This research was supported by the Hori Sciences and Arts Foundation, Nagoya (Japan).
1Understanding a city is a complex process, especially when it is huge and each era has left its mark on the urban heritage (Panerai, 1999). In the face of demographic growth, the concentration of the population in large centers and accelerated urbanization, cities continue to experience significant spatial growth, pushing back the limits of horizontal and vertical boundaries, spreading out in space by creating new networks with the immediate environment, modifying what already exists according to new needs and even creating new, non-existent forms.
2Architecture being a fact of culture, it is the representation of the city associated with creation and even innovation. Thus, the city, through its perpetual reinvention, contributes to enriching and diversifying the cultural dimension of urban heritage, and at the same time creates its own identity (Krier, 1980). Moreover, the cityscape is in constant evolution. Its identity changes over time, in each era. Through a historical urban analysis of the panorama that makes up the city, this study could identify the recurring identity, landscape elements and determine the items that influence the visual complexity of the city, past and present.
3The present paper approaches the city as a physical and spatial entity, which requires analysis and understanding before considering historical, sociological and psychological factors (Hillier, 1984). How to explain the recent transformations of urban fabrics and urban landscapes is the main question of this research.
4This research then deals with the evolution of urban fabrics and urban landscapes of three districts in three world capitals, namely Algiers, Tokyo and New York. It focuses on the analysis of the processes of transformation of urban fabrics and cityscapes using fractal analyses. According to Dupuy (2017), a fractal is an object deployed on an infinite range of scales with a unique fractal dimension. In physical systems, the range of scales is limited and the fractal dimension varies with the scale. Mandelbrot (1982) suggests that a turbulent flame has a multi-scale geometry. He proposes to consider a space as several scales. When a physical law such as the fall of bodies is independent of the scale at which it is considered, it is called “scale covariance” (Nottale, 1993, 2011). The term “scale” is then defined differently as: space, time, space-time, phase space, etc. (Dupuy, 2017). The study of the evolution of urban fabrics and cityscapes over time is essentially based on scale covariance and approaches scale as space-time.
5The corpus of this study is limited to a single district in each city. These districts have nearly the same size but not the same context. This limitation is primarily dictated by the size, time, and nature of the work.
6Due to the feasibility of the research and the availability of data, the idea of this study was to explore the spatial evolution of selected urban fabrics in different architectural and urban contexts in Algiers, Tokyo and New York. It also remains to be understood how the visual complexity of the selected urban landscapes has evolved. It should be noted that this paper studied all districts with their administrative boundaries. However, the district boundary measurement would correspond to a historical limit of stepwise urbanization (Figure 1).
7The district of Beb-El-Oued is located in the north of the capital Algiers. During the Ottoman period, the gate of Beb-El-Oued was one of the gates of Algiers, opening onto Oued M’kacel which flows from the heights of the Bouzareah district (Figure 1A). It was composed of a few houses and the Dey’s hospital. The district began to expand very rapidly during the French colonial period. It became a suburb and then an overcrowded neighborhood especially in the 1950s. Until 1962, Beb-El-Oued was the main popular European neighborhood of the city. After Algeria’s independence, it did not experience any real changes in the building, still presenting as the main popular neighborhood of the city (Chérif, 2017). It is possible to distinguish three main temporal phases for this study, that is to say: (A) the Ottoman phase (until 1830); (B) the colonial phase (until 1960) and finally the modern phase (from 1962).
8Japan is an archipelagic nation, and the role of trade and war is significant in explaining how Japanese modernization and industrialization were established, especially after World War II. Modernization began earlier than the American occupation (Ichikawa, 1994; Nakayama, 2019). Totman (2014) divides Japanese history into four phases: foraging (ca. 500B CE), agriculture (until 1890), imperial industrialism (1890-1945), and entrepreneurial industrialism (1945-2010). Chiyoda is a special district located in the heart of Tokyo; it was formed by the union of Kanda and Kōjimachi districts following the transformation of Tokyo City into Tokyo Metropolis (Figure 1B).
9Lower Manhattan is considered the town square of America since its discovery in 1524. The city developed very quickly in the 18th century. It remained in English hands until the end of the war in 1783. In 1820, New York became the most populous city in the United States. Throughout the 19th century, New York grew, modernized and industrialized. The city continued its dazzling development at the beginning of the 20th century, becoming a metropolis with worldwide influence. The flow of European immigrants made it not only an international capital but also a global cultural capital (Lockwood, 2014) (Figure 1C).
10Due to data availability and time limitations, the authors select three main time periods for all districts, i.e.: (A) before the 19th century, (B) between the 19th and 20th centuries, and (C) after the 20th century.
Figure 1: Situation of selected districts.
11The present study explored fractal characteristics by investigating: (A) the evolution of the morphological identity of urban fabrics, and (B) the evolution of urban landscapes. It focused on three districts: Beb-El-Oued (Algiers), Chiyoda-ku (Tokyo) and Lower Manhattan (New York). Each urban fabric and cityscape was analyzed according to three different time periods: (A) before the 19th century, (B) between the 19th and 20th centuries, and (C) after the 20th century.
12The first phase involved the extraction of urban morphological identities related to the fractal characteristics of three different urban fabrics and urban boundaries. The second phase focused on the fractality of edge textures of cityscape images using the box-counting method (Figure 2).
Figure 2: Research design.
13Two datasets were used in this study. The first dataset, used for the urban morphological analysis, consisted of nine maps. Three maps, representing each selected city, were collected from national planning agencies at different scales. They were then drawn using Archicad software, processed and rasterized into binary images in uncompressed format (*.tiff). All maps were saved at the same scale.
14Built-up areas were represented by black pixels and lacunas (empty spaces) by white pixels (Figure 3). The analyses were carried out using the "Fractalyse 2.4" software, developed in 2003 by the "City, Mobility, Territory" team of the ThéMA laboratory under the direction of Professor Pierre Frankhauser.
Figure 3: Urban fabrics and urban boundaries of each district.
15The second dataset, used for the cityscape analysis, consisted of 36 images. 12 images, representing each district, were collected from the web. Each time period was represented by four images (Figure 4). In order to analyze the fractal information of cityscapes, the images in the dataset were processed to the same resolution and size. They were then transformed into binary bitmap files (*.bmp) using the Sobel algorithm to detect their edges. The process was based on detecting white edges on a black background. The analyses were performed using "Benoit™ 1.3" fractal analysis software, developed in 2017 by TruSoft Int’l Inc.’s.
Figure 4: Evolution of urban landscapes.
16Fractal is a term derived from the Latin verb “frangere” and the adjective “fractus”, which means rough, irregular and fragmented (Mandelbrot, 1982).
17Fractal geometry could describe fractured shapes, which exhibit repeating patterns showing "scale invariance" or "self-similarity" at different fine magnifications. It could also measure shapes that are not entirely self-similar and that exhibit "scale dependence", which is our case (Nottale, 1989; 1992, Taylor, 1999a; 1999b; 2002; Gouyet, 1996). These descriptions are based on a parameter called fractal dimension. First used to study natural forms (Mandelbrot, 1975; 1982), fractal analyses have been applied to urban studies by linking urban hierarchy to fractal geometry. Other studies have explored the relationship between the fractal character of urban landscapes and environmental perception, such as preference, aesthetics, complexity, interest, etc.
18The fractal dimension in urban and architectural studies has been investigated by a number of authors in relation to different analytical objectives: (A) the spatial growth pattern of cities (Batty, 1994; Frankhauser, 1994; 2005), (B) the natural and urban skyline (Oku, 1990; Cooper, 2003; 2005; Heath, 2000), and (C) historic street plans (Mizuno, 1990; Rodin, 2000).
19One of the main features of any fractal object is its fractal dimension, denoted D. It measures its degree of irregularity and fragmentation across scales (Mandelbrot, 1975). The fractal dimension makes it possible to describe Euclidean structures, such as uniform surfaces, points or lines, and to characterize certain distributions for which traditional methods vary according to the reference surfaces. It also describes the hierarchical organization of spatial configurations. It allows the study and classification of urban fabrics through the study of their surfaces and urban boundaries and via several morphological indicators such as degrees of homogeneity, hierarchy, complexity, compactness, dendricity and roughness (Frankhauser, 1997; 1998; 2002; 2003; De-Keersmaecker, 2004; Badariotti, 2005; 2007; Tannier, 2011; Arrouf, 2015; Dupuy, 2017). It also allows the creation of a typology of urban fabrics from reference fractal models, such as Sierpinski’s carpet, Fournier’s dust and Teragon.
20Fractal morphological analysis is characterized by two descriptors, the fractal dimension, which measures the degree of homogeneity of the distribution of built-up spaces, and the boundary dendricity, which describes the smoothness of a boundary (Tannier, 2013). It offers a wide range of analysis methods such as correlation, dilation, and radial analyses as well as other measures that offer information about morphological identity. These methods are chosen based on their relevance and stability.
21The idea is to cover the structure in question with objects and geometric elements of a given size (e.g., squares of base length ε) and to determine the minimum number of such objects, necessary to cover the entire structure. For fractals constructed according to iteration, the fractal dimension "D" is defined by the following equation (Mandelbrot, 1982):
D=-logN/logr (1)
22According to Frankhauser (2003), it is possible to introduce a generalized fractal law N (ε) in the following form that includes a and c:
N(ε)= a × ε-D + c (2)
23Where a: is a constant called the factor or factor of form. It characterizes the general shape and size of the object, but is also related to deviations from the fractal law. Mathematically, it is the measure of the object.
24c: is a parameter that allows a better adjustment of the fractal curve by eliminating the deviations of the fractal law, often observed for distances at the scale of buildings.
25In addition to its wide range of analytical methods, fractal geometry offers a multitude of indicators that characterize urban fabrics and measure its morphological identity (Arrouf, 2015; Dupuy, 2017). The indicators used in this study are:
26- The degree of homogeneity of a surface (Dsurf): Fractal dimensions can be used to describe the pattern of space filling in the city development process. They depend on the measurement method and the extent (size, central location) of the area under study (Chen, 2017). By measuring the fractal correlation dimension D, which is a second-order multifractal dimension, we can obtain information about the homogeneity or heterogeneity of the distribution of built-up areas. A value close to two corresponds to a fairly homogeneous urban fabric, a value close to zero transcribes a strong heterogeneity in the distribution of built-up areas.
27- The degree of hierarchy: it provides information on the hierarchy of the distribution of built-up areas by measuring the fractal correlation dimension D and studying the scaling behavior. A value of D close to two with a slightly fluctuating scaling behavior corresponds to a weakly hierarchical urban fabric. A value of D close to zero and a fluctuating scaling behavior transcribes a strong hierarchy.
28- The degree of complexity (a): it gives information on the complexity of the urban fabric studied through the measurement of the form factor “a”. The higher the value, the more complex the urban fabric.
29- The degree of compactness (N): it provides information on the compactness of the urban fabric by the number of iterations resulting from the dilation analysis.
30- The degree of homogeneity of the border (Dbord): it provides information on the homogeneity or heterogeneity of the urban border by measuring the fractal dimension of correlation of the border “Dbord”.
31- The dendricity Index (δ): it gives information on the dendricity of the urban boundary and the sinuosity of the urban fabric. It represents the relationship between the built mass, the border and the way in which it is used. It is expressed by the index of dendricity “δ”:
δ=Dsurf/Dbord (3)
32- The roughness Index (Is): It gives information on the roughness of the urban fabric by means of the synthetic index of roughness “Is”. It is related to the Euclidean dimension and measures the difference to dimension 2 (Dsurf) and the difference to dimension 1 (Dbord). This index increases with the roughness and complexity of the urban fabric. For a Euclidean shape, homogeneous in surface and border, it is equal to zero (Is= 0). For a rougher and more complex form, it is between one and two (1 <Is <2). Its equation is as follows (Badariotti, 2005):
Is= (2 – Dsurf) – (1 – Dbord) (4)
33The fractal dimension of edge textures can be measured using different methods that depend on the objective of each research. However, all of these methods are based on a power law that generates scale invariant properties (Taylor, 2008). Box counting is the most commonly used mathematical method to estimate the fractal dimension of patterns demonstrating scale dependence (Kacha, 2013; 2015).
34In this study, a large grid was placed over each cityscape image. Each grid box was checked for the presence of white pixels. Then, the boxes containing white pixels were recorded. In the next step, a smaller scale grid was placed over the cityscape images and the same process was applied to look for possible white pixels (details) in the grid boxes. Finally, a comparison was made between the number of boxes with details in the first grid and the number of boxes with details in the second grid. This comparison was done by plotting a log-log plot (Richardson plot) for each grid size. By repeating this process on several grids of different scales, a log-log linear correlation between the number of boxes counted and the associated grid is obtained. This paper used the Benoit 1.3© software which defines the fractal dimension as the exponent D:
35Where N(d) is the number of boxes of linear size d, necessary to cover a dataset of points distributed in a two-dimensional plane.
36The degree of homogeneity of the built-up area is measured by the fractal dimension. It varies between zero and two. The more homogeneous the urban fabric, the more its value tends towards two. The more heterogeneous it is, the more its value tends towards zero.
37The urban pattern consists of a large number of individual buildings or groups of buildings and the distances between them vary considerably. Despite its highly heterogeneous appearance, it could follow a well-defined principle of spatial organization because it has fractal characteristics. The notion of "homogeneity" or "heterogeneity" then loses all meaning. The application of fractal geometry thus allows to find more coherent interpretations using fractal dimension.
38Granularity is used to refer to the way in which a city’s property is divided, particularly in terms of the lot sizes into which city blocks are divided. Older urban patterns are generally very fine-grained. While newer urban patterns tend to typically be very coarse-grained.
39In the case of the Beb-El-Oued district, the fractal dimensions vary between 1.737 and 1.844 respectively. These values indicate increasingly homogeneous morphologies. The fractal dimensions of the different urban fabrics across the different periods were positively correlated. The most correlated urban fabrics were those of 1912 and 1960, with an (r) value equal to 0.984. Those of 1960 and 2004 have the lowest correlation coefficient (r= 0.490). This confirms, first, the similarity between the morphological identities of the two urban fabrics (1912 and 1960), which belong to the same historical period (French colonization) of evolution of urban fabrics in the city of Algiers. Second, the dissimilarity between the morphological identities in 1960 and 2004, which represents a significant historical change in the evolution of urban fabrics in Algiers (post-independence population growth) (Figure 5a, 6a).
40In the case of New York City and especially the southern part of Lower Manhattan, fractal dimension values range from 1.739 to 1.807. These values indicate increasingly homogeneous morphologies. The urban fabric has nearly identical homogeneity values, the latter due to a rigorous planning of urban evolution. The surface fractal dimensions of the urban fabrics across the different periods are positively correlated. The most correlated urban fabrics are those of 1782 and 1934, with an (r) value equal to 0.758. The urban fabrics of 1782 and 2016 have the lowest correlation coefficient (r= 0.338). This confirms, on the one hand, the analogy between the morphological logic of the surfaces of the urban fabrics of 1782 and 2016, which represents the same historical period of the evolution of lower Manhattan. On the other hand, the divergence between the morphological logic of urban fabric surfaces in 1782 and 2016 presents a significant historical gap (Figure 5b, 6a).
41In the case of Chiyoda-Ku urban fabrics, the values of the average fractal correlation dimensions ranged from 1.845 to 1.836. The evolution of the urban fabrics showed almost identical and stable homogeneity values. The fractal dimensions of the urban fabrics across the different time periods were positively correlated. The most correlated urban fabrics were those of 1857 and 1929, with an (r) value equal to 0.887. The 1857 and 2016 urban fabrics, on the other hand, had the lowest correlation coefficient (r= 0.447) (Figure 5c, 6a).
42The high fractal dimensions are likely related to the type of data used in all cases of this study. These data obviously do not contain information about courtyards at the building block scale.
43Notice that for these three categories, p-values are higher than 0.0001.
Figure 5: The scaling behaviors of the analyzed urban fabrics.
44The scaling behavior curves allow us to understand the structure of urban fabrics and to measure the homogeneity of their distribution. Thus, they provide the means to differentiate the temporal evolution of urban fabrics.
45Figure 5 shows the scaling behaviors related to the correlation analyses of the evolution of urban fabrics. In Algiers, the curves of the scaling behaviors of periods A and B follow the same direction, while that of period C is more fluctuating. This means that periods A and B are characterized by an overly hierarchical structure compared to the following period, which presents a non-hierarchical morphology. These fluctuations correspond to lacunas that hierarchize the urban fabric (Figure 5.a). In New York, the most fluctuating curve is that of the second period (Figure 5.b). In contrast, in Tokyo, all the scale behavior curves followed the same direction with small fluctuations, reflecting little prioritization (Figure 5.c).
46The first and second periods were characterized by much more fluctuating (more hierarchical) curves compared to the last period, which was characterized by more compact and homogeneous urban fabrics in the three districts (Figure 5.d, 5.e, 5.f).
47The form factor “a” reflects the degree of complexity of urban fabrics. For the analyzed urban fabrics, the values of “a” vary between 0.947 and 2.777. The least complex urban fabrics are those of the last period in the three cities, with values of 1.60 and 1.67, respectively. The urban fabric of Lower Manhattan in its primary period had a higher value of “a” equal to 2.777, signifying very high complexity (Figure 6.b).
48The dilation analysis provides information on the degree of compactness of an urban fabric via the number of iterations required for its total dilation. Obtaining a single cluster for the Chiyoda-ku urban fabrics required between 35 and 45 iterations, revealing high compactness. The number of iterations for Beb-El-Oued urban fabrics showed their looseness compared to Chiyoda-ku and lower Manhattan (50 ~ 100 iterations). This could be explained by the large lacunas they contain (Figure 6.c).
49The morphology of urban boundaries was measured using two fractal descriptors: the fractal dimension of correlation and the dendricity index. The latter showed that the analyzed urban fabrics, whose boundaries correspond to stages of urban morphogenesis, had an inhomogeneous morphology (the edge varies between 1.078 and 1.436) (Figure 2).
50The most homogeneous border was that of Chiyoda-ku before the 19th century (Figure 6.d). In modern era, the three districts had the most dendritic urban fabrics (values close to two) (Figure 6.e). The indexes of the analyzed urban fabrics describe a sinuous situation that refers to a logic of the Teragon “1 <δ <2”.
51Using the synthetic roughness indicator (Is) based on two fractal dimensions of correlation, one on the surface and the other on the border, it is possible to measure the roughness of the urban fabric according to different eras. The roughness index of all urban fabrics was higher before the 19th century (Figure 6.f).
Figure 6: Fractal features of the analyzed districts.
52The results related to fractal dimension Db showed higher values for modern urban landscapes in general. Prior to the 19th century, the fractal dimensions of Beb-El-Oued cityscape images varied between 1.572 and 1.708 respectively. Lower Manhattan cityscapes ranged from 1.622 to 1.787. And Chiyoda cityscapes were between 1.536 and 1.736. Between the nineteenth and twentieth centuries, the fractal dimensions of Beb-El-Oued cityscape images varied between 1.673 and 1.751 respectively. Lower Manhattan cityscapes ranged from 1.642 to 1.799. And Chiyoda cityscapes were between 1.671 and 1.781. However, after the 20th century, the fractal dimensions of Beb-El-Oued cityscape images varied between 1.730 and 1.781 respectively. Lower Manhattan cityscapes ranged from 1.816 to 1.846. And Chiyoda cityscapes were between 1.839 and 1.861.
53In summary, American and Japanese urban landscapes are more complex compared to Algerian urban landscapes. This is due to the important number of details and elements (such as signage, vegetation, …etc.), which characterizes Japanese and American urban landscapes (Figure 7).
Figure 7: Fractal dimensions of urban landscape images over time.
54Descriptive statistical analysis of the collected data helped in checking the variables and understanding the characteristics of the results. Given a dataset of N items, where N= 63, authors generated the 9x7 fractal indicators to measure similar distances within three districts according to distance criterion. SPSS 21.0 software was used to analyze the clustering of all the results. Hierarchical clustering analysis based on Ward method helped in defining five groups within the dataset. These clusters emerged from two main clusters that have the biggest distances. The results revealed that, prior to the 19th century, the characteristics of both urban fabrics and cityscape images of the three districts cluster together, while the ones of the other two periods converge. This might reflect that the urban development of the three cities seems to follow similar logics. For urban fabrics, population growth helped in occupying more ground and making it more homogeneous (the relationship between the morphologies of built-up areas, open spaces and urban boundaries). On the other hand, the evolution of urban landscapes experienced an increase in the degree of complexity. This might be due to the increasing amount of street advertising signs, colors, textures, …etc. (Figure 8).
Figure 8: Hierarchical clustering of the analyzed datasets.
55This paper aims to categorize the evolution of the urban fabrics and urban landscapes in three districts (Beb-El-Oued, Chiyoda-ku and Lower Manhattan) from three different cities (Algiers, Tokyo and New York), according to their fractal characteristics. It contributes to a better understanding of the analysis of cities over time by:
56Fractal morphological analyses of urban fabrics were conducted using two main descriptors which are: the fractal dimension of built-up areas and the dendricity of boundaries. The results showed that all urban fabrics belonging to the same city have the same urban morphological logic. The comparison revealed that each urban fabric had increasingly higher degrees of homogeneity and boundary dendricity that correspond to Teragon’s logic. The results also confirm that the morphological logic of all urban fabrics is generally dependent on building logic and lifestyle. When comparing the characteristics of the three districts, it was found that their morphological identities were concordant but distinguishable through time. This is mainly due to the similarity of urbanization strategies due to population growth.
57A fractal analysis of urban landscapes was performed using the box-counting method. The results clearly indicated that complexity increases over time, especially in Tokyo and New York. This is due to the increasing number of details such as signage, architectural elements, skyline complexity, texture, etc. Some authors have already addressed the issue of excessive visual complexity in the urban landscape, which can cause sensory overload in pedestrians (Cavalcante, 2014).
58It is important to mention the limitations of this research. First, this study was only able to cover the evolution of three districts. This is due to time limitations and lack of resources such as historical maps and images of older urban landscapes. Second, this study only covered the intrinsic physical characteristics of the analyzed districts without taking into account extrinsic features such as social or cultural factors.
59Authors believe that studying other districts within the same cities would open the boundaries of this research to new perspectives. By taking into account other extrinsic social and psychological variables, it could help uncover hidden aspects such as the meaning of changing urban fabrics and urban landscape identities.
