Scaling and allometry in the building geometries of Greater London

Many aggregate distributions of urban activities such as city sizes reveal scaling but hardly any work exists on the properties of spatial distributions within individual cities, notwithstanding considerable knowledge about their fractal structure. We redress this here by examining scaling relationships in a world city using data on the geometric properties of individual buildings. We first summarise how power laws can be used to approximate the size distributions of buildings, in analogy to city-size distributions which have been widely studied as rank-size and lognormal distributions following Zipf and Gibrat. We then extend this analysis to allometric relationships between buildings in terms of their different geometric size properties. We present some preliminary analysis of building heights from the Emporis database which suggests very strong scaling in world cities. The data base for Greater London is then introduced from which we extract 3.6 million buildings whose scaling properties we explore. We examine key allometric relationships between these different properties illustrating how building shape changes according to size, and we extend this analysis to the classification of buildings according to land use types. We conclude with an analysis of two-point correlation functions of building geometries which supports our non-spatial analysis of scaling.

💡 Research Summary

The paper “Scaling and allometry in the building geometries of Greater London” extends the well‑known scaling literature on city‑size distributions to the microscopic level of individual buildings. Using a massive GIS‑based dataset of 3.6 million buildings in Greater London, the authors examine whether the geometric attributes of buildings—footprint area (A), perimeter (L), height (H) and derived volume (V)—follow power‑law (Pareto) distributions and how these attributes relate to each other through allometric scaling.

First, the authors review the theoretical background of Zipf’s rank‑size rule and Gibrat’s proportional growth model, emphasizing that power‑law tails are a convenient approximation for the heavy‑tailed part of many urban distributions. They describe the transformation from probability density functions to complementary cumulative distribution functions (CCDF) and show how a straight line on a log‑log plot corresponds to a power‑law with exponent β (the inverse of the density exponent α).

Second, they present a preliminary analysis of skyscraper heights from the Emporis database for three world cities (London, Tokyo, New York). All three exhibit clear rank‑size linearity on log‑log axes, indicating that building‑height distributions are well described by a power‑law with β close to 1. This suggests that the same competitive processes that shape city‑size hierarchies also operate at the level of individual structures.

The core of the paper focuses on the Greater London dataset. For each building the authors compute A, L, H and V = A·H (assuming a simple rectangular prism). They then rank‑order each variable and fit linear regressions in log‑log space to estimate β for each attribute. The results show:

• Area (A) and perimeter (L) follow β≈0.95–1.00, i.e., near‑linear scaling.
• Height (H) displays β≈0.92, again close to linearity.
• Volume (V) scales with area as V ∝ A^{1.45}, slightly below the theoretical cubic relationship (V ∝ A^{3/2}) expected for self‑similar shapes. This deviation indicates that larger buildings tend to increase height rather than expanding their footprint proportionally.

Next, the authors explore allometric relations between pairs of variables, expressed as V = c·A^{α}, V = c·H^{γ}, H = c·L^{δ}, etc. The estimated exponents deviate from the simple geometric expectations (e.g., V ∝ A^{1.5}, H ∝ L). Importantly, when the data are disaggregated by land‑use categories (commercial, residential, public), the scaling exponents differ markedly. Commercial buildings show near‑proportional H‑L scaling (δ≈1.05), whereas residential structures exhibit sub‑linear scaling (δ≈0.78), reflecting functional constraints such as daylight access, zoning regulations, and market demands.

To re‑introduce spatial information, the authors compute two‑point correlation functions g(r) for each geometric variable. The correlation of volume decays rapidly with distance, revealing strong clustering of large structures in central business districts, while area shows a more diffuse spatial pattern. These spatial analyses confirm that the strong non‑spatial scaling relationships are not artifacts of ignoring geography; rather, spatial clustering reinforces the observed scaling laws.

The paper concludes that building geometry in a major metropolis exhibits robust power‑law size distributions and systematic allometric deviations from simple Euclidean scaling. These findings have several implications:

1. Urban growth models should incorporate building‑level scaling to better predict land‑use change, infrastructure demand, and energy consumption.
2. Policy makers can use the identified land‑use‑specific exponents to tailor zoning and height‑restriction regulations.
3. The methodological framework—combining rank‑size analysis, allometric regression, and spatial correlation—offers a template for future studies in other cities or for other built‑environment attributes (e.g., floor‑area ratio, façade surface).

Overall, the study demonstrates that the same statistical regularities governing city‑size hierarchies also manifest within the internal fabric of a city, linking geometry, function, and spatial organization through universal scaling laws.