Mysterious disorder: the history of fractals and their areas of application. Concepts of fractal and fractal geometry

Hello everybody! My name is, Ribenek Valeria, Ulyanovsk and today I will post several of my scientific articles on the LCI website.

My first scientific article in this blog will be devoted to fractals. I will say right away that my articles are designed for almost any audience. Those. I hope they will be of interest to both schoolchildren and students.

I recently learned about these most interesting objects mathematical world as fractals. But they exist not only in mathematics. They surround us everywhere. Fractals are natural. I will talk about what fractals are, about the types of fractals, about examples of these objects and their applications in this article. To begin with, I’ll briefly tell you what a fractal is.

Fractal(Latin fractus - crushed, broken, broken) is a complex geometric figure that has the property of self-similarity, that is, composed of several parts, each of which is similar to the entire figure. In a broader sense, fractals are understood as sets of points in Euclidean space that have a fractional metric dimension (in the sense of Minkowski or Hausdorff), or a metric dimension different from the topological one. As an example, I will insert a picture depicting four different fractals.

I'll tell you a little about the history of fractals. The concepts of fractal and fractal geometry, which appeared in the late 70s, have become firmly established among mathematicians and programmers since the mid-80s. The word "fractal" was coined by Benoit Mandelbrot in 1975 to refer to the irregular but self-similar structures with which he was concerned. The birth of fractal geometry is usually associated with the publication of Mandelbrot’s book The Fractal Geometry of Nature in 1977. His works used the scientific results of other scientists who worked in the period 1875-1925 in the same field (Poincaré, Fatou, Julia, Cantor, Hausdorff). But only in our time has it been possible to combine their work into a single system.

There are a lot of examples of fractals, because, as I said, they surround us everywhere. In my opinion, even our entire Universe is one huge fractal. After all, everything in it, from the structure of the atom to the structure of the Universe itself, exactly repeats each other. But there is, of course, more specific examples fractals from different areas. Fractals, for example, are present in complex dynamics. There they naturally appear in the study of nonlinear dynamic systems. The most studied case is when the dynamic system is specified by iterations polynomial or holomorphic function of a complex of variables on surface. Some of the most famous fractals of this type are the Julia set, Mandelbrot set and Newton pools. Below, in order, the pictures depict each of the above fractals.

Another example of fractals is fractal curves. It is best to explain how to construct a fractal using the example of fractal curves. One of these curves is the so-called Koch Snowflake. There is a simple procedure for obtaining fractal curves on a plane. Let us define an arbitrary broken line with a finite number of links, called a generator. Next, we replace each segment in it with a generator (more precisely, a broken line similar to a generator). In the resulting broken line, we again replace each segment with a generator. Continuing to infinity, in the limit we get a fractal curve. Below is the Koch Snowflake (or Curve).

There are also a huge variety of fractal curves. The most famous of them are the already mentioned Koch Snowflake, as well as the Levy curve, the Minkowski curve, the Dragon's broken line, the Piano curve and the Pythagorean tree. I think you can easily find an image of these fractals and their history on Wikipedia if you wish.

The third example or type of fractals are stochastic fractals. Such fractals include the trajectory of Brownian motion on a plane and in space, the Schramm-Löwner evolution, various types of randomized fractals, that is, fractals obtained using a recursive procedure into which a random parameter is introduced at each step.

There are also purely mathematical fractals. These are, for example, the Cantor set, the Menger sponge, the Sierpinski Triangle and others.

But perhaps the most interesting fractals are natural ones. Natural fractals are objects in nature that have fractal properties. And here the list is already big. I won’t list everything, because it’s probably impossible to list them all, but I’ll tell you about some. For example, in living nature, such fractals include our circulatory system and lungs. And also the crowns and leaves of trees. This can also include starfish, sea ​​urchins, corals, sea shells, some plants such as cabbage or broccoli. Several such natural fractals from living nature are clearly shown below.

If we consider inanimate nature, then there are many more interesting examples there than in real life. Lightning, snowflakes, clouds, well-known to everyone, patterns on windows on frosty days, crystals, mountain ranges - all these are examples of natural fractals from inanimate nature.

We looked at examples and types of fractals. As for the use of fractals, they are used in a variety of fields of knowledge. In physics, fractals naturally arise when modeling nonlinear processes, such as turbulent fluid flow, complex diffusion-adsorption processes, flames, clouds, etc. Fractals are used when modeling porous materials, for example, in petrochemistry. In biology, they are used to model populations and to describe systems. internal organs(system of blood vessels). After the creation of the Koch curve, it was proposed to use it in calculating the length of the coastline. Fractals are also actively used in radio engineering, information science and computer technology, telecommunications and even economics. And, of course, fractal vision is actively used in modern art and architecture. Here is one example of fractal patterns:

And so, with this I think to complete my story about such an unusual mathematical phenomenon as a fractal. Today we learned about what a fractal is, how it appeared, about the types and examples of fractals. I also talked about their application and demonstrated some of the fractals visually. I hope you enjoyed this little excursion into the world of amazing and fascinating fractal objects.

The most ingenious discoveries in science can radically change human life. The invented vaccine can save millions of people; the creation of weapons, on the contrary, takes away these lives. More recently (on the scale of human evolution) we have learned to “tame” electricity - and now we cannot imagine life without all these convenient devices that use electricity. But there are also discoveries that few people attach importance to, although they also greatly influence our lives.

One of these “inconspicuous” discoveries is fractals. You've probably heard this catchy word before, but do you know what it means and how much interesting information is hidden in this term?

Every person has a natural curiosity, a desire to understand the world around him. And in this endeavor, a person tries to adhere to logic in judgments. Analyzing the processes taking place around him, he tries to find the logic of what is happening and derive some pattern. The greatest minds on the planet are busy with this task. Roughly speaking, scientists are looking for a pattern where there shouldn't be one. Nevertheless, even in chaos it is possible to find connections between events. And this connection is a fractal.

Our little daughter, four and a half years old, is now at that wonderful age when the number of questions “Why?” many times exceeds the number of answers that adults manage to give. Not long ago, while examining a branch raised from the ground, my daughter suddenly noticed that this branch, with its twigs and branches, itself looked like a tree. And, of course, what followed was the usual question “Why?”, to which parents had to look for a simple explanation that the child could understand.

The similarity of a single branch with a whole tree discovered by a child is a very accurate observation, which once again testifies to the principle of recursive self-similarity in nature. Many organic and inorganic forms in nature are formed in a similar way. Clouds, sea shells, a snail’s “house,” the bark and crown of trees, the circulatory system, and so on—the random shapes of all these objects can be described by a fractal algorithm.

⇡ Benoit Mandelbrot: father of fractal geometry

The word “fractal” itself appeared thanks to the brilliant scientist Benoit B. Mandelbrot.

He himself coined the term in the 1970s, borrowing the word fractus from Latin, where it literally means “broken” or “crushed.” What is it? Today, the word “fractal” most often means a graphic representation of a structure that, on a larger scale, is similar to itself.

The mathematical basis for the emergence of the theory of fractals was laid many years before the birth of Benoit Mandelbrot, but it could only develop with the advent of computing devices. At the beginning of its scientific activity Benoit worked in research center IBM company. At that time, the center's employees were working on transmitting data over a distance. During the research, scientists were faced with the problem of large losses arising from noise interference. Benoit faced a difficult and very important task— understand how to predict the occurrence of noise interference in electronic circuits when the statistical method turns out to be ineffective.

Looking through the results of noise measurements, Mandelbrot noticed one strange pattern - the noise graphs at different scales looked the same. An identical pattern was observed regardless of whether it was a noise graph for one day, a week, or an hour. It was necessary to change the scale of the graph, and the picture was repeated every time.

During his lifetime, Benoit Mandelbrot repeatedly said that he did not study formulas, but simply played with pictures. This man thought very figuratively, and any algebraic problem translated into the field of geometry, where, according to him, the correct answer is always obvious.

It is not surprising that it was a man with such a rich spatial imagination who became the father of fractal geometry. After all, awareness of the essence of fractals comes precisely when you begin to study the drawings and think about the meaning of strange swirl patterns.

A fractal pattern does not have identical elements, but is similar on any scale. Construct such an image with high degree manual detailing was previously simply impossible; it required a huge amount of calculations. For example, the French mathematician Pierre Joseph Louis Fatou described this set more than seventy years before Benoit Mandelbrot's discovery. If we talk about the principles of self-similarity, they were mentioned in the works of Leibniz and Georg Cantor.

One of the first fractal drawings was a graphical interpretation of the Mandelbrot set, which was born thanks to the research of Gaston Maurice Julia.

Gaston Julia (always wearing a mask - injury from World War I)

This French mathematician wondered what a set would look like if it were constructed from a simple formula iterated through a feedback loop. If we explain it “on our fingers,” this means that for a specific number we find a new value using the formula, after which we substitute it again into the formula and get another value. The result is a large sequence of numbers.

To get a complete picture of such a set, you need to do a huge number of calculations - hundreds, thousands, millions. It was simply impossible to do this manually. But when powerful computing devices became available to mathematicians, they were able to take a fresh look at formulas and expressions that had long been of interest. Mandelbrot was the first to use a computer to calculate a classical fractal. After processing a sequence consisting of a large number of values, Benoit plotted the results on a graph. That's what he got.

Subsequently, this image was colored (for example, one of the methods of coloring is by the number of iterations) and became one of the most popular images ever created by man.

As the ancient saying attributed to Heraclitus of Ephesus says, “You cannot step into the same river twice.” It is perfectly suited for interpreting the geometry of fractals. No matter how detailed we look at a fractal image, we will always see a similar pattern.

Those wishing to see what an image of Mandelbrot space would look like when zoomed in many times over can do so by downloading the animated GIF.

⇡ Lauren Carpenter: art created by nature

The theory of fractals soon found practical application. Since it is closely related to the visualization of self-similar images, it is not surprising that the first to adopt algorithms and principles for constructing unusual forms were artists.

The future co-founder of the legendary Pixar studio, Loren C. Carpenter, began working in 1967 at Boeing Computer Services, which was one of the divisions of the famous corporation developing new aircraft.

In 1977, he created presentations with prototype flying models. Loren's responsibilities included developing images of the aircraft being designed. He had to create pictures of new models, showing future aircraft from different angles. At some point, the future founder of Pixar Animation Studios came up with the creative idea of ​​using an image of mountains as a background. Today, any schoolchild can solve such a problem, but in the late seventies of the last century, computers could not cope with such complex calculations - there were no graphic editors, not to mention applications for 3D graphics. In 1978, Lauren accidentally saw Benoit Mandelbrot's book Fractals: Form, Chance and Dimension in a store. What caught his attention in this book was that Benoit gave a lot of examples of fractal shapes in real life and argued that they can be described by a mathematical expression.

This analogy was not chosen by the mathematician by chance. The fact is that as soon as he published his research, he had to face a whole barrage of criticism. The main thing that his colleagues reproached him for was the uselessness of the theory being developed. “Yes,” they said, “these are beautiful pictures, but nothing more. The theory of fractals has no practical value.” There were also those who generally believed that fractal patterns were simply a by-product of the work of the “devilish machines”, which in the late seventies seemed to many to be something too complex and unexplored to be completely trusted. Mandelbrot tried to find obvious applications for fractal theory, but in the grand scheme of things he didn't need to. Over the next 25 years, the followers of Benoit Mandelbrot proved the enormous benefits of such a “mathematical curiosity,” and Lauren Carpenter was one of the first to try the fractal method in practice.

After studying the book, the future animator seriously studied the principles of fractal geometry and began to look for a way to implement it in computer graphics. In just three days of work, Lauren was able to render a realistic image of the mountain system on his computer. In other words, he used formulas to paint a completely recognizable mountain landscape.

The principle Lauren used to achieve her goal was very simple. It consisted of dividing a larger geometric figure into small elements, and these, in turn, were divided into similar figures of a smaller size.

Using larger triangles, Carpenter split them into four smaller ones and then repeated this process over and over again until he had a realistic mountain landscape. Thus, he managed to become the first artist to use a fractal algorithm for constructing images in computer graphics. As soon as word of the work became known, enthusiasts around the world took up the idea and began using the fractal algorithm to imitate realistic natural shapes.

One of the first 3D visualizations using a fractal algorithm

Just a few years later, Lauren Carpenter was able to apply his developments in a much larger project. The animator created a two-minute demo of Vol Libre from them, which was shown on Siggraph in 1980. This video shocked everyone who saw it, and Lauren received an invitation from Lucasfilm.

The animation was rendered on a VAX-11/780 computer from Digital Equipment Corporation with a clock speed of five megahertz, and each frame took about half an hour to render.

Working for Lucasfilm Limited, the animator created 3D landscapes using the same scheme for the second full-length film in the Star Trek saga. In The Wrath of Khan, Carpenter was able to create an entire planet using the same principle of fractal surface modeling.

Currently, all popular applications for creating 3D landscapes use a similar principle for generating natural objects. Terragen, Bryce, Vue and other 3D editors rely on a fractal algorithm for modeling surfaces and textures.

⇡ Fractal antennas: less is more

Over the past half century, life has rapidly begun to change. Most of us accept achievements modern technologies for granted. You get used to everything that makes life more comfortable very quickly. Rarely does anyone ask the questions “Where did this come from?” and “How does it work?” A microwave heats up breakfast - great, a smartphone gives you the opportunity to talk to another person - great. This seems like an obvious possibility to us.

But life could have been completely different if a person had not sought an explanation for the events taking place. Take cell phones, for example. Remember the retractable antennas on the first models? They interfered, increased the size of the device, and in the end, often broke. We believe they have sunk into oblivion forever, and part of the reason for this is... fractals.

Fractal patterns fascinate with their patterns. They definitely resemble images of cosmic objects - nebulae, galaxy clusters, and so on. It is therefore quite natural that when Mandelbrot voiced his theory of fractals, his research aroused increased interest among those who studied astronomy. One of these amateurs named Nathan Cohen, after attending a lecture by Benoit Mandelbrot in Budapest, got the idea practical application acquired knowledge. True, he did this intuitively, and chance played an important role in his discovery. As a radio amateur, Nathan sought to create an antenna with the highest possible sensitivity.

The only way to improve the parameters of the antenna, which was known at that time, was to increase its geometric dimensions. However, the owner of the property in downtown Boston that Nathan rented was categorically against installing large devices on the roof. Then Nathan began experimenting with different antenna shapes, trying to get the maximum result with minimum sizes. Inspired by the idea of ​​fractal forms, Cohen, as they say, randomly made one of the most famous fractals from wire - the “Koch snowflake”. Swedish mathematician Helge von Koch came up with this curve back in 1904. It is obtained by dividing a segment into three parts and replacing the middle segment with an equilateral triangle without a side coinciding with this segment. The definition is a little difficult to understand, but in the figure everything is clear and simple.

There are also other variations of the Koch curve, but the approximate shape of the curve remains similar

When Nathan connected the antenna to the radio receiver, he was very surprised - the sensitivity increased dramatically. After a series of experiments future professor Boston University realized that an antenna made using a fractal pattern has high efficiency and covers a much wider frequency range compared to classical solutions. In addition, the shape of the antenna in the form of a fractal curve makes it possible to significantly reduce the geometric dimensions. Nathan Cohen even came up with a theorem proving that to create a broadband antenna, it is enough to give it the shape of a self-similar fractal curve.

The author patented his discovery and founded a company for the development and design of fractal antennas Fractal Antenna Systems, rightly believing that in the future, thanks to his discovery, cell phones will be able to get rid of bulky antennas and become more compact.

In principle, this is what happened. True, to this day Nathan is engaged in a legal battle with large corporations that are illegally using his discovery to produce compact communication devices. Some well-known mobile device manufacturers, such as Motorola, have already reached an amicable agreement with the inventor of the fractal antenna.

⇡ Fractal dimensions: you can’t understand it with your mind

Benoit borrowed this question from the famous American scientist Edward Kasner.

The latter, like many other famous mathematicians, loved to communicate with children, asking them questions and receiving unexpected answers. Sometimes this led to surprising consequences. For example, the nine-year-old nephew of Edward Kasner came up with the now well-known word “googol,” meaning one followed by one hundred zeros. But let's return to fractals. The American mathematician liked to ask the question how long is the US coastline. After listening to the opinion of his interlocutor, Edward himself spoke the correct answer. If you measure the length on a map using broken segments, the result will be inaccurate, because the coastline has a large number of unevenness. What happens if we measure as accurately as possible? You will have to take into account the length of each unevenness - you will need to measure every cape, every bay, rock, the length of a rocky ledge, a stone on it, a grain of sand, an atom, and so on. Since the number of irregularities tends to infinity, the measured length of the coastline will increase to infinity when measuring each new irregularity.

The smaller the measure when measuring, the longer the measured length

Interestingly, following Edward's prompts, the children were much faster than the adults in saying the correct solution, while the latter had trouble accepting such an incredible answer.

Using this problem as an example, Mandelbrot suggested using new approach to measurements. Since the coastline is close to a fractal curve, it means that a characterizing parameter can be applied to it - the so-called fractal dimension.

What a regular dimension is is clear to anyone. If the dimension is equal to one, we get a straight line, if two - flat figure, three - volume. However, this understanding of dimension in mathematics does not work with fractal curves, where this parameter has a fractional value. Fractal dimension in mathematics can be conventionally considered as a “roughness”. The higher the roughness of the curve, the greater its fractal dimension. A curve that, according to Mandelbrot, has a fractal dimension higher than its topological dimension has an approximate length that does not depend on the number of dimensions.

Currently, scientists are finding more and more areas to apply the theory of fractals. Using fractals, you can analyze fluctuations in stock exchange prices, study all sorts of natural processes, such as fluctuations in the number of species, or simulate the dynamics of flows. Fractal algorithms can be used for data compression, such as image compression. And by the way, to get a beautiful fractal on your computer screen, you don’t have to have a doctorate.

⇡ Fractal in the browser

Perhaps one of the easiest ways to get a fractal pattern is to use an online vector editor from the young talented programmer Toby Schachman. The tools of this simple graphic editor are based on the same principle of self-similarity.

At your disposal there are only two simplest shapes - a quadrangle and a circle. You can add them to the canvas, scale them (to scale along one of the axes, hold down the Shift key) and rotate them. Overlapping according to the principle of Boolean addition operations, these simplest elements form new, less trivial forms. These new shapes can then be added to the project, and the program will repeat generating these images ad infinitum. At any stage of working on a fractal, you can return to any component of a complex shape and edit its position and geometry. A fun activity, especially when you consider that the only tool you need to create is a browser. If you do not understand the principle of working with this recursive vector editor, we advise you to watch the video on the official website of the project, which shows in detail the entire process of creating a fractal.

⇡ XaoS: fractals for every taste

Many graphic editors have built-in tools for creating fractal patterns. However, these tools are usually secondary and do not allow fine tuning of the generated fractal pattern. In cases where it is necessary to construct a mathematically accurate fractal, the cross-platform editor XaoS will come to the rescue. This program makes it possible not only to build a self-similar image, but also to perform various manipulations with it. For example, in real time you can take a “walk” along a fractal by changing its scale. Animated movement along a fractal can be saved as an XAF file and then reproduced in the program itself.

XaoS can load a random set of parameters, and also use various image post-processing filters - add a blurred motion effect, smooth out sharp transitions between fractal points, simulate a 3D image, and so on.

⇡ Fractal Zoomer: compact fractal generator

Compared to other fractal image generators, it has several advantages. Firstly, it is very small in size and does not require installation. Secondly, it implements the ability to determine the color palette of a picture. You can choose shades in color models RGB, CMYK, HVS and HSL.

It is also very convenient to use the random selection option color shades and a function to invert all colors in the picture. To adjust the color, there is a function of cyclical selection of shades - when you turn on the corresponding mode, the program animates the image, cyclically changing the colors on it.

Fractal Zoomer can visualize 85 different fractal functions, and the formulas are clearly shown in the program menu. There are filters for image post-processing in the program, although in small quantities. Each assigned filter can be canceled at any time.

⇡ Mandelbulb3D: 3D fractal editor

When the term "fractal" is used, it most often refers to a flat, two-dimensional image. However, fractal geometry goes beyond the 2D dimension. In nature, you can find both examples of flat fractal forms, say, the geometry of lightning, and three-dimensional volumetric figures. Fractal surfaces can be three-dimensional, and one of the very clear illustrations of 3D fractals in Everyday life- head of cabbage. Perhaps the best way to see fractals is in the Romanesco variety, a hybrid of cauliflower and broccoli.

You can also eat this fractal

Create 3D objects with similar shape Mandelbulb3D program can do this. To obtain a 3D surface using a fractal algorithm, the authors of this application, Daniel White and Paul Nylander, converted the Mandelbrot set to spherical coordinates. The Mandelbulb3D program they created is a real three-dimensional editor that models fractal surfaces of different shapes. Since we often observe fractal patterns in nature, an artificially created fractal three-dimensional object seems incredibly realistic and even “alive.”

It may resemble a plant, it may resemble a strange animal, a planet, or something else. This effect is enhanced by an advanced rendering algorithm, which makes it possible to obtain realistic reflections, calculate transparency and shadows, simulate the effect of depth of field, and so on. Mandelbulb3D has a huge number of settings and rendering options. You can control the shades of light sources, select the background and level of detail of the simulated object.

The Incendia fractal editor supports double image smoothing, contains a library of fifty different three-dimensional fractals, and has a separate module for editing basic shapes.

The application uses fractal scripting, with which you can independently describe new types of fractal designs. Incendia has texture and material editors, and the rendering engine allows you to use volumetric fog effects and various shaders. The program implements the option of saving a buffer during long-term rendering, and supports the creation of animation.

Incendia allows you to export a fractal model to popular 3D graphics formats - OBJ and STL. Incendia includes a small utility called Geometrica, a special tool for setting up the export of a fractal surface to a 3D model. Using this utility, you can determine the resolution of a 3D surface and specify the number of fractal iterations. Exported models can be used in 3D projects when working with 3D editors such as Blender, 3ds max and others.

IN Lately work on the Incendia project has slowed down somewhat. On this moment the author is looking for sponsors to help him develop the program.

If you don’t have enough imagination to draw a beautiful three-dimensional fractal in this program, it doesn’t matter. Use the parameters library, which is located in the INCENDIA_EX\parameters folder. Using PAR files, you can quickly find the most unusual fractal shapes, including animated ones.

⇡ Aural: how fractals sing

We usually don’t talk about projects that are just being worked on, but in this case we have to make an exception, since this is a very unusual application. The project, called Aural, was invented by the same person who created Incendia. However, this time the program does not visualize the fractal set, but sounds it, turning it into electronic music. The idea is very interesting, especially considering unusual properties fractals. Aural is an audio editor that generates melodies using fractal algorithms, that is, in essence, it is an audio synthesizer-sequencer.

The sequence of sounds produced by this program is unusual and... beautiful. It may well be useful for writing modern rhythms and, it seems to us, is especially well suited for creating audio tracks to screensavers of television and radio programs, as well as “loops” of background music to computer games. Ramiro has not yet provided a demo of his program, but promises that when he does, in order to work with Aural, you will not need to study fractal theory - you will just need to play with the parameters of the algorithm for generating a sequence of notes. Listen to how fractals sound, and.

Fractals: musical break

In fact, fractals can help you write music even without software. But this can only be done by someone who is truly imbued with the idea of ​​natural harmony and who has not turned into an unfortunate “nerd.” It makes sense to take an example from a musician named Jonathan Coulton, who, among other things, writes compositions for Popular Science magazine. And unlike other performers, Colton publishes all of his works under a Creative Commons Attribution-Noncommercial license, which (when used for non-commercial purposes) provides for free copying, distribution, transfer of the work to others, as well as its modification (creation of derivative works) so that adapt it to your tasks.

Jonathan Colton, of course, has a song about fractals.

⇡ Conclusion

In everything that surrounds us, we often see chaos, but in fact this is not an accident, but an ideal form, which fractals help us to discern. Nature is the best architect, ideal builder and engineer. It is structured very logically, and if we don’t see a pattern somewhere, this means that we need to look for it on a different scale. People understand this better and better, trying to imitate natural forms in many ways. Engineers design Acustic systems in the form of a shell, they create antennas with the geometry of snowflakes and so on. We are sure that fractals still contain many secrets, and many of them have yet to be discovered by humans.

What do a tree, a seashore, a cloud, or the blood vessels in our hand have in common? At first glance, it may seem that all these objects have nothing in common. However, in fact, there is one property of structure that is inherent in all of the listed objects: they are self-similar. From a branch, like from a tree trunk, smaller shoots extend, from them even smaller ones, etc., that is, a branch is similar to the whole tree. The circulatory system is structured in a similar way: arterioles depart from the arteries, and from them the smallest capillaries through which oxygen enters the organs and tissues. Let's look at satellite images of the sea coast: we will see bays and peninsulas; Let's look at it, but from a bird's eye view: we will see bays and capes; Now imagine that we are standing on the beach and looking at our feet: there will always be pebbles that protrude further into the water than the rest. That is, the coastline, when zoomed in, remains similar to itself. The American mathematician (though he grew up in France) Benoit Mandelbrot called this property of objects fractality, and such objects themselves - fractals (from the Latin fractus - broken).

This concept does not have a strict definition. Therefore, the word "fractal" is not a mathematical term. Typically, a fractal is a geometric figure that satisfies one or more of the following properties: It has a complex structure at any increase in scale (unlike, for example, a straight line, any part of which is the simplest geometric figure - a segment). Is (approximately) self-similar. It has a fractional Hausdorff (fractal) dimension, which is larger than the topological one. Can be constructed using recursive procedures.

Geometry and algebra

Studying fractals on turn of the 19th century and XX centuries was more episodic than systematic, because previously mathematicians mainly studied “good” objects that could be studied using general methods and theories. In 1872, the German mathematician Karl Weierstrass constructed an example of a continuous function that is nowhere differentiable. However, its construction was entirely abstract and difficult to understand. Therefore, in 1904, the Swede Helge von Koch came up with a continuous curve that has no tangent anywhere, and is quite easy to draw. It turned out that it has the properties of a fractal. One variant of this curve is called the “Koch snowflake”.

The ideas of self-similarity of figures were picked up by the Frenchman Paul Pierre Levy, the future mentor of Benoit Mandelbrot. In 1938, his article “Plane and spatial curves and surfaces consisting of parts similar to the whole” was published, which described another fractal - the Levy C-curve. All of these fractals listed above can be conditionally classified as one class of constructive (geometric) fractals.

Another class is dynamic (algebraic) fractals, which include the Mandelbrot set. The first research in this direction began at the beginning of the 20th century and is associated with the names of the French mathematicians Gaston Julia and Pierre Fatou. In 1918, Julia published an almost two-hundred-page memoir on iterations of complex rational functions, which described Julia sets, a whole family of fractals closely related to the Mandelbrot set. This work was awarded a prize by the French Academy, but it did not contain a single illustration, so it was impossible to appreciate the beauty of the open objects. Despite the fact that this work made Julia famous among mathematicians of that time, it was quickly forgotten. Attention again turned to it only half a century later with the advent of computers: it was they who made visible the richness and beauty of the world of fractals.

Fractal dimensions

As you know, the dimension (number of dimensions) of a geometric figure is the number of coordinates necessary to determine the position of a point lying on this figure.
For example, the position of a point on a curve is determined by one coordinate, on a surface (not necessarily a plane) by two coordinates, and in three-dimensional space by three coordinates.
From a more general mathematical point of view, one can define the dimension in this way: an increase in linear dimensions, say, by a factor of two, for one-dimensional (from a topological point of view) objects (segment) leads to an increase in size (length) by a factor of two, for two-dimensional ones (a square ) the same increase in linear dimensions leads to an increase in size (area) by 4 times, for three-dimensional (cube) - by 8 times. That is, the “real” (so-called Hausdorff) dimension can be calculated as the ratio of the logarithm of the increase in the “size” of an object to the logarithm of the increase in its linear size. That is, for a segment D=log (2)/log (2)=1, for a plane D=log (4)/log (2)=2, for a volume D=log (8)/log (2)=3.
Let us now calculate the dimension of the Koch curve, to construct which a unit segment is divided into three equal parts and the middle interval is replaced by an equilateral triangle without this segment. When the linear dimensions of the minimum segment increase three times, the length of the Koch curve increases by log (4)/log (3) ~ 1.26. That is, the dimension of the Koch curve is fractional!

Science and art

In 1982, Mandelbrot’s book “Fractal Geometry of Nature” was published, in which the author collected and systematized almost all the information about fractals available at that time and presented it in an easy and accessible manner. Mandelbrot placed the main emphasis in his presentation not on heavy formulas and mathematical constructions, but on the geometric intuition of readers. Thanks to illustrations obtained using a computer and historical stories, with which the author skillfully diluted the scientific component of the monograph, the book became a bestseller, and fractals became known to the general public. Their success among non-mathematicians is largely due to the fact that with the help of very simple constructions and formulas that even a high school student can understand, images of amazing complexity and beauty are obtained. When personal computers became powerful enough, even a whole direction in art appeared - fractal painting, and almost any computer owner could do it. Now on the Internet you can easily find many sites devoted to this topic.

Scheme for obtaining the Koch curve

War and Peace

As noted above, one of the natural objects that have fractal properties is the coastline. There is one thing connected with it, or more precisely, with the attempt to measure its length. interesting story, which formed the basis scientific article Mandelbrot, and is also described in his book “Fractal Geometry of Nature”. We are talking about an experiment carried out by Lewis Richardson, a very talented and eccentric mathematician, physicist and meteorologist. One of the directions of his research was an attempt to find a mathematical description of the causes and likelihood of an armed conflict between two countries. Among the parameters that he took into account was the length of the common border of the two warring countries. When he collected data for numerical experiments, he discovered that data on the common border of Spain and Portugal differed greatly from different sources. This led him to the following discovery: the length of a country's borders depends on the ruler with which we measure them. The smaller the scale, the longer the border. This is due to the fact that with greater magnification it becomes possible to take into account more and more new bends of the coast, which were previously ignored due to the coarseness of the measurements. And if, with each increase in scale, previously unaccounted for bends of lines are revealed, then it turns out that the length of the boundaries is infinite! True, this does not actually happen - the accuracy of our measurements has a finite limit. This paradox is called the Richardson effect.

Constructive (geometric) fractals

The algorithm for constructing a constructive fractal in the general case is as follows. First of all, we need two suitable geometric shapes, let's call them the base and the fragment. At the first stage, the basis of the future fractal is depicted. Then some of its parts are replaced with a fragment taken at a suitable scale - this is the first iteration of the construction. Then the resulting figure again changes some parts to figures similar to the fragment, etc. If we continue this process ad infinitum, then in the limit we will get a fractal.

Let's look at this process using the Koch curve as an example (see sidebar on the previous page). Any curve can be taken as the basis for the Koch curve (for the “Koch snowflake” it is a triangle). But we will limit ourselves to the simplest case - a segment. The fragment is a broken line, shown at the top in the figure. After the first iteration of the algorithm, in this case the original segment will coincide with the fragment, then each of its constituent segments will itself be replaced by a broken line similar to the fragment, etc. The figure shows the first four steps of this process.

In the language of mathematics: dynamic (algebraic) fractals

Fractals of this type arise when studying nonlinear dynamic systems (hence the name). The behavior of such a system can be described by a complex nonlinear function (polynomial) f (z). Let's take some initial point z0 on the complex plane (see sidebar). Now consider such an infinite sequence of numbers on the complex plane, each next of which is obtained from the previous one: z0, z1=f (z0), z2=f (z1), ... zn+1=f (zn). Depending on the initial point z0, such a sequence can behave differently: tend to infinity as n -> ∞; converge to some end point; cyclically take a series of fixed values; More complex options are also possible.

Complex numbers

A complex number is a number consisting of two parts - real and imaginary, that is, the formal sum x + iy (x and y here are real numbers). i is the so-called imaginary unit, that is, that is, a number that satisfies the equation i^ 2 = -1. The basic mathematical operations on complex numbers are defined: addition, multiplication, division, subtraction (only the comparison operation is not defined). To display complex numbers, a geometric representation is often used - on the plane (it is called complex), the real part is plotted along the abscissa axis, and the imaginary part is plotted along the ordinate axis, and the complex number will correspond to a point with Cartesian coordinates x and y.

Thus, any point z of the complex plane has its own behavior during iterations of the function f (z), and the entire plane is divided into parts. Moreover, the points lying on the boundaries of these parts have the following property: with an arbitrarily small displacement, the nature of their behavior changes sharply (such points are called bifurcation points). So, it turns out that sets of points that have one specific type of behavior, as well as sets of bifurcation points, often have fractal properties. These are the Julia sets for the function f (z).

Dragon family

By varying the base and fragment, you can get a stunning variety of constructive fractals.
Moreover, similar operations can be performed in three-dimensional space. Examples of volumetric fractals include the “Menger sponge”, “Sierpinski pyramid” and others.
The dragon family is also considered to be a constructive fractal. Sometimes they are called by the name of their discoverers “Heavey-Harter dragons” (in their shape they resemble Chinese dragons). There are several ways to construct this curve. The simplest and most visual of them is this: you need to take a fairly long strip of paper (the thinner the paper, the better), and bend it in half. Then bend it in half again in the same direction as the first time. After several repetitions (usually after five or six folds the strip becomes too thick to be gently bent further), you need to bend the strip back, and try to create 90˚ angles at the folds. Then in profile you will get the curve of a dragon. Of course, this will only be an approximation, like all our attempts to depict fractal objects. The computer allows many more steps of this process to be depicted, and the result is a very beautiful figure.

The Mandelbrot set is constructed somewhat differently. Consider the function fc (z) = z 2 +c, where c is a complex number. Let's construct a sequence of this function with z0=0; depending on the parameter c, it can diverge to infinity or remain limited. Moreover, all values ​​of c for which this sequence is limited form the Mandelbrot set. It was studied in detail by Mandelbrot himself and other mathematicians, who discovered many interesting properties of this set.

It can be seen that the definitions of the Julia and Mandelbrot sets are similar to each other. In fact, these two sets are closely related. Namely, the Mandelbrot set is all values ​​of the complex parameter c for which the Julia set fc (z) is connected (a set is called connected if it cannot be divided into two disjoint parts, with some additional conditions).

Fractals and life

Nowadays, the theory of fractals finds wide application in various areas of human activity. In addition to a purely scientific object for research and the already mentioned fractal painting, fractals are used in information theory to compress graphic data (the self-similarity property of fractals is mainly used here - after all, to remember a small fragment of a picture and the transformations with which you can obtain the remaining parts, much less is required memory than for storing the entire file). By adding random disturbances to the formulas that define a fractal, you can obtain stochastic fractals that very plausibly convey some real objects - relief elements, the surface of reservoirs, some plants, which is successfully used in physics, geography and computer graphics to achieve greater similarity of simulated objects with real. In radio electronics, in the last decade, antennas with a fractal shape began to be produced. Taking up little space, they provide high-quality signal reception. Economists use fractals to describe currency fluctuation curves (this property was discovered by Mandelbrot more than 30 years ago). This concludes this short excursion into the amazingly beautiful and diverse world of fractals.

Chaos is order that needs to be deciphered.

Jose Saramago, "The Double"

“For future generations, the twentieth century will be remembered only for the creation of the theories of relativity, quantum mechanics and chaos... the theory of relativity did away with Newton’s illusions about absolute space-time, quantum mechanics dispelled the dream of the determinism of physical events, and, finally, chaos debunked Laplace’s fantasy of the complete predetermination of the development of systems.” These words of the famous American historian and popularizer of science James Gleick reflect the enormous importance of the issue, which is only briefly covered in the article brought to the attention of the reader. Our world arose from chaos. However, if chaos did not obey its own laws, if there was no special logic in it, it would not be able to generate anything.

New is well forgotten old

Let me quote one more from Gleick:

The thought of internal similarity, that the great can be embedded in the small, has long caressed the human soul... According to Leibniz, a drop of water contains the entire world sparkling with colors, where water splashes sparkle and other unknown universes live. “See the world in a grain of sand,” Blake called, and some scientists tried to follow his behest. The first researchers of seminal fluid tended to see in each sperm a kind of homunculus, that is, a tiny but fully formed person.

The retrospective of such views can be turned much further into history. One of the basic principles of magic - an integral stage of development of any society - is the postulate: a part is similar to the whole. It was manifested in such actions as burying the skull of an animal instead of the whole animal, a model of a chariot instead of the chariot itself, etc. By preserving the skull of an ancestor, relatives believed that he continued to live next to them and take part in their affairs.

Even the ancient Greek philosopher Anaxagoras considered the primary elements of the universe as particles similar to other particles of the whole and the whole itself, “infinite in both multitude and smallness.” Aristotle characterized the elements of Anaxagoras with the adjective “similar to parts”.

And our contemporary, American cyberneticist Ron Eglash, exploring the culture of African tribes and South American Indians, made a discovery: since ancient times, some of them have used fractal principles of construction in ornaments, patterns applied to clothing and household items, in jewelry, ritual ceremonies, and even in architecture. Thus, the structure of the villages of some African tribes is a circle in which there are small circles - houses, inside of which there are even smaller circles - houses of spirits. For other tribes, instead of circles, other figures serve as architectural elements, but they are also repeated on different scales, subordinated to a single structure. Moreover, these principles of construction were not a simple imitation of nature, but were consistent with the existing worldview and social organization.

Our civilization, it would seem, has moved far from primitive existence. However, we continue to live in the same world; we are still surrounded by nature, living according to its own laws, despite all human attempts to adapt it to our needs. And man himself (let’s not forget about this) remains part of this nature.

Gert Eilenberger, a German physicist who began studying nonlinearity, once remarked:

Why is the silhouette of a naked tree bent under the pressure of a storm wind against the background of a gloomy winter sky perceived as beautiful, but the outlines of a modern multifunctional building, despite all the efforts of the architect, do not seem so at all? It seems to me that... our sense of beauty is “fueled” by a harmonious combination of order and disorder, which can be observed in natural phenomena: clouds, trees, mountain ranges or snowflake crystals. All such contours are dynamic processes frozen in physical forms, and a combination of stability and chaos is typical for them.

At the origins of chaos theory

What do we mean by chaos? The inability to predict the behavior of the system, random jumps in different directions that will never turn into an orderly sequence.

The first researcher of chaos is the French mathematician, physicist and philosopher Henri Poincaré. Back at the end of the 19th century. While studying the behavior of a system with three bodies interacting gravitationally, he noticed that there could be non-periodic orbits that are constantly neither moving away from a specific point nor approaching it.

Traditional geometry methods, widely used in the natural sciences, are based on approximating the structure of the object under study geometric shapes, for example, lines, planes, spheres, the metric and topological dimensions of which are equal to each other. In most cases, the properties of the object under study and its interaction with the environment are described by integral thermodynamic characteristics, which leads to the loss of a significant part of information about the system and its replacement with a more or less adequate model. Most often, such a simplification is completely justified, but there are numerous situations where the use of topologically inadequate models is unacceptable. An example of such a discrepancy was given in his PhD thesis(now a Doctor of Chemical Sciences) Vladimir Konstantinovich Ivanov: it is detected by measuring the area of ​​​​a developed (for example, porous) surface solids using sorption methods that record adsorption isotherms. It turned out that the size of the area depends on the linear size of the “measuring” molecules not quadratically, which would be expected from the simplest geometric considerations, but with an exponent, sometimes very close to three.

Weather forecasting is one of the problems that humanity has been struggling with since ancient times. There is a well-known joke on this topic, where the weather forecast is transmitted along a chain from a shaman - to a reindeer herder, then to a geologist, then to the editor of a radio program, and finally the circle is closed, since it turns out that the shaman learned the forecast from the radio. The description of such a complex system as weather, with many variables, cannot be reduced to simple models. This problem began the use of computers for modeling nonlinear dynamic systems. One of the founders of chaos theory, American meteorologist and mathematician Edward Norton Lorenz devoted many years to the problem of weather forecasting. Back in the 60s of the last century, trying to understand the reasons for the unreliability of weather forecasts, he showed that the state of a complex dynamic system can greatly depend on the initial conditions: a slight change in one of many parameters can radically change the expected result. Lorenz called this dependence the butterfly effect: “The fluttering of a moth’s wings in Beijing today could cause a hurricane in New York in a month.” His work on the general circulation of the atmosphere brought him fame. Studying the system of equations with three variables describing the process, Lorenz graphically displayed the results of his analysis: the lines of the graph represent the coordinates of the points determined by the solutions in the space of these variables (Fig. 1). The resulting double helix, called Lorentz attractor(or “strange attractor”), looked like something endlessly confusing, but always located within certain boundaries and never repeating itself. Movement in an attractor is abstract (the variables can be speed, density, temperature, etc.), and yet it conveys the features of real physical phenomena, such as the movement of a water wheel, convection in a closed loop, radiation from a single-mode laser, dissipative harmonic vibrations(the parameters of which play the role of corresponding variables).

Of the thousands of publications that have made up the specialized literature on the problem of chaos, hardly any has been cited more often than Lorentz's 1963 paper “Deterministic Non-Periodic Flow”. Although computer modeling had already transformed weather forecasting from an “art to a science” at the time of this work, long-term forecasts were still unreliable and unreliable. The reason for this was the same butterfly effect.

In the same 60s, mathematician Stephen Smail from the University of California assembled a research group of young like-minded people at Berkeley. He was previously awarded the Fields Medal for his outstanding research in topology. Smale studied dynamic systems, in particular nonlinear chaotic oscillators. To reproduce all the disorder of the van der Pol oscillator in phase space, he created a structure known as a “horseshoe” - an example of a dynamical system that has chaotic dynamics.

“Horseshoe” (Fig. 2) is a precise and visible image of a strong dependence on initial conditions: you will never guess where the starting point will be after several iterations. This example was the impetus for the invention of “Anosov diffeomorphisms” by the Russian mathematician, a specialist in the theory of dynamical systems and differential equations, differential geometry and topology, Dmitry Viktorovich Anosov. Later, from these two works the theory of hyperbolic dynamical systems grew. It took a decade before Smale's work came to the attention of other disciplines. “When this did happen, physicists realized that Smail had turned an entire branch of mathematics to face real world» .

In 1972, University of Maryland mathematician James York read Lorentz's above-mentioned paper and it struck him as a surprise. York saw a living physical model in the article and considered it his sacred duty to convey to physicists what they had not seen in the works of Lorentz and Smail. He forwarded a copy of Lorenz's article to Smail. He was amazed to discover that an unknown meteorologist (Lorentz) ten years earlier had discovered the disorder that he himself had once considered mathematically incredible, and sent copies to all his colleagues.

Biologist Robert May, a friend of York's, was studying changes in animal populations. May followed in the footsteps of Pierre Verchlust, who back in 1845 drew attention to the unpredictability of changes in the number of animals and came to the conclusion that the population growth rate is not a constant value. In other words, the process turns out to be nonlinear. May tried to capture what happens to a population when fluctuations in the growth coefficient approach a certain critical point (bifurcation point). By varying the values ​​of this nonlinear parameter, he discovered that fundamental changes were possible in the very essence of the system: an increase in the parameter meant an increase in the degree of nonlinearity, which, in turn, changed not only the quantitative, but also the qualitative characteristics of the result. Such an operation influenced both the final value of the population size that was in equilibrium and its ability to generally achieve the latter. Under certain conditions, periodicity gave way to chaos, oscillations that never died down.

York mathematically analyzed the described phenomena in his work, proving that in any one-dimensional system the following happens: if a regular cycle appears with three waves (smooth rises and falls in the values ​​of any parameter), then in the future the system will begin to demonstrate how regular cycles of any other duration , and completely chaotic. (As it turned out several years after the article was published on international conference in East Berlin, the Soviet (Ukrainian) mathematician Alexander Nikolaevich Sharkovsky was somewhat ahead of York in his research). York wrote an article for the famous scientific publication American Mathematical Monthly. However, York achieved more than just a mathematical result: he demonstrated to physicists that chaos is omnipresent, stable and structured. He gave reason to believe that complex systems, traditionally described by difficult-to-solve differential equations, could be represented using visual graphs.

May tried to draw the attention of biologists to the fact that animal populations experience more than just ordered cycles. On the way to chaos, a whole cascade of period doubling arises. It is at the bifurcation points that a certain increase in the fertility of individuals could lead, for example, to a change in the four-year population cycle gypsy moth eight-year-old. American Mitchell Feigenbaum decided to start by calculating the exact values ​​of the parameter that gave rise to such changes. His calculations showed that it did not matter what the initial population was - it was still steadily approaching the attractor. Then, with the first doubling of periods, the attractor, like a dividing cell, bifurcated. Then the next multiplication of periods occurred, and each attractor point began to divide again. The number - an invariant obtained by Feigenbaum - allowed him to predict exactly when this would happen. The scientist discovered that he could predict this effect for the most complex attractor - at two, four, eight points... Speaking in the language of ecology, he could predict the actual number that is achieved in populations during annual fluctuations. So Feigenbaum discovered the “period doubling cascade” in 1976, building on May’s work and his research on turbulence. His theory reflected a natural law that applies to all systems experiencing a transition from an ordered state to chaos. York, May and Feigenbaum were the first in the West to fully understand the importance of period doubling and were able to convey this idea to everyone scientific community. May stated that chaos must be taught.

Soviet mathematicians and physicists advanced in their research independently of their foreign colleagues. The study of chaos began with the work of A. N. Kolmogorov in the 50s. But the ideas of foreign colleagues did not go unnoticed. The pioneers of chaos theory are considered to be the Soviet mathematicians Andrei Nikolaevich Kolmogorov and Vladimir Igorevich Arnold and the German mathematician Jurgen Moser, who built the chaos theory called KAM (Kolmogorov-Arnold-Moser theory). Another of our outstanding compatriots, the brilliant physicist and mathematician Yakov Grigorievich Sinai, applied considerations similar to the “Smale horseshoe” in thermodynamics. As soon as Western physicists became acquainted with Lorentz’s work in the 70s, it became famous in the USSR. In 1975, while York and May were still making considerable efforts to gain the attention of their colleagues, Sinai and his comrades organized a research group at Gorky to study this problem.

In the last century, when narrow specialization and separation between various disciplines became the norm in science, mathematicians, physicists, biologists, chemists, physiologists, and economists struggled with similar problems without hearing each other. Ideas that require a change in the usual worldview always find it difficult to find their way. However, it gradually became clear that such things as changes in animal populations, fluctuations in market prices, weather changes, the distribution of celestial bodies by size, and much, much more, are subject to the same patterns. “Awareness of this fact forced managers to reconsider their attitude towards insurance, astronomers - to look at solar system, politicians - to change their opinion about the causes of armed conflicts."

By the mid-80s the situation had changed greatly. The ideas of fractal geometry united scientists who were puzzled by their own observations and did not know how to interpret them. For chaos researchers, mathematics became an experimental science, and computers replaced laboratories. Graphic images have become of paramount importance. The new science gave the world a special language, new concepts: phase portrait, attractor, bifurcation, section of phase space, fractal...

Benoit Mandelbrot, drawing on the ideas and work of his predecessors and contemporaries, showed that such complex processes as the growth of a tree, the formation of clouds, variations in economic characteristics or the size of animal populations are governed by essentially similar laws of nature. These are certain patterns according to which chaos lives. From the point of view of natural self-organization, they are much simpler than the artificial forms familiar to civilized people. They can only be considered complex in the context of Euclidean geometry, since fractals are determined by specifying an algorithm, and therefore can be described using a small amount of information.

Fractal geometry of nature

Let's try to figure out what a fractal is and what it is eaten with. And you can actually eat some of them, such as the typical representative shown in the photograph.

Word fractal comes from Latin fractus - crushed, broken, broken into pieces. A fractal is a mathematical set that has the property of self-similarity, i.e., scale invariance.

The term "fractal" was coined by Mandelbrot in 1975 and gained widespread popularity with the publication of his 1977 book The Fractal Geometry of Nature. “Give the monster some cozy, homely name, and you will be surprised how much easier it will be to tame it!” - said Mandelbrot. This desire to make the objects under study (mathematical sets) close and understandable led to the birth of new mathematical terms, such as dust, cottage cheese, serum, clearly demonstrating their deep connection with natural processes.

The mathematical concept of a fractal identifies objects that have structures of various scales, both large and small, and thus reflects the hierarchical principle of organization. Of course, different branches of a tree, for example, cannot be exactly aligned with each other, but they can be considered similar in statistically speaking. In the same way, the shapes of clouds, the outlines of mountains, the line of the sea coast, the pattern of flames, the vascular system, ravines, lightning, viewed at different scales, look similar. Although this idealization may be a simplification of reality, it significantly increases the depth of the mathematical description of nature.

Mandelbrot introduced the concept of “natural fractal” to denote natural structures that can be described using fractal sets. These natural objects include an element of chance. The theory created by Mandelbrot makes it possible to quantitatively and qualitatively describe all those forms that were previously called tangled, wavy, rough, etc.

The dynamic processes discussed above, the so-called feedback processes, arise in various physical and mathematical problems. They all have one thing in common - competition between several centers (called “attractors”) for dominance on the plane. The state in which the system finds itself after a certain number of iterations depends on its “starting place.” Therefore, each attractor corresponds to a certain region of initial states, from which the system will necessarily fall into the final state under consideration. Thus, the phase space of the system (the abstract space of parameters associated with a specific dynamic system, the points in which uniquely characterize all its possible states) is divided into areas of attraction attractors. There is a peculiar return to the dynamics of Aristotle, according to which each body tends to its destined place. Simple boundaries between " adjacent territories" as a result of such rivalry rarely arise. It is in this border area that the transition from one form of existence to another occurs: from order to chaos. General form The expression for the dynamic law is very simple: x n+1 → f x n C . The whole difficulty lies in the nonlinear relationship between the initial value and the result. If you start an iterative process of the indicated type from some arbitrary value \(x_0\), then its result will be the sequence \(x_1\), \(x_2\), ..., which will either converge to some limiting value \(X\) , striving for a state of rest, it will either come to a certain cycle of values ​​that will be repeated again and again, or it will behave erratically and unpredictably all the time. It was precisely such processes that were studied by French mathematicians Gaston Julia and Pierre Fateau during the First World War.

Studying the sets they discovered, Mandelbrot in 1979 came to depict an image on the complex plane, which is, as will be clear from what follows, a kind of table of contents for a whole class of forms called Julia sets. The Julia set is a set of points arising as a result of iteration of the quadratic transformation: x n → x n−1 2 + C, the dynamics in the vicinity of which are unstable with respect to small perturbations of the initial position. Each successive value of \(x\) is obtained from the previous one; complex number \(C\) is called control parameter. The behavior of the sequence of numbers depends on the parameter \(C\) and the starting point \(x_0\). If we fix \(C\) and change \(x_0\) in the field of complex numbers, we get the Julia set. If we fix \(x_0\) = 0 and change \(C\), we obtain the Mandelbrot set (\(M\)). It tells us what kind of Julia set we should expect for a particular choice of \(C\). Each complex number \(C\) either belongs to the region \(M\) (black in Fig. 3) or does not. \(C\) belongs to \(M\) if and only if the “critical point” \(x_0\) = 0 does not tend to infinity. The set \(M\) consists of all points \(C\) that are associated with connected Julia sets, but if a point \(C\) lies outside the set \(M\), the Julia set associated with it is disconnected. The boundary of the set \(M\) determines the moment of the mathematical phase transition for the Julia sets x n → x n−1 2 + C . When the parameter \(C\) leaves \(M\), the Julia sets lose their connectivity, figuratively speaking, explode and turn into dust. The qualitative jump that occurs at the boundary \(M\) also affects the region adjacent to the boundary. The complex dynamic structure of the boundary region can be approximately shown by painting (conditionally) in different colors the zones with the same time of “running away to infinity of the initial point \(x_0\) = 0”. Those values ​​of \(C\) (one shade) for which the critical point requires a given number of iterations to be outside the circle of radius \(N\) fill the gap between the two lines. As we approach the boundary \(M\), the required number of iterations increases. The point is increasingly forced to wander along winding paths near the Julia set. The Mandelbrot set embodies the process of transition from order to chaos.

It is interesting to trace the path that Mandelbrot took to his discoveries. Benoit was born in Warsaw in 1924; in 1936 the family emigrated to Paris. After graduating from the Ecole Polytechnique and then the University in Paris, Mandelbrot moved to the USA, where he also studied at the California Institute of Technology. In 1958, he took a job at IBM's Yorktown research center. Despite the purely applied activities of the company, his position allowed him to conduct research in a variety of areas. Working in the field of economics, the young specialist began studying cotton price statistics over a long period of time (more than 100 years). Analyzing the symmetry of long-term and short-term price fluctuations, he noticed that these fluctuations during the day seemed random and unpredictable, but the sequence of such changes did not depend on the scale. To solve this problem, he first used his developments of the future fractal theory and graphic display processes under study.

Interested in a variety of areas of science, Mandelbrot turned to mathematical linguistics, then it was the turn of game theory. He also proposed his own approach to economics, pointing out the orderliness of scale in the spread of small and large cities. While studying a little-known work by the English scientist Lewis Richardson, published after the author's death, Mandelbrot encountered the phenomenon of the coastline. In the article "How long is the UK coastline?" he explores in detail this question, which few people have thought about before, and comes to unexpected conclusions: the length of the coastline is... infinity! The more accurately you try to measure it, the greater its value becomes!

To describe such phenomena, Mandelbrot came up with the idea of ​​​​dimension. The fractal dimension of an object serves as a quantitative characteristic of one of its features, namely, its filling of space.

The definition of the concept of fractal dimension dates back to the work of Felix Hausdorff, published in 1919, and was finally formulated by Abram Samoilovich Besikovich. Fractal dimension is a measure of detail, fracture, and unevenness of a fractal object. In Euclidean space, the topological dimension is always determined by an integer (the dimension of a point is 0, a line is 1, a plane is 2, a volumetric body is 3). If you trace, for example, the projection onto the plane of motion of a Brownian particle, which seems to consist of straight segments, i.e., have dimension 1, it will very soon turn out that its trace fills almost the entire plane. But the dimension of the plane is 2. The discrepancy between these quantities gives us the right to classify this “curve” as a fractal, and call its intermediate (fractional) dimension fractal. If we consider chaotic movement particles in the volume, the fractal dimension of the trajectory will be greater than 2, but less than 3. Human arteries, for example, have a fractal dimension of approximately 2.7. Ivanov’s results mentioned at the beginning of the article related to the measurement of the pore area of ​​silica gel, which cannot be interpreted within the framework of conventional Euclidean concepts, find a reasonable explanation when using the theory of fractals.

So, from a mathematical point of view, a fractal is a set for which the Hausdorff-Besicovich dimension is strictly greater than its topological dimension and can be (and most often is) fractional.

It must be especially emphasized that the fractal dimension of an object does not describe its shape, and objects that have the same dimension, but generated by different formation mechanisms, are often completely different from each other. Physical fractals are rather statistically self-similar.

Fractional measurement allows the calculation of characteristics that cannot be clearly determined otherwise: the degree of unevenness, discontinuity, roughness or instability of an object. For example, a winding coastline, despite its immeasurable length, has a roughness that is unique to it. Mandelbrot indicated ways to calculate fractional measurements of objects in the surrounding reality. In creating his geometry, he put forward a law about disordered forms that occur in nature. The law stated: the degree of instability is constant at different scales.

A special type of fractals are time fractals. In 1962, Mandelbrot was faced with the task of eliminating noise in telephone lines that was causing problems for computer modems. The quality of signal transmission depends on the probability of errors occurring. Engineers struggled with the problem of reducing noise, coming up with puzzling and expensive techniques, but did not get impressive results. Based on the work of the founder of set theory, Georg Cantor, Mandelbrot showed that the emergence of noise - the product of chaos - cannot be avoided in principle, therefore the proposed methods of dealing with them will not bring results. In search of a pattern in the occurrence of noise, he receives “Cantor dust” - a fractal sequence of events. Interestingly, the distribution of stars in the Galaxy follows the same patterns:

“Matter”, uniformly distributed along the initiator (a single segment of the time axis), is exposed to a centrifugal vortex, which “sweeps” it to the extreme thirds of the interval... Curdling can be called any cascade of unstable states, ultimately leading to a thickening of matter, and the term cottage cheese can determine the volume within which a certain physical characteristic becomes - as a result of curdling - extremely concentrated.

Chaotic phenomena such as atmospheric turbulence, crustal mobility, etc., exhibit similar behavior at different time scales, just as scale-invariant objects exhibit similar structural patterns at different spatial scales.

As an example, we will give several typical situations where it is useful to use ideas about fractal structure. Columbia University professor Christopher Scholz specialized in studying the shape and structure of the Earth's solid matter and studied earthquakes. In 1978, he read Mandelbrot's book Fractals: Shape, Randomness and Dimension » and attempted to apply the theory to the description, classification and measurement of geophysical objects. Scholz found that fractal geometry provided science with effective method descriptions of the specific lumpy landscape of the Earth. The fractal dimension of the planet's landscapes opens the door to understanding its most important characteristics. Metallurgists have discovered the same thing at another scale - on the surfaces of different types of steel. In particular, the fractal dimension of a metal surface often allows one to judge its strength. A huge number of fractal objects produce the phenomenon of crystallization. The most common type of fractals that arise during crystal growth are dendrites; they are extremely widespread in living nature. Ensembles of nanoparticles often demonstrate the implementation of "Lewy dust". These assemblies combine with absorbed solvent to form transparent compacts—Lewy glasses, potentially important photonics materials.

Since fractals are expressed not in primary geometric forms, but in algorithms, sets of mathematical procedures, it is clear that this area of ​​mathematics began to develop by leaps and bounds along with the advent and development of powerful computers. Chaos, in turn, gave rise to new computer technologies, special graphics technology that is capable of reproducing amazing structures of incredible complexity generated by certain types of disorder. In the age of the Internet and personal computers, what was quite difficult in Mandelbrot's time has become easily accessible to anyone. But the most important thing in his theory was, of course, not the creation beautiful pictures, but the conclusion is that this mathematical apparatus is suitable for describing complex natural phenomena and processes that had not previously been considered in science at all. The repertoire of algorithmic elements is inexhaustible.

Once you master the language of fractals, you can describe the shape of a cloud as clearly and simply as an architect describes a building using drawings that use the language of traditional geometry.<...>Only a few decades have passed since Benoit Mandelbrot declared: “The geometry of nature is fractal!” Today we can already assume much more, namely that fractality is the primary principle of construction of all natural objects without exception.

In conclusion, let me present to your attention a set of photographs illustrating this conclusion, and fractals constructed using computer program Fractal Explorer. Our next article will be devoted to the problem of using fractals in crystal physics.

Post Scriptum

From 1994 to 2013, a unique work of domestic scientists, “Atlas of Temporal Variations in Natural Anthropogenic and Social Processes,” was published in five volumes - an unparalleled source of materials that includes monitoring data of space, biosphere, lithosphere, atmosphere, hydrosphere, social and technogenic spheres and spheres related to human health and quality of life. The text provides details of the data and the results of their processing, and compares the features of the dynamics of time series and their fragments. A unified presentation of results makes it possible to obtain comparable results to identify common and individual features of the dynamics of processes and cause-and-effect relationships between them. Experimental material shows that processes in different areas are, firstly, similar, and secondly, more or less connected with each other.

So, the atlas summarized the results of interdisciplinary research and presented a comparative analysis of completely different data over a wide range of time and space. The book shows that “the processes occurring in the earthly spheres are caused by a large number interacting factors that cause different responses in different areas (and at different times), which speaks to “the need for an integrated approach to the analysis of geodynamic, space, social, economic and medical observations.” It remains to express hope that this fundamentally important work will be continued.

As it became clear in last decades(in connection with the development of the theory of self-organization), self-similarity is found in a wide variety of objects and phenomena. For example, self-similarity can be observed in the branches of trees and shrubs, during the division of a fertilized zygote, snowflakes, ice crystals, during the development economic systems, in the structure of mountain systems, clouds.

All of the listed objects and others similar to them are fractal in structure. That is, they have the properties of self-similarity, or scale invariance. This means that some fragments of their structure are strictly repeated at certain spatial intervals. It is obvious that these objects can be of any nature, and their appearance and shape remain unchanged regardless of scale. Both in nature and in society, self-repetition occurs on a fairly large scale. Thus, the cloud repeats its ragged structure from 10 4 m (10 km) to 10 -4 m (0.1 mm). Branching is repeated in trees from 10 -2 to 10 2 m. Collapsed materials that generate cracks also repeat their self-similarity on several scales. A snowflake that falls on your hand melts. During the period of melting, transition from one phase to another, a snowflake-drop is also a fractal.

A fractal is an object of infinite complexity, allowing you to see no less detail up close than from afar. A classic example of this is the Earth. From space it looks like a ball. As we approach it, we will discover oceans, continents, coastlines and mountain ranges. Later, finer details will appear: a piece of earth on the surface of the mountain, as complex and uneven as the mountain itself. Then tiny particles of soil will appear, each of which is itself a fractal object

A fractal is a nonlinear structure that maintains self-similarity when scaled up or down infinitely. Only at short lengths does nonlinearity transform into linearity. This is especially clearly manifested in the mathematical procedure of differentiation.

Thus, we can say that fractals as models are used in the case when a real object cannot be represented in the form of classical models. This means that we are dealing with nonlinear relationships and the non-deterministic nature of data. Nonlinearity in the ideological sense means multivariate development paths, the presence of a choice from alternative paths and a certain pace of evolution, as well as the irreversibility of evolutionary processes. In the mathematical sense, nonlinearity is a certain type of mathematical equations (nonlinear differential equations) containing the desired quantities in powers greater than one or coefficients depending on the properties of the medium. That is, when we apply classical models (for example, trend, regression, etc.), we say that the future of the object is uniquely determined. And we can predict it by knowing the past of the object (initial data for modeling). And fractals are used in the case when an object has several development options and the state of the system is determined by the position in which it is currently located. That is, we are trying to simulate chaotic development.

When they talk about the determinism of a certain system, they mean that its behavior is characterized by an unambiguous cause-and-effect relationship. That is, knowing the initial conditions and the law of motion of the system, you can accurately predict its future. It is this idea of ​​motion in the Universe that is characteristic of classical, Newtonian dynamics. Chaos, on the contrary, implies a disorderly, random process, when the course of events can neither be predicted nor reproduced.

Chaos is generated by the own dynamics of a nonlinear system - its ability to exponentially quickly separate arbitrarily close trajectories. As a result, the shape of the trajectories depends very much on the initial conditions. When studying systems that, at first glance, develop chaotically, the theory of fractals is often used, because It is this approach that allows us to see a certain pattern in the occurrence of “random” deviations in the development of the system.

The study of natural fractal structures gives us the opportunity to better understand the processes of self-organization and development of nonlinear systems. We have already found out that natural fractals of various, winding lines are found all around us. This is the seashore, trees, clouds, lightning strike, metal structure, human nervous or vascular system. These intricate lines and rough surfaces came into view scientific research, because nature showed us a completely different level of complexity than in ideal geometric systems. The structures under study turned out to be self-similar in spatiotemporal terms. They endlessly self-reproduced and repeated themselves on various length and time scales. Any nonlinear process ultimately leads to a fork. In this case, the system, at the branching point, chooses one path or another. The trajectory of the system’s development will look like a fractal, that is, a broken line, the shape of which can be described as a branched, intricate path that has its own logic and pattern.

The branching of a system can be compared to the branching of a tree, where each branch corresponds to a third of the entire system. Branching allows a linear structure to fill a volumetric space, or, to put it more precisely: a fractal structure coordinates different spaces. A fractal can grow, filling the surrounding space, just as a crystal grows in a supersaturated solution. In this case, the nature of the branching will be associated not with chance, but with a certain pattern.

The fractal structure is self-similarly repeated at other levels, at more high level organization of human life, for example, at the level of self-organization of a group or team. Self-organization of networks and forms moves from the micro level to the macro level. Taken together, they represent an integral unity, where the whole can be judged by the part. This course work examines the fractal properties of social processes as an example, which indicates the universality of the theory of fractals and its loyalty to different areas of science.

It is concluded that a fractal is a way of organized interaction of spaces of different dimensions and nature. To the above, it should be added that not only spatial, but also temporal. Then even human brain And neural networks will represent a fractal structure.

Nature loves fractal forms. A fractal object has a spreading, discharged structure. When observing such objects with increasing magnification, one can see that they exhibit a pattern that repeats at different levels. We have already said that a fractal object can look exactly the same regardless of whether we observe it on a meter, millimeter or micron scale (1:1,000,000 fractions of a meter scale). The property of symmetry of fractal objects manifests itself in invariance with respect to scale. Fractals are symmetrical about the center of stretching or scaling, just as round bodies are symmetrical about the axis of rotation.

A favorite image of nonlinear dynamics is fractal structures, in which, with a change in scale, the description is built according to the same rule. In real life, the implementation of this principle is possible with slight variations. For example, in physics, when moving from level to level (from atomic to nuclear processes, from nuclear to elementary particles) patterns, models, methods of description change. We observe the same thing in biology (the population level of an organism, tissue, cell, etc.) The future of synergetics depends on the extent to which nonlinear science can help in describing this structural heterogeneity and various “interlevel” phenomena. Currently, most scientific disciplines do not have reliable fractal conceptual models.

Today, developments within the framework of the theory of fractals are carried out in any special science - physics, sociology, psychology, linguistics, etc. Then society, social institutions, language, and even thought are fractals.

In the discussions that have unfolded in recent years among scientists and philosophers around the concept of fractals, the most controversial issue is as follows: is it possible to talk about the universality of fractals, that every natural object contains a fractal or goes through a fractal stage? Two groups of scientists have emerged to answer the question this question in exactly the opposite way. The first group (“radicals”, innovators) supports the thesis about the universality of fractals. The second group (“conservatives”) denies this thesis, but still claims that not every object of Nature has a fractal, but in every area of ​​Nature a fractal can be found.

Modern science has quite successfully adapted the theory of fractals for different fields of knowledge. Thus, in economics, the theory of fractals is used in the technical analysis of financial markets that exist in developed countries world for more than one hundred years. For the first time, it is possible to predict the future behavior of stock prices if its direction for some recent period is known, noted C. Dow. In the nineties of the 19th century, having published a number of articles, Dow noted that stock prices are subject to cyclical fluctuations: after a long rise, there is a long fall, then again rise and fall.

In the middle of the 20th century, when the entire scientific world was captivated by the newly emerging theory of fractals, another famous American financier R. Elliot proposed his theory of the behavior of stock prices, which was based on the use of the theory of fractals. Elliott proceeded from the fact that the geometry of fractals occurs not only in living nature, but also in social processes. He also included trading in shares on the stock exchange as a social process.

The basis of the theory is the so-called wave diagram. This theory makes it possible to predict the further behavior of a price trend, based on knowledge of the background of its behavior and following the rules for the development of mass psychological behavior.

The theory of fractals has also found application in biology. Many, if not all, biological structures and systems of plants, animals and humans have a fractal nature, some semblance of it: nervous system, pulmonary system, circulatory and lymphatic systems, etc. Evidence has emerged that the development malignant tumor It also follows the fractal principle. Taking into account the principle of self-affinity and congruence of a fractal, a number of intractable problems of evolution can be explained organic world. Fractal objects are also characterized by such a feature as the manifestation of complementarity. Complementarity in biochemistry is the mutual correspondence in the chemical structure of two macromolecules, ensuring their interaction - the pairing of two strands of DNA, the connection of an enzyme with a substrate, an antigen with an antibody. Complementary structures fit together like a key to a lock (Encyclopedia of Cyril and Methodius). Polynucleotide chains of DNA have this property.

Some of the most powerful applications of fractals lie in computer graphics. Firstly, this is fractal compression of images, and secondly, the construction of landscapes, trees, plants and the generation of fractal textures. At the same time, to compress and record information, a self-similar increase in the fractal is necessary, and to read it, accordingly, a self-similar increase is required.

The advantages of fractal image compression algorithms are the very small size of the packed file and short image recovery time. Fractal packed images can be scaled without causing pixelation. But the compression process takes a long time and sometimes lasts for hours. The fractal lossy packaging algorithm allows you to set the compression level, similar to the jpeg format. The algorithm is based on search large parts images like some small parts. And only information about the similarity of one part to another is recorded in the output file. When compressing, a square grid is usually used (pieces are squares), which leads to a slight angularity when restoring the image; a hexagonal grid does not have this drawback.

Among literary works find those that have a textual, structural or semantic fractal nature. Textual fractals potentially repeat elements of text indefinitely. Textual fractals include a non-branching infinite tree, identical to themselves from any iteration (“The priest had a dog...”, “The parable of the philosopher who dreams that he is a butterfly who dreams that she is a philosopher who dreams...”, “The statement is false , that the statement is true, that the statement is false..."); non-branching endless texts with variations (“Peggy had a funny goose…”) and texts with extensions (“The House That Jack Built”).

In structural fractals, the layout of the text is potentially fractal. Texts with such a structure are arranged according to the following principles: a wreath of sonnets (15 poems), a wreath of wreaths of sonnets (211 poems), a wreath of wreaths of sonnets (2455 poems); “stories within a story” (“The Book of One Thousand and One Nights”, J. Pototsky “Manuscript Found in Saragossa”); prefaces that hide the authorship (U. Eco “The Name of the Rose”).