How buildings change the fundamental allometry

We demonstrate that the original fundamental allometry alone cannot accurately describe the relationship between urban area and population size. Instead, building height is a third factor that interplays with area and population. To illustrate this, we propose a straightforward model based on the idea that city area is the result of people’s desire to live close to one another while also having sufficient living space . This leads to a more general form of fundamental allometry (relating area, population, and building height). We support our theoretical analysis with an empirical examination of data from 32 countries. Our findings indicate that building height influences the residuals of the fundamental allometric relationship: cities with taller buildings generally fall below the regression curve of area versus population, while those with shorter buildings tend to be above it. This suggests that building height is not merely a natural outcome of our derivation but also extends beyond the fundamental allometry, playing an additional role in shaping urban structure.