# 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

**L**_{1}-distance is the

distance between two points measured along axes at right angles. In a

plane with p

_{1} at (x

_{1}, y

_{1}) and p

_{2} at (x

_{2}, y

_{2}), the Manhattan distance is:

- , when m = 1.

(One can note that the L

_{2}-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.

See also: