You can "disprove" the Pythagorean equation the same way, by taking the limit of the Manhattan distance between two points as the grid size approaches zero. It

You can "disprove" the Pythagorean equation the same way, by taking the limit of the Manhattan distance between two points as the grid size approaches zero. It approaches d_x + d_y, not sqrt(d_x^2 + d_y^2).