Thinking outside the 10-dimensional box

An explanation of high-dimensional spheres and the counterintuitive ways they fit inside boxes as dimensions increase.

Visualizing high-dimensional spheres to understand a surprising puzzle. Help fund future projects: patreon.com/3blue1brown This video was sponsored by Brilliant: brilliant.org/3b1b An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 3b1b.co/high-d-thanks Home page: 3blue1brown.com/ Podcast! benbenandblue.com/ Check out Ben Eater's channel: youtube.com/user/eaterbc ------------------ Animations largely made using manim, a scrappy open source python library. github.com/3b1b/manim If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind. Music by Vincent Rubinetti. Download the music on Bandcamp: vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people. ------------------ 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, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: 3b1b.co/recommended Various social media stuffs: Website: 3blue1brown.com Twitter: twitter.com/3Blue1Brown Patreon: patreon.com/3blue1brown Facebook: facebook.com/3blue1brown Reddit: reddit.com/r/3Blue1Brown