Fractal - geometric learning in nature

zhaozj2021-02-16  51

Fractal - geometric learning in nature

B.B.Mandelbrot

Fractal geometry played two roles. Its technology determines the geometry of chaos, and it is also a geometry describing mountains, clouds and galaxies.

Natural science and geometry are always working together. In the 17th century, Cople discovered the use of an ellipse to describe the planet around the sun. This inspires Newton to explain these elliptical tracks. Similarly, the ideal pending reciprocating can be represented by a sinusoidal waveform. Simple dynamics often links simple geometry. This mathematical image suggests that there is a smooth relationship between the shape of the object and the force acting on it. In the example of the planet and the pendulum, it also implies that physics is determined, and the future can predict the future.

Two new scientific progress deeply affects the geometry. The first is that since it recognizes that nature is full of things called the chaotic chaotic. Many surfaces in the universe seem to obey the simple physical system of the law, and its behavior is still unpredictable. For example, it is protected by two force. The idea of ​​using the decisions cannot predict the movement, which makes most people surprised.

The second progress comes from the most irregular and complex phenomena of our surroundings: the shape of the mountain and cloud groups, the distribution of the galaxy in the universe, off-home, the fluctuation of financial market prices, etc. The result. A way to get this mathematical description is to find the "model." In other words, it is necessary to conceive or discover some mathematical rules, making it "mathematically forged" on certain parts of the realization - a photo of the mountain or cloud group, the most deep space of the manda, the newspaper financial version, etc. .

In fact, Galileo has declared that "the great books of nature are written in mathematical language", but also added, "The characteristics are triangular, circular and other geometric patterns, without these geometric graphics can only do in the dark maze No results wandering ". However, regardless of the simulation decision chaotic or simulated irregular system, these European miles have not been used. These phenomena needed to be far from triangles and circles. They need non-European miles to structural - especially new geometries that need to be called fractal geometry.

In 1975, I created a fractal (Fractal) by the Latino Fractus describing the gravel. The fractal is geometric profile, which is opposite to European miles, there is no rule. First, they have no rules everywhere. Second, they have the same degree of irregularities in various scale. No matter from the distance, it is still from nearby, and the fractal object looks like a look - it is self-similar. The overall small piece, from the distance, is not shaped at a small point, and it is close to find that it has become profiled, and its shape is substantially similar to the overall shape of the previously observed.

Natural world provides many fractal examples. For example, mutual plants, cauliflowers and broccoli, as well as many other plants, each of which is very similar to its overall. Its generation rules ensure the characteristics of a small scale in a small scale.

Fractal crafts processed with obvious mathematical models are Sierpinski gaskets. Take a black triangle and divide it into four smaller triangles, take off the fourth triangle of the central portion, leave a white triangle. Each new triangle also repeats the above practice, it can get the scale constantly narrowed with the same form of structure, and the edge is always divided by the previous step length. When the part of the object is completely similar, it can be said that the object is linear. However, most important fractals and linear self-simplification are still different. Some of them are fractals that describe ordinary randomness, and others can describe chaos, or nonlinear systems (the factors that work on system behavior in such systems, the degree of effect is not proportional to the effect of its effect). Let us hold an example for both situations.

Our fractal is known for their falsification of coastlines, mountains and clouds. Another example is some scenes produced for "Star Travel II".

Our fractal simulation works from a small amount of human wisdom and a large number of museum knowledge. Human wisdom starts from observing some things, and observed like a three-dimensional painter. "The cloud group is not a spherical, the hill is not a cone, the coastline is not round, the bark is not light, and the lightning does not travel along the straight line." All of these natural structures have irregular shapes, they are self-similar. In other words, we have found that the overall part of the entire portion can further reveal its deep structure, and it is almost the original structure of the original structure we handled. The musen knowledge involves the collection and classification of the facts of natural structure. For example, when you measure the coastline of a country, the less fine it is, the longer the length of the coastline, because you have to get into the coastal length of less and smaller irregularities. Lewis. Richardson has found an experience law that describes this length increase.

In order to make the fractal geometric meaning, we have to find a method, from the number of points to express the complexity of the shape, like the geometric reference angle, length, area, curvature, and one-dimensional, two-dimensional, Three dimensional these concepts.

For complex geometries, the concept of normal dimension may vary with the scale. For example, a ball having a thickness of 10 cm in diameter is made of 1 mm thick thread. From a distance, the ball is a bit. It is far from 10 cm, and the line is three-dimensional. At 1 0 mm, it is a one-dimensional line. At 1 mm, each line becomes a cylinder, and the overall change is one-dimensional, so, the dimension "cross" repeatedly from a value to another. When the ball represents a limited number of ineherry like atoms, it becomes zero. The dimension corresponding to the fractal, and the normal dimension (0, 1, 2, 3) is called the fractal dimension. Usually, their dimension is not integer.

The simplest fractal dimension is similar to the similar dimensional DS only gives the common dimension required to describe the object - 0, 1, 2, and 3, respectively. What should I look at a curve linear self-similar fractal? Such a curve can be seen from a very smooth one-dimensional line, which means that the wire is wrapped too much, so that every part of it seems to be a region on the face, it is almost two dimension. The corresponding DS value is within a range greater than 1 and less than 2. This will make DS as a measure of this curve complexity. More general, DS is a measure of fractal shape complexity or coarseness.

Another simple fractal dimension is a quality dimension. The increase in quality of one dimensional straight rod is proportional to the length, and it is 2R. The increase in the quality of the two-dimensional disk of the radius R is proportional to the circular area πr ^ 2. The increase in ball quality is proportional to the spheroid volume 4 / 3π r ^ 3. In this way, when the dimension is further increased, the increase in quality increases and the corresponding dimension of R.

In the fractal, the increase in quality is the role of normal dimension of the DM of R, since the fractal dimension is natural. Fortunately, in all simple situations, DS and DM (and other definitions of fractal dimensions) strictly take the same value. If it is not the simplest case, their values ​​can be different.

The next step in the model is to envision the simplest geometry, which is the same as the generated natural structure. In fact, we have gathered and continuously enrich the structural tools that can be used for fractal geometry. To test whether such mathematical tools are appropriate, we compare the numerical features of the model and real things - for example, the fractal dimension of the mountains. However, this is not enough, we have to use a computer to test this mathematical tool. In the end, we wish to produce a theory from the fractal simulation method of the mountain to describe the terrain of the Earth's surface.

Since the fractal can be used to describe the complex nature profile, the behavior of fractal can describe the complex kinetics system is not surprising. As previously shown in the chaotic series, the dynamics of liquid turbulence, weather, or insect groups are non-linear, and there is typical determinism of chaotic properties. If you own iterations for these equations - testing their solutions during long-term evolution - we found that many mathematical properties, especially when doing calculation, it shows itself. My most famous contribution in nonlinear fractal is proposed Mandelbrot Set. This set is formed by iteration from a relatively simple equation. It shows exotic graphics, very complicated. Some people are called nonlinear fractal geometric portraits.

The Mandbrot set does not only produce beautiful images. If we look at a large number of images very carefully, it will find that countless experimental observations can be reproduced in the form of mathematical speculation. Many of them have formed a brilliance and prove. It also encourages mathematics to adopt a new method, using computer screens.

Mathematical speculation is usually derived from the foregoing theorem. In recent decades, there is no need to enter from physics or draw diagram, which means some areas of pure mathematics, such as the theory of iterative theory (Mandbrott, which belongs to this theory), has lost power. Making a fractal image on a computer re-enable the iterative theory. Use mutually related images to provide deep information for mathematically new discovery. The study of Mandbrotte has already got a lot of speculation, which is simple, but it is difficult to prove. Analysis and research have produced many interesting pairs.

Naturally, many related fresses will produce beautifully interested graphics. In fact, some shapes that are considered fractal today have been discovered as many years. Some of this mathematics is published in 1875 to 1925, During the French math, Henry Ponached, Pierre Map and Caston Julia et al. But no one realized that they used the tools described as the image and their important significance to these two aspects of real world physics.

A pattern describing the true world random fracture is called aggregation diffusion (DLA) random generating form. Here, there is a confusing form of intricate forms like a tree. DLA can simulate the formation of ash, the leakage of water in the rock, the expansion of solid cracks and the burst of lightning.

In order to see how it formed, take a very large chess board, and she is not allowed to move in the center of the chessboard. The soldiers allow it to move in any of the four directions on the board, starting from the edge of the chessboard, starting with the indication, complete random, or drunken. The direction of each step is selected from the direction of four equal chances. When a soldier arrives in the original queen, it will become a new queen, and you can't move further. Finally, a dendritic, rather than a network of Queen group is gradually formed, called "Witten Sander DLA CLUSTER".

It is completely not expected that large-scale computer simulations have shown that the DLA family is fractal; they are almost similar. Its very little part and a large part are reduced in the form of a form and it. However, people and randomly formed linear self-similarity is still different, and we will propose some of the interesting topics. The DLA fractal growth process is characterized by it very clearly showing smoothly changing parameters to produce uneven effects. To this end, let us re-express the original structure by the theory of electrostatic potential energy. Imagine a large box of DLA, placed in a positive potential field, the target, that is, the original queen, placed in the center, and its potential energy is zero. So what is the potential energy value in other positions in the box?

Scientists have long known that when the shape of the center is a smooth curve, or there is a small amount of folding (like a triangle or square) how the potential energy is calculated. These classic analysis calculations determine some curves equal to equal potential. These equal potential curves are smooth, and they are in a fixed box and the center fixed object boundary, which reflects the gradual changes in the development. Secondly, assume that the boundary of the center fixed object is like a needle, then the equal potential curve around the needle will be very intensive. The decline in potential energy should be very sharp, causing discharge: Needs to make the same role like a lightning needle. When the center of the center is a DLA, its boundary is full of needles, and the lightning is to attack these needles in exposure. It has finally emerged for a new thing in need: the mechanism of DLA and the expansion or bifurcation of lightning after the needle is flashing. The DLA experiment made us realize that when the boundary is allowed to move, the DLA family has developed into a larger DLA structure. This means that we can establish a fractal fractal view from the equation that has a smooth characteristic of the epitope line. In this sense, the fractal geometry has been the new topic and new research fields.

Fractal geometry is also to describe other complex phenomena in nature. One of its most outcomes is the study of turbulence, not only studying how it will appear - the kinetics displayed on the phase map is fractal - also study the complex shape of the turbulent structure. In this way, the jet and tail of the water and cloud groups were originally fractal. This is due to the role of the fluid motion equation (Navier - Stokes equation). Linking the shape of the shape and the dynamics that produces profiles is far from solving. Drawing this relationship map will become an important step in understanding turbulence.

Fractal can be appropriately described in the field is the living thing and the universe, although in all cases, the fractal description will fail in very small scale and very large scale. Trees or roads do not have to be branched without limitation, and the entire trees will not be part of a super tree. The distribution of the universe is the contrary. A number of galaxies can see the small scale extends at least 15 million to 30 million light. There is more and more powerful evidence that there is a large blank area with more than 300,000 light years. This blank area is expected in the fractal distribution.

What is the importance of fractal? Like chaotic theory, it is very time to say what is still too late, but its prospects are optimistic. Many fresses have had an important impact on culture and have been seen as the results of new art forms. Some fractals are real simulations, while others are completely fictional and abstraction. Mathematics and artists unexpectedly saw such a cultural interaction.

In the view, fractal art seems to be magic. But there will be no mathematicians to neglect its structure and significance. Many of the basic equations are considered to be part of pure mathematics. There is no use of the real world, and the true natural phenomenon it represents has never seen it. Most importantly, as mentioned, the most active field of applying fractal is in physics, and they have helped handle some very old problems and solve certain new difficult problems.

The final by-product of fractal diagrams is its attraction of young people that is rescuing their interest in science. Mandbrroet sets and other fractals now appear on T Blut and Placards. Many people hope that this will make young people feel the beauty and performance of mathematics, and feel the relationship between them and the real world.

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