It turns out that π² ≈ g is not some mystical coincidence. When the first scientists contemplated the definition of the meter, there was one elegant proposal: to make the meter equal to the length of a pendulum that takes exactly one second to swing from one side to the other.
For a mathematical pendulum, the period of oscillation is calculated by the formula: T = 2π √(L / g). If we take the length L = 1 meter and set the full period T = 2 seconds (so that it takes exactly one second for each half swing), the equation implies: g = π² (m/s²).
The definition of the meter was later changed: it was tied to one ten-millionth of the distance from the equator to the North Pole along the meridian passing through Paris. However, this geodetic definition was inspired by the earlier idea with the pendulum. Notably, both approaches match up with an accuracy of 1%. Essentially, since the old “pendulum” definition was the main candidate for a long time, values were adjusted so that the new meter was convenient and close to the measurements customary at that time.
It is also interesting that the number of seconds in a year roughly corresponds to the number of pi * 10^7. Earth’s orbital speed is about v = 30 km/s. The distance from the Sun to Earth is approximately r = 150,000,000 km. Thus, over a year, Earth travels a path of about d = 2 * π * r. Then, the orbital period equals T = d/v = π * 2 * r/v = π * 10⁷ seconds.
Hi, I'm Rauf Aliev. 