Sierpinski Triangle-like fractals are revealed in Pascal's triangle by coloring the entries based on their divisibility. Why are these patterns hiding here? What does it have to do with the prime rows of Pascal's triangle?

Sonification of the first 25 rows of Pascal's triangle mod 5 by Marc Evanstein / music.py
Thank you Marc!!! Check out his videos on the music of these fractal patterns!

- "The Hidden Music of Pascal's Triangle"    • The Secret Music of Pascal's Triangle  

- Playlist: "Hearing Pascal's Triangle"    • Hearing Pascal's Triangle  

Other related videos and links
Eddie Woo "Why are the prime rows in Pascal's Triangle so special?"    • Why are the prime rows in Pascal's Tr...   (This video is how I originally learned abou tthis and started working on this video!)

Lucas's Theorem: The fracals you see are basically equivalent to this theorem that tells you that you can figure out n choose k mod p by writing n in base p. The argument in the video is essentially an induction proof of Lucas's theorem. See https://en.wikipedia.org/wiki/Lucas%2... and also this very nice thesis on this topic https://digitalcommons.unl.edu/cgi/vi...

A Tale of Three Triangles: Two other points of view on the Sierpinski Triangle in addition to the Pascal's Triangle view using "Waterfall Coordinates"    • The Tale of Three Triangles  

Nice article on this topic from the MAA
https://maa.org/press/periodicals/loc...

Background music
Background by 2TECH-AUDIO | https://2tech-audio.bandcamp.com
Music promoted by https://www.free-stock-music.com
Creative Commons / Attribution 3.0 Unported License (CC BY 3.0)
https://creativecommons.org/licenses/...

Chapters
00:00 Sierpinski Triangle Teaser
00:05 Pascal's Triangle even/odd is Sierpinski
00:34 Fractals using divisibility by 3 4 and 5
01:10 Detailed 5 color example
02:15 3+3 =1 (mod 5)
02:45 Miracle row of all zeros in row 5
03:54 5 by 5 triangle stencil
05:44 Another miracle row in row 25
06:04 Why zeros?
06:56 Prime rows of Pascal's triangle
08:51 Proof using binomial coefficient formula
11:58 Music of Pascal's Triangle (by Marc Evanstein)

Made in Manim
The animations for this project were made in manim and the voiceover for the animations was made in manim-voiceover. The (very messy) code for things can be found here https://github.com/mcnica89/manim/blo...