The challenge of a science of cities is to understand the links between urban morphogenesis, efficiency and resilience. Mathematical regularities emerge in resilient cities, coming from the scale-free properties of complex systems that present the same level of complexity across their different scales. They take the form of inverse power laws that are the « signature » of complexity. In living cities, these mathematical regularities derive from historical layering over millennia (Paris) or from intense market forces (New York). In complex, living and resilient cities, the distribution of elements and connections does not obey Gaussian laws but scale-free inverse power laws. Understanding the universality of this structure which also characterizes natural phenomena and living systems, and which has been violated by modernist city planning, would allow planning more efficient and resilient cities. The paper shows how initial breaks of symmetry fostered the emergence of scale-free structures in Paris and New York, with long-range time correlations, and how a break of symmetry in the spatial layout created a highly differentiated socio-economic structure in Barcelona.

Keywords: urban morphology; symmetries; scaling.