Some mathematical order amidst the chaos of turbulence. Vortex rings with Physics Girl: https://www.youtube.com/watch?v=N7d_RWyOv20 Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: http://3b1b.co/turbulence-thanks Home page: https://www.3blue1brown.com/ Thanks to Dan Walsh for many great ideas, and thanks to Mike Hansen for many helpful conversations. Error correction: I meant to describe Kolmogorov as a “20th-century mathematician” not “19th-century”. Whoops! I think during the narration I must have made the classic 1900s vs. 19th-century mix up. Anyone aware of his work is more than aware of what century he lived in, which apparently applies to quite a few commenters. And while we're here, the animation at 9:55 misleadingly has the vortex kind of speed up during the transition, which it should not do. Really, the velocity scales of eddies decrease for lower length scales. Introduction to turbulence: http://www.astronomy.ohio-state.edu/~ryden/ast825/ch7.pdf More details on vortex stretching: https://www.math.nyu.edu/faculty/tabak/vorticity.pdf Video on NightHawkInLight with a similar demo: https://www.youtube.com/watch?v=K94Cc21KEIA Music by Vincent Rubinetti: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com/ Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown_animations/ Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown