Is $\pi^2 \approx g$ a coincidence?

Question

In spite of their different dimensions, the numerical values of $\pi^2$ and $g$ in SI units are surprisingly similar, $$\frac{\pi^2}{g}\approx 1.00642$$

After some searching, I thought that this fact isn't a coincidence, but an inevitable result of the definition of a metre, which was possibly once based on a pendulum with a one-second period.

However, the definition of a metre has changed and is no longer related to a pendulum (which is reasonable as $g$ varies from place to place), but $\pi^2 \approx g$ still holds true after this vital change. This confused me: is $\pi^2 \approx g$ a coincidence?

My question isn't about numerology, and I don't think the similarity between the constant $\pi^2$ and $g$ of the planet we live on reflects divine power or anything alike - I consider it the outcome of the definitions of SI units. This question is, as @Jay and @NorbertSchuch pointed out in their comments below, mainly about units and somewhat related to the history of physics.

Answer 1

The differential equation for a pendulum is

$$\ddot{\phi}(t) = -\frac{g}{l}\cdot\sin{\phi(t)}$$

If you solve this, you will get $$\omega = \sqrt{\frac{g}{l}}$$ or $$T_{1/2}=\pi\sqrt{\frac{l}{g}}$$ $$g=\pi^2\frac{l}{T_{1/2}^2}$$

If you define one metre as the length of a pendulum with $T_{1/2}=1\,\mathrm{s}$ this will lead you inevitably to $g=\pi^2$.

This was actually proposed, but the French Academy of Sciences chose to define one metre as one ten-millionth of the length of a quadrant along the Earth's meridian. See Wikipedia’s article about the metre. That these two values are so close to each other is pure coincidence. (Well, if you don't take into account that the French Academy of Sciences could have chosen any fraction of the quadrant and probably took one matching the one second pendulum.)

Besides that, $\pi$ has the same value in every unit system, because it is just the ratio between a circle’s diameter and its circumference, while $g$ depends on the chosen units for length and time.

Answer 2

It's annoyingly unclear how far it's a coincidence, but at any rate it isn't completely a coincidence.

As you can see in e.g. the Wikipedia article about the metre, a unit almost equal to the metre but derived from a pendulum was first proposed in 1670 and the idea was in the air when revolutionary France decided to make a new set of units.

This pendulum-derived unit, if adopted, would have made $g$ equal to $\pi^2\, \mathrm{m}/\mathrm{s}^2$ by definition. The proof of this is very easy, and can be found in other answers here, so I shan't repeat it.

(So if that definition had been adopted, the answer to the question here would be an unequivocal yes.)

Here is a link to an English translation of the report of the commission appointed by the French Academy of Sciences. They explain that the pendulum-based definition is very nice but has the drawback that it depends on the second which is a rather arbitrary unit. So instead they propose to take $10^{-7}$ of a quarter of a meridian of the earth.

Now, this unit they've adopted is (1) almost exactly equal to the pendulum-based unit, but also (2) derived in a conspicuously simple way from the dimensions of our planet. So it's overdetermined. Did Borda, Lagrange, Laplace, Mongé and Condorcet (an impressive list of names indeed, by the way!) choose the particular earth-based definition they did because of its closeness to the pendulum-based definition, or not? That's what's annoyingly unclear.

I find two useful clues in their report. They point in opposite directions.

First, they say

in adopting the unit of measure which we have proposed, a general system may be formed, in which all the divisions may follow the arithmetical Scale, and no part of it embarras our habitual usages: we shall only say at present that this ten millionth part of a quadrant of the meridian which will constitute our common unit of measure will not differ from the simple pendulum but about a hundred and forty fifth part; and that thus the one and the other unit leads to systems of measure absolutely similar in their consequences.

which makes it plain that they knew how close the two were, and were glad to take advantage of it. But, second, they say

We might, indeed, avoid this latter inconvenience by taking for unit the hypothetical pendulum which should make but a single vibration in a day, a length which divided into ten thousand millions of parts would give an unit for common measure, of about twenty seven inches; and this unit would correspond with a pendulum which should make one hundred thousand vibrations in a day: but still the inconvenience would remain of admitting a heterogeneous element, and of employing time to determine an unit of length, or which is the same in this case, the intensity of the force of gravity at the surface of the earth.

so they were clearly prepared to countenance the possibility of a somewhat different unit, and the argument they give against this one has nothing to do with its disagreement with the pendulum-based unit that's so close to the metre.

(Though ... if they had ended up plumping for that definition, they might also have proposed redefining the second to be $10^{-5}$ days, in which case they would again have made $g=\pi^2$ by definition.)

I think the first of those passages is enough to make it clear that it isn't a total coincidence that $g\simeq\pi^2$. Borda et al knew that their definition was close to the pendulum-based one, and offered that fact as a (minor) reason for accepting it. But I think the second is enough to suggest that it could easily have been otherwise: my feeling is that if the definition based on meridian length had been, say, 5% different from the pendulum-based one, they would still have preferred it.

In comments below, user Pulsar has found an interesting article about this whose conclusions are roughly the same as mine: it sure looks as if the pendulum-based second was a motivation for the choice of $1/(4\times10^{7})$ of a meridional great circle, but nothing here is altogether clear and we have to rely on conjecture about the motivations of the scientists involved.

Answer 3

In addition to Anedar's answer, I'll try to address things from a bigger perspective.

When they made the SI unit system, they chose units that are convenient for humans. From a scientific perspective, it would make sense to use for example Gaussian units, but for the man on the street, it is more useful to have the units more or less agree to things humans encounter in daily life.

So for a length scale, you want something that is roughly the same size as a human body. And for a time scale, you want something that roughly represents how fast humans can count.

In the SI unit, the meter and second were chosen in such a way that:

• 1 m is roughly the typical distance of an adult's leg.
• 1 s is roughly the typical time it takes to walk two steps.

Because there is a relation between leg length, time between steps and gravity, this corresponds to a gravity constant in this unit system of roughly $\pi^2$.

They could have chosen different anthropocentric units. The meter could have been defined twice as long or twice as short, and the second could have been defined twice as long or twice as short. But not a factor thousand longer or shorter, then it would not have been accepted by the community, because the units would have been very inconvenient.

So I claim that on every planet with life which developed a system of units, their local gravity constant in their system of units is between 1 and 100.

(This answer explains why the numeric value of $g$ in SI units is not 10000000. It does not explain why the numeric value of $g$ in SI units is not 13.)

Answer 4

$g$ is a value with units, and $\pi$ is a dimensionless number. If you consider a unit system that uses miles, days, and grams as the units of length, time and mass, you can see that $g$ will be quite different.

Answer 5

Assume the values of meter and second to be fixed. The equation for half a period in Anedar's answer $$T_{1/2}=\pi \sqrt{\frac{l}{g}}$$ would return the values of $\pi^2$ and $g$ to be equal if one measures a meter long pendulum to complete a half period in 1 second.

One just simply need to find a place on earth where this would be the case since $g$ is not really a constant. In that place one would indeed measure $g=\pi^2$. So in that place it would be a coincidence.

Answer 6

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ 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 Newton’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 Hilbert’s celebrated adage, wir mussen wissen, wir werden wissen, 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. 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.

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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 $1.8$ metres almost exactly, with an error of no more than a few millimetres, making the above length my own personal rod; indeed, I am a rather metric person, since even my own height towers at just slightly over $1.7$ metres, and does not exceed $171\rm~cm$ — but I digress $\ldots$

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Some of the above relations are $($easily$)$ explained $($away$)$ by means of basic arithmetic, such as, for instance, the fact that $3\cdot7^3\simeq2^{10}\simeq10^3$, or $2^7\simeq5^3\simeq11^2$, and $2^8\simeq3^5$, the latter two “culprits” being responsible for the beautiful approximation $3000_{12}\simeq5000_{10}$, or, equivalently, $12^4\simeq2\cdot10^4$, which relate duodecimal thousands and myriads to their decimal counterparts; others, however, are $($much$)$ harder to dispel. Nevertheless, this is precisely what we shall endeavour to achieve !

Let us therefore fearlessly approach the most awe-inspiring of all the above-listed coincidences, and merrily $($and mercilessly$)$ debunk the life out of it $-$ in the name of science ! :-$)$

Now, the way I see it, if the ratio in question were truly $3:4$, then dividing the distance traveled by light in a day’s time $($since this is the smallest naturally occurring time unit which is also easily observable by man$)$ to the length of an Earth’s meridian should yield a result of exactly $648~000.~$ However, by employing the most accurate measurements known to date, namely that of $$c=299~792~458~\rm\dfrac ms~,$$ and a quarter of a terrestrial meridian being $\ell\simeq10~001~965~.~7293\rm~m$, we ultimately arrive at the dull and uninspiring figure of $~\dfrac{24\cdot60^2\cdot c}\ell~\simeq~647~424~\dfrac49,~$ which is roughly $~575~\dfrac59~$ times less than expected.

In other words, by enhancing the resolution of our lengths and ratios, the ghosts of modern superstitions are forever shattered in the cold light of day by the power of reason, and our minds can finally rest assured that the whole thing was nothing more than a tempest in a teapot,
or much ado about nothing, as Shakespeare so wonderfully put it all those centuries ago ! Now
all that is left to do is praying no one notices that the previous ratio can also be expressed as $27~27\rm~BB$ in base $12$, with an error of less than one and a half units. :-$)$

On a more serious note, it all boils down to divisors $($usually powers of $2,~3,$ and $5)$ and numeration systems. $-~$ Doesn’t it ?$\ldots$ In the words of Thomas More, I trust I make myself obscure. :-$)$

Answer 7

Pi is invariant across most Newtonian geometry systems. g however changes drastically around the planet as well as when you change units. But if you like coincidences, there is about π * 10^7 sec/year.

Answer 8

Given our knowledge of physics, this must surely be a coincidence.

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$\pi$ arises from investigation of certain mathematical relations. Let's consider the ratio of a circle's radius and circumference: This is not the most interesting occurrence of $\pi$ (the relations studied by Euler and after are sometimes considered more noteworthy) but it is a nice simple example.

The circle is a set of points equidistant from an origin. The constraints only allow a single arrangement of such points in certain kinds of space. $\pi$ characterizes this unique arrangement. Note how this was arrived at with purely mathematical reasoning, without any recourse whatsoever to the physical world. Aliens in an entirely different parallel universe, or demons in Hell could have reasoned likewise, and discovered the same $\pi$, regardless of how different the physical laws governing them are. Mathematics is unaware of reality, it doesn't care about what happens in the so-called real world. All it is, is logical deduction of consequences arising from a set of axioms.

It so happens that $\pi$ can be experimentally observed, for instance by constructing circles out of wire. But there is a very biased causality here: The wire loop exhibits $\pi$ because our world is like the Platonic ideal of Euclidean space, not the other way around. Although it is worth noting that there is of course a reason why Euclid happened to start with exactly that kind of space, and not another.

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$g$ arises from the action of masses on each other. For unclear reasons, the world which we inhabit contains masses. The way these masses behave appears to follow certain rules. These rules were deduced by observation of the physical world. Analysis of the rules yielded $g$.

Aliens in Universe X, or demons in Hell, could not themselves find $g$ without observing our universe. By methods analogous to ours, they can find $g_{alien}$ or $g_{hell}$. These can easily be different from our $g_{earth}$, but they're not prohibited from coincidentally equaling it either. The numbers have absolutely nothing to do with each other, being parts of disjoint physical systems.

Note that physics is itself an abstract construct, there is no reason to believe that the universe obeys our laws of physics, it has merely never been observed to act in contradiction to those laws which have not yet been refuted (the circularity is meaningful). Unlike mathematics, the construct of physics relies on not only a priori assumptions, but also a posteriori observations of our physical world.

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You don't have to accept that $\pi_{earth}=\pi_{hell}=\pi_{alien}$. You can, for instance, make the disturbing but reasonable objection that mathematics is nothing but an artifact of the human brain, and is not universal but a posteriori. In a sense this position is weak, because we have observed animals to have a similar understanding of mathematics, but of course that is only circumstantial evidence, not proof.

If you do accept that $\pi_{earth}=\pi_{hell}=\pi_{alien}$, being that $\pi$ is obtained with no input from the physical world: Then whereas everyone's $\pi$ is necessarily equal, everyone's $g$ need not be. Thus the relation you observe would only hold in our universe, not in hell or Universe X. In other words, our universe "easily could have had" a different $g$ - it is unclear whether the laws of physics we know had no choice but to be the way they are, or if there was some kind of dice rolling to conjure up a bunch of random laws, and we "could have" ended up with a different set. It is not even clear if the laws have always held, or will hold in the future. Although we have never observed them not holding so far - except for the ones we did, but we don't talk about those anymore...

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One can observe that while mathematics doesn't care about the world, the world does appear to obey mathematics. We have never observed the real world contradict mathematical logic. So, it is not impossible that one day, the true nature of $g$ will be understood, and it will turn out to have a geometrical origin (for instance), and we will find out that your observation is in fact meaningful, not mere coincidence. But as far as I know, no such geometrical explanation exists. I doubt this will ever happen either, because the relation works only on Earth, and not even everywhere on Earth.

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Note 1: In this answer, I took a philosophical position regarding the nature of mathematics, that is not understood to be necessarily true. There are valid objections to it. I personally feel that my position is prima facie congruent, so I wrote this answer. If you have a radically different concept of mathematics, perhaps other answers can be given to your original question.

Note 2: I didn't want to be mean and give you the boring answer right off the bat. For the sake of completeness, here it is: $\pi^2$ is like $g$... only if you use meters and seconds, two explicitly arbitrary units. In Planck units the relation does not exist. In fact, with the right units, you can make $g$ be like $e$, or your age, or your ZIP code, or any other number you desire.