What is the optimal shape of a city?

If one defines the distance between two points as the Manhattan distance (the sum of the horizontal distance along streets and the vertical distance along avenues) then one can define a city as being optimal if the average distance between pairs of points is a minimum. In this paper a nonlinear differential equation for the boundary curve of such a city is determined. The problem solved here is the continuous version of an optimization problem on how to design efficient allocation algorithms for massively parallel supercomputers. In the language of continuum mechanics, the shape of the optimal city is that taken by a blob of incompressible fluid composed of molecules whose pairwise interactions are described by an attractive potential proportional to the Manhattan distance between the particles.