The animated mosaics use 12 different types of animated squares. Each one is made up of the same 20 frames as each other, although each of the 12 squares start and end at different frames (keeping the same sequence, though). The first square changes its frames every 4 seconds, the next one every 4.25 seconds, etc right up to the 12th which changes every 6.75 seconds.

So here's the problem, the differences in animation rates seem to keep the patterns always changing and they never seem to repeat, but surely they must at some point. How long will it take until it starts repeating?

Here's the answer (thanks to my brother Jeremy):

First, some trivia: there are 20 ^ 12 (20 to the exponent 12) different patterns possible, which works out to 4e15 (a 4 with 15 digits following). It could take a long time to show them all. However they do not all get shown, because your timings are not prime.

By "repeat", I assume you mean the exact same frames in the exact same spots, and the exact same timing.

The fastest spot repeats its whole series every 20f x 4s/f = 80s (f = frame, s = seconds). The next fastest spot repeats its whole series every 20f x 4.25s/f = 85s. And so on up to the slowest, at 135s. What you are looking for is the "lowest common multiple" of these numbers 80, 85, ..., 135. The LCM is the smallest number that all of these values divide into evenly. To figure it out, we factor out each number into primes, and gather up all the unique primes. Sorry, no quick formula. Here goes:

Multiply out the unique primes: 2^4 x 3^3 x 5^3 x 7 x 11 x 13 x 17 x 19 x 23. My calculator says 4e11 seconds.

So not to worry, the pattern will not fully repeat for roughly 12700 years.

There are some disclaimers here. That 12700 year figure is for the whole pattern to repeat from the beginning -- including timing and order. It may be possible that some specific picture would repeat earlier, for a split second.