What's the average area of a cube's shadow?
Numberphile video on Bertrand's paradox:    • Bertrand's Paradox (with 3blue1brown)...  
Help fund future projects:   / 3blue1brown  
An equally valuable form of support is to simply share the videos.

There's a small error at 19:30, I say "Divide the total by 1/2", but of course meant to say "Multiply..."

Curious why a sphere's surface area is exactly four times its shadow?
   • But why is a sphere's surface area fo...  

If you liked this topic you'll also enjoy Mathologer's videos on very interesting cube shadow facts:
Part 1:    • The cube shadow theorem (pt.1): Princ...  
Part 2:    • The cube shadow theorem (pt.2): The b...  

I first heard this puzzle in a problemsolving seminar at Stanford, but the general result about all convex solids was originally proved by Cauchy.
Mémoire sur la rectification des courbes et la quadrature des surfaces courbes par M. Augustin Cauchy
https://ia600208.us.archive.org/27/it...

The artwork in this video was done by Kurt Bruns

Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Hindi: rajeshwarpandey



Timestamps
0:00 The players
5:22 How to start
9:12 Alice's initial thoughts
13:37 Piecing together the cube
22:11 Bob's conclusion
29:58 Alice's conclusion
34:09 Which is better?
38:59 Homework



These animations are largely made using a custom python library, manim. See the FAQ comments here:
https://www.3blue1brown.com/faq#manim'>https://www.3blue1brown.com/faq#manim
https://github.com/3b1b/manim
https://github.com/ManimCommunity/manim/

You can find code for specific videos and projects here:
https://github.com/3b1b/videos/

Music by Vincent Rubinetti.
https://www.vincentrubinetti.com/

Download the music on Bandcamp:
https://vincerubinetti.bandcamp.com/a...

Stream the music on Spotify:
https://open.spotify.com/album/1dVyjw...



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:   / 3blue1brown  
Reddit:   / 3blue1brown  
Instagram:   / 3blue1brown_animations  
Patreon:   / 3blue1brown  
Facebook:   / 3blue1brown