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Suppose that we have a unit square and are interested in the distance between two opposite corners.

The euclidean distance is $\sqrt{2}$.

The manhattan distance is $1 + 1 = 2$.

Suppose we subdivide the square by divide both the width and height in half.

The euclidean distance remains $\sqrt{2}$.

The manhattan distance of $n$ subdivisions is $n^2 \frac{1}{n^2} + n^2 \frac{1}{n^2} = 2$.

Then, $\lim_{n \to \inf} = 2$.

That said, if $n$ is large I could be presented with a unit square with such high resolution that I wouldn't be able to distinguish the one with a distance of $2$ and $\sqrt{2}$. This seems strange to me. Why doesn't $\lim_{n \to \inf} = \sqrt{2}$ intuitively? I'm assuming it has to do with the fact that no matter how small the resolution, the hypothenuse of each subdivision will always be shorter than the sides, but it's still a bit strange to me.

Cedric Martens
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    Ultimately, it is because the "length of this curve" function is not a continuous function. While the sequence of curves you generate does "approach" the desired curve in some sense, it does not mean that their lengths should.

    This is the core flaw behind many, more elementary, "fake proofs" in math, like claims that $\pi=4$ or $\sqrt 2 = 2$.

    Some more discussion is here and elsewhere

     – PrincessEev Jun 13 '23 at 21:46
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    At the end of the day, the manhattan distance still travels up 1 unit and right 1 unit, regardless of how much it zig zags in between. – Shou Jun 13 '23 at 22:01

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