Tau day 2026: Pi square is nearly g and other coincidences

This will be the last article about \(\pi\) for this year, but I could not miss today, given that \(\pi \approx 22/7\) and in most countries of the world today’s date is 22/7, the pi approximation day.

In the spirit of approximations, and to continue the other article let’s discuss about the \(\pi^2 \approx g\) were \(g\) is the acceleration due to gravity at sea level on Earth.

During the French Revolution, one common version for the unit of length was equal to the length of wire needed for a simple pendulum to have a period of 2 seconds – so that it swings from one side to the other in just one second.

Since the period of the pendulum of length \(l\) is \(T_l\) given by

\[ T_l = 2\pi \sqrt{\frac{l}{g}} \]

we get

\[ \pi^2 = \frac{gT_l^2}{4l} \]

If we now set \(T_l = 2s\) and \(l = 1m\) we would get exactly \(\pi^2 = g [s^2/m]\) (where the \([s^2/m]\) factor makes the right hand side dimensionless, as it should be).

We can stop here, we understand why we have the coincidence we approached.

But, do we? We know \(\pi^2\) is not exactly \(g\), we know \(g\) changes on the surface of the Earth from place to place, and we know that the meter did not end up being defined this way.

In fact, the formula for \(T_l\) is assuming the small angle approximation. If I recall correctly, for an angle \(\theta\) the error is of the order \(\mathcal{O}\left(\sin^2\left(\frac{\theta}{2}\right)\right)\).

This could be an error we can manage. More importantly, though, the bob at the bottom of the pendulum’s wire matters. The formula above considers the mass to be all concentrated at a single point, but that’s not the reality. In reality we need to look at the moment of inertia for the bob. Rather than doing all the math here, have a look at the solution to a mystery: the definition of a unit of measure based on the second pendulum first showed up a century before the meter was defined. But, it used a mysterious correction factor. I really recommend reading those articles, in fact the entire blog that these 2 belong to.

For our article, let’s continue with the historical development. The meter was first formally defined as one ten-millionth of the shortest distance between the North Pole and the Equator (i.e., a fraction of the length of a meridian), as it passes through Paris. This distinction at the end is important, if we assume that the Earth is not a regular smooth sphere.

For this article, let’s just assume that the Earth is a sphere of radius \(R\). In this case, we assume that all meridians have a length of 40,000 km, by the definition of the meter (4 times the distance from the pole to the Equator).

Here, we have two options. We can try to use formulas for \(g\) taking into account the density of the Earth (assumed constant here), the volume, the definition of the meter and maybe reach a formula for \(\pi^2 / g\) that we could approximate to 1 (times the units of measure). In fact, this is what I had in mind when I first hinted at this article.

That would be a coincidence that is only valid for our planet. Change the density, and you get a different ratio. I’ll leave getting to this formula and seeing what changes from planet to planet as something that a dedicated reader could attempt as an exercise.

Instead, let’s also look at how we defined our second, as this will lead us to a new coincidence.

Our ancestors had to define units of measurement for time and space. They had two “natural” units they could use: the length of the day (\(T\)), and – if they knew about Earth being round – the circumference of the Earth (\(L\)). The Babylonians split \(T\) into hours, minutes and seconds, making 2 seconds (the period of our second pendulum) be \(T / 43200\). The French defined the kilometer as \(L / 40000\), as we have seen.

I think the 43200 and 40000 being so close together is a nice coincidence, albeit, this one is also tied to our planet. There is no relationship that I know that would link the duration of a day on a planet and the planet’s radius.

Which means, this article is just a post about historical coincidences. Since then, we have moved to defining our units in terms of quantum phenomena and the speed of light, but we still have nations that use anything but the metric system – a small boulder the size of a big boulder comes to mind.