Four Corners, Nebraska is located on a flat plain, and its streets are layed out in a perfect rectangular grid. Main Street runs east and west, and Central Avenue runs north and south. Their intersection is the origin of the coordinate system, and the city blocks are precisely 100 feet square.
So if you travel 1000 feet, or ten blocks,due east, you are at the intersection of Main St and 10th Av. If you then turn north on 10th Av and travel ten blocks you will arrive at the intersection of 10th and 10th, and you will have traveled a total distance of 20 blocks, or 2000 feet.

However, the Pythagorean Theorem tells us that the straight line distance along the diagonal from Main and Central to 10th and 10th is 1414 feet and some change. So if a crow wants to fly from the origin to 10th and 10th, he can save himself some time by going in a straight line between the two, instead of going along the base and height of the triangle. Everybody knows this.

A flightless pedestrian, however, is constrained to stay on the street, so he travels the full 2000 feet. But what if he wants to shorten the distance by approximating the diagonal distance? If he walks 500 feet (5 blocks) east, ten blocks north, then 5 more blocks east, he gets to his destination but walks the same distance--2000 feet. The same thing happens if he goes 5 blocks, east 5 north, then 5 more east, and another 5 north. Try it again, going two blocks at a time, you still wind up walking 1000 feet east and 1000 north total: just add up all the little jogs. The same happens if you hug the diagonal by going 1 block east, 2 north, then 2 blocks east and so on. As long as you are alternating left and right turns, you will always wind up traveling 2000 feet. You don't need calculus to tell you that you don't approach the limit of the diagonal distance along the hypotenuse as the number of jogs increases without limit. In fact, you wind up walking 2000 feet even if each jog is only a micron in length. We know this is a mathematical fact, but why it should be so is very difficult to comprehend intuitively, it's almost as if space itself is compressed along the diagonal.

One practical consequence of this is that you don't shorten your route a bit by taking a diagonal route across town, zig-zagging across the diagonal; the distance remains the same unless you can find a boulevard allowing you to travel in the diagonal direction. Unless there is one, you're better off just finding the north and south roads you can make the best time on and ignore trying to cut the corner, it does you no good. Since all towns are layed out more or less as grids, simply measuring the distance between two points on a map will always be shorter, by a considerable amount, than the actual distance you travel.

We are assuming, of course, that Four Corners is perfectly flat, but it is actually on the surface of a very large sphere, the earth. Now the problem becomes even more interesting....