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 ?
