# Why are the shortest distances in Euclidean geometry not taxicab? [duplicate]

I am of course well aware that distances in Euclidean geometry are calculated from the Pythagorean theorem. This is more of a conceptual question.

My question may also be formulated as follows:

If we approach the diagonal of a square (of side length 1) with a set of segments parallel to the sides, to total length of this set of segments is 2 (= taxicab distance), no matter the number of segments. If the number of segments tends towards infinity, the total length is still 2, but the set of segments looks like the diagonal of the square (whose length is $$\sqrt{2}$$).

What is it about our world and Euclidean geometry that makes taxicab distance not the shortest distance?

Sorry if what I'm saying is not strictly correct, I hope you can still understand what I mean :)

• As to the question of "why our world and Euclidean geometry is like this": in the first case, it's an empirical question, and in the second, it's stipulative. Euclidean geometry is pretty much defined by the pythagorean metric. Commented Jun 29, 2020 at 18:07