Manhattan would be squarer if it had diagonal streets too

There are different ways of measuring distance - that's the abstraction of metric spaces.

The usual distance that people use looks like this:

Points at a certain distance form a circle around the point that you're measuring from.

Another metric is the Manhattan distance, where rather than moving in any direction you can only move along one axis at once. This leads to diamond contours instead of circles.

Those are both well known metrics. I wondered what the Manhattan distance would look like if you were allowed to move diagonally as well as along the axes. The distance seems to be max(abs(dx),abs(dy)), and that looks like this, all square: