[_Jim]

re: "Has always driven me nuts that pi is an approximation."

You don't EVEN want to think about the "Fine-structure constant" then ...

“The strength of the familiar electromagnetic force between two electrons, for example, is expressed in physics in terms of a constant known as the fine structure constant. The value of this constant, almost exactly 1/137, has puzzled many generations of physicists. A joke made about the famous English physicist Paul Dirac (1902-1984), one of the founders of quantum mechanics, says that upon arrival to heaven he was allowed to ask God one question. His question was: "Why 1/137?”

― Mario Livio, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number

[Kartographer]

Why 1/137??
Because otherwise my son we would not be standing here having this conversation.

[HangnJudge]

Well, the Fine-structure constant is not quite 1/137

α^-1 = 137.035999139 approximately

[EEGator]

Long term ratio of a Fibonacci sequence.
1.618...