This year for Pi Day, I have a demo of π's appearance in bouncing balls, also known as Galperin's method due to a paper published in 2003 (Playing Pool with π (the Number π from a Billiard Point of View", Regular and Chaotic Dynamics, V. 8, No 4, 2003). Here is a youtube video demonstrating the phenomenon:
Numberphile also produced a video about it, and the NY Times Blog also gave a different derivation.
Here's a link discussing the math.
Basically, conservation of momentum and conservation of energy produce a system of equations that reduce down to stating that the the cosine of the number of collisions will change from positive to negative at an integer equal to [10^N*π], so the number of collisions will produce digits of π!