I noticed that all the "circles" for alternative metrics are aligned with the coordinate system. For example, the one for the Manhattan distance has its corners

I noticed that all the "circles" for alternative metrics are aligned with the coordinate system. For example, the one for the Manhattan distance has its corners on the coordinate axes. What if we added an additional condition that a distance metric should not change when the orientation of the coordinate system is changed? Could we still have different values for the pi constant then?