Abstract

Is it possible to draw a circle in Manhattan, using only its discrete network of streets and boulevards? In this study, we will explore the construction and properties of circular paths on an integer lattice, a discrete space where the distance between two points is not governed by the familiar Euclidean metric, but the Manhattan or taxicab distance, a metric linear in its coordinates. In order to achieve consistency with the continuous ideal, we need to abandon Euclid's very original definition of the circle in favour of a parametric construction. Somewhat unexpectedly, we find that the Euclidean circle's defining constant $\pi$ can be recovered in such a discrete setting.