Perhaps I could more easily answer your question if you tried to explain why you think the length of the zig-zag line should approach the length of the straight line. It's zig-zagging - why should it end up the same length? You say that in the Manhattan Distance every point is a step towards the goal, but it's not directly towards the goal, therefore you always get a bit more than you want.
The number of points isn't relevant - the Cantor Set (also known as Cantor Dust) is an uncountably large subset of the real line, and yet its length is zero.
I guess the only insight I can offer at this point is that length has nothing to do with points.
I think he is not asking what he wants to know. He wants to know how we know why Archimedes' solution does converge to the right length. Couldn't there be a problem with his solution that we're just not seeing yet? i.e. what makes Archimedes right and the troll wrong?
Here's a thought: Archimedes's approximation approaches the target. That is, the perimeter of each subsequent approximation is closer to that of the circle than all that came before.
The side of a regular polygon is √(sin²(τ/n)+(1−cos(τ/n))²)=√(2(1−cos(τ/n))), while the corresponding arc is τ/n.
d/dn τ/n−√(2(1−cos(τ/n))) = −(τ/n²+sin(τ/n)/√(2(1−cos(τ/n)))) < 0, so the difference is strictly decreasing (note that sin(τ/n)>0 when n>2).
So there's your lower bound for the difference. You'd still have to prove the limit is at 0, though.
That doesn't explain why. Why is it that if tangents approach the circle (whatever that means precisely) that the length of the approximation approaches the length of the circle?
Loosely speaking, the total path length is defined as the sum of a bunch of infintesimal segments that have both position and direction. For the sum (integral) of those infintesimals to approach the sum of the similar infintisemals around the actual circle, those of the approximation path have to approach those of the circle in the limit, and of course this means in both position and direction.
The number of points isn't relevant - the Cantor Set (also known as Cantor Dust) is an uncountably large subset of the real line, and yet its length is zero.
I guess the only insight I can offer at this point is that length has nothing to do with points.