Doesn't the relationship hold if we change units? It seems like it must.
When I worked with electric water pumps I loved that power can be easily calculates from electrical, mechanical, and fluid measurements in the same way if you use the right units. VoltsAmps, torquerad/sec, pressure*flow_rate all give watts.
Nope, it completely vanishes in other units. If you do all your distance measurements in feet, for example, the value of pi is still about 3.14 but the acceleration due to gravity at the earth's surface is about 32 feet s^(-2). If you do your distance measurements in furlongs and your time measurements in hours then the acceleration due to gravity becomes about 630,000 furlongs per hour squared and pi (of course) doesn't change.
If I come up with my own measuring unit, let's call it the sneezle (whatever the actual length I assign to it) I will be able to also define a duration unit (say, the snifflebeat) based on the time it takes for a pendulum one sneezle long to complete a full oscillation, and vice versa I can define the sneezle by adjusting the length of a pendulum so that it oscillates in two snifflebeats. Here are the maths:
T = 2π√(l/g)
T/2π = √(l/g)
(T/2π)^2=l/g
g = l/(T/2π)^2
g = l/(T^2/4π^2) = 4π^2xl/T^2
Now substitue T with 2 and l with 1 an you get
g = 4π^2x1/2^2 = π^2
It doesn't matter what the pair of units assigned to T and l are. However, they'll be interrelated.
There is nothing arbitrary, and no coincidences behind g =~ π^2. It just requires to do some history of metrology and some basic maths/physics.
If you want to discuss coincidences, may I suggest you to comment on this remark I made and which hasn't received any attention yet ?
This is not quite the same situation, as you are calculating a value having a dimension (that of power, or energy per second) three different ways using a single consistent system of units, and getting a result demonstrating / conforming to the conservation of energy. If you were to perform one of these calculations in British imperial units (such as from pressure in stones per square hand and rate of flow in slugs per fortnight) you would get a different numerical value (I think!) that nonetheless represents the same power expressed in different units. The article, however, is discussing a dimensionless ratio between a dimensionless constant and a physical measurement that is specific to one particular planet.
A more natural way to say it is that equality requires that the unit of length is the length of an arbitrary pendulum and the unit of time is the half-period of the same pendulum.
The pendulum is a device that relates pi to gravity.
The arbitrary length pendulum with a period of 2 seconds which is your unit of length, (or 1 Catholic meter) is much shorter on the moon.
In local Catholic meters gravity would be pi squared Catholic meters / second.
As it would on any planet.
Either way works to approximately pi. There is a particular length where it works out exactly to pi which is about 3.2 feet, or about 1 meter. My point was that equations like that remain true regardless of units.
The reason pi squared is approximately g is that the L required for a pendulum of 2 seconds period is approximately 1 meter.
When I worked with electric water pumps I loved that power can be easily calculates from electrical, mechanical, and fluid measurements in the same way if you use the right units. VoltsAmps, torquerad/sec, pressure*flow_rate all give watts.