Abstract
Urban scaling laws relate socio-economic, behavioural and physical variables to the population size of cities. They allow for a new paradigm of city planning and for an understanding of urban resilience and economics. The emergence of these power-law relations is still unclear. Improving our understanding of their origin will help us to better apply them in practical applications and further research their properties. In this work, we derive the basic exponents for spatially distributed variables from fundamental fractal geometric relations in cities. Sub-linear scaling arises as the ratio of the fractal dimension of the road network and of the distribution of the population embedded in three dimensions. Super-linear scaling emerges from human interactions that are constrained by the geometry of a city. We demonstrate the validity of the framework with data from 4750 European cities. We make several testable predictions, including the relation of average height of cities and population size, and the existence of a critical density above which growth changes from horizontal densification to three-dimensional growth.