Fractal theory is a very popular and active new theory and discipline. The concept of fractal was first put forward by Benoit B. Mandelbrot, an American mathematician. The mathematical basis of fractal theory is fractal geometry, that is, fractal information, fractal design and fractal art are all derived from fractal geometry.
The most basic feature of fractal theory is to describe and study objective things from the perspective of fractal dimension and mathematical methods, that is, to describe and study objective things with mathematical tools of fractal dimension. It jumps out of the traditional barriers of one-dimensional line, two-dimensional surface, three-dimensional space-time and even four-dimensional space-time, and is closer to the description of the real attributes and States of complex systems, which is more in line with the diversity and complexity of objective things.
Chinese name
Fractal theory
Foreign name
Fractal theory
presenter
Benoit Mandelbrot te
Applied discipline
Fractal information, fractal design, fractal art
area of application
Fractal information, fractal design, fractal art
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Significance of fractal function of main fractal model
definition
1967, Mandelbrot published an article entitled "How long is the British coastline? In the authoritative American magazine Science. Statistical self-similarity and fractal dimension (how long is the coast of Britain? Statistical self-similarity and fractal dimension). As a curve, the coastline is characterized by extremely irregular and uneven, showing extremely tortuous and complicated changes. We can't distinguish this part of the coast from that part of the coast in shape and structure. This almost equal degree of irregularity and complexity shows that the coastline is self-similar, that is, the local morphology and the overall state are similar. Without buildings or other things as a reference, the coastline of 10 km photographed in the air will look very similar to the two enlarged coastline photos of 10 km. In fact, forms with self-similarity widely exist in nature, such as continuous mountains and rivers, floating clouds, cracks in rocks, Brownian motion of particles, tree crowns, cauliflower and cerebral cortex ............................... ……Mandelbrot called these forms which are similar to the whole to some extent fractal. 1975, he founded fractal geometry. On this basis, the science of studying fractal properties and its application is formed, which is called fractal theory.
principle
Linear fractal is also called self-similar fractal. Self-similarity principle and iterative generation principle are important principles of fractal theory. It means that fractal is invariant under the usual geometric transformation, that is, scale independence. Self-similarity is based on symmetry of different scales, that is, recursion. The self-similarity in fractal can be the same or similar in statistical sense. The standard self-similar fractal is a mathematical abstraction, which iteratively generates infinitely fine structures, such as Koch snowflakes and Scherbinsky carpets. There are only a few such regular fractals, most of which are statistical random fractals
The classification of fractal is further introduced here. According to the degree of self-similarity, fractal can be divided into regular fractal and random fractal. Regular fractal refers to fractal with strict self-similarity, which can be described by simple mathematical models, such as three-point Cantor set and Koch curve. Random fractal refers to fractal with statistical self-similarity, such as winding coastline and floating clouds.
Fractal model
Cantor triple diversity
1883, the German mathematician G.Cantor proposed the now widely known three-point Cantor set. Sancantor set is easy to construct, but it shows many typical fractal characteristics. It is constructed by the unit interval, and then by the process of continuously removing some sub-intervals from this interval (as shown in the right picture). The concrete construction process is as follows: Step 1, divide the closed interval [0, 1] into three segments, and remove the middle 1/3 segment, leaving only two closed intervals [0, 1/3] and [2/3, 1]. Step 2, divide the remaining two closed intervals into three segments evenly, and also remove the middle segment, so there are four closed intervals: [0, 1/9], [2/9, 1/3], [2/3, 7/9] and [8/9, 6544]. Third, the middle 1/3 paragraph between each cell is deleted repeatedly. With this continuous division, the last remaining part between communities formed a three-point Cantor set. The Hausdorff dimension of the three-point Cantor set is 0.6309.
Construction process of three-point cantor set
Koch curve
1904, the Swedish mathematician robert koch constructed the "Koch curve" geometry. Koch curve is greater than one dimension, and its length is infinite, but less than two dimensions. It is a typical fractal like a three-point Cantor set. According to the different fractal time, there are many kinds of Koch curves, such as cubic Koch curve and quartic Koch curve. Taking cubic Koch curve as an example, this paper introduces the construction method of Koch curve, and others can be analogized. The construction process of cubic Koch curve is mainly divided into three steps: the first step is to give an initial figure-a line segment; Step 2, fold the middle of this line segment 1/3 outward; Step 3: Fold 1/3 in the middle of each line segment outward according to the method of step 2. If this continues, the Koch curve will eventually be constructed. The construction process of its legend is as shown on the right (a graph that has been iterated for 5 times).
Generation process of Koch curve
Joulia set
Julia set is obtained by French mathematicians Gaston Julia and Pierre Faton after developing the basic theory of complex variable iteration. Julia set is also a typical fractal, but its expression is quite complicated and it is difficult to describe it by classical mathematical methods. Julia set consists of a complex function.
Where c is a constant.
Joulia set
Although this complex variable function looks simple, it can generate very complex fractal graphics.
The picture on the right is a graph generated by Julia set. Because c can be any value, when c takes different values, the generated graphs are different.
Fractal dimension effect
Fractal dimension, also called fractal dimension or fractal dimension, as a quantitative representation and basic parameter of fractal, is another important principle of fractal theory. For a long time, people are used to defining points as zero dimension, straight lines as one dimension, planes as two dimensions and spaces as three dimensions. Einstein introduced the time dimension into the theory of relativity, thus forming a four-dimensional space-time. Considering a problem in many aspects, we can build a high-dimensional space, but all of them are integer dimensions. Mathematically, the geometric objects in Euclidean space are constantly stretched, compressed and twisted, and the dimension remains unchanged, which is the topological dimension. However, this traditional view of dimension has been challenged. Mandelbrot once described the dimension of a rope ball: observing the rope ball from a distance can be regarded as a point (zero dimension); Seen from a close distance, it is full of a spherical space (three dimensions); Closer, you will see the rope (one dimension); Microscopically, the rope becomes a three-dimensional column, which can be decomposed into one-dimensional fibers. So, what about the intermediate state between these observation points?
Obviously, there is no exact boundary between a rope ball and a three-dimensional object. Mathematician Hausdorff put forward the concept of continuous space in 19 19, that is, the spatial dimension can change continuously, and it can be a natural number, a positive rational number or a positive irrational number, which is called Hausdorff dimension. Written as Df, the general expression is: k = l df, also known as k = (1/l) (-df). Take the natural logarithm, and get Df=lnK/lnL, where L is the multiple of an object unfolding in each independent direction and K is the multiple of the original object. Df is generally not necessarily a natural number. Therefore, Mandelbrot also defines fractal as a set with Hausdorff dimension greater than or equal to topological dimension. Why can't the British coastline be accurately measured? Because Euclid's one-dimensional measure is inconsistent with the dimension of coastline. According to Mandelbrot's calculation, the dimension of British coastline is 1.26. Using fractal dimension, the length of coastline can be determined.
meaning
The "fractal fever" that began in the early 1980s lasted for a long time. As a new concept and method, fractal is being applied in many fields. John wheeler, an American physicist, said: Whoever is not familiar with fractals in the future cannot be called scientific literacy. This shows the importance of fractal. Professor Zhou Haizhong, a famous scholar in China, believes that fractal geometry not only shows the beauty of mathematics, but also reveals the essence of the world and changes the way people understand the mysteries of nature. It can be said that fractal geometry is a kind of geometry that truly describes nature, and the research on it has greatly expanded the cognitive field of human beings. As a very popular and active new theory and discipline in today's world, fractal geometry makes people re-examine the world: the world is nonlinear and fractals are everywhere. Fractal geometry not only makes people realize the integration of science and art, the unity of mathematics and artistic aesthetics, but also has its profound scientific methodology significance.
Note: Fractal theory is like holding a microscope to see how long a kilometer is, and it is only applicable to scientific research. For the length in study and life, we still adopt the agreement under ideal conditions.
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