Like the apparently backward-turning wheels (especially wagon wheels) you can see on TV or in movies, this has a lot to do with the extended Euclidean Algorithm.

ax + by = gcd(a,b) indeed.

Wait. ax + by = gcd(a,b) how? Not that I doubt your statement, wofdog. I just don't understand it!

This works best by example so suggest me a pair of moderately-sized positive integers a & b to demonstrate with.

OK. a=56 and b=987

The video was beautiful, great choice of music. As a non-science person, I appreciaetd it for its aesthetic qualities.

Great. The gcd (greatest common divisor; the largest number that goes into both) of 56 and 987 is 7. And we can express 7 as an integer combination of multiples of 56 and 987:

7=56(53)+987(-3)

So in the notation above, we can take x=53 and y=-3.

The interesting thing is that the gcd of a & b -- in this case 7 -- is the SMALLEST number that can be expressed as an integer combination of a & b. The full details of Euclid's algorithm for finding the gcd both prove that is true and give a rapid method for finding the numbers x and y.

You can actually see this occurring if you mark out a number line with all the multiples of a and b indicated on it. The smallest gap between two marks is the gcd. Take 10 and 4, for instance. Here's a little line with all the multiples of 10 and 4 marked (the first dot is 1; the first emphasized position is at 4):

...X...X.X.X...X...X...X...X.X.X...X...X

The minimum distance between X's on this line is 2 (as between 8 and 10, for instance), corresponding to the fact that gcd(4,10)=2.

Now, imagine that multiples of A correspond to time ticks when a water drop emerges, and multiples of B correspond to a flash of the strobe. Or a wheel spoke passing the top of the wheel and the camera capturing a frame. It's really the gcd of A & B - together with the human visual system's propensity for continuous interpolation - that determines what the resulting visual effect will be when you sample a phenomenon that occurs B times per second at a rate of A times per second. By choosing your frame rate carefully, you can get any desired effect.

For instance if I have drops coming at 10 per second (100 ms between drops), and I strobe every 99 ms, the effect I'll get is as if the drops were moving very slowly upward. The visual continuity I get is actually between one drop (x) and the drop behind it (x+1) - at the next strobe, (x+1) will be just a tiny bit above where x, so the perceived motion is just slightly upward.

If I have very fine control over the rates of strobing then I can get any desired effect because I can make the frame rate prime to the drip rate, and from there get any multiple of the gcd, which is 1. If I don't have sufficiently good control over the strobe light, I can only get apparent rates which are multiples of the gcd's I can achieve.

Gee, Wolfdog! I have no idea what you just said, but I'm pretty sure I want to either marry you or adopt you, you silver-tongued devil.

Very, very cool, pi!

time is so pwned!

Thanks WD! I'm not sure I followed that completely, but I get the gist. I think.