But that connection actually is a coincidence. From what I can tell, when they standardized the meter, they were specifically going for something close to half of a toise, which was the unit defined as two pendulum seconds. So they searched about for something that could be measured repeatably and land on something close to a power of ten multiple of their target unit. The relationship to a circle there doesn’t have anything to do with the pi^2 thing.
Not a coincidence. They defined the meter from the second using the pendulum formula, and the pandulum formula has a pi in it, so pi is going to appear somewhere. The reason there is pi is probably because a pendulum is defined by its length and follows a circular motion that has the length as its radius.
We could imagine removing pi from the pendulum equation, but that would mean putting it back elsewhere, which would be inconvenient.
Right, that connection is not a coincidence. The connection the previous commenter drew between the meter, pi, and the circumference of the earth is a coincidence.
> The reason there is pi is probably because a pendulum is defined by its length and follows a circular motion that has the length as its radius.
It’s not quite that easy: For small excursions x the equation of motion boils down to x’’+(g/L)x=0. There is not a π in sight there! But the solution has the form x=cos(√(g/L)t+φ), with a half period T=π√(L/g), thus bringing π back in the picture. So indeed not a coincidence.
It was news to me, but that's what the article says, and it is supported by by Wikipedia, at least. [1]
In addition, I feel the article glosses over the definition of the second. At the time, it was a subdivision of the rotational period of the earth (mostly, with about 1% contribution from the earth's orbital period, resulting in the sidereal and and solar days being slightly different.) Clearly, the Earth's rotational period can (and does) vary independently of the factors (mass and radius) determining the magnitude of g.
The adoption of the current definition of the second in terms of cesium atom transitions looks like a parallel case of finding a standard that could be measured repeatably (with accuracy) and be close to the target unit - though it is, of course, a much more universal measure than is the meridional meter.
