What should the value of pi be in taxicab geometry?

The value of pi in taxicab geometry technically does not exist as any taxicab shape would consist of right angles. A circle does not contain any right angles, therefore circles do not exist in taxicab geometry (so neither does the value pi).

However, if you were to place a square around a circle with diameter 1, where each side is a tangent to the circle and then removed the corners of the square repeatedly without ever removing any or the area of the circle (see Figure 1), the value of 'pi' in that case would be 4 (as pi = diameter/circumference (or in this case perimeter)).

Figure 1

Here the problem arises, as perimeter and circumference are not exactly the same thing. Circumference only pertains to a circle (and circles do not exist in taxicab geometry) whereas perimeter pertains to any geometric shape (whether in taxicab geometry or Euclidean geometry).