How to calculate the fractal dimension
I. Introduction. Fractal theory is an important branch of modern mathematics, which mainly studies irregular, complex, and self-similar geometric structures in nature. The fractal dimension is a key parameter in fractal theory to describe the complexity of the fractal structure. This article will introduce the calculation methods of fractal dimension in detail, including box dimension, similarity dimension, and information dimension.
Second, the calculation method of box dimension.
Box dimension is the most commonly used method for calculating fractal dimension. The basic idea is to cover the fractal structure to be measured with a square box with an edge length r, and as r decreases, the number of boxes required n(r) changes. The box dimension d is defined as:
d = lim(r→0) log n(r) / log(1/r)
In the actual calculation, the discretization method is usually adopted, a series of gradually decreasing r values are selected to calculate the corresponding n(r), and then the scatter plot of n(r) and 1 r is plotted on the double logarithmic coordinate graph, and the estimated value of the box dimension is obtained by linear regression.
3. Calculation method of similarity dimension.
The similarity dimension is suitable for fractal structures with self-similarity. Assuming that the fractal structure consists of n similar parts, each of which is r times the size of the original structure, then the similarity dimension d is defined as:
d = log n / log(1/r)
In the actual calculation, it is necessary to determine the self-similarity of the fractal structure, find out the values of n and r, and then substitute the formula to calculate the similarity dimension.
Fourth, the method of calculating the dimension of information.
Information dimension is a fractal dimension calculation method based on probability distributions. Assuming that the fractal structure can be divided into a series of non-overlapping subsets, each with a certain probability distribution p(i), then the information dimension d is defined as:
d = lim(ε→0) σp(i) *log p(i)] / log ε
where is the scale that divides the subset. In the actual calculation, it is necessary to determine the probability distribution of the fractal structure, and select a series of gradually decreasing values to calculate the corresponding information dimension.
5. Application examples.
Taking the Sierpinski triangle on a two-dimensional plane as an example, the calculation method of fractal dimension is introduced. The Sierpinski triangle is a typical self-similar fractal structure that can be iteratively generated. When calculating the box dimension, we can choose a series of decreasing r values, cover the sierpinski triangle with a square box, and count the number of boxes needed n(r). Then, the scatter plot of n(r) and 1 r was plotted on the double logarithmic coordinate plot, and the estimated value of the box dimension was obtained by linear regression. When calculating the similarity dimension, we can observe that the Sierpinski triangle consists of three similar small triangles, each of which is 1 2 times larger than the original triangle. Therefore, n=3, r=1 2, the value of the similar dimension can be obtained by substituting the formula. When calculating the dimension of information, we need to determine the probability distribution of the Sierpinski triangle. Assuming that each small triangle has the same probability, then p(i)=1 3. Then, a series of decreasing values are selected to calculate the corresponding information dimension.
VI. Conclusions. There are many methods for calculating fractal dimension, including box dimension, similarity dimension, and information dimension. In practical application, the appropriate calculation method should be selected according to the characteristics and requirements of the fractal structure. By calculating the fractal dimension, we can better understand and describe the complexity and self-similarity of fractal structures, which provides strong support for research in related fields.
The above is a detailed introduction to the calculation method of fractal dimension, including box dimension, similarity dimension and information dimension. It is hoped that these contents will be helpful to readers and provide valuable references for further research on fractal theory and application of fractal dimensionality.