The Topology of Cities

Is there an ideal city form? As cities proliferate worldwide this has become a central question underpinning sustainable development and economic opportunity for billions of people. We provide extensive empirical evidence, mathematical analysis and a set of theorems to show that the answer to this question is topological, not geometric. We show that cities can be decomposed into two types of networked spaces – accesses and places and prove that these spaces display universal topological characteristics common to all cities, provided specific mathematical conditions are met. While exceptions to these conditions are rare in developed cities, many urban slums fall into a different topological class. This expresses the central difficulty of developing cities as a rigorous mathematical problem that we show how to solve optimally through the introduction of infrastructure networks into city blocks at minimal disruption and cost. One sentence Summary: All cities are shown mathematically to share a common topology, manifested by transformations between any two street layouts and the gradual evolution of their neighborhoods.