#### Re: Can someone explain . . .

*The magic of geometry*

And some even simpler linear algebra. It's like* if you have two buses that serve the same bus stop, one that arrives every 40 minutes, another every 45 minutes. The time between instances where both buses arrive at once is the least common multiple of the two times, which in this case would be 360 minutes. Accounting for the wobble is like saying that you only visit the bus stop every, say, 50 minutes so you're only interested in times that you're actually there. Again, you use the LCM. The LCM of 360 and 50 (or of 40, 45 and 50, if you want to combine all three values at once) is 1800, so the time between the coincidences is 1800 minutes or 30 hours.

* Obviously, this is a simplification. The bus stops would be moving, for one thing, since we're interested in colinearity rather than when planets are at fixed points. In two or three dimensions with elliptical orbits, the calculations are a bit more involved, but the basic ideas of periodicity still hold (as far as I know; please correct me if I'm wrong). The reason I'm talking about the simpler case is that it helps to understand that the LCM is fundamental to combining periods. Most notably, if the periods being combined are relatively prime, then the combined period is the product of each of the individual periods, which might be a surprising result if you didn't know about the LCM.

Incidentally, they reckon that cicadas are so successful because the period of their life cycle is relatively prime to that of the predators that keep them under control. This means that they get the maximal period between "busts" in their predator-prey cycle.