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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 :)

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    $\begingroup$ 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. $\endgroup$ Commented Jun 29, 2020 at 18:07

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Note that the "taxicab distance" between two points depends very much on the orientation of your "taxicab grid". When traversing between opposite corners of a square that's aligned to the taxicab grid, the overall Euclidean distance vector does not align with the taxicab distance vectors, so you wind up traveling further. If, however, you rotate the square by 45 degrees, the Euclidean distance vector lines up perfectly with the taxicab grid, and the Euclidean and taxicab distances between the corners are identical.

From this, you can see that there is no "preferred orientation" of the world. Depending on how you align your grid, you will get very different answers for the taxicab distance between two points, and none of them are any more "correct" than any others. In the real world, there is not a single taxicab distance between two points, so we clearly canot define the minimum distance based on an arbitrary selection of the taxicab grid's alignment.

The taxicab distance only allows you to move along certain pre-defined axes which do not have any fixed analogue in the real world - those axes are simply a mathematical construct. The Euclidean distance, on the other hand, does not prefer or stipulate anything with regard to the axes, as it allows you to move in multiple dimensions simultaneously, effectively making the axes irrelevant. Euclidean geometry does not care how you define or orient the coordinate system.

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