The pedestrian travel distance formula would essentially be the same as the taxi-distance formula in that you can only move along a grid pattern. However, instead of your grid being defined by roads and intersections. It would be defined by sidewalks and crosswalks. It also means you might not take the most direct route because your route is dependent on the location of the nearest crosswalk. (I'm also assuming you want to always take the shortest route between two points.)[br]Metric Axiom 1 is satisfied. By definition the distance must always be positive and the only way for the distance to equal zero is if the two locations are on top of each other, or they are the same location. [br]Metric Axiom 2 should also hold. Logically, you want to take the shortest walk between two locations, so while there may be more than one way to walk between two locations, you will always pick the shortest route, and because crosswalks and sidewalks are not one way, you would take the same route going from A to B as you would from B to A. The only instance you might not take the same route is if there is another route of equal distance, essentially a four-way cross walk at the intersection of two roads. For example, the intersection of 5th and broad street downtown. [br]As far as I can tell, Metric Axiom 3 still holds. Consider the figure below. First, assume PC and QR are opposide sides of a street. [br]Assume that there is a crosswalk connecting both P to R and C to Q. Then, PQ+QR=2+1=3>1=PR. [br]Assume that there is a crosswalk only connecting C to Q. Then PQ+QR=2+1=3=PR. [br]Assume that there is a crosswalk only connecting P to R. Then PQ+QR=2+1=3>1=PR. This logic can be expanded to any route between 3 points. Thus, the triangle inequality holds.