Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 165 (2016) 1478 – 1482 15th International scientific conference “Underground Urbanisation as a Prerequisite for Sustainable Development” Fractal dimensions of urban border as a criterion for space management Marija Jevrica, Marina Romanovich b,* a University of Montenegro, Faculty of Civil Engineering, Podgorica, Montenegro Peter the Great St Petersburg Polytechnic University, Polytechnicheskaya 29, St Petersburg, 195251, Russia b Abstract Tendency of the big cities population to be in the vicinity of open space and/or the environment, for reasons of reducing stress, noise, pollution, traffic jams, etc., is more and more evident It was noted that such a tendency exists even in Podgorica (Montenegro) and through a questionnaire conducted was confirmed Information on the fractal dimensions of urban border and urban area can be used as a parameter for decision making on spatial development In this way, the peripheral areas of the city that give shape to the urban border, could be densified but not at the expense of their contact with open space or greenery Click here and insert your abstract text © 2016 The Authors Published by Elsevier Ltd This is an open access article under the CC BY-NC-ND license © 2016 The Authors Published by Elsevier Ltd (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the 15th International scientific conference “Underground Peer-review under responsibility of the scientific committee of the 15th International scientific conference “Underground Urbanisation as a Urbanisation as a Prerequisite for Sustainable Development Prerequisite for Sustainable Development Keywords: urban border, urban area, fractal dimension, decision making Introduction New knowledge about the complex systems contributed to the new view of the city that it is not seen anymore as a simple, organized system with rectilinear geometry and structure, but as a complex organism that manifests a certain order through its form, at different scales of observation Because of that, the researches argue that "cities are * Corresponding author Tel.: +79213189799 E-mail address: p198320@yandex.ru 1877-7058 © 2016 The Authors Published by Elsevier Ltd This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the 15th International scientific conference “Underground Urbanisation as a Prerequisite for Sustainable Development doi:10.1016/j.proeng.2016.11.882 Marija Jevric and Marina Romanovich / Procedia Engineering 165 (2016) 1478 – 1482 1479 ideal candidates for fractal analysis" [1] This implies the fact that cities are formed through the gradual effect of local factors, which generate regular urban patterns Therefore, the consideration of small changes at the local level, which can be significant on a global level, is very important for urban pattern generation We are witnessing the alienation of man's cities and its conflict with the natural laws and environmental phenomena Physical structure of the city could make it possible to maintain the essential relationship of man and the environment [2] , The latest researches reveal parameters related to complexity theory and fractals as one of criteria in making decisions in the planning process [3] - [9] The reason for this study is a questionnaire that was conducted in Podgorica, the capital of Montenegro, in order to assess the impact of certain factors of the built environment on the quality of life in neighborhoods The survey was conducted on a sample of 193 inhabitants (engineers) Their opinion is that public utility infrastructure, with a share of 32%, has the major importance to the quality of life in the urban units; within the assessment of the urban unit location, the green area vicinity is of the most importance, while city center vicinity is far less significant The reason for such result can be interpreted in two ways: generally improved conditions of transport to the center, as well as good network of urban centers, but also the desire and need of the population for closer contact with nature Also, the vicinity to recreational areas has a great importance to the space users wellbeing, so if we unite these segments, we get the important parameters for further decision making and urban planning In the second part of the questionnaire, there were two questions: Which of the following objects of urban greenery you consider necessary for a good quality of life in the urban unit and could vicinity of a park or park forest be the reason to you for moving to peripheral areas of the city? A large percentage of respondents (96%) consider the proximity to the park, forest or even both, necessary for satisfying quality of life Only 4% of respondents would be satisfied with the existence of alleys, lawns and similar greenery From the answer to the second question (close to the half of the respondents answered affirmatively), we can make conclusion about importance of a man’s contact with nature, even on a daily basis This information may affect the management of the built environment in two ways: firstly, the population has a need to be close to open spaces, not rushing towards the city center; Secondly, in order to resettlement there must be provided a good quality of the other factors, such as good public utility infrastructure, urban facilities availability and satisfying transport links This information is of great importance for making decisions about the built environment and it points to the need for such built environment design that will bring greenery facilities, even parks and park forests, closer to the largest population Fractal geometry may offer solutions [10] Theoretical background and methodology Fractals are complex, hierarchically arranged structures that show feature of self-similarity across different scale of observation This and other fractal properties indicate significant similarities with built environment patterns, urban borders or land use maps [9] Fractal geometry provides more adequate models to describe the spatial shapes than Euclidean geometry because the heterogeneity, self-similarity and hierarchy are the essential attributes of fractal structures [11] In this regard, the fractal analysis of the built environment form can be implemented in two ways: through the analysis of spatial distribution of built areas and through the analysis of urban border Fractal theory basics Complexity theory and fractals can be applied to many natural and artificial systems whose dynamics, ie interaction of local factors, generates ordered patterns Fractal theory provides an explanation for the complexity that arises in the form and structure of such systems The term fractal describes the objects which, at first glance, have no regular geometric form and Euclidean geometry such objects described only as amorphous They most often occur in nature Fractals are found in the microstructure of DNA, crystals, plants, certain organs of humans and animals; the mega-structures of clouds, mountains, river flows, lightning, coast, galaxies; in the regularity of fluctuations in heart rate, stock flows, weather conditions, earthquakes etc The basic properties of fractals are: generation through the iteration (Figure 1), selfsimilarity of its parts with the whole and fractal dimension 1480 Marija Jevric and Marina Romanovich / Procedia Engineering 165 (2016) 1478 – 1482 Figure Hilbert Curve generation through five iterations The Euclidean dimension of object is known to be linked to the coordinate system of object, or the number of its coordinate axes So, a point has no dimension (0), a line has dimension equals 1, a square has dimension 2, a cube Although the human mind used to operate with integer dimension for more than two millennia, most objects that surround us can be described by a dimension that is not a whole number, fractal dimension It is more suitable for the analysis of so-called organic forms, then integer Euclidean dimension Meaning of the rational fractal dimension is essential to understanding the fractal theory and its applications The idea of fractal dimension could be shown in the example of Hilbert curve that originates by suitable bending of straight line (dimension equals 1) so it can fill the whole square (dimension equals 2) For the fractal dimension could be said that it is the number that "measures" how well an object fills the space in which it is located To the date, several definitions of fractal dimension are proposed, through various researches In scientific purposes, box-counting dimension is commonly used Fractal analysis tool and methodology One or more methods of fractal dimension measuring can be relevant for the analysis of urban pattern All methods are based on the same principle of logarithmic equations and try to formulate the relationship between the dimensions of the object and a different scale on which the fractal dimension is measured In this study, we used Fractalyse software and two methods in FracLac tool: SLAC method (sliding box counting) and the BC method (box counting) Fractalyse is a software designed for measuring fractality of objects It requires binary images in raster format, with black pixels represent the built environment The analysis is carried out iteratively The so-called window is the area within which the counting is done The number of counted pixels (N) contained in the "window" of a size (ε) increases from iteration to iteration and thus artificially alter the level of image analysis, ie extent to which it is observed The dependence of the number of pixels on window size is represented on the graph, where y-axis is represented by number of pixels (N), and x-axis by window size (ε), to give an empirical curve If the empirical curve follows a fractal law, it takes the form of parabolic or hyperbolic function Exponent D is the fractal dimension ܰ ൌ ܽߝ ܿ (1) In Fractalyse software, matching these two curves is estimated using a adjusted correlation coefficient If a matching is bad, then the pattern that we examine is not fractal or it is a multi-fractal In the latter case, the empirical curve should be divided into several parts, each of them corresponds to a different curve (for each part of the curve, nonlinear regression gives different values for the three parameters a, c and D) The minimal coefficient which confirms the structure as a fractal is not unambiguously defined in the literature Although probability and statistics requires a value of the correlation coefficient r greater than 0,999, some authors dealing with the fractal analysis of the urban areas [12] proposed correlation coefficient value higher than 0.996 as acceptable, while the others claim that it can be even less [13] FracLac, plug-in for Image J (Java applet for image processing), also calculates the fractal dimension based on box-counting methods and procedure is similar to Fractalyse Marija Jevric and Marina Romanovich / Procedia Engineering 165 (2016) 1478 – 1482 1481 Fractal dimension of urban border The uniqueness of each urban form can be identified by measuring the level of complexity that it shows According to Batty [1], Frankhauser [5] and Salingaros [3], fractals have the ability to summarize the complexity, compactness and spatial heterogeneity of spatial distribution in a single value - fractal dimension (Df), which is independent of the scale Fractal analysis of urban pattern can mathematically identify the level of complexity of its structure It can be expected that the regular city plans are more homogeneous as so as more dispersed settlements in whose planning has a lot of repetition Heterogeneous pattern may appears as a result of planning concept or less controlled development For example, the development along the river valleys and the main roads, causes lower values of fractal dimensions than in the case of compact urban areas Fractal analysis of the urban area of Podgorica was performed using the software tool Fractalyse Selected method of analysis was dilatation method The program analyzes the image (ortho-photo) by replacing each occupied point with a square (sides ε), and that surface is considered to be fully occupied The size of these squares is gradually increased and total occupied area A (ε) is measured at every step After certain number of iterations, all details of the image less than ε disappear, to give an approximation of the original form If this total surface is divided by the area of a square ε2, the number of elements N (ε) needed to cover the whole image is obtained The rest of the procedure is ordinary in the methods of fractal analysis: plotting logarithmic graphic and comparing the empirical curve with the analytic curve etc., as described in 2.2 Table Fractal dimension values for urban border of Podgorica Number of dilations Fractal dimension Correlation coefficient curves matching 1.313 0.99969 good 1.402 0.99998 excellent 1.379 0.99996 excellent 10 1.32 0.99991 excellent 30 1.293 0.99990 excellent Df Depending on the number of the original images dilations, different values of the fractal dimension are obtained, but also two curves matching is not the same, ie the correlation coefficient varies For this reason, more analysis is done to check the optimal number of dilations of the image In the first case, number of dilation equals 3, empirical curve shows a linear regression (correlation coefficient 0.99969), indicating a fractal structure In other cases, experimental curve is compared to general non-linear function The matching is better with generalized fractal law and fractal dimension of 1.4 for the area of Podgorica can be adopted (by retaining four dilations) As expected, the value of the fractal dimension is not high, due to the existence of large areas that are not built or urbanized (Table 1) The morphology of the urban border is an indicator of urban dispersion Fractal dimension of the urban border was also investigated The urban border was isolated and then analyzed by box-counting method Previously, the area map was transferred to binary image by Java applet for image processing After that, enclosed empty spaces within urban boundaries were dilated and filled Selected scale was such that the image size was 10000x7000 pixels maximal size that FracLac could handle with When it comes to scale and number of dilation, many combinations were tried and the highest quality of fractal dimension assessment obtained for 50 dilations Fractal dimension of urban border of Podgorica is 1.406 Comparative analysis of European urban agglomerations shows that the fractal dimension of the urban area equals 1.60 - 1.87, and fractal dimensions of urban border equals 1.2 - 1.55 Overall, it can be concluded that the fractal dimension of approximately 1.7 and boundaries around 1.4 - 1.5 provide good articulation of urban and green areas Green surfaces are easily accessible, while the pattern preserves a certain degree of compactness [14] Fractal dimensions of urban border and area of Podgorica could be higher and this fact should be kept in mind through the urban planning Also, low fractal dimension of the border implies one of two following facts: that there is no interest or no conditions for settlement in peripheral areas of the city In combination with the results of a 1482 Marija Jevric and Marina Romanovich / Procedia Engineering 165 (2016) 1478 – 1482 questionnaire, on this level of the problem analysis, that can be concluded that low quality of public infrastructure is the reason Conclusion There is tendency of large cities population to be in the vicinity of open space and /or the natural environment, for reasons of reducing stress, noise, pollution, and often heat, crowds, etc which are common in urban area Such a tendency is evident in Podgorica, that is confirmed through a questionnaire Information on the fractal dimensions of urban border and urban area can be used as a parameter for decision making in the spatial development field For example, in the case of new residential areas planning, additional undeveloped areas hiring for further spatial development often happens, which is not in accordance with the principles of rational use of space Analyzing these two values of fractal dimension of the observed area, some conclusions about the pattern fragmentation or its compactness can be made Increasing fractal dimension of urban area and reducing fractal dimension of urban border on the peripheral areas of the city, indicates denser, homogeneous patterns with the tendency of filling existing empty spaces The opposite trend indicates a process of urban sprawl, which is characterized by fragmented development Increasing the fractal dimension of urban border can be a way to carry out a local increase in settlement density without quality of life reducing, ie peripheral areas of the city that give shape to the urban border, can be denser but not at the expense of their contact with open or green spaces The questionnaire results show a great interest of citizens in the settlement of such areas and indicate possible directions for decision-making in this regard Bearing in mind the previous results, we can conclude that analysis and concepts based on fractal analysis, combined with computer programs and tools, can provide new criteria when it comes to decision making, and thus the management of the built environment References [1] P Longley, M Batty, Fractal Cities: A Geometry of Form and Function, CA and London: Academic Pressю, San Diego, 1994 [2] R Radoviс, Forma grada Berograd: Orion Art, 2005 [3] N Salingaros, Ecology and the Fractal Mind in the New Architecture, www.fractal.org [4] T Haghani Fractal, Geometry, Complexity and the Nature of Urban Morphological Evolution, www fractalmorphology.com [5] P Frankhauser, I Thomas, The morphology of built-up landscapes in Wallonia: a classification using fractal indices [6] URL:http://www.math.univ-lille1.fr/ [7] N Salingaros, Connecting a fractal City Principles of Urban Structure, Techne Press, Amsterdam, 2005 [8] M Jevriс, J Kaleziс, Fractals in urban space Mongeometry, Proceedings of the Int conference of Geometry and Graphics, Belgrade, 2010, pp.186-195 [9] P Frankhauser, Approaching urban patterns by fractal geometry: From theory to application, URL: http://cddthema.univ-fcomte.fr/ [10] M Batty, Cities as Complex Systems: Scaling, Interactions, Networks, Dynamics and Urban Morphologies, URL: http://www.casa.ucl.ac.uk/ [11] M Jevriс et al, Application of fractal geometry in urban 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Applied Mechanics and Materials, Vol 627 (2014) 276-282 ... dimensions than in the case of compact urban areas Fractal analysis of the urban area of Podgorica was performed using the software tool Fractalyse Selected method of analysis was dilatation method... ways: through the analysis of spatial distribution of built areas and through the analysis of urban border Fractal theory basics Complexity theory and fractals can be applied to many natural and... a questionnaire Information on the fractal dimensions of urban border and urban area can be used as a parameter for decision making in the spatial development field For example, in the case of