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I would. Pi is (indeed always) the circumference of a circle divided by its diameter. To calculate the perimeter of a 2d convex hull, my impulse would be to mod

I would. Pi is (indeed always) the circumference of a circle divided by its diameter. To calculate the perimeter of a 2d convex hull, my impulse would be to model the shape in polar coordinates and integrate over theta. The fact that this calculation relies on polar coordinates strongly implies a relationship to circles -- specifically, L = 2 * pi * r_mean turns out to be unsurprising, since you're basically treating each slice of the hull as a slice of a circle, and calculation based on that circle's radius.

drdeca 10y ago
by usually I meant that is usually how it is defined. Of course it is always true that it is that value, I just meant, like, what description of the value is used to define what value we mean. I was unclear on that part. I meant that I think that in most cases, pi = circumference / diameter is the definition of pi, rather than pi being defined in some other way, and it being proven as a theorem that pi = circumference / diameter. Like, suppose I defined pi as sqrt(6*Zeta(2)) , this would still be the same value, so it would work as a definition of pi, but I don't think it is often defined in a way like this. I think I mostly wrote ^ because I felt like the way I worded my previous comment could be misinterpreted in a way that I feel would make me look dumb, so I wanted to clarify what I actually meant. Your point about considering it in polar coordinates seems like a good explanation. I hadn't thought of that, I like it, thank you for that. I don't yet see how to make that rigorous, but it sounds like it makes sense in a nice way. Thank you.