[dylukes]

I don't find the first point surprising. (Our) pi is the one tied to the only metric where the unit circle is perfectly continuous, differentiable, etc.

The 2-norm is very special for many reasons I won't enumerate... and it seems apropos that its corresponding constant (pi)... for relating a distance from a point (wlog 0,0) to the result of integrating a constant around the path those points occupy/form/consist in... would itself tend to be found more than others.

Perhaps this is simply because without that continuity and differentiability everywhere of the corresponding path generated by the metric's unit circle, many other pieces would fall like dominoes.

There is something uniquely central about a concise relation between a point, a distance, and a path.