Hidden Fractal Geometry Explains the Strange Scaling Laws of Cities

Cities are strange mathematical objects. They grow, expand, and interconnect in ways that stubbornly refuse to obey the tidy geometry taught in school, and a new study published in Heliyon argues that the reason lies hidden in fractal dimensions. Yanguang Chen, a researcher devoted to the quantitative science of cities, has systematically demonstrated that the exponents of urban allometric growth, the famous power laws linking different measures of a city such as population, area, and the number of settlements in a region, are best understood as ratios of fractal dimensions rather than simple Euclidean ones. The finding resolves a puzzle that has haunted urban science since the mid-twentieth century, when researchers first tried and failed to explain scaling exponents with the ordinary geometry of lengths, areas, and volumes.

Allometric growth is a concept borrowed from biology, where it describes how, for example, the surface area of an animal scales with its body volume. In its classical mathematical form, the law states that the relative growth rate of one measure is proportional to the relative growth rate of another, with a constant coefficient. The solution to this differential equation is a power function, and the exponent of that function is the allometric scaling exponent. When John Q. Stewart and later Naroll and von Bertalanffy imported these ideas into the study of cities and urbanization, scientists assumed the exponents should be explainable by Euclidean dimensions. Urban population was treated as a three-dimensional measure and urban area as a two-dimensional one, predicting a scaling exponent of two-thirds. Where two measures shared the same dimension, the expected exponent was exactly one.

Observations refused to cooperate. Decades of empirical calculations on real cities produced scaling exponents that were neither integers nor simple ratios of integers, and this mismatch created what the literature calls the dimensional dilemma. Scientists faced an unpalatable choice: either abandon the law of allometric growth, which clearly described real patterns in the data, or find a more sophisticated geometry to explain it. The escape route, Chen argues, was opened by the arrival of fractal geometry in the 1970s and 1980s, through the work of Benoit Mandelbrot and, in urban contexts, Michael Batty and Pierre Frankhauser. Once cities were recognized as fractal objects, sprawling, self-similar structures whose detailed shapes repeat across scales, the exotic values of scaling exponents suddenly became interpretable.

The theoretical core of the new study rests on the principle of dimensional consistency, an idea stretching back to ancient Greek mathematics. Measures of different dimensions, such as length, area, and volume, cannot form simple proportional relationships; to construct such relationships, the dimensions must be brought into consistency through exponents. When two measures of a complex system are proportional, the scaling exponent connecting them must equal the ratio of their dimensions. If those dimensions are Euclidean, the exponent should be a recognizable fraction like two-thirds or three-halves. But if the exponent is not such a ratio, then at least one of the underlying measures must be fractal, possessing a dimension greater than its topological dimension and typically non-integer. Chen turns this logic into a diagnostic table: integer or simple-fraction exponents consistent with Euclidean geometry point to ordinary measures, while values such as three-quarters or 0.85 betray the presence of fractal structure.

To make the argument concrete, Chen assembled empirical evidence from Chinese cities, drawing on census data from 2000 and 2010 covering the thirty-one regions of the Chinese mainland. Two allometric relationships were examined: the scaling between the total urban population of each region and the number of cities it contains, and the scaling between total urban population and the population of each region’s central city. In every case, the estimated exponents fell well below one, clustering around values from roughly 0.64 to 0.72 depending on the year, the relationship, and the estimation method. None of these numbers can be produced by any ratio of Euclidean dimensions. Under the dimensional consistency principle, the only coherent conclusion is that the measures involved are fractal, and the exponents are ratios of fractal dimensions.

The analysis is careful about statistics in ways that matter for reproducibility. Chen distinguishes between the three worlds of science, the real world of cities, the mathematical world of deductive reasoning, and the computational world of data and algorithms, and shows that parameter estimates depend on which method is used. The least squares regression of population on city number yields a different exponent from the regression of city number on population, and the product of the two exponents equals the goodness of fit, a relationship confirmed in the data. To handle this asymmetry, the study employs the reduced major axis method, which averages the two directions of regression and yields modified exponents. Municipalities directly under the central government, which contain only a single city and therefore behave unlike multi-city provinces, were identified as outliers using standardized residuals, scatter plots, and K-means clustering, and were excluded from the fitting in a documented and principled way.

Beyond the headline relationships, the study connects allometric scaling to other cornerstones of urban science. Zipf’s law, the celebrated rank-size rule stating that city populations follow a power-law distribution, turns out to be mathematically intertwined with cross-sectional allometry: a pair of correlated Zipf distributions for city population and city area derives the transversal allometric model, and the Pareto exponent of the size distribution can itself be read as a fractal dimension. In this framework, cross-sectional allometric scaling is a secondary law flowing from rank-order scaling, and the allometric exponent between two size measures equals the ratio of their two Pareto-derived fractal dimensions. Spatial allometry, measured through concentric circles drawn around a city center, similarly yields exponents that are ratios of radial fractal dimensions describing how population density and built-up land fall away from the core.

The work does not stop at geometry; it also ventures into why scaling laws exist at all. Chen reviews candidate mechanisms, including self-organized criticality, proportional random growth, and preferential attachment, and then advocates a dual explanation grounded in two mathematically equivalent principles: entropy maximization at the macro level and utility maximization at the micro level. A single entropy-maximizing process generates an exponential distribution, but a pair of coupled entropy-maximizing processes, one governing the growing number of cities and one governing the growth of each city, combines two exponential functions into a power law. The same structure emerges from microeconomic reasoning, where individuals and organizations maximizing their satisfaction produce the same aggregate pattern. Intriguingly, this equivalence suggests that the socio-economic character of a city is partly legible through its physical, fractal properties.

The research even carries policy implications. In the multivariate Cobb-Douglas analysis, the contribution of city number to total urban population grew stronger between 2000 and 2010, while the contribution of central-city population weakened, indicating that many small and medium-sized cities collectively drive urbanization more than a few giant metropolises. This is the long tail effect of power-law distributions: when the fractal dimension of the city-size distribution exceeds one, the numerous small settlements at the tail of the hierarchy can absorb enormous urban populations. For Chen, the practical lesson is that developing small and medium-sized cities first, and then tuning the largest cities, is the mathematically informed path to raising urbanization, a conclusion that elevates a seemingly abstract fractal ratio into a tool for planning the urban future. The study’s remaining open question, the possible connection between allometric growth and the third defining property of fractals, entropy conservation, is flagged honestly as unsolved, leaving an inviting frontier for the next chapter of urban science.

Subject of Research: Fractal dimension as the mathematical basis of allometric scaling exponents in urban systems

Article Title: Fractal dimension accounts for allometric scaling exponents of cities

Article References: Chen, Y. (2026). Fractal dimension accounts for allometric scaling exponents of cities. Heliyon, 12(15), Article e45409. https://doi.org/10.1016/j.heliyon.2026.e45409

Image Credits: AI Generated

DOI: 10.1016/j.heliyon.2026.e45409

Keywords: fractal dimension, allometric scaling, urban science, power laws, Zipf's law, Chinese cities, urbanization, dimensional consistency, entropy maximization, city size distribution, Heliyon, complex systems