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Fractal geometry is not just a chapter in mathematics, it also helps ordinary people see the same world in different ways. -- Benoit Mandelbrot

What is fractal? Fractal is a mode of "infinite cycle", which repeats itself on different scales. This nature of constant repetition is called self-similarity. Fractals are considered to be infinitely complex, which means that if you can constantly enlarge a fractal image (below), you can always see the same structure.

The term fractal, coined by Mandelbrot in 1975, comes from the Latin word "fractus", meaning "broken" or "broken".

Because of the "infinite complexity" of fractals, you may think that fractals are difficult to make, but it is a very simple process. To make a fractal, you just need to repeat the same process over and over again. In mathematical terms, mathematical fractal is an iterative (a form of recursion) equation.

The most famous fractal in mathematics is the Mandelbrot set, which comes from the complex set c. Mathematician Adrien Douady defined the following functions

To pay homage to Mandelbrot and named it the Mandelbrot Collection. When iterating from zonal 0, it does not diverge to infinity.

Another famous example of fractal is the Serpinski triangle, named after the Polish mathematician Serbinski, which consists of three steps:

Start with equilateral triangles

Divide the triangle into four equilateral triangles and remove the middle triangle

Repeat step 2 indefinitely for each remaining small triangle.

Newton fractal is the result of solving polynomial equation by Newton method. The above picture is the result of finding the three roots of the polynomial z ^ 3-1 by Newton's method.

Fractals in Nature

A computer model of the lung the human lung and cortical neurons are perfect examples of fractal organs.

The neurons in the female brain that send information-- called axons, the application of fractals in Harvard Lichtman Lab.

The applications of fractal fluid distribution techniques include chromatography, ion exchange, distillation and other applications that require plug flow characteristics to maximize performance.

Fractal Antenna Internal View Fractal Antenna Systems (FAS) has created fractal antennas, fractal metamaterials, fractal batteries to fractal circuit design, and their fractal technology innovation is endless. Fractal antennas made by FAS are entering mobile phones and other devices. Their fractal antennas have a clear technological advantage over their competitors in four key aspects:

The inherent broadband / multi-band means that only one antenna is needed, reducing the number of overall antennas, thus reducing costs.

Smaller, lighter and requires less equipment support.

Compared with traditional antennas, fewer components are needed to achieve the same performance.

Compared with the non-fractal antenna, the gain is higher.

Fractal geometry has been applied to many heat exchangers, including radiators, fuel cells, microreactors, distributors, collectors, shell-and-tube heat exchangers, spindles and silicon / germanium nanocomposites. Because the fractal structure has the advantage of temperature uniformity, engineers have done a lot of optimization research on the chip heat sink.

Fractal analysis of CT images is being used to understand and solve various lung diseases. The method of detecting lung abnormalities follows a simple algorithm based on two-dimensional and three-dimensional fractal dimensions. The algorithm is used to scan high-resolution CT images and classify normal and abnormal lungs according to their overall fractal dimension.

The schematic diagram of self-similar structure and self-similar fluctuation indicates that the study of heart rate regulation based on fractal model is under way. The image above depicts how heart rate regulation, similar to the fractal time process, fluctuates on different time scales that are statistically self-similar.

Fractals not only have artistic and mathematical value, but also have great applications in computer graphics, computer file compression system, Internet architecture and medicine. Because fractal is still a relatively new field in mathematics, there are still more applications undiscovered.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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