My question today is spurred by something I came across in chess called The Crooked Path. The idea is simple enough and essentially comes down to this: if the king wants to move say 6 squares up the board, it can either move those 6 squares up in a straight line, or it can instead move 3 squares diagonally up and left, then 3 squares diagonally up and to the right. Note that these two choices of paths (i.e. the straight line or the two consecutive diagonals) both take the same amount of time for the king, 6 turns.
To me this seems to violate the triangle inequality that says if some triangle has sides of length x, y and z, then we must have x+y > z, i.e. that the sum of two side lengths has to be greater than the length of the third side.
Would I be wrong to assert that the discrete geometry of the chess board is what is responsible for violating the triangle inequality? If it is the discrete geometry that is the cause, does there exist a general principle speaking to this phenomena?
I am not too familiar with the principles and general behaviour of discrete geometry, but this certainly struck me as interesting. So any insight on this behaviour would be much appreciated!
Thanks to discussion from @Somos and @311411 in the comments I have found the answer to my own question. It was pointed out that my question essentially is meaningless unless I define what metric I'm giving to the chess board geometry to make it into a metric space.
I had been thinking of the number of moves the king makes from some starting point as the values the metric takes on. It turns out this metric is exactly the Taxi Cab metric. It also turns out, as the linked article describes, that the Taxi Cab metric is just the metric induced from the $\ell^{1}$-norm, which is known to satisfy the triangle inequality with equality.
Something I cannot speak too is perhaps the "illusion" that the king in the problem seems to travel a longer distance in the same amount of moves by taking the diagonal route, rather than taking the straight route.