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The most beautiful formula not enough people understand 61 min 
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I Cooled my PC with a SLUSHIE MACHINE 25 min On the volumes of higher-dimensional spheres Explore the 3b1b virtual career fair: See https://3b1b.co/talent Become a supporter for early views of new videos: https://3b1b.co/support An equally valuable form of support is to simply share the videos. Home page: https://www.3blue1brown.com Thanks to UC Santa Cruz for letting me film there, and special thanks to Pedro Morales-Almazan for arranging everything. My video on Numberphile with a fun application of this problem: https://youtu.be/6_yU9eJ0NxA Timestamps: 0:00 - Introduction 1:01 - Random puzzle 6:16 - Outside the box 14:35 - Setting up the volume grid 21:14 - Why 4πr^2 25:21 - Archimedes in higher dimensions 36:17 - The general formula 40:40 - 1/2 factorial 44:58 - Why 5D spheres are the biggest 50:16 - Concentration at the surface 54:27 - A unit-free interpretation 57:50 - 3b1b Talent 59:13 - Explaining the intro animation ------------------ These animations are largely made using a custom Python library, manim. See the FAQ comments here: https://3b1b.co/faq#manim Music by Vincent Rubinetti. https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. If you're reading the bottom of a video description, I'm guessing you're more interested than the average viewer in lessons here. It would mean a lot to me if you chose to stay up to date on new ones, either by subscribing here on YouTube or otherwise following on whichever platform below you check most regularly. Mailing list: https://3blue1brown.substack.com Twitter: https://twitter.com/3blue1brown Bluesky: https://bsky.app/profile/3blue1brown.com Instagram: https://www.instagram.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Facebook: https://www.facebook.com/3blue1brown Patreon: https://patreon.com/3blue1brown Website: https://www.3blue1brown.com
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