At each iteration step, new built-up pixels appear under the constraint that their location is compatible with the fractal nature of the simulated pattern. Other non fractal constraints have been integrated into the model, accelerating, slowing down or preventing the apparition of the built-up areas rivers, slopes declivity, exposure….
But locally, the simulated and the real patterns could be very different. Following the same direction, it would be particularly interesting to provide several models of fractal growth allowing the simulation of urban patterns with well differentiated characteristics. Thus, it could be possible to simulate different conceivable evolutions of an original urban pattern, each of the simulations corresponding to a particular vision of the urbanisation process e.
Such a modelling uses the fractal approach to replace the Euclidean spatial representation of the city i. Reitel 2 in the framework of a contract directed by P. Frankhauser 3 for the French Ministry of the Public Works 4. It deals mainly with the morphological evolution of the urban area of Basle 5 in the course of last century. The available data are images of the urban pattern at three dates — — Appendix 1, 2 and 3.
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The analysis of the images aims to explore the ability of fractal measures to characterise the process of urban sprawl. The ambition is to provide a set of analyses which may be used for comparing the urban realities of a variety of countries by using a unique methodological tool. Fractalyse offers different methods to measure the fractal dimension of an image.
But, whatever the chosen method, the general principles are always the same: This image is composed of two types of pixels: At each iteration step, the analysis involved counting the number of black pixels built-up pixels contained in a counting window. From one step to the next, the size of the counting window is enlarged. By doing that, we artificially change the level of analysis of the image.
So, for each analysis we have two elements varying according to the counting step iteration step i: How to calculate the fractal dimension of an image. The next stage of the analysis is to fit this empirical curve with another one, the estimated curve. If the empirical curve follows a fractal law, the estimated curve has the form of a power law parabolic or hyperbolic. The parameter c corresponds to the point of origin on the Y-axis. Its absolute value may be very high. It gives a synthetic indication of the local deviations from the estimated fractal law. In the case of a mathematical fractal structure a should be equal to 1.
In some cases a is equal to 0. If its value goes over 10 or beyond 0. Indeed, it is possible to obtain a great variety of estimations of the fractal dimension stemming from a unique empirical curve. Different methodological choices lead to different estimations of the fractal dimension. This has to be taken into account when analysing the results. The first method is the calculation of the fractal dimension of the images by using the correlation analysis.
The second one is based on an iterative transformation of the images step by step dilation and a representation of some information about the transformed images on a two-dimension graph for each step of the iteration. This second approach provides no calculation of fractal dimension, but results from a multi-scalar reasoning on a typical fractal nature.
The number of occupied points inside each window is enumerated. This allows the mean number of points per window of that given size to be calculated. The same operation is applied for windows of increasing sizes. In a multi-fractal theoretical framework, this correlation dimension should be extended to a series of three, four or more points. It is interesting to estimate not only the global fractal dimension of each image, but also the fractal dimensions for several portions of the empirical curves 9. Actually, whereas the fractality of a structure is clear when the adjustment between the empirical curve and the estimated curve is good, a structure is characterised by the combination of different types of fractal behaviour when the fit between the two curves remains good after having segmented the curve into several portions.
During the first dilation, each pixel is surrounded by a border of one pixel width. Then, the reference element is a square of 3 2 pixels size. At the second iteration step, each pixel is surrounded by a border of two pixels width. The structuring element is then a square of 5 2 pixels size. And so on… As the size of the squares gradually increases, the details smaller than the size of the structuring element are overlooked. Thus, we gradually obtain an approximation of the original form.
The graphs represent either the evolution of the length of the border of the built-up area at each step of the dilation, or the evolution of the number of clusters of built-up pixels through the dilations. The fractal dimensions result from the correlation analysis of the border of the built-up area and from the correlation analysis of the built-up surface of the urban area.
In addition, the high fluctuations of the fractal dimensions when changing the limits of the zone under study i.
Fractal correlation dimensions - Borders of the urban area. The spatial extension of the urban area happened mostly in the valleys and along the main transportation axis tramways and railway. Thus, the border has become tentacular and covers more space than in Between and this trend was only slightly reinforced, which explains that the fractal dimensions are very similar at the two dates. The general form of the border in is very close to the one in in a general context of a higher consumption of space. The only difference is the estimation of the fractal dimension of 1.
The border has become so tortuous, that it covers the space just as a surface does. It indicates a more pronounced urban sprawl in than in The first point on the X-axis is 4. For this value of 4. The first dilation step is characterised by an extension of the border for each of the three curves: Clear differences may be observed between the shape of the curves of and on the one hand, and the shape of the curve of on the other hand.
But, the differences dwindle in the course of the dilations. Evolution of the length of the urban border with the dilations. The general morphology of the whole urban area, although it was expanding, did not change much between these two dates.
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As early as the second step of dilatation, many built-up elements are aggregated and the length of the border sharply decreases, while above the 85 m threshold, the buildings are more distant from each other and do not aggregate so rapidly. The longer the initial border, the less compact is a town. A steep curve slope indicates that numerous settlements are close enough for aggregating at further steps of the analysis and coins therefore urban sprawl. A variety of shapes of curves could be related to different types of urban growth. Fractal correlation dimensions — Built-up surface of the urban area.
The number of clusters varies according to the steps of dilation. At the beginning, it is much lower in clusters than in 34 clusters , while the number of clusters in was in between 19 This corresponds to the number of non contiguous buildings which has increased in the recent periods, following a growing trend to urban sprawl. For the three dates, a sharp decrease in the number of clusters can be observed after the first steps of dilation, with slightly different thresholds corresponding to the mean size of neighbourhoods at the time.
The slowing down of the decreasing curve is less pronounced for the more recent periods, due to a larger fraction of space being occupied by non compact built-up zones. Number of clusters of built-up pixels at each step of the dilation. This description is in accordance with the observations made about the border of the urban area. It shows that the general form of the agglomeration was already shaped in , the consecutive evolution being merely a space filling process around the existing built-up cores.
Considering tables 1 and 2, it appears that fractal dimensions of the border and of the built-up area are similar in and , while results are more different in the case of The relationship between surface and border changed over time. The results obtained should now be interpreted thoroughly in order to identify the substantive meaning of the identified thresholds as well as the substantive meaning of the intersection of the curves which appeared.
But it seems useful here to sum up the morphological properties of urban patterns which can be identified through the analysis presented and which manifest themselves in the existence of urban sprawl: Such an objective could be attained mainly through systematic comparisons with other urban areas. Below, we briefly review a list of remaining questions for urban geography which could be solved by intensifying comparative research.
The reference to fractals is relatively recent in geographical literature, the first appeared less than twenty years ago, and probably deeper insights will be gained as studies become more numerous and more systematic. The main advantage of fractal geometry is to provide a model of reference which seems more adapted than Euclidean geometry to the description of spatial forms created by societies: When comparing observed spatial patterns to Euclidean geometry, these properties appear as major deviations and anomalies specifying social systems, whereas direct comparison to fractal models may reveal specific features which have not been noticed yet.
Another very important although not yet fully explored property of fractals is their relation to underlying non linear generative mechanisms. The design and use in simulation of models which would explicitly connect individual behaviour or micro processes to the emergence of fractal morphologies at upper levels of observation would greatly improve our understanding of the genesis of such forms and allow a more systematic exploration of their stability, limits and rationales.
What would be an index of the fractality of cities? We know that because of its homogeneity, a perfectly compact city is not fractal, neither are suburbs which would be homogeneously scattered. In between, how should be the variations in the degree of fractality interpreted? Fractal dimensions can be compared but they are very concise summaries of entire urban structures which may differ in other ways while exhibiting the same fractal dimension.
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To include a comma in your tag, surround the tag with double quotes. Skip to content Skip to search. Language English View all editions Prev Next edition 2 of 6. Check copyright status Cite this Title City and society: Ronald John , Published Harmondsworth [etc. Urban regions Geographical aspects. View online Borrow Buy Freely available Show 0 more links Set up My libraries How do I set up "My libraries"? These 21 locations in All: Australian National University Library.
Open to the public. Open to the public ; Federation University Australia - Gippsland campus library. Flinders University Central Library. La Trobe University Library. Geographic approaches to spatial representation are closely linked to a set of core spatial concepts including location, region, distribution, spatial interaction, scale, and change that implicity constrain and shape how geographers represent what they observe. In effect, these concepts become a priori assumptions underlying geographic perspectives and shaping decisions by geographers about how to represent their data and what they choose to represent.
Geographers approach spatial representation in a number of ways to study space and place at a variety of scales. Tangible representations of geographic space may be visual, verbal, mathematical, digital, cognitive, or some combination of these. Reliance on representation is of particular importance when geographic research addresses intangible phenomena e. Tangible representations and links among them also provide a framework within which synthesis can take place. Geographers also study cognitive spatial representations—for example, mental models of geographic environments—in an effort to understand how knowledge of the environment influences peoples' behavior in that environment and make use of this knowledge of cognitive representation in developing approaches to other forms of representation.
Visual representation of geographic space through maps was a cornerstone of geographic inquiry long before its formal recognition as an academic area of research, yet conventional maps are not the only visual form used in geographic research. This continuum can be defined by a dimension scale, which ranges from atomic to cosmological, and abstractness level, which ranges from images to line drawings.
Due to the centrality of geographic maps as a means for spatial representation, however, concepts developed for mapping have had an impact on all forms of spatial representation. This role as a model and catalyst for visual representation throughout the sciences is clear in Hall's recent popular account of mapping as a research tool used throughout science, as well as the recognition by computer scientists that maps are a fundamental source of many concepts used in scientific visualization Collins, An active field of geographic research on spatial representation involves formalizing the ''language" for visual geographic representation.
Another important field of research involves improved depiction of the Earth's surface. A notable example is the recent advance in matching computational techniques for terrain shading with digital elevation databases covering the conterminous United States see Sidebar 3. The conventional map is one of many visual representations of space used by geographers and other scientists.
As one of a continuum of spatial representations, maps occupy a "fuzzy" category defined by an "abstractness level" horizontal axis and a "scale dimension" vertical axis. After MacEachren , Figure 4. Verbal representation refers to attempts to evoke landscapes through a carefully constructed description in words. Some of the geographers who have become best known outside the discipline rely almost exclusively on this form of representation. Geographers have drawn new attention to the power of both verbal and visual representations, exploring the premise that every representation has multiple, potentially hidden, and perhaps duplicitous, meanings Gregory, A current field of research linking verbal and visual forms of spatial representation concerns hypermedia documents designed for both research and instructional applications.
The concept of a geographic script analogous to a movie script has been proposed as a strategy for leading people through a complex web of maps, graphics, pictures, and descriptions developed to provide information about a particular issue Monmonier, Mathematical representations include models of space, which emphasize location, regions, and distributions; models of functional association; and models of process, which emphasize spatial interaction and change in place.
Visual maps, of course, are grounded in mathematical models of space, and it can be demonstrated that all map depictions of geographic position are, in essence, mathematical transformations from the Earth to the plane surface of the page or. The combination of visual and mathematical representation draws on advantages inherent in each see Plate 2. A good example of the link between mathematical and visual representation is provided by the Global Demography Project Tobler et al.
In this project more than 19, digitized administrative polygons and associated population counts covering the entire world were extrapolated to and then converted to spherical cells. Cognitive representation is the way individuals mentally represent information about their environment. Human cognitive representations of space have been studied in geography for more than 25 years. They range from attempts to derive "mental maps" of residential desirability to assessing ways in which knowledge of spatial position is mentally organized, the mechanisms through which this knowledge expands with behavior in environments, and the ways in which environmental knowledge can be used to support behavior in space.
The resulting wealth of knowledge about spatial cognition is now being linked with visual and digital forms of spatial representation. This link is critical in such research fields as designing interfaces for geographic information systems GISs and developing structures for digital geographic databases. Recent efforts to apply the approaches of cognitive science to modeling human spatial decision making have opened promising research avenues related to way finding, spatial choice, and the development of GIS-based spatial decision support systems.
In addition, research about how children at various stages of cognitive development cope with maps and other forms of spatial representation is a key component in efforts to improve geography education. Digital representation is perhaps the most active and influential focus of representational research because of the widespread use of GISs and computer mapping. Geographers have played a central role in the development of the representational schemes underpinning GISs and computer mapping systems. Geographers working with mathematicians at the U. Census Bureau in the s were among the first to recognize the benefits of topological structures for vector-based digital representations of spatial data.
It has been adapted to computer mapping through an innovative system for linking topological and metrical geographic representations. Related work by geographers and other scientists at the U. Geographers working in GIS research have investigated new approaches to raster grid-based data structures. Raster structures are compatible with the structure of data in remote sensing images, which continue to be a significant source of input data for GIS and other geographic applications.
Raster structures are also useful for overlying spatial data. Developments in vector and raster data structures have been linked through an integrated conceptual model that, in effect, is eliminating the raster-vector dichotomy Peuquet, This research is particularly important because solutions to key generalization problems are required before the rapidly increasing array of digital georeferenced data can be integrated through GISs to support multiscale geographic analysis. Generalization in the digital realm has proved to be a difficult problem because different scales of analysis demand not only more or less detailed information but also different kinds of information represented in fundamentally different ways.
Increasingly, the aspects of spatial representation discussed above are being linked through digital representations.
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Transformations from one representation to another e. This reliance on digital representation as a framework for other forms of representation brings with it new questions concerning the impact of digital representation on the construction of geographic knowledge.
One recent outgrowth of the spatial representation traditions of geography is a multidisciplinary effort in geographic information science. This field emphasizes coordination and collaboration among the many disciplines for which geographic information and the rapidly emerging technologies associated with it are of central importance. The University Consortium for Geographic Information Science UCGIS , a nonprofit organization of universities and other research institutions, was formed to facilitate this interdisciplinary effort.
UCGIS is dedicated to advancing the understanding of geographic processes and spatial relationships through improved theory, methods, technology, and data. This survey of geography's perspectives illustrates the variety of topics pursued by geography as a scientific discipline, broadly construed. The methods and approaches that geographers have used to generate knowledge and understanding of the world about them—that is, its epistemologies—are similarly broad.
The post-World War II surge in theoretical and conceptual geography, work. Extensive use is still made of this approach, especially in studying environmental dynamics but also in spatial analysis and representation. It is now recognized, however, that the practice of such research frequently diverges from the ideals of positivism. Many of these ideals—particularly those of value neutrality and of the objectivity of validating theories by hypothesis testing—are in fact unattainable Cloke et al.
Recognition of such limitations has opened up an intense debate among geographers about the relative merits of a range of epistemologies that continue to enliven the field Gregory, Of particular interest, at various points in this debate, have been the following:. Geographers debate the philosophical foundations of their research in ways similar to debates among other natural scientists, social scientists, and humanists, although with a particular emphasis on geographical views of the world and on representation.
These debates have not been restricted to the philosophical realm but have had very practical consequences for substantive research, often resulting in contrasting theoretical interpretations of the same phenomenon. For example, neopositivist and structural accounts of the development of settlement systems have evolved through active engagement with one another, and debates about how to assess the environmental consequences of human action have ranged from quantitative cost-benefit calculations to attempts to compare and contrast instrumental with local and indigenous interpretations of the meaning and significance of nature.
In subsequent chapters we have not attempted to mark these different perspectives, choosing instead to stress the phenomena studied rather than the approaches taken. We attempt selectively to include leading researchers from different perspectives working on a particular topic, to the extent that their work can be constituted as scientific in the broad sense that we use that term see Sidebar 1. While we recognize that different perspectives frequently lead to intense debates engaging very different views of the same phenomenon, there is no space in this report to detail these debates.
Such often vigorous interchanges and differences strengthen geography as both a subject and a discipline, however, reminding researchers that different approaches may be relevant for different kinds of questions and that the selection of any approach shapes both the kind of research questions asked and the form the answers take, as well as the answers themselves. As political, economic, and environmental issues increasingly spread across the globe, the science of geography is being rediscovered by scientists, policymakers, and educators alike. Geography has been made a core subject in U.
Rediscovering Geography presents a broad overview of geography's renewed importance in a changing world. Through discussions and highlighted case studies, this book illustrates geography's impact on international trade, environmental change, population growth, information infrastructure, the condition of cities, the spread of AIDS, and much more.
The committee examines some of the more significant tools for data collection, storage, analysis, and display, with examples of major contributions made by geographers. Rediscovering Geography provides a blueprint for the future of the discipline, recommending how to strengthen its intellectual and institutional foundation and meet the demand for geographic expertise among professionals and the public. Based on feedback from you, our users, we've made some improvements that make it easier than ever to read thousands of publications on our website.
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