The Solutions to the Uncertainty Problem of Urban Fractal Dimension Calculation

Chen, Yanguang

doi:10.3390/e21050453

Cited by 14 publications

(8 citation statements)

References 66 publications

(136 reference statements)

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“…Since the length of coastline cannot be effectively measured, Mandelbrot put forward the concept of fractal dimension [ 5 ]. Fractal dimension can be defined on the basis of entropy and correlation function [ 3 , 4 , 6 ]. It is actually the invariant quantity in scaling transform and thus a parameter indicating symmetry.…”

Section: Introductionmentioning

confidence: 99%

Fractal Modeling and Fractal Dimension Description of Urban Morphology

Chen

2020

Entropy

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The conventional mathematical methods are based on characteristic length, while urban form has no characteristic length in many aspects. Urban area is a scale-dependence measure, which indicates the scale-free distribution of urban patterns. Thus, the urban description based on characteristic lengths should be replaced by urban characterization based on scaling. Fractal geometry is one powerful tool for the scaling analysis of cities. Fractal parameters can be defined by entropy and correlation functions. However, the question of how to understand city fractals is still pending. By means of logic deduction and ideas from fractal theory, this paper is devoted to discussing fractals and fractal dimensions of urban landscape. The main points of this work are as follows. Firstly, urban form can be treated as pre-fractals rather than real fractals, and fractal properties of cities are only valid within certain scaling ranges. Secondly, the topological dimension of city fractals based on the urban area is 0; thus, the minimum fractal dimension value of fractal cities is equal to or greater than 0. Thirdly, the fractal dimension of urban form is used to substitute the urban area, and it is better to define city fractals in a two-dimensional embedding space; thus, the maximum fractal dimension value of urban form is 2. A conclusion can be reached that urban form can be explored as fractals within certain ranges of scales and fractal geometry can be applied to the spatial analysis of the scale-free aspects of urban morphology.

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Section: Introductionmentioning

confidence: 99%

Fractal Modeling and Fractal Dimension Description of Urban Morphology

Chen

2020

Entropy

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“…Different combinations of fractal elements result in different fractal dimensions for the terrain. The spatial analysis of a site can be conducted by obtaining comparable fractal dimension values [43]. The current study compared the fractal dimension of the restored terrain of Sparkling Lake generated by combining fractal elements with the surrounding landscape terrain to determine whether the restored Sparkling Lake terrain is similar to the surrounding terrain and whether the combination form of fractal elements under this mode is appropriate.…”

Section: Combination Of Fractal Elementsmentioning

confidence: 99%

“…Fractal dimension characteristics can also be seen in artificial landscapes, such as terraces [29,30], artificial land use [31], building environments [32][33][34], traffic networks [35,36] and others. The practice of fractal geometry in landscape restoration of quarries, open-pit mines, and other damaged environments [37,38], as well as river shoreline restoration [39], urban skyline construction [40,41], urban spatial organization structure [42][43][44][45][46][47], and landscape restoration [11], has confirmed that the balance, congruence, and symmetry of fractal images enhance pleasure in perception and cognition. Therefore, any change in fractal dimensions may affect people's visual judgment and preference for a landscape [48][49][50].…”

Section: Introductionmentioning

confidence: 99%

Fractal Features in Terrain Restoration of Jiuzhai Valley, a World Natural Heritage Site in China

Zou,

Du,

Song

2023

Fractal Fract

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Jiuzhai Valley, a World Natural Heritage Site, was significantly damaged by an earthquake in 2017. However, case studies on the restoration of World Natural Heritage sites are lacking. This study aimed to use the box-counting method to analyze fractal characteristics of the terrain in Shuzheng Valley. Research data were used to conduct artificial intervention restoration of the earthquake-damaged terrain. Our results showed that (i) the travertine terrain shows self-similarity at different scales. The fractal dimension was related to terrain complexity: the more complex the terrain, the higher the fractal-dimension value; (ii) a combined form of fractal generator elements at the same scale was related to terrain complexity—differences in the spatial combination of the fractal generator elements can be compared based on fractal dimension; and (iii) the newly restored dam terrain also showed fractal characteristics whose spatial combination form was similar to that of the surrounding terrain. The complexity of the terrain’s fractal element combination may be related to the influence of surrounding environmental factors and the different ecological functional requirements. This study provides basic data for the near natural restoration of the Sparkling Lake travertine terrain after an earthquake and proposes new concepts and strategies for restoring World Natural Heritage Site terrains.

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“…Since real fractals often have more than one scaling process, a single fractal dimension is insufficient to characterize them. A set of comparable parameters is required, which is provided by the multifractal spectrum [ 16 ].…”

Section: Introductionmentioning

confidence: 99%

“…The spatio-temporal evolution of urban morphology is a critical focus area within geography, but existing multifractal research is not extensive enough. Urban multifractal research can be classified into three spaces: real space, order space, and phase space [ 16 ]. Real space corresponds to the spatial subdivision of urban land [ 17 ], transportation [ 18 , 19 , 20 ], population [ 21 , 22 , 23 , 24 ], and other elements [ 25 ]; order space refers to the hierarchical subdivision of urban system [ 26 , 27 ]; and phase space corresponds to dynamic processes.…”

Section: Introductionmentioning

confidence: 99%

Characterizing the Spatio-Temporal Variations of Urban Growth with Multifractal Spectra

Chen

2023

Entropy

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Urban morphology exhibits fractal characteristics, which can be described by multifractal scaling. Multifractal parameters under positive moment orders primarily capture information about central areas characterized by relatively stable growth, while those under negative moment orders mainly reflect information about marginal areas that experience more active growth. However, effectively utilizing multifractal spectra to uncover the spatio-temporal variations of urban growth remains a challenge. To addresses this issue, this paper proposes a multifractal measurement by combining theoretical principles and empirical analysis. To capture the difference between growth stability in central areas and growth activity in marginal areas, an index based on generalized correlation dimension Dq is defined. This index takes the growth rate of Dq at extreme negative moment order as the numerator and that at extreme positive moment order as the denominator. During the stable stage of urban growth, the index demonstrates a consistent pattern over time, while during the active stage, the index may exhibit abnormal fluctuations or even jumps. This indicates that the index can reveal spatio-temporal information about urban evolution that cannot be directly observed through multifractal spectra alone. By integrating this index with multifractal spectra, we can more comprehensively characterize the evolutionary characteristics of urban spatial structure.

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The Solutions to the Uncertainty Problem of Urban Fractal Dimension Calculation

Cited by 14 publications

References 66 publications

Fractal Modeling and Fractal Dimension Description of Urban Morphology

Fractal Modeling and Fractal Dimension Description of Urban Morphology

Fractal Features in Terrain Restoration of Jiuzhai Valley, a World Natural Heritage Site in China

Characterizing the Spatio-Temporal Variations of Urban Growth with Multifractal Spectra

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