$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ Is $~\pi^2\approx g~$ a coincidence ?
Some have answered yes, others said no, and yet others considered both $(!)$ as perfectly viable options. Personally, I cannot help but chuckle, as this question reminds me of <PERSON>’s famous disc, which can be said to be both white and colored at the same time, depending on whether it is either rotating, or at rest. To add even more to the already mystifying fog of confusion, I shall hereby venture yet a fourth opinion :
$\qquad\qquad\qquad\qquad\qquad\quad$ We don’t know, and we never shall !
Granted, such a statement, when taken at face value, would undoubtedly appear as an impious affront to <PERSON>’s celebrated adage, <PERSON>, <PERSON>, but before anyone accuses me of embracing either philosophical pessimism or epistemological agnosticism, let me assure you, dear reader, that such is simply not the case; rather, I am basing this short assertion purely on mathematical foundations. 346
Basically, there are four main ways in which a measuring unit can be created, that is both practical or anthropocentric, as well as universally meaningful, at the same time $($not to mention reproducible$)$ :
* the length of the pendulum with a half-period of exactly one second, since the length of a pendulum with a half-period of one minute will be exceedingly long;
* the ten-millionth, the hundred-millionth, or even the billionth part of either a terrestrial meridian, or the Earth’s equator, since the other two adjacent options, i.e., the millionth and the ten-billionth part, would be either way too big, or way too small;
* the distance traveled by light in the hundred-millionth, the billionth, or even the ten- billionth part of a second; again, the other two adjacent options i.e., the ten-millionth and the hundred-billionth part, would be either way too long, or way too short;
* the length of a so-called third $($ i.e., the sixtieth part of a second $)$ of the Earth’s meridian or equator.
Of course, someone might, at this point, easily be tempted to say that I have committed a hideous and unpardonable abuse by painstakingly enumerating all those powers of ten listed above, since the metric system, as we have it today, is coincidentally decimal, but such would not necessarily have been the case, given an alternate course of human history $($thus, for instance, if one were to take the distance traveled by light in $10^{-9}$ seconds, such a length could easily have been interpreted as representing a “new foot”, to be further subdivided into $12$ “new inches”, ultimately yielding a “new yard” of $0.9$ metres$)$.
Now, the shocking surprise, which astounded many at the time of its first discovery, and still does so even today, is as follows : the ratio of the first three units is $1:4:3$, almost exactly, the sheer “niceness” of the numbers involved being utterly uncanny, to say the very least. $($Spooky, thought-provoking, challenging, bewildering, and mesmerizing also come to mind$)$. Adding insult to injury, as the proverb goes, we also notice that twice the value of the latter unit, representing the $3~600^\text{th}$ part of a nautical mile, equals $103$ centimetres, with an error of less than $\pm1$ millimetre; speak- ing of which, the thousandth part of a nautical mile is also conspicuously close to the length of a fathom, measuring the distance between the fingertips of a man’s outstretched arms.
Furthermore, even if one were quite purposefully to go out of one’s way, and intentionally try to avoid the two coincidences above, by $($repeatedly$)$ dividing, based purely on number-theoretical principles, the aforementioned non-metric unit into, say, sevenths, $($since the powers of all other previous primes already appear abundantly in its sexagesimal creation$)$, one would arrive at the eerie conclusion that it adds up to $5.4$ metres, with an error of less than half a millimetre.
As an aside, as $($even further$)$ coincidence would have it, my own personal fathom is $1. 840