Two Sets of Simple Formulae to Estimating Fractal Dimension of Irregular Boundaries

Irregular boundary lines can be characterized by fractal dimension, which provides important information for spatial analysis of complex geographical phenomena such as cities. However, it is difficult to calculate fractal dimension of boundaries systematically when image data is limited. An approximation estimation formulae of boundary dimension based on square is widely applied in urban and ecological studies. However, the boundary dimension is sometimes overestimated. This paper is devoted to developing a series of practicable formulae for boundary dimension estimation using ideas from fractals. A number of regular figures are employed as reference shapes, from which the corresponding geometric measure relations are constructed; from these measure relations, two sets of fractal dimension estimation formulae are derived for describing fractal-like boundaries. Correspondingly, a group of shape indexes can be defined. A finding is that different formulae have different merits and spheres of application, and the second set of boundary dimensions is a function of the shape indexes. Under condition of data shortage, these formulae can be utilized to estimate boundary dimension values rapidly. Moreover, the relationships between boundary dimension and shape indexes are instructive to understand the association and differences between characteristic scales and scaling. The formulae may be useful for the pre-fractal studies in geography, geomorphology, ecology, landscape science, and especially, urban science.Comment: 28 pages, 2 figures, 9 table

Fractal dimension evolution and spatial replacement dynamics of urban growth

This paper presents a new perspective of looking at the relation between fractals and chaos by means of cities. Especially, a principle of space filling and spatial replacement is proposed to explain the fractal dimension of urban form. The fractal dimension evolution of urban growth can be empirically modeled with Boltzmann's equation. For the normalized data, Boltzmann's equation is equivalent to the logistic function. The logistic equation can be transformed into the well-known 1-dimensional logistic map, which is based on a 2-dimensional map suggesting spatial replacement dynamics of city development. The 2-dimensional recurrence relations can be employed to generate the nonlinear dynamical behaviors such as bifurcation and chaos. A discovery is made that, for the fractal dimension growth following the logistic curve, the normalized dimension value is the ratio of space filling. If the rate of spatial replacement (urban growth) is too high, the periodic oscillations and chaos will arise, and the city system will fall into disorder. The spatial replacement dynamics can be extended to general replacement dynamics, and bifurcation and chaos seem to be related with some kind of replacement process.Comment: 17 pages, 5 figures, 2 table

Dynamics of Urban Growth: Modeling the Fractal Dimension of the City of Irbid , Jordan

The emergence of fractal geometry engendered a shift from the common view that sees cities as simple, ordered and structured, expressed by smooth lines and shapes, towards a view that cities are complex organisms evolving according to local rules and conditions. The main objectives of this study are first to prove the fractality of the geometry of Irbid, Jordan as a case study and second to provide mathematical procedure and tool (fractal geometry) for urban analysis. The research simulates the growth of Irbid across many scales and times. The main hypothesis asserted here is that Irbid is growing systematically and factually, even though its growth demographically and geometrically hasn’t been strictly regular .This supports the argument that cities globally or locally would produce fractal growth at every level of their hierarchy

Evolution and Dynamics of Fractal Growth of the Urban Green Spaces in Seville (Spain)

Like urban growth, the increase in the area of urban green spaces can be described using fractal design, a measure, of the dynamic evolution of public space of leisure and recreation of the citizens, associated with the growth form of the city. Throughout the history, the city of Seville has been a good example of sustainability and eco-design in spite of the enormous physical transformations carried out both in the city center and in the periphery. In essence, the evolutionary process of the city is both technical and social giving rise to a landscape that is transformed and remains. In this work, from the evolution of green area per inhabitant from 1842 to 2016, a prediction model capable of characterizing the changes of fractal dimension associated with the growth of Seville is proposed. This prediction model can be used to estimate the growth rate of the fractal dimension, and therefore to reveal the spatiotemporal process and pattern of Seville growth. Especially, the model lays a foundation for researching the correlation between urban form and urbanization and for developing the theory of spatial replacement dynamics