Definition of fractal topography to essential understanding of scale-invariance

Yi Jin,Ying Wu,Mengyu Zhao,Jienan Pan,Hui Li

doi:10.1038/srep46672

Abstract

Fractal behavior is scale-invariant and widely characterized by fractal dimension. However, the cor-respondence between them is that fractal behavior uniquely determines a fractal dimension while a fractal dimension can be related to many possible fractal behaviors. Therefore, fractal behavior is independent of the fractal generator and its geometries, spatial pattern, and statistical properties in addition to scale. To mathematically describe fractal behavior, we propose a novel concept of fractal topography defined by two scale-invariant parameters, scaling lacunarity (P) and scaling coverage (F). The scaling lacunarity is defined as the scale ratio between two successive fractal generators, whereas the scaling coverage is defined as the number ratio between them. Consequently, a strictly scale-invariant definition for self-similar fractals can be derived as D = log F /log P. To reflect the direction-dependence of fractal behaviors, we introduce another parameter Hxy, a general Hurst exponent, which is analytically expressed by Hxy = log Px/log Py where Px and Py are the scaling lacunarities in the x and y directions, respectively. Thus, a unified definition of fractal dimension is proposed for arbitrary self-similar and self-affine fractals by averaging the fractal dimensions of all directions in a d-dimensional space, which . Our definitions provide a theoretical, mechanistic basis for understanding the essentials of the scale-invariant property that reduces the complexity of modeling fractals.

Highlights

Fractal behavior is scale-invariant and widely characterized by fractal dimension
Fractals were originally introduced by Mandelbrot[1] to describe the fractal behaviors of similar geometries in disordered and irregular objects such as the natural coastlines[1,2,3], phenomena in natural and artificial materials[4,5,6], porous media[7,8,9,10], biological structures[11], rough surfaces[12,13,14,15], as well as novel application of factuality to complex networks and brain systems[16,17,18]
The first one is what parameters determine fractal behavior? For convenience, we call what defines fractal behavior fractal topography because of the scale and size background, and that natural structures are often hierarchical, for example, with a sponge-like topology[28]. It suffices to exhibit structure in a fractal object using a variant of the Sierpinski gasket

Summary

Methods and Discussion

Mechanistic basis for understanding the property of scale-invariance, we must mathematically define it per the key requirements we have previously laid out. There are two scale-invariant parameters that determine the fractal behavior of fractal generator G: the ratio of the sizes of two successive scaling objects (li/li+1) and the ratio of their number (N(G(li+1))/N(G(li))). To mathematically define fractal topography, we first propose two notations: Scaling lacunarity (P): The unit ratio between two successive fractal generators G(li) and G(li+1), with the characteristic dimensions li and li+1 in a fractal object, as. Topography information and defines fractal dimension in a strictly scale-invariant manner, other than what is implied in the number-size relationship. (3) scaling lacunarity and scaling coverage are real scale-invariant dimensionless parameters different from the scale l and number N(G(l)), and they are independent of the fractal generator G(l0) and its geometries, spatial patterns, and statistical properties.

Fractals Koch curve Sierpinski carpet Sierpinski gasket Menger Sponge

Conclusion

Py x

Additional Information