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Figure 6. An example of the computer-generated fractals (black and white) viewed by the subjects for the eye-tracking results shown in Table 2. The red lines are the eye trajectories.

Figure 8. Visual preference for computer-generated fractals: The vertical axis in each panel corresponds to the percentage of trails for which patterns of a given D value were chosen as a function of fractal dimension (D). Each of the four different panels uses a different fractal configuration to investigate this visual preference. The fractal images are shown as insets in each panel. The main effect of fractal dimension (D) on visual preference was significant for all four types of fractal images: F8,19 = 22.16, p F8,19 = 38.01, p F8,19 = 15.68, p F8,19 = 1.54, p

In particular, the blood circulatory system distributes the blood in the whole body starting from an organ of large size and reaching individual cells. To achieve this, the distribution network starts from a large artery leaving the heart and branches out in many levels. At each level, the number of branches increases while their diameter decreases in order to reach and feed all cells. This is a mathematical property of fractals in 3D which are constructed as space filling objects .

Abstract: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.Keywords: fractal; fractal dimension; pre-fractal; multifractals; scaling range; entropy; spatial correlation; fractal cities

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 are limited. An approximation estimation formula of boundary dimension based on square is widely applied in urban and ecological studies. But 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 prefractal studies in geography, geomorphology, ecology, landscape science, and especially, urban science.

This formula is familiar to many scholars who like geographical and ecological fractals because it was once derived by Olsen et al.  in another way. If the shape of a natural system is similar to a square, or a system has four growing directions, the boundary dimension of the system shape can be estimated by equation (10).

Several points of explanations should be provided for the exceptional values in the fractal dimension estimation results. In theory, the boundary dimension defined in a 2-dimensional embedding space is supposed to come between 0 and 2 [3, 9, 12, 13, 35, 36]. The reasonable values vary from 1 to 1.5. However, the following causes often result in overestimation of boundary dimension. First, the boundary dimension calculation is based on the geometric measure relation deriving from regular real fractals in the mathematical world. A real fractal has no scaling range, or, in other words, the scaling range of a real fractal is infinite. Applying the fractal measure relations proceeding from regular real fractals to the random prefractals gives rise to significant bias in many cases . Second, if the image resolution is high enough, the length of the boundary line may be very long, but the area within the boundary curve is certain. This phenomenon can be illustrated by a regular fractal termed Koch lake (see Appendix). The Koch lake can be treated as models of lakes, islands, urban region, and so on. Third, compared with the second set of approximate formulae, the first set of approximate formulae enlarges the ratio of the perimeter logarithm to the area logarithm of a shape relatively. For example, for the formulae based on square, in equation (10), the circumference is reduced to a quarter of the original length, while in equation (30), the area is enlarged to 16 times the original one. As a result, the boundary dimension value of a shape based on equation (10) is significantly greater than the value based on equation (30). Generally speaking, the estimated value of a boundary dimension is not less than 1. However, if a figure is near a Euclidean shape, the fractal dimension estimation result may be an outlier and less than 1 because the formulae are designed for prefractals rather than for Euclidean shapes.

The aforementioned results are based on power-law relations, and a power law represents a geometric measure relation and reflects a proportional relationship. A power function has two parameters: one is the proportionality coefficient and the other is the power exponent. In the framework of Euclidean geometry, the power exponent is always a known constant and bears little useful information. Thus, we can construct various shape indexes based on proportionality coefficients. A proportional constant is always a dimensionless parameter reflecting a ratio of one measure to another measure. On the contrary, in the framework of fractal geometry, the proportionality coefficient bears little information, but the power exponent is unknown parameter and possesses spatial information. A simple system has characteristic scale and can be described with the mathematical method based on Euclidean geometry, while a complex system has no characteristic scale and cannot be effectively described by conventional mathematical methods. In this case, it is necessary to replace the characteristic scale concept with scaling idea. The power exponent is known as scaling exponent. Fractal geometry is a powerful tool for scaling analysis of complex systems, and the fractal dimension is an important scaling exponent. Based on the notion of fractals, various fractal indexes can be defined to characterize fractal-like phenomena.

This is one of the characteristics of fractals: infinite filling in a finite space. In this case, regardless of the formula in Table 1, the fractal dimension is a variable dependent on m instead of a constant. For example, based on square, the fractal indexes and shape index are as follows: 041b061a72