If one lives in a city or other urban area whose streets are arranged in a roughly grid-like fashion, they know that the absolute distance to some place on that grid is not equal to the driving distance. Take, for example, the diagram below.

            + — + — + — + — + < end           
            |   |   |   | ⁄ |                 
            + — + — + — + — +                 
            |   |   | ⁄ |   |   + — + = 1 unit
            + — + — + — + — +                 
            |   | ⁄ |   |   |                 
            + — + — + — + — +                 
            | ⁄ |   |   |   |                 
    start > + — + — + — + — +                 

Considering that the difference between any two connected + signs is one unit, and all the orthogonal lines are drivable streets, the shortest absolute distance (as the crow flies) from the start to the end is along the diagonal line, i.e. 4√2 or ~5.66 units. However, any driving route to the destination would have to be along the grid lines, so any path that never goes West or South would get there on the shortest possible route (barring construction detours and the like) which is 8 units. One such route is shown in bold: a tally of the number of horizontal and/or vertical strokes it travels over can be trivially confirmed as 8.

This metric for measuring distances is called the taxicab metric, where the only differences from the situation described above are that the resolution of the grid is infinitesimally small, so as to allow components in the real numbers, and that it exists in an arbitrary number of dimensions, not simply two. There are many alternate names, such as the Manhattan distance, city block distance, and rectilinear distance. The idea was first considered by Hermann Minkowski, the same mathematician to give us Minkowski's question mark function, Minkowski addition, and Minkowski spacetime.

The general idea is that distances between points are considered to be equal to the sums of the rectangular components of the vector between them, i.e. the sum of the absolute values of the differences between the coordinates. Mathematically, the distance between vectors p and q in n-dimensional space is

d(p, q) = i | pi - qi |

This leads to a number of interesting quirks. For instance, if you subscribe to the definition of a circle as "a collection of points which are all equidistant from the center", then the figure of a circle is a diamond shape, i.e. the circles here have corners! (See this diagram.) A neat side effect of this is that the taxicab value for pi is exactly 4: the diameter is 2r for a radius r, and the circumference is composed of 4 diagonal lines, themselves each of taxicab length 2r!

The taxicab metric happens to be an Lp space, for p = 1. For comparison, traditional Euclidean space has p = 2.
