On Urban Growth & Form

Allometry is the study of the relationship between size and shape, first outlined by Otto Snell in 1892 and Julian Huxley in 1932. It has been a concern for mathematical biologists since D’Arcy Thompson’s investigations collected in On Growth and Form.

The allometric hypothesis suggests that there are critical ratios between geometric attributes that are fixed by the functioning of the element in question and if the element changes in size, these ratios need to remain fixed for the element to still function. 1

Bettencourt & West’s work shows clear macroscopic correlations for a number of structural, social and economic characteristics of cities (GDP, gasoline stations, electrical consumption), each scaling with population size N as a power law of the form y=N^β. These provide a mathematical basis for Aristotle’s metaphor of the city as a living organism, as biologists have also shown this to be true for metabolic functions in living beings.

A number of infrastructural features - such as road surface area and gas stations - scale with an exponent β< 1, indicating economies of scale, whilst the majority of wealth indicators - such as average wage, or number of patents - scale with β> 1, indicating increasing returns at larger sizes. There is evidence here that larger cities are more structurally efficient than smaller ones - a trend also echoed in organisms. These findings are complementary to observations of geometric scaling and fractal structure in cities (Batty & Longley, 1974)

Claims of a unified theory or a solution to the city are misleading, as many of these relationships are in effect universal allometric scaling laws evident in urban settlements. These are no more ‘solutions’ to the range of phenomena exhibited by cities than the allometric relation of metabolism to body mass - Kleiber’s Law - provides an understanding of the inner workings of a living organism.

Correlation in data-driven research such as this clearly does not in itself provide a theory. Association of these characteristics is weakly argued when a “universal social dynamic” is posited, “inextricably linking” the correlated phenomena in a “dynamical network” 2. Replacing simplified notions of supply and demand (or of causation) with a complex network model seems both timely and useful, but mere correlations of variables with size don’t prove this to be a more faithful conceptual model for a city than any other.

In referring to the relation between the size and shape of things, allometry must in an urban context be considered with respect to the energy capacity of the human body - or the enhanced body, if we consider various prostheses in the form of vehicles.

A famous case study in the allometry of buildings is Baron Haussmann’s Paris. Building heights were set during the re-development to a maximum eaves height of 17.5m, or 5-6 storeys, creating a homogenous landscape which is still visible today. This was considered at the time a comfortable distance a human would climb unaided multiple times a day. Berlin settled on 22 meters for its Mietskaserne (tenements) in 1853, with this figure shaped by the concerns of the Fire Department of the time.

In modern urban development, the advent of the automobile, rail and subways has produced a large sprawl of low-level (2-3 storey) housing stock, dramatically enlarging the surface area of the urban landscape while lowering its density; a trend seen in most Western cities. This is in sharp contrast to the multi-storey apartments that pre-date the industrial revolution and still inhabit the compact cores of most European settlements.

Meanwhile, the introduction of megastructures and skyscrapers, enabled by a range of technologies (most notably the triad of elevators, escalators and air-conditioning), has produced new allometric relations in buildings. The pace of technological innovation means that the processes that might describe an urban ‘metabolism’ must be considered to be evolving from decade to decade. For example, since the introduction of skyscrapers, rank-size building height distributions also produce clear scaling laws. This phenomenon, as well as geometric scaling in cities are discussed further in (Batty et al, 2007).

These shifts in construction and transportation produce new surface to volume ratios in our growing cities, which in turn have repercussions for the energy consumption of urban regions. If we take as a simple example an ideal sphere, heat dissipation is proportional to its surface to volume ratio, which scales inversely with size, resulting in more efficient retention of heat at larger scales. The same kind of relations, when aggregated from the building to urban scale, suggest that discontinuities in building size and density significantly alter the energy footprint of our cities.

At the urban scale, transportation, along with the spatial distribution of land-use, will continue to be key factors governing allometric relationships in cities. A continuing research challenge exists in taking historical data for particular urban cases to show how a range of variables from land-use ratios to the range of infrastructural features explored by Bettencourt & West change during the long-term growth that typifies the evolution of an urban settlement.

See Also: Flowprint, Morphogenesis, Fractal, Isochronic, Expressway, Dwelling\Moriyama

• 1 Batty et Al, Geometric Scaling and Allometry in Large Cities (International Space Syntax Symposium, İstanbul, 2007)
• 2 Bettencourt et al, Growth, innovation, scaling, and the pace of life in cities, PNAS, 2007, Vol. 104(170):7301-7306