Question about approximation of straight line

Question

Please forgive me for my bad drawing, so every line on this diagram is meant to be a straight line.

Anyway, here is my problem, on the first diagram(the one on the far left) we see a two lines each of length 1 unit. If we make some steps to reach from the top to the bottom(as indicated on the diagram), then the total lengths of each step(both vertical and horizontal), add up to 2. If we make these steps shorter and shorter, they will always keep on adding to 2. In some sense, these steps are approximating a straight line if we make them infinitely small. But, the straight line connecting those lines would be of length sqrt(2), but how does the value go from 2 to sqrt{2} ?

My intuition is that these steps will never be equal to the straight line even if we the take the limit of the number of steps to be infinite. But I cannot prove this assertion, any help or insights would be much appreciated.

Answer 1

Using only vertical and horizontal steps gives the "taxicab" distance (see wiki). That is, $d(P,Q)=|\Delta x|+|\Delta y|,$ where the deltas are changes in $x$ or $y$ coordinates. As your picture shows, adding these up always gives $2$ (as long as the points go along the diagonal one at a time, no doubling back). But the usual distance to sum is $d(P,Q)= [(\Delta x)^2+(\Delta y)^2]^{1/2}.$

Answer 2

We've made the following observations:

1. After successive iterations, the shape of your figure gets arbitrarily close to the shape of a straight line.
2. The length of the shape, however, remains constant at 2.
3. The length of the straight line is different—it's $\sqrt{2}$.

This is a result of length being a discontinuous quality. Discontinuity means that you can find shapes that are as similar to each other as you like, and yet they still have very different length measurements. In this case, the stairstep figure gets as similar as you like to the straight line, while still having a measurement of 2 instead of the straight line's measurement of $\sqrt{2}$.

Another example would be that you can take a straight line and perturb it with various imperceptible wiggles and increase its length arbitrarily.

Most qualities are not as strange as length. In most cases, we expect the measurements of successive stages of a process to tend toward the measurement of the end result. Length is an exception to this tendency.