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!