The red points at the border of the square below are all at a constant distance from the square's center. That is, if the distance function is the taxicab metric, also known as L1 norm or Manhattan distance. $$ d_1(p, q) = |p_1 - q_1| + |p_2 - q_2| $$
So technically, this square is a circle.
If the radius is 1, then the circumference must be 8, the diameter 2, and thus \(\pi = 4\)! Edward J. Goodwin was right after all. 😛