Taxi Cab Geometry: Correct way to describe the discrepancy between "stair step" & diagonal lines? The taxi-cab geometry problem is described here: < In what way is a diagonal line different than a zig-zag line following its path as described in taxi-cab-geometry? Is a diagonal line considered to be "off grid", in another dimension? What is an accurate way to describe the difference/relationship? ![enter image description here]( In the above image all the total length of the zig-zag line is equal to $CA+AB$ The diagonal is equal to $\sqrt{CA^2 + AB^2}$ You can approximate the irrational number $pi$ like so: ![enter image description here]( But the same does the apply the first example. In what way are these two examples different? Both start off with apparent low-resolution approximations, but one does approximate an irrational number while the other does not?

It means that any method you use to try to formalize the idea of how curves can vary and relate to one another will satisfy one (or both!) of the following two properties:

* Stairsteps do not converge to the diagonal
* Length is not a continuous function on curves



For example, one of the things we might want to insist on for convergence is for both points and slopes to converge, rather than just points. Clearly, the slope of the stairsteps does not converge to the slope of the diagonal, since the former are always (when defined) in the set $\\{ 0, \infty \\}$, but the slopes of the latter are always $1$.