Preliminary evidence for a theory of the fractal city

Spatial Structure of an Urban Park System Based on Fractal Theory: A Case Study of Fuzhou, China

The rationality and efficiency of the spatial structure of an urban park system are critical in building a livable urban environment. Fractal theory is currently treated as the frontier theory for exploring the law of complex systems; however, it has rarely been applied to urban park systems. This study applied the aggregation, grid and correlation dimension models of fractal theory in Fuzhou, China. The spatial structure and driving factors of the urban park system were analyzed and an innovative model was proposed. The evidence shows that the spatial structure of the park system has fractal characteristics, although self-organization and optimization have not yet been fully formed, revealing a multi-core nesting pattern. Moreover, the core is cluster of four popular parks with weakening adsorption, and the emerging Baima River Park is located at the geometric center, which is likely to be further developed. The system structure is primarily driven by geographical conditions, planning policies, and transportation networks. Against this backdrop, an innovative model for the park system was proposed. The central park has heterogeneity and synergistic development, relying on the kinds of flow which can lead to the formation of a park city, a variation of a garden city. At the regional scale, relying on the geographical lines, the formation of a regional park zone could be realized. These findings provide new perspectives to reveal the spatial structure of urban park systems. The information derived can assist policy makers and planners in formulating more scientific plans, and may contribute to building a balanced and efficient urban park system.

An Overview of Fractal Geometry Applied to Urban Planning

Since computing advances in the last 30 years have allowed automated calculation of fractal dimensions, fractals have been established as ubiquitous signatures of urban form and socioeconomic function. Yet, applications of fractal concepts in urban planning have lagged the evolution of technical analysis methods. Through a narrative literature review around a series of “big questions” and automated bibliometric analysis, we offer a primer on fractal applications in urban planning, targeted to urban scholars and participatory planners. We find that developing evidence demonstrates linkages between urban history, planning context, and urban form and between “ideal” fractal dimension values and urban aesthetics. However, we identify gaps in the literature around findings that directly link planning regulations to fractal patterns, from both positive and normative lenses. We also find an increasing trend of most literature on fractals in planning being published outside of planning. We hypothesize that this trend results from communication gaps between technical analysts and applied planners, and hope that our overview will help to bridge that gap.

A Three-Dimensional Box-Counting Method to Study the Fractal Characteristics of Urban Areas in Shenyang, Northeast China

Modern expansion and three-dimensional growth are rapidly altering the morphological features of traditional cities. This morphological phenomenon fully reflects the internal organization mode and composition rules of modern cities. This study draws on the research method of three-dimensional fractal, focusing on the situation where there is less research on the fractal form at the block scale, and conducts a fractal research on the three-dimensional form of a city at the meso and micro scales, in order to reveal the fractal characteristics of modern urban density. Based on fractal theory, the urban form of Shenyang, Northeast China, was quantitatively analyzed using the box-counting (Minkowski–Bouligand) method to calculate the two-dimensional (2D) and three-dimensional (3D) box dimensions of urban areas. Next, by analyzing the correlations between morphological indicators and 2D and 3D fractal dimensions, this study proposes cluster features of the correlation between the 3D fractal dimension and floor-area ratio. Then, this study summarizes the fractal characteristics of Shenyang’s urban form, based on the cluster analysis and spatial features of various urban areas within the city. The analysis results show the fractal dimension of Shenyang’s urban form to have characteristic expected values; fractal dimension clusters reflect spatial differences in the forms of different urban areas. The 3D D value of architectural morphology fractal in urban areas of Shenyang is between 2.41 and 2.70. From this, the representative characteristics of Shenyang’s urban form were obtained: first, it has the fractal characteristics of morphological hierarchy and system embeddedness; second, under unified and standardized management, its basic urban form structure displays the fractal characteristics of morphological similarity and system hierarchy; and third, its 3D urban form characteristics include the spatial accumulation of clusters and morphological patches, creating a patchwork of different building heights and densities, with the spatial clustering of density form highly correlated with the fractal dimension. The results of this research will provide reference samples for the morphological identification, design control, and design review of modern cities, and enrich the research results of the application of fractal theory to urban morphology at the meso and micro scales.

Fractal-Based Modeling and Spatial Analysis of Urban Form and Growth: A Case Study of Shenzhen in China

Fractal dimension curves of urban growth can be modeled with sigmoid functions, including logistic function and quadratic logistic function. Different types of logistic functions indicate different spatial dynamics. The fractal dimension curves of urban growth in Western countries follow the common logistic function, while the fractal dimension growth curves of cities in northern China follow the quadratic logistic function. Now, we want to investigate whether other Chinese cities, especially cities in South China, follow the same rules of urban evolution and attempt to analyze the reasons. This paper is devoted to exploring the fractals and fractal dimension properties of the city of Shenzhen in southern China. The urban region is divided into four subareas using ArcGIS technology, the box-counting method is adopted to extract spatial datasets, and the least squares regression method is employed to estimate fractal parameters. The results show that (1) the urban form of Shenzhen city has a clear fractal structure, but fractal dimension values of different subareas are different; (2) the fractal dimension growth curves of all the four study areas can only be modeled by the common logistic function, and the goodness of fit increases over time; (3) the peak of urban growth in Shenzhen had passed before 1986 and the fractal dimension growth is approaching its maximum capacity. It can be concluded that the urban form of Shenzhen bears characteristics of multifractals and the fractal structure has been becoming better, gradually, through self-organization, but its land resources are reaching the limits of growth. The fractal dimension curves of Shenzhen’s urban growth are similar to those of European and American cities but differ from those of cities in northern China. This suggests that there are subtle different dynamic mechanisms of city development between northern and southern China.

Fractal Dimension of a Transportation Network and its Relationship with Urban Growth: A Study of the Dallas-Fort Worth Area

A city and its transportation network are both complicated systems. Fractal geometry provides an effective way to describe the complex property of geographical features. This paper uses a modified box-counting method to describe the fractal property of urban transportation networks. Assuming that human settlements of different sizes are all operated by the same growth procedure, this paper investigates the relationship between the mass size of cities and the complexity of their road systems. The results confirm that, as cities grow from small to large, their transportation networks generally become more complicated—the urban spaces are filled up more densely by city roads and the locations within a city are more accessible. The quantitative relationship identified between the complexity of urban transportation network and city size provides an empirical guide for the planning and policymaking of urban development and road construction.

The Dynamics of the Tel Aviv Morphology

In this paper we suggest an approach for understanding the spatial behavior and structure of cities. It views cities as physical objects and is based on urban morphology alone. The units of examination are urban clusters instead of municipalities defined by politically determined boundaries. Clusters are defined as contiguous built-up urban areas. We present characteristics of clusters, including their morphology. Previous work that analyzed urban clusters focused on the Pareto distribution of clusters and on the behavior of the biggest cluster. Our work presents a more thorough description of the characteristics of urban clusters. By means of historic data of the Tel Aviv metropolis we present cluster statistics and we study their dynamics. We present characteristics of the clusters from 1935 to 2000, including their number, rank-size distribution, and morphology through the area–perimeter relation. These indicators present important anomalies in 1964 and 1985. Our study suggests that the urban cluster approach can be used as a tool to study urban phenomena and we hope that through them we shall be able to investigate economic and social phenomena as well.

Assuming it is all about conditions. Framing a simulation model for complex, adaptive urban space

In this paper, we explore the route beyond the conventional, linear attitude within planning and its rationality debate. We combine our theoretical reasoning with a multiscale approach and with fractal-like argumentation which results in a frame of conditions which is supported by the outline of a theoretical conceptual simulation model which would also allow non-linear, iterative simulations of the urban space. The understanding of autonomous non-linear spatial development has a direct impact on planning. Addressing the underlying thinking behind Haken’s synergetics we develop a framework within which the interdependencies between different levels of scale are key. We are aware that bottom-up and top-down processes often have a mutual influence on one another. We therefore propose a conceptual simulation model for planning where conditions have an impact at various levels of scale. In coherence with the idea of the ‘dynamic behaviour of the system after a planning decision was made’, this feedback gives us information on the surviving and non-surviving planning scenarios and decisions and is reminiscent of systems which are open to self-organizing pattern formation. Our reasoning with regard to planning and decision-making and their multilevel consequences is strongly influenced by the arguments presented in complexity studies.

Cities and regions in Britain through hierarchical percolation

Urban systems present hierarchical structures at many different scales. These are observed as administrative regional delimitations which are the outcome of complex geographical, political and historical processes which leave almost indelible footprints on infrastructure such as the street network. In this work, we uncover a set of hierarchies in Britain at different scales using percolation theory on the street network and on its intersections which are the primary points of interaction and urban agglomeration. At the larger scales, the observed hierarchical structures can be interpreted as regional fractures of Britain, observed in various forms, from natural boundaries, such as National Parks, to regional divisions based on social class and wealth such as the well-known North-South divide. At smaller scales, cities are generated through recursive percolations on each of the emerging regional clusters. We examine the evolution of the morphology of the system as a whole, by measuring the fractal dimension of the clusters at each distance threshold in the percolation. We observe that this reaches a maximum plateau at a specific distance. The clusters defined at this distance threshold are in excellent correspondence with the boundaries of cities recovered from satellite images, and from previous methods using population density.

Multifractal to monofractal evolution of the London street network

We perform a multifractal analysis of the evolution of London's street network from 1786 to 2010. First, we show that a single fractal dimension, commonly associated with the morphological description of cities, does not suffice to capture the dynamics of the system. Instead, for a proper characterization of such a dynamics, the multifractal spectrum needs to be considered. Our analysis reveals that London evolves from an inhomogeneous fractal structure, which can be described in terms of a multifractal, to a homogeneous one, which converges to monofractality. We argue that London's multifractal to monofractal evolution might be a special outcome of the constraint imposed on its growth by a green belt. Through a series of simulations, we show that multifractal objects, constructed through diffusion limited aggregation, evolve toward monofractality if their growth is constrained by a nonpermeable boundary.

Fractal cartography of urban areas

In a world in which the pace of cities is increasing, prompt access to relevant information is crucial to the understanding and regulation of land use and its evolution in time. In spite of this, characterization and regulation of urban areas remains a complex process, requiring expert human intervention, analysis and judgment. Here we carry out a spatio-temporal fractal analysis of a metropolitan area, based on which we develop a model which generates a cartographic representation and classification of built-up areas, identifying (and even predicting) those areas requiring the most proximate planning and regulation. Furthermore, we show how different types of urban areas identified by the model co-evolve with the city, requiring policy regulation to be flexible and adaptive, acting just in time. The algorithmic implementation of the model is applicable to any built-up area and simple enough to pave the way for the automatic classification of urban areas worldwide.

Retrieving Spatial Policy Parameters from an Alternative Plan Using Constrained Cellular Automata and Regionalized Sensitivity Analysis

In this paper we propose an approach to identify the spatial policy parameters (termed the implementation intensity reflecting planning controls on corresponding spatial constraint) associated with a predefined alternative plan, namely, a predefined-binary urban form. During plan implementation, the alternative plan cannot be fully realized in some cases due to practical urban growth driven by both institutional forces and market incentives, which are comprehensive and complex. Few researchers have investigated spatial policies appropriate for an alternative plan. We aim to propose a novel approach incorporating constrained cellular automata and regionalized sensitivity analysis, a method for global sensitivity analysis to calculate the realization possibility and identify the spatial policy parameters for an alternative plan. This approach is first tested in a virtual space with four predefined urban forms and various point, line, and polygon spatial constraints, with both positive and negative impacts on urban growth. Finally, the approach is also tested in the Beijing Metropolitan Area to identify the required spatial policy parameters for four alternative plans with seven spatial constraints.

Mining the Urban Sprawl Pattern: A Case Study on Sunan, China

China's urbanization is going into a fast development stage. This paper focuses on the recent evolution of an urbanized area – Sunan, the southern part of Jiangsu province in the Yangtze River Delta in China – by means of complementary approaches, especially different fractal and autocorrelation measures. The research shows that Sunan's urban clusters are becoming more and more homogenous and compact and are growing up along the important transportation axes. The enriching discussion of the findings establishes the links between the morphology of urban sprawl and recent socio-economic changes in China.