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Introduction to Fractal Analysis and Modeling in Human Geography

Cécile TANNIER and François SÉMÉCURBE

Laboratoire ThéMA, CNRS-Université de Franche-Comté, Besançon, France

(…) geography foregrounds the organization of space, its functioning; it seeks to explain its diversity or rather its diversification, at different scales; it prioritizes the question of location (why is it there and not elsewhere); starting from space, it questions reality, and especially society, in relation to its material basis, inherited both from the actions of past generations and from the forces of nature

(Durand-Dastès 1989).

Describing and understanding processes of concentration and dispersion of the spatial distribution of human settlements is at the heart of research in geography. A human settlement consists of the materialization in space, the physical inscription, of one or more activities. It is characterized by a certain level of durability. It may comprise a building, a group of a few buildings or a larger set of thousands of buildings. Each human settlement (a building or similar) is the location of one or more functions: residential, commercial, industrial, artisanal, religious, politics, logistics and so on. The durability of a building does not imply the temporal continuity of the activities occurring within it. For example, the transformation of a former farm into a second home creates a change in activity, which in this case becomes seasonal or more occasional. At the extreme, we may consider ruins no longer associated with an activity or a caravan parked in the same place and unoccupied for several years to be human settlements.

A human settlement is found within a more or less artificial unbuilt environment (parks, parking lots, orchards, fields). It is connected to other buildings, near or far, with the same or different functions, by means of transport and communication networks. All this – buildings, unbuilt environment, networks – constitutes a container for human activities. The contents associated with this container are individuals and groups performing activities, pursuing spatial practices and social relationships. In this work, we are only interested in the shapes of the content or the container, which can be geometrically assimilated to sets of points (centroids of buildings, nodes of road networks, built pixels and so on). Other types of spatial configurations, for example, flows or movement trajectories, will not be discussed.

Human settlements present varying degrees of concentration or dispersion in space and time: at the local scale, town centers or village centers, peri-central districts, peripheral extensions (business areas, housing estates); at the scale of an urban area, monocentric or polycentric agglomerations, compact or sprawling; at the regional scale, more or less diffuse and more or less hierarchical population systems. Concentrations and dispersions, considered as states or processes, vary according to the level of aggregation of the entities considered (Openshaw 1983; Fotheringham and Wong 1991; Madelin et al. 2009) and the spatial resolution selected (Müller 1978; Goodchild 1980; Toger et al. 2016). Spatial resolution corresponds to the fineness of the detail with which the elements of the container (buildings, unbuilt environment, networks) are represented. For example, the built-up fabric of an urban agglomeration may be mapped building by building, each building having its own spatial footprint delimited with high precision (metric or even infra-metric). In this case, the spatial resolution is fine. The same built-up fabric can be mapped in the form of urban patches roughly delimiting the envelope of different groups of buildings. In this case, the spatial resolution is coarser. A built-up fabric can also be mapped in the form of 10 m-sided square cells (fine spatial resolution) or 200 m-sided square cells (coarser spatial resolution). The level of aggregation of the analyzed or modeled entities is different from their spatial resolution. For example, we may study a district as a whole, without disaggregating it, or consider it as an aggregate of built blocks or even buildings. The fineness of the spatial resolution of the elementary entities under consideration partially determines the range of possible levels of aggregation.

When, in order to model a given phenomenon, we take into account several levels of aggregation (Grasland 2003; Sanders 2007) or a series of nested spatial resolutions (Goodchild and Mark 1987; Lam and Quattrochi 1992; Tate and Atkinson 2001), or even ranges (neighborhoods) of different sizes (White 2005; van Vliet et al. 2009; Mathian and Sanders 2014), the modeling is multi-scale1. When we represent variations of the phenomenon across levels of aggregation or spatial resolutions, the modeling is cross-scale. It is particularly interesting to study the concentration and dispersion of human settlements using multi- or cross-scale models, especially fractals, as this makes it possible to explicitly account for spatial hierarchies (e.g. the fact that an urban agglomeration has a large number of small built-up clusters, a smaller number of medium-sized aggregates and an even smaller number of large-sized clusters) and functional hierarchies (e.g. different levels of functional centrality). Multi- or cross-scale models allow us to represent and characterize variations in the concentration and dispersion of human settlements across scales; in other words, to determine in what way and how the concentration of human settlements varies from one level of analysis to another, and to identify possible invariances between scales, the degree of generality of which may vary in space and time.

1.1. Geometric models of the spatial distribution of human settlements: from the use of smooth and regular shapes to the adoption of fractal shapes

Many spatial models, which can be graphically represented, have been designed to describe and explain the shapes of the spatial distribution of human settlements in relation to the functioning of people and societies. As early as 1826, J. von Thünen proposed a model of radio-concentric organization of cultures around a center according to a principle of increasing transport costs from the center, where consumers are located, to the periphery. A century later, Burgess (1925) envisaged a model of city growth as the gradual addition of concentric rings of decreasing land rent from the center to the periphery. Subsequently, Christaller (1933) and Lösch (1940)2 designed two models for the organization of hierarchically nested central areas, while Stewart (1947) and Clark (1951) proposed a mathematical model of decreasing density of the population of cities from the center to the periphery. Still a little later, Thünen’s model was applied to the modeling of the spatial organization of urban spaces according to a principle of decreasing land rent from the center to the periphery by Alonso (1964), Mills (1967) and Muth (1969). All these founding models have been the subject of numerous tests and applications, and have led to multiple developments. The geometric shapes of reference that underlie these models – circles, rings, triangles – are smooth and regular.

However, the shapes of the spatial distribution of human settlements are fundamentally non-uniform and very irregular. In the suburban fringes of contemporary European cities, for example, built-up areas have extremely variable spacings between buildings (Figure 1.1). The same is true when looking at cities as a whole: the size of the built-up clusters is highly variable, as are the distances between them (Figure 1.2).

Figure 1.1. Two examples of the built-up fabrics characteristic of peri-urban fringes: black outlines represent the footprint of buildings of different sizes and shapes

This irregularity in the shapes observed is not unique to geography; it is found in many other fields of both natural (physics, biology, chemistry, etc.) and social science (finance, architecture, graphic arts, etc.). Natural objects in particular (coastlines, mountains, clouds, plants, cellular tissues, etc.) very often display such irregularities. Historically, mathematicians have neglected the study of shapes, whose irregularity and complexity meant they could not be described or analyzed using classical mathematical tools. Infinitesimal calculus, introduced by Newton and Leibniz in the second half of the 17th century, was indeed an ideal tool for the analysis of smooth objects forming part of the framework of classical Euclidean geometry, but not for objects displaying complex and irregular shapes.

Figure 1.2. A built-up urban fabric: the agglomeration of Basel in 1957. Source: topographic maps of the Bundesamt für Landestopographie, 3084 Wabern, Germany. Manual digitization performed by the “Image, ville, environnement” laboratory, Strasbourg, France

It was not until the end of the 1960s that the systematic study of such forms developed within mathematics, notably due to the efforts of Benoît Mandelbrot (1924–2010), often recognized as “The Father of Fractals” (Falconer 2013). Since the 1980s, all areas of science have been examined from a fractal perspective: biology, medicine, image compression, signal theory,...