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

The study drew on data from a six-year, school-based physical activity initiative conducted in south-eastern Norway. Of 351 children whose parents provided consent, 330 completed at least one valid maximal treadmill test, an unusually rigorous approach in pediatric research where field estimates of fitness are far more common than direct measurement. Because valid peak oxygen uptake data from 2016 were available from only three of the nine participating schools following a technical data loss, the researchers based their inferential analyses on measurements collected in 2015, 2017, 2018, 2019 and 2020, providing a longitudinal window that spans most of primary school.

At the heart of the research lies a methodological question that has divided pediatric exercise scientists for decades: how should oxygen uptake be expressed when comparing children of different sizes? Raw absolute values in liters per minute naturally favor older, heavier children. Dividing by total body mass is the most common correction, but it penalizes children with higher body fat, who contribute mass but little metabolic machinery for running. Allometric scaling, which uses exponentials of body mass, and scaling to fat-free mass, which isolates metabolically active tissue, offer alternatives. The HOPP team, led by Asgeir Mamen of Kristiania University College, together with Julianna Buer of the Norwegian School of Sport Sciences and Per Morten Fredriksen of the University of Inland Norway, designated fat-free-mass-scaled peak oxygen uptake as their prespecified primary outcome and tracked all four expressions simultaneously.

The statistical architecture reflected the complexity of the design. Linear mixed models included test year, school allocation group, sex, and the interaction between test year and group as fixed effects, with participant identity as a random intercept and test year as the repeated factor. The primary model incorporated 904 observations from 314 children, a sample that gives the analysis substantial power to detect developmental trends. School-level models were run as sensitivity analyses to probe whether results were being driven by the clustering of children within individual schools, a known hazard of school-based interventions.

The headline result was unambiguous in direction, if nuanced in interpretation. Fat-free-mass-scaled peak oxygen uptake was significantly associated with test year, allocation group, sex, and the test year by group interaction, with all p-values at or below 0.030. From fifth to sixth grade, the adjusted mean fell by 5.73 milliliters per kilogram of fat-free mass per minute in the intervention group and by 5.11 in the control group. Critically, the difference between those group-specific changes was just 0.62 milliliters, with a 95 percent confidence interval spanning from minus 4.25 to 3.00 and a p-value of 0.736. In other words, the school-based physical activity program appeared to make no measurable difference to the final-year decline.

That equivalence between intervention and control schools is perhaps the study’s most sobering finding. The HOPP project was designed as a pedagogical experiment in which schools adopted enhanced physical activity programming, and the reasonable hope was that extra activity would buffer children against fitness losses as they approach adolescence. Instead, the data suggest the late-primary-school decline was resistant to the intervention as delivered. Body-mass-relative and allometrically scaled oxygen uptake also fell significantly in both groups during the final year, while absolute peak oxygen uptake, unadjusted for growth, held roughly steady, hinting that increases in raw aerobic capacity were simply failing to keep pace with children’s rapid physical development.

Sex differences emerged across every expression of fitness, with boys showing higher adjusted values than girls. Yet the magnitude of that gap depended heavily on how the data were scaled. When oxygen uptake was normalized to fat-free mass, which strips away the influence of differences in body composition between the sexes, the sex difference was substantially attenuated. This observation reinforces a growing consensus in pediatric physiology: apparent fitness gaps between boys and girls are partly artifacts of scaling choices and body composition rather than true differences in the oxidative capacity of muscle tissue.

The school-level sensitivity analyses added another layer of nuance. Changes in the three body-size-adjusted expressions of fitness were negative at all nine schools, suggesting the final-year decline was a pervasive phenomenon rather than a quirk of any particular classroom, neighborhood or teaching staff. Such consistency strengthens the biological plausibility of the trend, while also underscoring that whatever combination of growth, maturation and behavior drives the decline, it operates broadly across the study population and overwhelms school-level variation in programming.

The authors are candid about the limitations that temper firm causal conclusions. The study design could not separate growth-related developmental changes from shifts in behavior associated with the COVID-19 pandemic, which overlapped with the later measurement years and is known to have disrupted children’s physical activity worldwide, nor from the possibility that the composition of participants changed over time. A retrospective trial registration, dated 20 June 2015 on ClinicalTrials.gov as NCT02495714, and the 2016 data gap further complicate the inferential picture. Still, the convergence of negative findings across multiple scaling methods and all nine schools makes the late-primary decline difficult to dismiss as statistical noise.

For researchers, the message is methodological: conclusions about childhood aerobic fitness hinge materially on scaling method, body composition and school-level heterogeneity, and studies that report only a single expression of peak oxygen uptake may tell an incomplete or even misleading story. For parents, educators and policymakers, the findings add to mounting evidence that the transition out of primary school is a vulnerable window for children’s cardiovascular health, one that conventional school-based activity interventions, at least as implemented in HOPP, did not shield against. Identifying what does work during those years, and distinguishing genuine developmental biology from modifiable behavior, now stands as an urgent task for pediatric exercise science.

Subject of Research: Longitudinal development of peak oxygen uptake in 6- to 12-year-old children

Article Title: Longitudinal development of oxygen uptake in 6- to 12-year-old children: the Health Oriented Pedagogical Project (HOPP)

Article References: Mamen, A., Buer, J., & Fredriksen, P. M. (2026). Longitudinal development of oxygen uptake in 6- to 12-year-old children: the Health Oriented Pedagogical Project (HOPP). BMC Pediatrics. https://doi.org/10.1186/s12887-026-07631-7

Image Credits: AI Generated

DOI: 10.1186/s12887-026-07631-7

Keywords: peak oxygen uptake, aerobic fitness, children, fat-free mass, allometric scaling, treadmill running, primary school, physical activity, longitudinal study, pediatric health, HOPP, Norway