Taxicab geometry

Taxicab geometry, considered by Hermann Minkowski in the 19th century, is a form of geometry in which the usual metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the (absolute) differences of their coordinates. More formally, we can define the Manhattan distance, also known as the L₁-distance is the distance between two points measured along axes at right angles. In a plane with p₁ at (x₁, y₁) and p₂ at (x₂, y₂), the Manhattan distance is:

, when m = 1.

(One can note that the L₂-distance is the normal Euclidean distance.)

Manhattan distance is also known as city block distance. It is so named because it is the distance a car would drive in a city laid out in square blocks, like Manhattan (discounting the facts that in Manhattan there are one-way and oblique streets and that real streets only exist at the edges of blocks - there is no 3.14th Avenue). Any route from a corner to another one that is 3 blocks East and 6 blocks North, will cover at least 9 blocks.