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.

城市规模定律将社会经济，行为和身体变量与城市人口规模联系起来。它们为城市规划提供了新的范式，并使人们对城市的韧性和经济性有了更深入的了解。这些权力法关系的出现还不清楚。增进对它们起源的理解将有助于我们更好地将它们应用到实际应用中，并进一步研究它们的特性。在这项工作中，我们从城市中的基本分形几何关系中得出空间分布变量的基本指数。亚线性标度是道路网络的分形维数与嵌入三个维数的人口分布之比得出的。超线性缩放来自受城市几何形状约束的人与人之间的互动。我们用来自4750个欧洲城市的数据证明了该框架的有效性。我们做出了一些可检验的预测，包括城市平均高度与人口规模之间的关系，以及临界密度的存在，在该临界密度以上，增长从水平密实变为三维增长。