Long time, no see! I’ve been busy with other projects, including my rewrite (with some new ideas from collaborators) of my PhD program. But enough of that, some maths!

So my colleagues brought up a conundrum that we didn’t resolve satisfactorily. I’m no good at analysis, so maybe this is a known issue.

We’re going to get the length of a line going from the origin to (1,1):

By Pythagoras (aka the Euclidean metric), we know the length of this line is . We want to work this out via an approximation. We start with a dumb approximation, a line from (0,0) to (0,1) and a line from (0,1) to (1,1):

This approximation has length 2. Not a bad approximation for a first go. So we refine it:

This is closer to the goal. If we add up the length, it’s still 2. Hmmm. Well, it’s touching the true line at more places, so let’s keep going.

Closer again, but if you add up the verticals, that’s always 1. And the same for horizontals, so the distance is still 2. Again, more points intersect the line, but the approximation gets no better. Even with a much finer approximation:

The length is still 2, not ! (Apologies for my dodgy drawing, it’s slightly off, but you get the idea). Following this logic, no matter how many times we iterate this, we never get any closer to the true value.

So what’s the deal? As a sequence of points, this appears to converge to the line, but the distance never does. Is this basically saying that the taxicab metric never converges to the Euclidean metric? Is the infinite iterate of this process some kind of Cantor set?

Anyone got a good answer?