This pair of videos includes several really surprising relationships, between binary/ternary counting, the Towers of Hanoi puzzle, and Sierpinski's triangle. I was also pretty excited to be able to do a project with Keith Schwarz, a phenomenal CS lecturer at Stanford. Anyone curious should look at some of his projects at http://www.keithschwarz.com/
This is also surprisingly relevant to the space filling curves video I did some time ago. In those, I included a brief animation of a curve that fills Sierpinski's triangle, but I'll admit that it was rather unintuitive to me. This here was the first time that I felt a deep intuitive reason for why such a curve should exist, and can naturally be defined.