The Double Pendulum Fractal
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- Опубликовано: 7 сен 2024
- The Double Pendulum Fractal & It's Chaotic Beauty
Dive into the mesmerizing world of chaotic systems with "The Double Pendulum Fractal." This video explores the double pendulum, a fascinating example of chaos theory where small differences in initial conditions lead to dramatically different outcomes.
Motions in chaotic behavor is based on nonlinearity of the mechnical systems. However, chaos is not a random motion. As you have seen, the motion can be described with a specific nested structure, which is called fractal.
While the chaotic behavior of the double pendulum is well-known, its fractal nature based on initial conditions remains a relatively uncharted territory.
Using advanced numerical simulations, we reveal the intricate fractal patterns that emerge over long timescales in the double pendulum's time evolution. You'll discover how energetics shape the gross structure of these fractals, exhibiting quasi-self-similar properties reminiscent of classic fractals like the Mandelbrot and Julia sets.
Let's unravel the dynamic pendulum's secrets and the beautiful, chaotic fractals hidden within. Whether you're intrigued by the butterfly effect, dynamic pendulums, or the satisfying gradients of periodic motion, this video is a simple explenation through the double pendulum chaos and the stunning fractal landscapes it creates.
#chaostheory #pendulum #fractal #SoMEπ #SoMEpi
How on earth does this only have 1.000 views? Everything from the editing, to his voice, to the way things are demonstrated and explained is so well done
Thank you so much 🙏🏻 Appreciate it!
full disclosure: As I hate recording my voice (english second language yk) I always edit my video’s with an AI generated voice, so that by the time everything is edited I know every word of the voice over & at what pace I should say them. I only have to replace it with my voice recording, at the very last. Unfortunately I uploaded the wrong version & only realised the mistake a few hours after publishing 😅😅
¡Great animacion! Video need more detail.
• ¿Pèndulums in grid all have equal length?
• Range x and y axis is -π; π
• ¿What is start velocity?
• ¿How select color?
• Range x and y for last grid is 0; 2π
Should compare equal level geography:
• Butterfly in Brasil; tornado in US
or
• Butterfly in Amazonas; tornado in Texas
Probably posted from Mars 😂
Breaking computers by asking questions too hard for them is one of my favorite things to do.
the divide by 0 is still the one and original
The production on this video is SO! high! 💪
Appreciate it! Something specific that you liked? 🙏
Superb video, very nicely explained ! I would have loved to see a zoom into the fractal !
Thank you so much 🙏🏻 might create some extra visual content next week, i’ll keep you posted 🙆♂️
Visualizing the varying angles progression as a fractal evolving over time is such a creative way to view chaos theory that I haven't seen anywhere else before. And it looks so beautiful.
Thank you so much 🙏🏻
Its beautifully smooth
It's like splitting hairs when the differences accumulate to the point the conditions part ways and become no longer similar
I can't tell if this is an AI generated voice or a real voice becuase it never changes in tone (thats a compliment)
@@calculator_gaming As I hate recording my voice (english second language yk) I always edit my video’s with an AI generated voice, so that by the time everything is edited I know every word of the voice over & at what pace I should say them. I only have to replace it with my voice recording, at the very last. Unfortunately I uploaded the wrong version & only realised the mistake a few hours after publishing 😅😅
this is so good, it deserves a place in the yt trends ngl
Thank you so much broo 🙏🏻
perfect video everyone watch this.
Wonderful! Thanks for making this!
Thanks Jeffrey! Appreciate it!
I've seen this fractal on Sam Maksimovich's channel before, but less well explained. This is a beautiful way to depict how that fractal is formed, my only criticism is the video transcript feels verbose without adding much, like it's been generated with chatgpt. It also doesn't help that the voice is AI generated.
Thank you so much 🙏🏻 I don’t see it as criticism, but as valuable feedback! I used AI to help with the writing of my intro and outro otherwise only spelling was corrected as English is my third language. Thanks for pointing that out! As I hate recording my voice I always edit my video’s with an AI generated voice, so that by the time everything is edited I know every word of the voice over and the pace I should say them. I only have to replace it with my voice recording, at the very last. Unfortunately I uploaded the wrong version & only realised the mistake a few hours after publishing 😅😅
Anything in particular that stood out for you on how I explained it?
Thanks in advance for the feedback! 🙏🙆♂️
Wow really great video - didn´t know these kinds of fractals - loving it! 👍
Thank you so much Tobias 🙏🏻 Anything in particular that stood out for you?
Very well made video!!! Super interesting. I'd love to see a tutorial of how to visualize things like this on computer.
Thank you so much 🙏🏻 something in particular that stood for you?
I created this mostly by coding it myself in Javascript utilising the p5js library. I would recommend checking out “The Coding Train” for full tutorials on how to make these kind of visuals :🙏✨✨🙆♂️
@@nicogsplayground I find the probability heat map overtime especially inspiring. Its almost like blending together spatial-temporal dimensions into a new realm of its own.
Beautiful representation :)
I have always wondered whether the double pendulum would retain any sort of continuity after having heard of chaos theory.
But hey, it did (somehow)! and that is a nice looking turbulence.
Thank you so much 🙏🏻 appreciate it!
Wow, this is cool. Looks very much like photoelasticity.
Thank you so much 🙏🏻 I did not know the term photoelasticity, had to google it, but you are 100% right, looks a lot like these holographic reflections!
Time to update my milkdrop plugin. Superb visuals here.
At least 20y since I’ve heard of Milkdrop waw haha! & Thank you so much 🙏🏻 anything in particular that you liked? 🙆♂️
Bro just updated chaos theory.
Great video. Sub'd. I intellectually love this sort of thing most deeply of all.
It's a category error to say "...how chaos theory shapes our world." Chaos Theory involves our attempt to model chaotic phenomena. It's not the phenomena itself, which is what shapes our world. That statement conflates the map with the territory.
Thank you so much Clark 🙏🏻
But doesn’t our attempt to model the phenomenon shape our world too? Take The three body problem, this shapes our understanding of the world and therefore shapes our world no? 🙆♂️ (my apologies if I understood the Q wrong)
But apart from the error anything in particular that you loved?🙆♂️✨
Awesome video! I think you didn't explain an important detail though, what do the colors represent?
Converted the angles, which we mapped on the grid to colors, rgb values. (01:57) But you are right that I should've probably put more emphasise on that!
@@nicogsplayground Makes sense. Since this looks like a continuous map and angles need to jump back to 0 if they go past 2π, you would need to map the sine or dot product or something like that to RGB space. Do you know the exact details?
@@Miaumiau3333have to double check but I think:
R: 255/2 + 255/2 * Sin(angle 1) * cos(angle 2)
G: 255/2 + 255/2 * Sin(angle 1) * sin(angle 2)
B: 255/2 + 255/2 * cos(angle 1)
@@nicogsplayground This looks good, thanks Nicog!
Everybody gangsta until triple pendulum
I was already sweating now tbh
😳 triple angles, 3D-version of the fractal. But how to calculate the color in this case. For 2 angles I assume sin(a,b) and 0=red, 1=violet. Is there stereoscopic variant 🤔
Bro what the fuck. What a masterpiece
Ayeeee appreciate it a lot! 😂🙆♂️🙏
This video is better than all the ones I've seen so far.
Thank you so much 🙏🏻 Anything specific that it made it better than all the other ones? Feedback is much appreciated!!!
@@nicogsplaygroundIn my opinion your videos are perfect. If you want more views to begin sharing your content, I would try to get promoted by similar youtubers. I saw that some of popular youtubers actually promoted small creators.
@@nicogsplaygroundoh, and a lot of tags
Would this be considered a fractal? I thought fractals require self-similarity... Wouldn't this be a subset of samples of the phase space of the system evolving as you scroll forward through time? Sorry if this is a dumb question.
No such thing as dumb questions, I had to google it to be sure, but Self similiarity includes Multifractal scaling
What is chaos? All I've ever seen are different states of order.
Motions in chaotic behavor is based on nonlinearity of the mechnical systems. However, chaos is not a random motion. As you have seen, the motion can be described with a specific nested structure, which is called fractal.
@@nicogsplayground Thanks for your answer. Nothing is random, therefore chaos doesn't exist. In fact, I can't even grasp what chaos would be, or how even hypothetically could exist...
@@ConnoisseurOfExistence that’s why it’s called Chaos Theory. The idea isn’t that something behaves randomly, one can predict it’s behaviour given the initial conditions. But when we try to determine the initial conditions based on it’s behaviour, it get’s pretty hard because of the “chaos”.
@@nicogsplayground Maybe we can't even determine its exact behavior (future state, position and momentum) from the initial conditions, without actually letting it play out. They call it computational irreducibility. Many complex systems are like that. Beautiful video, by the way. I shared it in many places. I love fractals!
@ConnoisseurOfExistence Plenty of things are random. Statistics and Thermodynamics rely on randomness existing to be accurate sciences.
what determines the color of the pendulum in this simulation?
The only variables that evolve over time: the angles made by the legs of the pendulum, hence why I mapped it on a grid 😅🙆♂️
Awesome
Thank you so much Evan 🙏🏻 Something in particular that you enjoyed? Can always use the feedback
@@nicogsplayground animations are quality
@@Evan---- Thanks!!! They often take a lot of time. Keyframing and tweaking curves so the speed and motion fits the whole composition and story line, can take hours for maybe 0.8 seconds of motion. Knowing that it doesn't go unnoticed by others really makes my day, appreciate it!
Almost looks like the outside of a Julia set at
I did not notice that, but now that you say so….
Orrrrrrr the Julia set looks like the inside of the Double Pendulum Fractal
Your channel rocks. Subscribed. This is so useful of a visualization of the notorious double pendulum!
Thank you so much Chase 🙏🏻 Glad you find it useful!!!! Makes my day
Can you do it with n-pendulum?
@@adrienledoux357 In theory yes, but then we should have an n-axis grid, which wouldn’t create anything interesting I guess haha
Even 3D glasses wouldn't be enough for that. You'd need nD glasses, not to mention a biology that could make sense of it.
Waoh! Can you run the simulation longer?
@@adrienledoux357 will upload some extra content, with some zoom in’s etc next week 🙆♂️ will keep you posted ✨
@@nicogsplayground what's the softwares used?
@@HarishBabuM Coded in Javascript using the p5js library 🙆♂️ Will share the github online version soon probably ✨anything in particular that stood out for you?
I’d love to see this fractal projected onto a torus
It’s 3am and I’m somewhere here
3am vibez guaranteed
How did you animated this? o:
All of the simulations where created using Javascript & the p5js library. Then I put everything together in Premiere Pro & animated the B-roll + some of the explanations. For example the grid was simulated in JS but I animated the x-axis & y-axis transition in After Effects :))
@@nicogsplayground you are f#cking talented bro
@@nachoalk ayeee appreciate it!!!!
lol i thought you were a tts
Absurd, what have you used to create this?
Thank you so much 🙏🏻 Using Javascript & the p5js library for all the simulations 🙆♂️🙆♂️
"length and mass are constant variables" doesn't make much sense but the video was good
Appreciate it! When simulating a double pendulum the mass of the two balls and the length of the legs connecting them stay constant and won’t change over time, that’s what I meant with “constant variables”. Meanwhile the position of the balls or the angles made by the legs are variables that change. I Should’ve said that indeed, thanks for the feedback!! 🙏🙏
I see symmetrical pattern, not chaos
Motions in chaotic behavor is based on nonlinearity of the mechnical systems. However, chaos is not a random motion. As you have seen, the motion can be described with a specific nested structure, which is called fractal.
around 02:52, the edges, every pixel represents a pendulum that had less than 0.001 deg in difference, but the vast amount of different colors shows the chaos that emerges from smaal initial differences. Many other chaotic systems look symmetrical btw 🙆♂
I wonder how much AI you used in the video making process. The wording of some sentences sounds kind of... non human.
English is not my first, nor my second language. With dutch & frensh being my motherlanguages, some of the sentence structures might sound strange, except from some suggestions by NotionAI for better words & grammar, the script is written by le me. 🙆♂️ first time someone called my writing non-human, but I’ll take the compliment as “out of this world” 😅❤️
Unless you where talking about the voice over, that’s actually a mistake 💀 As I hate recording my voice (accent, english yk) I always edit my video’s with an AI generated voice, so that by the time everything is edited I know every word of the voice over & at what pace I should say them. I only have to replace it with my voice recording, at the very last. Unfortunately I uploaded the wrong version & only realised the mistake a few hours after publishing 😅😅
its*