Pushing Simulations to the LIMIT to Find Order in Chaos

Finding the order in chaos by releasing 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 me so I can keep making videos like this?   / drewscampfire   For one-time payments: https://buymeacoffee.com/drewscampfire Or if you prefer PayPal: https://paypal.me/drewscampfire The animations were made with Blender and Manim CE. Both are free, open-source, and supported by awesome communities. https://docs.manim.community/en/stable/ https://www.blender.org/download/ The code used to generate the animations can be found here: https://github.com/drewscampfire/Drew... 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.    • GPU Accelerated Python   Freya Holmer - Freya's videos on splines and Bézier curves contributed a lot to my animation style and movement.    • The Continuity of Splines      • The Beauty of Bézier Curves   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.    • The Double Pendulum Fractal   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 (https://skfb.ly/69ICS) "Grandfather Clock" (modified) by Carl-HeinzLangley (https://skfb.ly/6z9II) "Italian Kitchen" by Francesco Coldesina (https://skfb.ly/6tPpI) "Kitchen table" (modified) by tahax (https://skfb.ly/6Goxs) "moon" by RenderX (https://skfb.ly/oFRLK) "Tennis Ball" by Arman.Abgaryan (https://skfb.ly/6YLwv) "Wall picture" by grafgrial (https://skfb.ly/oqnRZ)