The Double Pendulum Fractal

[@carykh]

wow this is the perfect blend between math and art! Now we need a solid 3D colored block for triple pendulum

[@dlrss1v274]

0:06 It's actually fascinating how the texts morph

[@bendank9762]

Why is this comment so high up with only 6 Likes

[@ffoundersvods9510]

@@bendank9762 verified

[@BananaPlaysGames]

@@dlrss1v274 I want the text transitions now lol

[@Spiffier]

Wow it’s carykh

[@momomigogoc3943]

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

[@um9643]

💫synth toilet💫

[@rly6078]

Best comment

[@official-obama]

r/brandnewsentence

[@my_d00m3]

synth toilet is my new favorite phrase

[@datonecaleb860]

✨synth toilet✨

[@bacanher9588]

0:28 literally my legs when I see a cockroach on the floor while I'm taking a shower.

[@peddr.o]

l i t e r a l l y

[@reddmst]

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

[@macdowntwo]

don't slip!

[@axelrios276]

Reddy what country are you from?

[@macdowntwo]

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

[@indibiningingman6510]

Me trying to understand: "well at least it looks cool"

[@suddeneevee9441]

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.

[@someperson6672]

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

[@hritikvaishnav603]

@@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?

[@TAKTlmao]

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

[@AtomicShrimp]

Really beautiful visualisations - thank you!

[@phoneticalballsack]

Is it You that doesnt understand english?

[@avocadospicedlatte1142]

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

[@DoryTheCat]

oh awesome to seeu here

[@raquelsanchez4129]

It looks very similar to 3b1b

[@zcqm]

Hello, glad to see you here!

[@umloginqualquer]

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.

[@EmmaJohnsonShenanigans]

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

[@randommax7481]

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

[@TiaTista]

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

[@arpyzero]

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.

[@828burke]

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.

[@TheDetonadoBR]

the relatively stable region also contains those branches

[@APozzi]

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

[@marijansmetko6619]

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

[@TheDetonadoBR]

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

[@eric.from.statefarm]

never thought id see the day that someone would capture what it feels like to stand up too quickly in digital form

[@SkeletonWithTrumpet]

Buddy that IS what happens

[@rosuav]

This is an amazing piece of chaotic beauty. Six hours of rendering? Worth it.

[@sammaksimovich2654]

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.

[@wissamkadamani]

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

[@incription]

@@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 :)

[@codenamelambda]

@@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.

[@jacksonhall206]

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

[@mysteriumxarxes3990]

Ive learned from the action lab that even in chaotic random behavior a trend can be stabilished and thats why statistics works

[@kattenelvis1778]

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.

[@jacobchateau6191]

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.

[@EmersonPeters]

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

[@zbnmth]

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

[@kattenelvis1778]

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

[@kailenpatel7254]

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.

[@awesomechaos4034]

It’s currently 1:50 AM and I just got through cleaning for several hours. I pull up KZbin 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.

[@TylerTheDevourer]

go to sleep

[@pikchassis]

@TylerTheDevourer3 hours ago

[@TylerTheDevourer]

@@pikchassis 6 hours ago

[@Flesh_Wizard]

​@TylerTheDevourer3 days ago

[@TylerTheDevourer]

@@Flesh_Wizard 6 hours ago

[@Speed_Of_Light186]

I like thinking about the orange and blue pendulums as legs.

[@highonlife2323]

you can do anything with I̵̼͖̪͌͂M̵̲͖̺̳̯̀̂͆̾̈́̐̚͝Ă̵̪̙͛̚͜Ģ̴̧̲̭̳͑͛͗́̏̏̓̅̽̆I̴̯̽̄N̵̡̨̳̣̑Ą̷̡̮̱͍̹̻̈́Ţ̸͕̖̫͚̃̃͗̄̉̉̏̍͐Ï̵̻̬̱̬͒̄̎̈O̶̗̼̱͌N̴̹̂́

[@StellarCrackhead42]

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

[@DannySullivanMusic]

Very cool how it turns to noise as it strays from the center. Awesome video!

[@Tenduere]

dang, amazing, it can turn into an album cover

[@leighton2504]

My thoughts exactly

[@PeptoAbismol]

took a screenshot of it for this reason

[@geetargato]

@@PeptoAbismol that's plagiarism bud

[@PeptoAbismol]

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

[@jimmynutrition7858]

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

[@juliuszkocinski7478]

I suddenly realised that no matter the apparent color noise this color change is continuous and my brain started to melt.

[@TheFinagle]

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.

[@ThewOrldIssqUare]

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

[@nikitakipriyanov7260]

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.

[@supersonictumbleweed]

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

[@datonecaleb860]

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

[@justinstewart9014]

this is what the chosen one speaks of. it all makes sense now.

[@joeltovarramos2250]

What a cool and creative way to plot the behavior of a double pendulum. It is art

[@berni_schmorg]

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

[@bengineer8]

Subscribed. I am here from 3b1b.

[@theawantikamishra]

Me too

[@meghanto]

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

[@dreamer097]

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.

[@rio__]

@@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/

[@comeycallate9959]

It remembered me the colors that appear over a soap bubble before it pops, when it goes dark (gray in the video) it pops

[@Benny_Blue]

This is EXACTLY what I needed to finish off my brief fascination with double pendulums. Thank you so much! This was awesome.

[@Vantitnav]

“Alright, who tore the fabric of reality?” 2:03

[@LucienOmalley]

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.

[@BlastinRope]

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.

[@LucienOmalley]

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

[@zecuse]

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).

[@mr_noobx3009]

I can't believe this is from a month ago, this looks like something that came out of 2008/2009 youtube

[@ryanjohnson3861]

it makes you realise that if this simulation went on long enough the whole screen would become the centre blue

[@tigertalar]

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

[@tlqy]

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

[@RobotAnimals]

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

[@MightySpaceman]

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.

[@amanuelgetachew634]

This is awesome!!! Please put more stuff like this on the internet.

[@hiimapop7755]

I fcking love the 3blue1brown art and animation. Amazing job on this.

[@PolyRocketMatt]

I love how it seemingly creates an ellipsoid shape but pierced with a stick

[@zcqm]

This is honestly my fav video, nice calm music, and beautiful visuals. Thank u so much ❤️

[@artemetra3262]

would be interesting to see this same effect being applied to other images for creative warping/distortion effects. really cool video!

[@3DAnaglyphGames]

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.

[@marcuslobstein3179]

excellent job

[@nzmons]

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!

[@dy7296]

1:46 an absolute GPU torture

[@catbucket8566]

Causally just made one of the hardest album covers of all time

[@maustank5224]

1:53 *Enter the portal to The Land of Fractals*

[@nikolaimikuszeit3204]

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.

[@olbluelips]

Beautiful! This reminds me of the fractal created by the basins of attraction of 3 magnets!

[@thedofflin]

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.

[@st0rm_982]

2:22 Samsung's Smartphone wallpapers be like:

[@floatingshlob]

Like someone desperately trying to walk

[@sinny5404]

Why does the thumbnail look so familiar even though I'm sure I've never seen it before in my life-

[@GeekBrony]

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

[@Quozzie]

@@GeekBrony tame impala?

[@mcvibing2785]

Fractals have a unique property that just make them easily recognizable

[@cornoffthecob1018]

If tame impala made watch the throne

[@scarlettestanley3391]

Thank you for every minute of that 6 hours...it made my night, that was beautiful ;)

[@sirbutterbutter8792]

Three videos in and all the visuals are awsome, amazing work im definitely relaxing with some of these

[@islaadele1212]

I wish I hadn't been so terrified of math at school. This is beautiful and fascinating. Thank you!

[@creepermasher]

The way this flows is so psychedelic. Math is art !

[@KILOPOWER]

woah. KZbin just recommended me this thing. I have no idea what is this or what's happenning, but it looks amazing

[@skg9905]

This is what I'd imagine acid to be like

[@raeus571]

the first time he shows the pathways it looks like a human heart

[@jessicawatson891]

i love the effect the text makes when it morphs into another set of text, it's mesmerizing

[@dubbyplays]

00:50 now I understand how god created the heart

[@prathameshsundaram7509]

Brilliant! Love the visualization and the concept! Totally worth it.

[@vaishnavchandra5220]

I can't believe I am watching this to entertain myself

[@glukaise]

i don't understand a single thing but the visual is dope

[@ranchu8385]

mom: **comes in my room** me: **switches tab to video game**

[@CalebSalstrom]

I really liked the way you presented the educational info. Thank you!

[@diophantine1598]

I wish you had more videos like this, I instantly thought of the other video when I saw the thumbnail!

[@comradepeter87]

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.

[@maxdavison914]

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

[@comradepeter87]

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

[@syngos98]

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...

[@maxdavison914]

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

[@PeptoAbismol]

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

[@Fogmeister]

Which properties of the pendulum were mapped onto which components of the colour? This is really interesting! Thanks 👍🏻

[@juniorsilvabroadcast]

Beautiful waves going on... this can be used on games or 3D rendering for water/weather?

[@austinpeete]

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

[@imgonnatellmom3245]

"You cant run away from your problems" Me: 0:29

[@tennohack6704]

In video editing, How did you do this 1:17 without manually setting the color each frame?

[@samuelwaller4924]

the whole video is animated from a program. So the square is colored at the same time as the fractal. The library is called Manim

[@infinitelyexplosive4131]

This is both very intuitive and something I never would have thought of without seeing it.

[@kyototo.]

°Even in chaos, there is order。 Also thank you for your interest and skill。

[@benjaminjohnson5372]

Since the chaos of two revealed a commonality represented by a vesica piscis, would a triple pendulum produce commonalities representable by something resembling a triquetra? There's something harmonic appearing here despite the chaos. I suspect a vesica piscis would appear in many similar scenarios involving two component chaos. I wonder if the border of the central shape changes to match the equation. Very fun stuff, worth further investigation. Thank you.

[@lock_ray]

I'm curious what that little stable peninsula sticking out at each side is... What periodic motion does that correspond to?

[@sammaksimovich2654]

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

[@firefly618]

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.

[@kamo7293]

the font and everything made it look like a 3 blue 1 brown video. man that's some amazing stuff

[@eternalcogn1to595]

2:11 kind of looks like a human heart.... bro....

[@Concrete-Society]

Seek medical attention.

[@ЕленаПоддубная-с6ш]

Aww, how beautiful is this fractal! 😙

[@bobtheblob728]

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

[@sammaksimovich2654]

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

[@bobtheblob728]

@@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

[@adarshsingh2777]

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

[@Manoj_b]

It's periodic

[@perialis2970]

why is it when we try to make math into pictures its always like we’re on lsd

[@WoolyCow]

as someone who knows nothing about fractals... the colours were pwettty :)

[@PaulPlaystheLegend]

pov: you’re opening up an interdimensinal portal: 1:41

[@liliththefirehawk796]

“It’s behavior is chaotic” HA! So is my wife’s. **waits around for fellow boomer validation**

[@cubixthree3495]

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

[@liliththefirehawk796]

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

[@cubixthree3495]

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

[@MintyArisato]

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

[@freescape08]

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?

[@NickGerr445]

damn i thought fulcrum was boutta hit a double penjamin blinker

[@howardlim9676]

"Small changes to the initial conditions results in wildly different behaviour" Just like telling your girlfriend you need some space.

[@EbbndFl0w]

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.

[@yahya2925]

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).

[@EbbndFl0w]

@@yahya2925 ahh, lol everything you said went way over my head! I do not study the Kabbalah, But thank you anyway.

[@joeltovarramos2250]

Man you can do a NFT with that, think about it

[@O_Obsidious]

the edges almost look like you're staring at something psychedelic, getting more n more complex until it's just noise in the corner of your vision

[@Account32548]

Just amazing and beautiful. Thank you for sharing this with us!

[@hammadmajid9508]

See you again in 5 years when youtube recommend this again

[@phytoplankton7003]

Crazy how this was made a month ago. It reminds me of those old science videos from 2009 but with much higher quality.

[@Questiala124]

1:58 this gave me an idea. Because of how absolutely chaotic the fractal is, it will shuffle through every possible combination it can, however it I see limited to a few colours and we only have a few pixels. There is a chance (thanks to it’s limited pixels) that they will all fall into the same pixel colour with every position to be the same. The chances are near 0 but not impossible.once it happens everything will be one colour, still unpredictable but much easier to calculate.

[@robiaster]

I think one way of thinking of this is that all starting points that theoretically are able to reach a state (meaning position and momentum of all elements) of another possible entry, they are just a phase shifted version of that state. From what I understood, this method can only compare position, but not inertia. It would be interesting to see whether a starting point will result in periodic movement (in the long, long term, or even something shorter) or whether a specific point results in no periodicity, and if the former, how they could be related to other input states in therms of phase. To me, it looks like the parts in the middle are a similar function offset in phase, while the outer ones are either not periodic, or have so long periods that any slight change in position offsets it to such a huge degree that it looses all relation to its neighbours.

[@kjekelle96]

Awesome! Trippy stuff like this is party the reason I got interested in math!

[@fatimaalaa2659]

So a mathematician gets to draw scribbles on a paper and call it science, but when _I_ do it I'm told to "stop ruining the textbook"?!

[@caseyriley1014]

After a short while it starts to look like a shattered lcd from a phone

[@supptk]

Well

[@jshemms]

very interesting, it’s cool when you showed how everything in it worked. keep up the cool videos!

[@quint3ssent1a]

That moment when rainbow lines run from central point reminds me of seeing transparent stiff plastic piece being bent and deformations distorting the light

[@shazibhaseen2614]

The way it evolves seems so much like a special version of the mandelbrot

[@gragaloth6237]

The fact that the graph just ends up as static perfectly shows the butterfly effect

[@WuffRobotica]

Stuff like this makes it easier to imagine an entire universe composed of waves of energy

[@davebdot6713]

Can someone explain why it went from double pendulum to "dude, the walls are melting"?

[@chromaflow9313]

This is amazing, thanks for sharing! What would be interesting would be to then zoom into the more unstable regions and see if there are little “pockets” of higher stability.

[@TheStaggpaul]

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