for a really interesting and terrifying video game boss that is/ has an eye like this
See you again in 5 years when youtube recommend this again
When you rub your eyes too hard
The way it evolves seems so much like a special version of the mandelbrot
Crazy how this was made a month ago. It reminds me of those old science videos from 2009 but with much higher quality.
The movements of the pair of double pendulums looks eerily like the movement of something "alive".
It looks like it's layering, like a sand blanket
Wonderful demonstration, my congratulations to the autor of such a video.
i love the effect the text makes when it morphs into another set of text, it's mesmerizing
It shows very intuitively how chaos produces randomness. The white noise-like outer regions are where the chaos is strong
Simple gets complicated in a hurry. Really nice.
That is so freaking cool watching entropy take over as it becomes more and more disordered
Aww, how beautiful is this fractal! 😙
Brilliant! Love the visualization and the concept!
Totally worth it.