/sci/ - Science & Math

>>11281244
this.

>>11229149
There's an obvious way to measure the lengths of line segments using the Pythagorean theorem, and by extension a curve composed of finitely many line segments. You run into trouble when the curve is more general.
Mathematicians define a curve to be rectifiable if as you approximate the curve with line segments and calculate the lengths of the approximate curves composed of line segments, the lengths converge to some number as the number of line segments and the maximum length of a line segment goes to infinity.
A circle defined by the equation x^2 + y^2 = 1 has been proven to be rectifiable, and the limit of the lengths that the approximating curves approach is proven to be an irrational number somewhere between 3 and 3.2. We call that number pi.

>>11171853
It's true with respect to the taxicab metric d((x1, y1), (x2, y2))=|x1 - x2|+ |y1 - y2|
>>11172060
True.
>>11172080
pi is defined by approximating it with figures that have finitely many corners. Your reasoning for why OPs pic is wrong is wrong.
>>11172220
Take the straight line and approximate it with straight lines. The sum doesn't change, but you still reach the smallest upper limit.
>>11172721
More 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...).
>>11172767
True.
>>11172768
Because 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 area
It 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.
>>11172839
Yes.
>>11172847
What do you mean?

The circumference of the unit circle IS 4 with the taxicab metric.

I made a version of this meme earlier with an explanation. It all comes down to how you want to define length. With a taxicab metric d(a,b) = |a_x - b_x| + |a_y - b_y|, the perimeter of an euclidean circle would indeed be 8.