Mathematics and Morphogenesis of the City: A Geometrical Approach

TL;DR

This paper introduces a comprehensive mathematical model for city morphogenesis that integrates geometrical, topological, and dynamical aspects to explain and generate diverse city layouts.

Contribution

It presents a novel geometrical and dynamical framework for modeling city development, unifying static analysis with a generative process.

Findings

Cities follow a division/extension logic in their development

The model reproduces diverse city shapes using simple rules

Static analysis of French towns supports the model's assumptions

Abstract

Cities are living organisms. They are out of equilibrium, open systems that never stop developing and sometimes die. The local geography can be compared to a shell constraining its development. In brief, a city's current layout is a step in a running morphogenesis process. Thus cities display a huge diversity of shapes and none of traditional models from random graphs, complex networks theory or stochastic geometry takes into account geometrical, functional and dynamical aspects of a city in the same framework. We present here a global mathematical model dedicated to cities that permits describing, manipulating and explaining cities' overall shape and layout of their street systems. This street-based framework conciliates the topological and geometrical sides of the problem. From the static analysis of several French towns (topology of first and second order, anisotropy, streets scaling) we make the hypothesis that the development of a city follows a logic of division / extension of space. We propose a dynamical model that mimics this logic and which from simple general rules and a few parameters succeeds in generating a large diversity of cities and in reproducing the general features the static analysis has pointed out.