Fractal cartography of urban areas

Fractal dimension complexity of gravitation fractals in central place theory

Settlement centers of various types, including cities, produce basins of attraction whose shape can be regular or complexly irregular (from the point of view of geometry). This complexity depends in part on properties of the space surrounding a settlement. This paper demonstrates that by introducing a dynamic approach to space and by including an equation of motion and space resistance, a dramatic change in the stylized static CPT (Central Place Theory) image occurs. As a result of the interplay of gravitational forces, basins of attraction arise around cities, whose boundaries appear to be fractals. This study provides a wealth of spatial fractal complex images which may change the traditional understanding of CPT.

From cognitive maps to spatial schemas

A schema refers to a structured body of prior knowledge that captures common patterns across related experiences. Schemas have been studied separately in the realms of episodic memory and spatial navigation across different species and have been grounded in theories of memory consolidation, but there has been little attempt to integrate our understanding across domains, particularly in humans. We propose that experiences during navigation with many similarly structured environments give rise to the formation of spatial schemas (for example, the expected layout of modern cities) that share properties with but are distinct from cognitive maps (for example, the memory of a modern city) and event schemas (such as expected events in a modern city) at both cognitive and neural levels. We describe earlier theoretical frameworks and empirical findings relevant to spatial schemas, along with more targeted investigations of spatial schemas in human and non-human animals. Consideration of architecture and urban analytics, including the influence of scale and regionalization, on different properties of spatial schemas may provide a powerful approach to advance our understanding of spatial schemas.

Characterizing Fractal Genetic Variation in the Human Genome from the Hapmap Project

Over the last decades, the exuberant development of next-generation sequencing has revolutionized gene discovery. These technologies have boosted the mapping of single nucleotide polymorphisms (SNPs) across the human genome, providing a complex universe of heterogeneity characterizing individuals worldwide. Fractal dimension (FD) measures the degree of geometric irregularity, quantifying how "complex" a self-similar natural phenomenon is. We compared two FD algorithms, box-counting dimension (BCD) and Higuchi's fractal dimension (HFD), to characterize genome-wide patterns of SNPs extracted from the HapMap data set, which includes data from 1184 healthy subjects of eleven populations. In addition, we have used cluster and classification analysis to relate the genetic distances within chromosomes based on FD similarities to the geographical distances among the 11 global populations. We found that HFD outperformed BCD at both grand average clusterization analysis by the cophenetic correlation coefficient, in which the closest value to 1 represents the most accurate clustering solution (0.981 for the HFD and 0.956 for the BCD) and classification (79.0% accuracy, 61.7% sensitivity, and 96.4% specificity for the HFD with respect to 69.1% accuracy, 43.2% sensitivity, and 94.9% specificity for the BCD) of the 11 populations present in the HapMap data set. These results support the evidence that HFD is a reliable measure helpful in representing individual variations within all chromosomes and categorizing individuals and global populations.

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.

Emergence of urban growth patterns from human mobility behavior

Cities grow in a bottom-up manner, leading to fractal-like urban morphologies characterized by scaling laws. The correlated percolation model has succeeded in modeling urban geometries by imposing strong spatial correlations; however, the origin of the underlying mechanisms behind spatially correlated urban growth remains largely unknown. Our understanding of human movements has recently been revolutionized thanks to the increasing availability of large-scale human mobility data. This paper introduces a computational urban growth model that captures spatially correlated urban growth with a micro-foundation in human mobility behavior. We compare the proposed model with three empirical datasets, discovering that strong social interactions and long-term memory effects in human movements are two fundamental principles responsible for fractal-like urban morphology, along with the three important laws of urban growth. Our model connects the empirical findings in urban growth patterns and human mobility behavior.

Analysis of Spatial and Temporal Changes and Expansion Patterns in Mainland Chinese Urban Land between 1995 and 2015

China has experienced greater and faster urbanization than any other country, and while coordinated regional development has been promoted, urbanization has also introduced various problems, such as an increased scarcity of land resources, uncontrolled demand for urban land, and disorderly development of urban fringes. Based on GIS, remote sensing data, and spatial statistics covering the period 1995–2015, this study identified the patterns, as well as spatial and temporal changes, with respect to urban land expansion in 367 mainland Chinese cities. Over this study period, the area of urban land in mainland China increased from 3.05 to 5.07 million km2, at an average annual growth rate of 2.56%. This urban land expansion typically occurred the fastest in medium-sized cities, followed by large cities, and then small cities, with megacities and megalopolises exhibiting the slowest expansion rates. Nearly 70% of the new urban land came from arable land, 11% from other built land, such as pre-existing rural settlements, and 15% from forests and grasslands. When considering marginal-, enclave-, and infill-type expansion patterns, growth in >80% of the 367 cities surveyed was dominated by marginal expansion patterns. Marginal and enclave expansion patterns were found to be becoming more prevalent, with infill-type expansion being seen less. The results of this study provide a theoretical basis and data support for urban spatial planning, the protection of farmland, and the promotion of urban land use efficiency, and can be used as guidance for regional urbanization planning.

Rural Morphology and Forces Driving Change in Rapidly Urbanizing Areas: A Case Study in Fujian, China

Rapid urbanization in China has transformed many rural areas from agriculture-dominated to diverse systems, but studies of rural morphology are limited compared to studies of urban settlement. This paper uses a fractal dimension (FD) value to analyze the change in rural morphology in Fujian Province, a region with a long history of rural settlement and rapid recent urbanization, and to explore the factors that influenced this change. We found that the rural FD value increased from 2000 to 2012 and that rural morphology was spatially heterogeneous. FD was generally lower than in urban areas but very close to a typical urban area value in the southeast coastal region. A structural equation model was used to identify key factors influencing rural morphology, which were natural conditions, rurality and economic development, while historic administration had the smallest positive effect. With a long history and unique administrative system, the spatial morphology of Chinese rural areas has shown characteristics distinct from compact urban or scattered rural areas. The urban planning method adopted by rural planners is not suitable in rural regions, because the planning potential of rural areas with high and low FD values varies. Although rural planning currently uses a very similar approach to urban planning, it should use a local, flexible and adaptive policy based on rural morphological characteristics.

Living Structure as an Empirical Measurement of City Morphology

Human actions and interactions are shaped in part by our direct environment. The studies of Christopher Alexander show that objects and structures can inhibit natural properties and characteristics; this is measured in living structure. He also found that we have better connection and feeling with more natural structures, as they more closely resemble ourselves. These theories are applied in this study to analyze and compare the urban morphology within different cities. The main aim of the study is to measure the living structure in cities. By identifying the living structure within cities, comparisons can be made between different types of cities, artificial and historical, and an estimation of what kind of effect this has on our wellbeing can be made. To do this, natural cities and natural streets are identified following a bottom-up data-driven methodology based on the underlying structures present in OpenStreetMap (OSM) road data. The naturally defined city edges (natural cities) based on intersection density and naturally occurring connected roads (natural streets) based on good continuity between road segments in the road data are extracted and then analyzed together. Thereafter, historical cities are compared with artificial cities to investigate the differences in living structure; it is found that historical cities generally consist of far more living structure than artificial cities. This research finds that the current usage of concrete, steel, and glass combined with very fast development speeds is detrimental to the living structure within cities. Newer city developments should be performed in symbiosis with older city structures as a whole, and the structure of the development should inhibit scaling as well as the buildings themselves.

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.

The Effects of Land Use and Land Cover Geoinformation Raster Generalization in the Analysis of LUCC in Portugal

Multiple land use and land cover (LUC) datasets are available for the analysis of LUC changes (LUCC) in distinct territories. Sometimes, different LUCC results are produced to characterize these changes for the same territory and the same period. These differences reflect: (1) The different properties of LUC geoinformation (GI) used in the LUCC assessment, and (2) different criteria used for vector-to-raster conversion, namely, those deriving from outputs with different spatial resolutions. In this research, we analyze LUCC in mainland Portugal using two LUC datasets with different properties: Corine Land Cover (CLC 2006 and 2012) and LUC official maps of Portugal (Carta de Ocupação do Solo, COS 2007 and 2010) provided by the European Environment Agency (EEA) and the General Directorate for Territorial Development (DGT). Each LUC dataset has undergone vector-to-raster conversion, with different resolutions (10, 25, 50, 100, and 200 m). LUCC were analyzed based on the vector GI of each LUC dataset, and with LUC raster outputs using different resolutions. Initially, it was observed that the areas with different LUC types in two LUC datasets in vector format were not similar—a fact explained by the different properties of this type of GI. When using raster GI to perform the analysis of LUCC, it was observed that at high resolutions, the results are identical to the results obtained when using vector GI, but this ratio decreases with increased cell size. In the analysis of LUCC results obtained with raster LUC GI, the outputs with pixel size greater than 100 m do not follow the same trend of LUCC obtained with high raster resolutions or using LUCC obtained with vector GI. These results point out the importance of the factor form and the area of the polygons, and different effects of amalgamation and dilation in the vector-to-raster conversion process, more evident at low resolutions. These findings are important for future evaluations of LUCC that integrate raster GI and vector/raster conversions, because the different LUC GI resolution in line with accuracy can explain the different results obtained in the evaluation of LUCC. The present work demonstrates this fact, i.e., the effects of vector-to-raster conversions using various resolutions culminated in different results of LUCC.

The role of city size and urban form in the surface urban heat island

Urban climate is determined by a variety of factors, whose knowledge can help to attenuate heat stress in the context of ongoing urbanization and climate change. We study the influence of city size and urban form on the Urban Heat Island (UHI) phenomenon in Europe and find a complex interplay between UHI intensity and city size, fractality, and anisometry. Due to correlations among these urban factors, interactions in the multi-linear regression need to be taken into account. We find that among the largest 5,000 cities, the UHI intensity increases with the logarithm of the city size and with the fractal dimension, but decreases with the logarithm of the anisometry. Typically, the size has the strongest influence, followed by the compactness, and the smallest is the influence of the degree to which the cities stretch. Accordingly, from the point of view of UHI alleviation, small, disperse, and stretched cities are preferable. However, such recommendations need to be balanced against e.g. positive agglomeration effects of large cities. Therefore, trade-offs must be made regarding local and global aims.

Navigation performance in virtual environments varies with fractal dimension of landscape

Fractal geometry has been used to describe natural and built environments, but has yet to be studied in navigational research. In order to establish a relationship between the fractal dimension (D) of a natural environment and humans' ability to navigate such spaces, we conducted two experiments using virtual environments that simulate the fractal properties of nature. In Experiment 1, participants completed a goal-driven search task either with or without a map in landscapes that varied in D. In Experiment 2, participants completed a map-reading and location-judgment task in separate sets of fractal landscapes. In both experiments, task performance was highest at the low-to-mid range of D, which was previously reported as most preferred and discriminable in studies of fractal aesthetics and discrimination, respectively, supporting a theory of visual fluency. The applicability of these findings to architecture, urban planning, and the general design of constructed spaces is discussed.

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.