Why is Pi here? | Half factorial without Gamma function

At 3:27 it should be (x+1)! = (x+1)x!
At 4:14 the angle on the right is 120°
At 11:11 B1 should be 2R

A video question that I had for a long time and found almost no answer to. So I made one. We find a general formula for the volume of n-dimensional hyperspheres and use that to extend the factorial to half integers.

Hint: snıpɐɹ uı ǝƃuɐɥɔ ʎuıʇ
Further material:    • But why is a sphere's surface area fo...  

===Chapters===
0:00 Introduction
3:43 n-Spheres and n-Balls
6:11 Deriving the recursive formulas
9:31 Putting it all together
12:42 Conclusion

===Made with===
Manim (https://www.manim.community/)

===Music===
In order of Appearance:

Home - We're finally landing
  / home-2001  

Music by Vincent Rubinetti
Download the music on Bandcamp:
https://vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
https://open.spotify.com/playlist/3zN...

C418 - Beton, AMS
https://c418.org/
https://c418.bandcamp.com/

===Tags===