Suppose we have the following shape in the complex plane: The EXTERIOR of the semi-infinite strip bounded by 0 < y < 1 and x > 0. The two physical angles making up the rectangle have interior angles of 3*pi/2 and thus exterior angles of -pi/2. Now, because the sum of the exterior angles of a polygon have to sum to 2pi, we can claim that the exterior angle at infinity is simply 2pi - (-pi/2 - pi/2) = 3pi. However, I'm having trouble justifying this geometrically. I've looked at projecting the surface onto the Riemann sphere, but the exterior angle at infinity simply seems to be either pi or -pi. I see no reason why it has to do an additional orbit of 2pi.