>>11030874>>11030921No, that's not a good explanation for what's going on. You can inscribe a regular n-polygon in the circle and as n increases, the perimeter will approach that of the circle (pi). The shapes will also never be exactly the same distance, so your "thus" doesn't follow at all.
What's actually happening is that in the euclidean space, we know how to calculate the distance of line segments but not of arbitrary shapes. We do have, however, a nice class of curves that we can calculate the length of in a very simple way - called rectifiable curves. We just connect points of the curve by lines and take the supremum of lengths such simpler curves. The circle is a rectifiable curve and the length turns out to be 2 pi r from this definition. The OP is an example of how careful you have to actually be about defining the length of a complex curve (that's not made out of line segments), and how different ways to define it lead to different lengths.
Also, now that I think about it, the taxicab sort of length is probably also well defined for rectifiable curves, which will also be a type of length. But we don't like this definition because it doesn't coincide with our intuition for simple line segments. So in fact, as long as we have a well-defined concept of a length of a curve for simple enough curves which coincides with our intuition for line segments and satisfies a certain number of intuitive properties, the length will always come out as 2 pi r.
OP is a beautiful example of how we need to be careful about our definitions, just like the examples of continuous yet nowhere differentiable functions and all sorts of weird topological spaces.
