Pi Visits Manhattan

TL;DR

This paper investigates the possibility of defining and constructing circles within a Manhattan (taxicab) metric space, revealing that the Euclidean constant π can be recovered in a discrete lattice setting.

Contribution

It introduces a parametric approach to define circles in Manhattan space and demonstrates the emergence of π in this discrete context, bridging continuous and discrete geometries.

Findings

Euclidean π can be recovered in Manhattan geometry

A parametric construction of circles in discrete spaces

Insights into discrete analogs of continuous geometric concepts

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.