Thanks to Pat Ballew of Pat's Blog for turning me on to this wonderful video of 15 pendulums of different lengths. To say this is mesmerizing and will watch it more than once is an understatement.
I sure it will go viral in short order. I'd love to figure out the mathematics involved.
"How it works: The period of one complete cycle of the dance is 60 seconds. The length of the longest pendulum has been adjusted so that it executes 51 oscillations in this 60 second period. The length of each successive shorter pendulum is carefully adjusted so that it executes one additional oscillation in this period. Thus, the 15th pendulum (shortest) undergoes 65 oscillations. When all 15 pendulums are started together, they quickly fall out of sync--their relative phases continuously change because of their different periods of oscillation. However, after 60 seconds they will all have executed an integral number of oscillations and be back in sync again at that instant, ready to repeat the dance."
... Harvard Natural Science