>>11171853It's true with respect to the taxicab metric d((x1, y1), (x2, y2))=|x1 - x2|+ |y1 - y2|
>>11172060True.
>>11172080pi is defined by approximating it with figures that have finitely many corners. Your reasoning for why OPs pic is wrong is wrong.
>>11172220Take the straight line and approximate it with straight lines. The sum doesn't change, but you still reach the smallest upper limit.
>>11172721More like, why definitions are important. Defining the length of a curve wrt. the taxicab metric gives you the result pi=4 (with pi here being the perimeter of the radius 0.5 euclidean circle wrt taxicab metric, I'm not claiming 4=3.14...).
>>11172767True.
>>11172768Because area and length are two fundamentally different concepts, as OPs pic demonstrates. You can't do what you do in measuring area to measure length - it's not well-defined.
>>11172770>because it does not converge in perimeter, only in areaIt does converge in perimeter to 4, clearly. It doesn't converge to 3.14..., but you'd have to ask why would you want it to converge to that number in the first place, how do you know the length is actually 3.14..? You need a definition of length.
>>11172839Yes.
>>11172847What do you mean?
