Why metric defined on $\mathbb{R}^2\times \mathbb{R}^2$ by $(a,b)\mapsto | a_1 - b_1| +| a_2 - b_2| $ is known as taxicab metric? $\mathbb{R}^2$ with the function defined on $\mathbb{R}^2\times \mathbb{R}^2$ by $(a,b)\mapsto | a_1 - b_1| +| a_2 - b_2|  $ where $a = (a_1, a_2)$ and $b = (b_1, b_2)$ is a metric. I wonder why it is known as taxicab metric. Could anyone explain me?
Thank you very much
 A: Imagine a big city laid out so that the streets all run either north-south or east-west and are spaced at regular intervals. In order to go from one point to another by taxicab, you must travel along the streets, so you can only travel north-south or east-west. The distance travelled is therefore the sum of the north-south separation and the east-west separation between you and your destination. The taxicab metric is the distance as the cab would travel instead of the straight-line, or Euclidean distance distance. (The latter is sometimes described in English as as the crow flies, so we could call the Euclidean metric the crow metric, but no one does!)
A: A picture is definitely the way to see this most easily. From a top down view the metric looks like a city, and the paths an object in this metric must take are the same as a taxicab must take in such a city (not being able to fly or drive through buildings for instance).
A: The idea is that it reflects the distance between two points if you only travel parallel to the axes. So to get from $(7,3)$ to $(4,8)$ you first travel 3 units left to $(4,3)$ then 5 up to $(4,8)$, for a total distance of eight. Hence the distance is the sum of the vertical distance and the horizontal distance.
It's meant to be evocative of the distance a taxicab would travel in a city where streets are in a grid pattern, so that you can only get places by going, say, east-west or north-south, rather than travelling directly towards them.
A: See

how the streets of Manhattan look, as opposed to other places nearby...
