How the geometry of cities explains urban scaling laws and determines their exponents.
Urban scaling laws relate socio-economic, behavioral, and physical variables to the population size of cities and allow for a new paradigm of city planning, and an understanding of urban resilience and economies. Independently of culture and climate, almost all cities exhibit two fundamental scaling exponents, one sub-linear and one super-linear that are related. Here we show that based on fundamental fractal geometric relations of cities we derive both exponents and their relation. Sub-linear scaling arises as the ratio of the fractal dimensions of the road network and the distribution of the population in 3D. Super-linear scaling emerges from human interactions that are constrained by the city geometry. We demonstrate the validity of the framework with data on 4750 European cities. We make several testable predictions, including the relation of average height of cities with population size, and that at a critical population size, growth changes from horizontal densification to three-dimensional growth.