Pushing Simulation to the Limit to Find Order in Chaos

Description:

Let's uncover order in chaos by running millions of double pendulums and generating a new fractal in the process. Timestamps: 0:00 Intro 0:49 3D Simulations of Order and Chaos 5:55 Laying the Groundwork and on Determinism 7:05 Plotting Order and Chaos 11:43 Visualizing All Possible Initial Positions 13:52 Pixel Visualization and the Islands of Stability 19:28 Flips and the Hidden Order in Chaos 25:24 Only a Tourist’s Guide Want to support the channel? For one-time payments: Or if you prefer PayPal: The animations were made with Blender and Manim CE. Both are free, open-source, and supported by awesome communities. The code used to generate the animations can be found here: All double pendulum simulations in the video were calculated using an 8th-order Dormand-Prince Runge-Kutta method, which is designed for solving ordinary differential equations with very high accuracy using double-precision floating-point calculations. 24:21: I have validated the existence of these unexpected elements by varying gravity, rod-length ratios, pendulum masses, and the _rtol_ and _atol_ values of the torchdiffeq’s dopri8 ODE solver. While the exact shapes and locations shift with these parameters, the following qualitative features persist: (1) regions where double pendulums fall into apparent visual chaos without ever flipping, and (2) distinct islands within which double pendulums flip repeatedly yet maintain a high degree of order over long durations. Fun(?) facts about the creation of this video: 1. The zooming shot at 24:45 was created by simulating a total of 5.04 billion double pendulums (42 seconds * 30 fps * 2000 px * 2000 px), each running for 70 seconds. My RTX 3060 laptop would have taken ~105 days to process the calculations for this shot alone, not even including the rendering of the animation. So, I rented a more powerful GPU instead. 2. In the process of making this video, I had to render tens of thousands of 4k-quality PNG images. My _Final Image Renders_ folder, at some point, had a size of 852 GB. *SPECIAL THANKS* *Mr. P Solver* - Special thanks for Luke's _GPU Accelerated Python_ series, which showed me how this video was computationally possible in the first place. Otherwise, some animations in this video would have taken me two orders of magnitude more time to process and render. *Freya Holmer* - Freya's videos on splines and Bézier curves contributed a lot to my animation style and movement. *Sam Maksimovich* - Finally, Sam's video on the double pendulum--the first, as far as I know, to have come up with the ingenious idea of turning individual double pendulums into pixels. *MUSIC CREDITS* _Above the Clouds_ by Theatre of Delays _Black Hole_ by Stephen Keech _Chasing Shadows_ by DVWLX _darklouds_ by Out of Flux _Deepest Dive_ by Tiko Tiko _First Try_ by Neon Ridge _Full Access_ by Jimmy Svensson _Game Over_ by 2050 _In Obit_ by Ian Post _Just Watch - Instrumental version_ by WEARETHEGOODS _Odd Numbers_ by Curtis Cole _The Art of Connection_ by Ardie Son _The Swan_ by Alon Peretz, Artlist Classics, Camille Saint-Saëns _Transition_ by Notize _Whodunit_ by FableForte *ASSET LICENSES* The following 3D models from Sketchfab are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0): "Big Mac" by Rev1Tech ( "Grandfather Clock" (modified) by Carl-HeinzLangley ( "Italian Kitchen" by Francesco Coldesina ( "Kitchen table" (modified) by tahax ( "moon" by RenderX ( "Tennis Ball" by Arman.Abgaryan ( "Wall picture" by grafgrial (