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

A fractal (lat. fractus - crushed, broken, broken) is a complex geometric figure that has the property of self-similarity, that is, it is composed of several parts, each of which is similar to the whole figure as a whole. 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 other than topological. For example, I will insert a picture of four different fractals.

Let me 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 in the everyday life of mathematicians and programmers since the mid-80s. The word "fractal" was introduced by Benoit Mandelbrot in 1975 to refer to the irregular but self-similar structures that he studied. The birth of fractal geometry is usually associated with the publication in 1977 of Mandelbrot's book The Fractal Geometry of Nature. His works used the scientific results of other scientists who worked in the period 1875-1925 in the same field (Poincaré, Fatou, Julia, Kantor, Hausdorff). But only in our time it was possible to combine their work into a single system.

There are many 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 are, of course, more specific examples of fractals from different areas. Fractals, for example, are present in complex dynamics. They are there appear naturally in the study of nonlinear dynamic systems. The most studied case is when the dynamical system is defined by iterations of a polynomial or a holomorphic system. function of a complex of variables on surface. Some of the most famous fractals of this type are the Julia set, the Mandelbrot set and the Newton basins. Below, in order, the pictures show each of the above fractals.

Another example of fractals are fractal curves. It is best to explain how to build a fractal using the example of fractal curves. One such curve is the so-called Koch Snowflake. There is a simpleprocedure for obtaining fractal curves on a plane. We 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. Shown below is a Koch snowflake (or curve).

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

The third example or kind of fractals are stochastic fractals. Such fractals include the trajectory of Brownian motion on a plane and in space, Schramm-Löwner evolutions, various types of randomized fractals, that is, fractals obtained using a recursive procedure, in 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 there is already a big list. I will not list everything, because, probably, I cannot list all of them, but I will tell about some. For example, in living nature, such fractals include our circulatory system and lungs. And also the crowns and leaves of trees. Also here you can include starfish, sea urchins, corals, sea shells, some plants, such as cabbage or broccoli. Below, several such natural fractals from wildlife are clearly shown.

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

We have considered examples and types of fractals. As for the use of fractals, they are used in various 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 of 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. Also, fractals are actively used in radio engineering, in computer science and computer technology, telecommunications and even economics. And, of course, fractal vision is actively used in contemporary art and architecture. Here is one example of fractal paintings:

And so, on 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. And I also talked about their application and demonstrated some of the fractals clearly. I hope you enjoyed this short excursion into the world of amazing and bewitching 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 these lives. More recently (on the scale of human evolution) we have learned to "tame" electricity - and now we can not 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 “imperceptible” discoveries is fractals. You have probably heard this catchy word, but do you know what it means and how many interesting things are hidden in this term?

Every person has a natural curiosity, a desire to learn about the world around him. And in this aspiration, 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 deduce some regularity. The biggest minds on the planet are busy with this task. Roughly speaking, scientists are looking for a pattern where it should not be. Nevertheless, even in chaos, one can find a connection 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 greater than the number of answers that adults have time to give. Not so long ago, looking at a branch raised from the ground, my daughter suddenly noticed that this branch, with knots and branches, itself looked like a tree. And, of course, the usual question “Why?” followed, for which the 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. Very many organic and inorganic forms in nature are formed similarly. Clouds, sea shells, the "house" of a snail, 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: the father of fractal geometry

The very word "fractal" appeared thanks to the brilliant scientist Benoît B. Mandelbrot.

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

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 his scientific career, Benoit worked at the IBM research center. At that time, the center's employees were working on data transmission over a distance. In the course of research, scientists were faced with the problem of large losses arising from noise interference. Benoit faced a difficult and very important task - to understand how to predict the occurrence of noise interference in electronic circuits when the statistical method is ineffective.

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

During his lifetime, Benoit Mandelbrot repeatedly said that he did not deal with formulas, but simply played with pictures. This man thought very figuratively, and translated any algebraic problem 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, the realization of the essence of fractals comes precisely when you begin to study drawings and think about the meaning of strange swirl patterns.

A fractal pattern does not have identical elements, but has similarity at any scale. To build such an image with a high degree of detail manually was previously simply impossible, it required great amount computing. For example, 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, then they were mentioned in the works of Leibniz and Georg Cantor.

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

Gaston Julia (always masked - WWI injury)

This French mathematician wondered what a set would look like if it were constructed from a simple formula iterated by a feedback loop. If explained “on the 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 amount of calculations - hundreds, thousands, millions. It was simply impossible to do it manually. But when powerful computing devices appeared at the disposal of 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 the classical fractal. Having processed a sequence consisting of a large number of values, Benoit transferred the results to a graph. Here's what he got.

Subsequently, this image was colored (for example, one of the ways to color 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 enter the same river twice." It is the best suited for interpreting the geometry of fractals. No matter how detailed we examine a fractal image, we will always see a similar pattern.

Those wishing to see how an image of Mandelbrot space would look like when magnified many times over can do so by uploading an 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 well-known corporation engaged in the development of new aircraft.

In 1977, he created presentations with prototypes of flying models. Lauren was responsible for 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 to use an image of mountains as a background. Today, any schoolchild can solve such a problem, but at the end of the seventies of the last century, computers could not cope with such complex calculations - there were no graphic editors, not to mention applications for three-dimensional graphics. In 1978, Lauren accidentally saw Benoit Mandelbrot's book Fractals: Form, Randomness and Dimension in a store. What caught his attention in this book was that Benoist gave many examples of fractal forms in real life and proved that they can be described by a mathematical expression.

This analogy was chosen by the mathematician not by chance. The fact is that as soon as he published his research, he had to face a whole flurry of criticism. The main thing that his colleagues reproached him with was the uselessness of the developed theory. “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 "devil machines", which in the late seventies seemed to many to be something too complicated and unexplored to be completely trusted. Mandelbrot tried to find an obvious application of the theory of fractals, but, by and large, he did not need to do this. The followers of Benoit Mandelbrot over the next 25 years proved to be of great use to such a "mathematical curiosity", and Lauren Carpenter was one of the first to put the fractal method into practice.

Having studied 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 visualize a realistic image of the mountain system on his computer. In other words, with the help of formulas, he painted a completely recognizable mountain landscape.

The principle that Lauren used to achieve her goal was very simple. It consisted in 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 broke them up into four smaller ones and then repeated this procedure over and over again until he had a realistic mountain landscape. Thus, he managed to become the first artist to use a fractal algorithm in computer graphics to build images. As soon as it became known about the work done, enthusiasts around the world picked up this idea and began to use the fractal algorithm to simulate realistic natural forms.

One of the first 3D renderings using the fractal algorithm

Just a few years later, Lauren Carpenter was able to apply his achievements in a much larger project. The animator based them on a two-minute demo, Vol Libre, 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 at a clock speed of five megahertz, and each frame took about half an hour to draw.

Working for Lucasfilm Limited, the animator created the same 3D landscapes for the second feature 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 the same principle of generating natural objects. Terragen, Bryce, Vue and other 3D editors rely on a fractal surface and texture modeling algorithm.

⇡ Fractal antennas: less is better, but better

Over the past half century, life has changed rapidly. Most of us accept achievement modern technologies for granted. Everything that makes life more comfortable, you get used to very quickly. Rarely does anyone ask the questions “Where did this come from?” and "How does it work?". A microwave oven warms up breakfast - well, great, a smartphone allows you to talk to another person - great. This seems like an obvious possibility to us.

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

Fractal drawings fascinate with their patterns. They definitely resemble images of space objects - nebulae, galaxy clusters, and so on. Therefore, it is quite natural that when Mandelbrot voiced his theory of fractals, his research aroused increased interest among those who studied astronomy. One such amateur named Nathan Cohen, after attending a lecture by Benoit Mandelbrot in Budapest, was inspired by the idea of ​​​​practical application of the knowledge gained. True, he did it 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 Nathan's downtown Boston apartment was adamantly opposed to installing large rooftop devices. Then Nathan began to experiment with various forms of antennas, trying to get the maximum result with the minimum size. Fired up with the idea of ​​fractal forms, Cohen, as they say, randomly made one of the most famous fractals out of wire - the “Koch snowflake”. The Swedish mathematician Helge von Koch came up with this curve back in 1904. It is obtained by dividing the segment into three parts and replacing the middle segment with an equilateral triangle without a side coinciding with this segment. The definition is a bit difficult to understand, but the figure is clear and simple.

There are also other varieties 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, the future professor at Boston University realized that an antenna made according to a fractal pattern has a 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 can significantly reduce the geometric dimensions. Nathan Cohen even developed 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 firm 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.

Basically, that's what happened. True, to this day, Nathan is in a lawsuit with large corporations that illegally use his discovery to produce compact communication devices. Some well-known mobile device manufacturers, such as Motorola, have already reached a peace agreement with the inventor of the fractal antenna.

⇡ Fractal dimensions: the mind does not understand

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

The latter, like many other famous mathematicians, was very fond of communicating with children, asking them questions and getting unexpected answers. Sometimes this led to surprising results. So, for example, the nine-year-old nephew of Edward Kasner came up with the now well-known word "googol", denoting a unit with one hundred zeros. But back to fractals. The American mathematician liked to ask how long the US coastline is. After listening to the opinion of the interlocutor, Edward himself spoke the correct answer. If you measure the length on the map with broken segments, then the result will be inaccurate, because the coastline has a large number of irregularities. And what happens if you measure as accurately as possible? You will have to take into account the length of each unevenness - you will need to measure each cape, each 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 with each new irregularity.

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

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

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

What is the usual dimension is clear to anyone. If the dimension is equal to one, we get a straight line, if two - a flat figure, three - volume. However, such an understanding of dimension in mathematics does not work with fractal curves, where this parameter has a fractional value. The fractal dimension in mathematics can be conditionally considered as "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 for the application of fractal theory. With the help of fractals, you can analyze fluctuations in stock prices, explore all kinds 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, for example for image compression. And by the way, to get a beautiful fractal on your computer screen, you don't have to have a doctoral degree.

⇡ Fractal in the browser

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

There are only two simple shapes at your disposal - a square and a circle. You can add them to the canvas, scale (to scale along one of the axes, hold down the Shift key) and rotate. Overlapping on the principle of Boolean addition operations, these simplest elements form new, less trivial forms. Further, these new forms can be added to the project, and the program will repeat the generation of these images indefinitely. At any stage of working on a fractal, you can return to any component of a complex shape and edit its position and geometry. It's a lot of fun, especially when you consider that the only tool you need to be creative 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 you to fine-tune the generated fractal pattern. In cases where it is necessary to build a mathematically accurate fractal, the XaoS cross-platform editor 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 “walk” through a fractal by changing its scale. Animated movement along a fractal can be saved as an XAF file and then played back in the program itself.

XaoS can load a random set of parameters, as well as 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 quite small in size and does not require installation. Secondly, it implements the ability to define the color palette of the picture. You can choose shades in RGB, CMYK, HVS and HSL color models.

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

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

⇡ Mandelbulb3D: 3D fractal editor

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

And this fractal can be eaten

The Mandelbulb3D program can create three-dimensional objects with a similar shape. To obtain a 3D surface using the 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 various shapes. Since we often observe fractal patterns in nature, an artificially created fractal three-dimensional object seems incredibly realistic and even “alive”.

It may look like a plant, it may resemble a strange animal, a planet, or something else. This effect is enhanced by an advanced rendering algorithm that 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 amount of settings and rendering options. You can control the shades of light sources, choose the background and the level of detail of the modeled 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 structures. Incendia has texture and material editors, and a rendering engine that allows you to use volumetric fog effects and various shaders. The program has an option to save the buffer during long-term rendering, animation creation is supported.

Incendia allows you to export a fractal model to popular 3D graphics formats - OBJ and STL. Incendia includes a small Geometrica utility - a special tool for setting up the export of a fractal surface to a three-dimensional model. Using this utility, you can determine the resolution of a 3D surface, 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.

Recently, work on the Incendia project has slowed down somewhat. At the moment, the author is looking for sponsors who would help him develop the program.

If you do not have enough imagination to draw a beautiful three-dimensional fractal in this program, it does not matter. Use the parameter library, which is located in the INCENDIA_EX\parameters folder. With the help of PAR files, you can quickly find the most unusual fractal shapes, including animated ones.

⇡ Aural: how fractals sing

We usually do not talk about projects that are just being worked on, but in this case we have to make an exception, this is a very unusual application. A project called Aural came up with the same person as Incendia. True, this time the program does not visualize the fractal set, but voices it, turning it into electronic music. The idea is very interesting, especially considering the unusual properties of fractals. Aural is an audio editor that generates melodies using fractal algorithms, that is, in fact, it is an audio synthesizer-sequencer.

The sequence of sounds given out by this program is unusual and ... beautiful. It may well come in handy for writing modern rhythms and, in our opinion, is especially well suited for creating soundtracks for the intros of television and radio programs, as well as "loops" of background music for computer games. Ramiro has not yet provided demo version of his program, but promises that when he does, in order to work with Aural, he will not need to study the theory of fractals - just play around with the parameters of the algorithm for generating a sequence of notes. Listen to how fractals sound, and.

Fractals: musical pause

In fact, fractals can help write music even without software. But this can only be done by someone who is truly imbued with the idea of ​​natural harmony and at the same time has not turned into an unfortunate “nerd”. It makes sense to take a cue from a musician named Jonathan Coulton, who, among other things, writes compositions for Popular Science magazine. And unlike other artists, 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 work to others, as well as its modification (creation of derivative works) in order to adapt it to your needs.

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, the ideal builder and engineer. It is arranged very logically, and if somewhere we do not see patterns, 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 speaker systems in the form of a shell, create antennas with snowflake geometry, and so on. We are sure that fractals still keep a lot of secrets, and many of them have yet to be discovered by man.

Fractals have been known for almost a century, are well studied and have numerous applications in life. However, this phenomenon is based on a very simple idea: an infinite number of figures in beauty and variety can be obtained from relatively simple structures using just two operations - copying and scaling.

What do a tree, a seashore, a cloud, or 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 the structure that is inherent in all the listed objects: they are self-similar. From the branch, as well as from the trunk of a tree, smaller processes depart, from them - even smaller ones, etc., that is, a branch is similar to the whole tree. The circulatory system is arranged in a similar way: arterioles depart from the arteries, and from them - the smallest capillaries through which oxygen enters organs and tissues. Let's look at satellite images of the sea coast: we will see bays and peninsulas; let's take a 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 remains similar to itself when zoomed in. American mathematician 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. Usually, a fractal is a geometric figure that satisfies one or more of the following properties: It has a complex structure at any magnification (unlike, for example, a straight line, any part of which is the simplest geometric figure - a segment). It is (approximately) self-similar. It has a fractional Hausdorff (fractal) dimension, which is larger than the topological one. Can be built with recursive procedures.

Geometry and Algebra

The study of fractals at the turn of the 19th and 20th centuries was more episodic than systematic, because earlier mathematicians mainly studied “good” objects that could be investigated using general methods and theories. In 1872, German mathematician Karl Weierstrass builds 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 it is quite simple to draw it. It turned out that it has the properties of a fractal. One variation 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, in which another fractal is described - the Lévy C-curve. All these fractals listed above can be conditionally attributed to 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 almost two hundred pages of memoir dedicated to iterations of complex rational functions, which describes Julia sets, a whole family of fractals closely related to the Mandelbrot set. This work was awarded the prize of the French Academy, but it did not contain a single illustration, so it was impossible to appreciate the beauty of the discovered objects. Despite the fact that this work made Julia famous among the mathematicians of the time, it was quickly forgotten. Again, attention 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 measurements) 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, in three-dimensional space by three coordinates.
From a more general mathematical point of view, dimension can be defined as follows: an increase in linear dimensions, say, twice, 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 (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, for the construction of which the unit segment is divided into three equal parts and replaced average interval an equilateral triangle without this segment. With an increase in the linear dimensions of the minimum segment three times, the length of the Koch curve increases in log (4) / log (3) ~ 1.26. That is, the dimension of the Koch curve is fractional!

Science and art

In 1982, Mandelbrot's book "The 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 made the main emphasis in his presentation not on ponderous formulas and mathematical constructions, but on the geometric intuition of readers. Thanks to computer generated illustrations and historical stories, with which the author skillfully diluted the scientific component of the monograph, the book became a bestseller, and the 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 trend in art appeared - fractal painting, and almost any computer owner could do it. Now on the Internet you can easily find many sites dedicated 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. With it, or rather, with an attempt to measure its length, one interesting story, which formed the basis of Mandelbrot's scientific article, and is also described in his book "The Fractal Geometry of Nature". We are talking about an experiment that was set up 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 between the two warring countries. When he collected data for numerical experiments, he found that in different sources the data on the common border of Spain and Portugal differ greatly. This led him to the following discovery: the length of the country's borders depends on the ruler with which we measure them. The smaller the scale, the longer the border will be. This is due to the fact that at higher magnification it becomes possible to take into account more and more bends of the coast, which were previously ignored due to the roughness of measurements. And if, with each zoom, previously unaccounted bends of lines are opened, then it turns out that the length of the borders is infinite! True, in fact this does not 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 by a fragment taken in a suitable scale - this is the first iteration of the construction. Then, in the resulting figure, some parts again change to figures similar to a fragment, and so on. If we continue this process indefinitely, then in the limit we get a fractal.

Consider this process using the example of the Koch curve (see sidebar on the previous page). Any curve can be taken as the basis of the Koch curve (for the Koch snowflake, this is a triangle). But we confine ourselves to the simplest case - a segment. The fragment is a broken line shown on the top of 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, and so on. The figure shows the first four steps of this process.

The language of mathematics: dynamic (algebraic) fractals

Fractals of this type arise in the study of nonlinear dynamical systems (hence the name). The behavior of such a system can be described by a complex nonlinear function (polynomial) f(z). Let us take some initial point z0 on the complex plane (see sidebar). Now consider such an infinite sequence of numbers on the complex plane, each 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 number of fixed values; more complex options are 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. Over complex numbers, the basic mathematical operations 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 along the ordinate axis, while 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 character of 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: for an arbitrarily small displacement, the nature of their behavior changes dramatically (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 are "Menger's sponge", "Sierpinski's pyramid" and others.
The family of dragons is also referred to constructive fractals. They are sometimes referred to by the name of the discoverers as the "dragons of Heiwei-Harter" (they resemble Chinese dragons in their shape). There are several ways to construct this curve. The simplest and most obvious of them is this: you need to take a sufficiently long strip of paper (the thinner the paper, the better), and bend it in half. Then again bend it in half in the same direction as the first time. After several repetitions (usually after five or six folds the strip becomes too thick to be carefully bent further), you need to straighten the strip back, and try to form 90˚ angles at the folds. Then the curve of the dragon will turn out in profile. Of course, this will only be an approximation, like all our attempts to depict fractal objects. The computer allows you to depict many more steps in this process, 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 us construct a sequence of this function with z0=0, depending on the parameter c, it can diverge to infinity or remain bounded. Moreover, all values ​​of c for which this sequence is bounded 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 non-intersecting parts, with some additional conditions).

fractals and life

Nowadays, the theory of fractals is widely used in various fields 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 (here, the self-similarity property of fractals is mainly used - after all, in order to remember a small fragment of a drawing and transformations with which you can get the rest of the parts, it takes much less memory than to store the entire file). By adding random perturbations to the formulas that define the fractal, one can obtain stochastic fractals that very plausibly convey some real objects - relief elements, the surface of water bodies, 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, they began to produce antennas that have a fractal shape. Taking up little space, they provide quite high-quality signal reception. Economists use fractals to describe currency fluctuation curves (this property was discovered by Mandelbrot over 30 years ago). This concludes this short excursion into the world of fractals, amazing in its beauty and diversity.

Back forward

Attention! The slide preview is for informational purposes only and may not represent the full extent of the presentation. If you are interested in this work, please download the full version.

Authors:
Bekbulatova Alina,
Getmanova Sofia

Leaders:
Mogutova Tatyana Mikhailovna,
Deryushkina Oksana Valerievna

Introduction.

Theoretical part of the project:

• The history of the development of fractal geometry.
• The concept of a fractal.
• Types of fractals:

a) geometric fractals, examples of geometric fractals;
b) algebraic fractals, examples of algebraic fractals;
c) stochastic fractals, examples.

• natural fractals.
• Practical application of fractals:
• in literature;
• in telecommunications;
• in medicine;
• in architecture;
• in design;
• in economics;
• games, movies, music
• in natural sciences
• in physics;
• in biology
• fractals for housewives
• modern paintings- fractal graphics.
• Fractal graphics.
• The role of fractal geometry in life is a hymn to fractals!

Practical part of the project

• Creation of the scientific work "Journey into the world of fractals"
• Placement on the Internet.
• Participation in olympiads, competitions.
• Create your own fractals.
• Creation of the brochure "The Wonderful World of Fractals"
• Holding the festival “The Amazing World of Fractals.

Introduction

Geometry is often referred to as cold and dry. One of the reasons is its inability to describe everything that surrounds us: the shape of a cloud, a mountain, a tree, or a seashore. Clouds are not spheres, mountains are not cones, coastlines are not circles, and the crust is not smooth, and lightning does not travel in a straight line. With great joy for us, we learned that in the modern world there is a new geometry - the geometry of fractals.

The discovery of fractals has revolutionized not only geometry, but also physics, chemistry, biology, in all areas of our lives.

Project relevance:

• The role of fractals in the modern world is quite large
• Convincing arguments in favor of the relevance of the study of fractals is the breadth of their application.

Research hypothesis:

Fractal geometry is a modern, very interesting area of ​​human knowledge. The appearance of fractal geometry is evidence of the ongoing evolution of man and the expansion of his ways of knowing the world.

Objective of the project:

To study the theory of fractals to create the scientific work "The Amazing World of Fractals" and the development and implementation on a computer of algorithms for drawing fractals on a plane.

Project objectives:

• To get acquainted with the history of the emergence and development of fractal geometry;
• To study the types of fractals, their application in the modern world.
• Run programs for creating fractals in Pascal and Logo programming languages
• Create a scientific work on fractals, publish it on the Internet.
• Create a brochure "The Wonderful World of Fractals"
• To hold the festival "The Amazing World of Fractals" in order to familiarize school students with the results of our work.

We worked on the project for 4 months.

The main stages of our work:

• Gathering the necessary information: using the Internet, books, publications on this topic. (2 weeks)
• Sorting information by topic: systematization and determination of the order of writing the work. The work took 2 weeks.
• Compilation of a text work: writing a text, partial design of systematized information. It took one month.
• Creating a presentation: compressing systematized information, determining the structure of the presentation, its creation and design, and took place within a month.
• Learning a fractal creation program and creating your own fractals in the Pascal and Logo programming languages ​​(until today)

Theoretical part of the project

We have studied the history of the creation of fractal geometry.

Interest in fractal objects revived in the mid-70s of the 20th century.

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, Kantor, Hausdorff But only in our time it was possible to combine their works into a single system.

So what is a fractal?

Fractal - a geometric figure composed of several parts, each of which is similar to the whole figure as a whole.

A small part of a fractal contains information about the entire fractal. Today, the word “fractal” is most often used to mean a graphic representation of a structure that is similar to itself on a larger scale.

Fractals are divided into geometric, geometric and stochastic.

Geometric fractals are otherwise called classical. They are the most visual, since they have the so-called rigid self-similarity, which does not change when the scale changes. This means that no matter how close you zoom into the fractal, you still see the same pattern.

Here are the most famous examples of geometric fractals.

Snowflake Koch.

Invented in 1904 by the German mathematician Helge von Koch.

To build it, a single segment is taken, divided into three equal parts, and the middle link is replaced by an equilateral triangle without this link. At the next step, we repeat the operation for each of the four resulting segments. As a result of the infinite repetition of this procedure, a fractal curve is obtained.

Durer's pentagon.

A fractal looks like a bunch of pentagons squeezed together. In fact, it is formed by using a pentagon as an initiator and isosceles triangles, the ratio of the largest side to the smallest in which is exactly equal to the so-called golden ratio. These triangles are cut from the middle of each pentagon, resulting in a figure similar to 5 small pentagons glued to one big.

Sierpinski napkin.

In 1915, the Polish mathematician Vaclav Sierpinski came up with an interesting object.

To construct it, a solid equilateral triangle is taken. In the first step, an inverted equilateral triangle is removed from the center. The second step removes three inverted triangles from the three remaining triangles, and so on.

Dragon Curve.

Invented by the Italian mathematician Giuseppe Peano.

Sierpinski carpet.

A square is taken, divided into nine equal squares, the middle of which is thrown out, and the same operation is repeated with the rest ad infinitum.

The second type of fractals is algebraic fractals.

They got their name because they are built on the basis of algebraic formulas. As a result of mathematical processing of this formula, a point of a certain color is displayed on the screen. The result is a strange figure in which straight lines turn into curves, self-similarity effects appear at various scale levels. Almost every dot on the computer screen is like a separate fractal.

Examples of the most famous algebraic fractals.

Mandelbrot set.

Mandelbrot sets are the most common among algebraic fractals. It can be found in many scientific journals, book covers, postcards, and computer screen savers. This fractal is reminiscent of a carding machine with glowing tree and circle areas attached to it.

Julia set.

The Julia set was invented by the French mathematician Gaston Julia. No less famous algebraic fractal.

Newton's pools.

Stochastic fractals.

Fractals, during the construction of which some parameters randomly change in an iterative system, are called stochastic. The term "stochastic" comes from the Greek word for "guess".

This results in objects very similar to natural ones - asymmetrical trees, indented coastlines, etc. Two-dimensional stochastic fractals are used in modeling the terrain and sea surface.

These fractals are used in modeling the terrain and the surface of the seas, the electrolysis process. This group of fractals has become widespread thanks to the work of Michael Barnsley of Institute of Technology state of Georgia.
A typical representative of this class of fractals is "Plasma".

The most understandable for us are the so-called natural fractals.

"The Great Book of Nature is written in the language of geometry" (Galileo Galilei).

natural fractals.

• In wildlife:
• Sea stars and urchins
• Flowers and plants (broccoli, cabbage)
• Tree crowns and plant leaves
• Fruit (pineapple)
• The circulatory system and bronchi of humans and animals
• In inanimate nature:
• Borders of geographical objects (countries, regions, cities)
• Frosty patterns on window panes
• Stalactites, stalagmites, helictites.

Almost all natural formations: tree crowns, clouds, mountains, coastlines have a fractal structure.
What does it mean?

If you look at a fractal object as a whole, then at a part of it on an enlarged scale, then at a part of this part, it is easy to see that they look the same.

Sea fractals.

Octopus is a sea bottom animal from the order of cephalopods.

Its bodies and suckers on all eight tentacles of this animal have a fractal structure.

Another typical representative of the fractal underwater world is coral.

Over 3500 varieties of corals are known in nature.

Green fractal - fern leaves.

Fern leaves have the shape of a fractal figure - they are self-similar.

Bow is a fractal that makes you cry. Of course, it is a simple fractal: ordinary circles of different diameters, one might even say a primitive fractal.

A striking example of a fractal in nature is "Romanescu”, she is also “Romance broccoli” or “cauliflower coral”.

Cauliflower- typical fractal.

Consider the structure of cauliflower.

If you cut one of the flowers, it is obvious that the same cauliflower remains in the hands, only of a smaller size. We can keep cutting over and over again, even under a microscope - but all we get is tiny copies of the cauliflower.

Matryoshka - souvenir toy is a typical fractal. The principle of fractality is obvious when all the figures of a wooden toy are lined up in a row, and not nested in each other.

Man is a fractal.

A child is born, grows, and this process is accompanied by the principle of "self-similarity", fractality.

The scope of fractals is wide.

Fractals in literature

Among literary works there are those that have a textual, structural or fractal nature. In literary fractals, elements of the text are endlessly repeated:

The priest had a dog
he loved her.
She ate a piece of meat
he killed her.
Buried in the ground
The inscription wrote:
The priest had a dog...

“Here is the house.
that Jack built.
And here is the wheat.

In the House,
that Jack built
And here is a cheerful tit bird,
Which deftly steals wheat,
Which is stored in a dark closet
In the House,
that Jack built... .

Fractals in telecommunications.

To transmit data over distances, fractal-shaped antennas are used, which greatly reduces their size and weight.

Fractals in medicine.

At present, fractals are widely used in medicine. By itself, the human body consists of many fractal structures: the circulatory system, muscles, bronchi, bronchial pathways in the lungs, arteries.

The theory of fractals is applied to the analysis of electrocardiograms.

Estimation of the magnitude and rhythms of the fractal dimension allows at an earlier stage and with greater accuracy and informativeness to judge homeostasis disorders and the development of specific heart diseases.

X-ray images processed using fractal algorithms give a better picture, and, accordingly, better diagnostics !!

Another area of ​​active application of fractals is gastroenterology.

A new research method in medicine, electrogastroenterography is a research method that allows you to evaluate the bioelectric activity of the stomach, duodenum and other parts of the gastrointestinal tract.

Fractals in architecture.

The fractal principle of the development of natural and geometric objects penetrates deep into architecture both as an image of the external solution of an object and as an internal principle of architectural shaping.

Designers from all over the world started to use in their work wonderful fractal structures, only recently described by prominent mathematicians.

The use of fractals has taken almost all areas of modern design to a new level.

The introduction of fractal structures has increased in many cases both the visual and functional aspects of the design.

Designer Takeshi Miyakawa dreamed of becoming a mathematician as a child.

How else to explain this piece of furniture: the Fractal 23 bedside table contains 23 drawers of various sizes and proportions, which somehow manage to get along with each other inside the cubic body, filling almost all the space available to them.

Fractals in economics.

Recently, fractals have become popular with economists for analyzing the course of stock exchanges, currency and trading markets.
Fractals appear on the market quite often.

Fractals in games.

Today, in many games (perhaps the most striking example of Minecraft), where there are various kinds of natural landscapes, fractal algorithms are used in one way or another. A large number of programs for generating landscapes and landscapes based on fractal algorithms have been created.

Fractals in cinema.

In cinema, a fractal algorithm is used to create various fantastic landscapes. Fractal geometry allows VFX artists to easily create objects such as clouds, smoke, flames, starry skies, and more. What can we say about fractal animation then, it's a real amazing sight.

Electonic music.

The spectacle of fractal animation is successfully used by VJs. Especially often such video installations are used at concerts of electronic music performers.

Natural Sciences.

Very often fractals are used in geology and geophysics. It is no secret that the coasts of islands and continents have a certain fractal dimension, knowing which you can very accurately calculate the length of the coasts.

The study of fault tectonics and seismicity is sometimes also studied using fractal algorithms.

Geophysics uses fractals and fractal analysis to study magnetic field anomalies, to study the propagation of waves and oscillations in elastic media, to study climate, and many other things.

Fractals in physics.

Fractals are widely used in physics. In solid state physics, fractal algorithms make it possible to accurately describe and predict the properties of solid, porous, spongy bodies, and aerogels. This helps in creating new materials with unusual and useful properties.
An example of a solid body is crystals.

The study of turbulence in flows adapts very well to fractals.

The transition to a fractal representation facilitates the work of engineers and physicists, allowing them to better understand the dynamics of complex systems.
Flames can also be modeled using fractals.

Fractals in biology.

In biology, they are used to model populations and to describe systems of internal organs (system of blood vessels). After the creation of the Koch curve, it was proposed to use it when calculating the length of the coastline.

Fractals for housewives.

It is easy to transfer the theory of fractals to the home, including the kitchen.

Anything can be the result of the application: fractal earrings, fractal tasty liver and much more. You only need to connect knowledge and ingenuity!

Fractal graphics are widely used in the modern world. Pictures are popular - the result of fractal graphics.

And this is no coincidence. Admire the beauty of fractal graphics!

Practical part of the project

• Created the scientific work "Journey into the world of fractals"
• Studied programs for creating fractals in the programming languages ​​Pascal and Logo
• Create your own fractals.
• We made "Sierpinski's Napkin" and "Sierpinski's Carpet" with our own hands
• Made "Fractal Earrings"
• Created a series of paintings "Wonders of fractal graphics"
• Published the work "Journey into the world of fractals" on the Internet.
• Participated with the work "Journey into the world of fractals" in VII All-Russian Olympiad schoolchildren and students "Science 2.0" in the subject "Mathematics". They took first place.
• They took part with the work "Journey to the world of fractals" in the All-Russian competition "Great discoveries and inventions". They took first place.
• They took part with the work "Journey to the world of fractals" in the VIII All-Russian Olympiad for schoolchildren and students "I am a researcher" in the subject of Mathematics. They took first place.
• Created a presentation "The amazing world of fractals"
• Created brochures "Application of fractals" and "Fractals around us"
• We held the festival "The Amazing World of Fractals" for students in grades 8-11"

So, we can say with full confidence about the huge practical application of fractals and fractal algorithms today.

The range of areas where fractals are used is very extensive and varied.

And for sure, in the near future, fractals, fractal geometry, will become close and understandable to each of us. We can't do without them in our lives!

Let's hope that the emergence of fractal geometry is evidence of the ongoing evolution of man and the expansion of his ways of knowing and understanding the world. Perhaps our children will also easily and meaningfully operate with the concepts of fractals and nonlinear dynamics, as we operate with the concepts of classical physics, Euclidean geometry.

Results of the project

• We got acquainted with the history of the emergence and development of fractal geometry;
• We studied the types of fractals, their application in the modern world.
• We created our own fractals in Pascal and Logo programming languages
• Created a scientific work on fractals.
• Created brochures "Fractals around us" and "Application of fractals"
• We held the festival "The Amazing World of Fractals" for students in grades 8-11.

When I do not understand everything in what I read, I am not particularly upset. If the topic does not come across to me later, then it is not very important (at least for me). If the topic meets again, for the third time, I will have new chances to better understand it. Fractals are among such topics. I first learned about them from a book by Nassim Taleb, and then in more detail from a book by Benoit Mandelbrot. Today, on the request "fractal" on the site, you can get 20 notes.

Part I. A JOURNEY TO THE ORIGINS

TO NAME IS TO KNOW. As far back as the beginning of the 20th century, Henri Poincaré remarked: “You are surprised at the power that one word can have. Here is an object about which nothing could be said until it was baptized. It was enough to give him a name for a miracle to happen ”(see also). And so it happened when, in 1975, the French mathematician of Polish origin, Benoit Mandelbrot, collected the Word. From Latin words frangere(break) and fractus(discontinuous, discrete, fractional) a fractal has formed. Mandelbrot skillfully promoted and propagated the fractal as a brand based on emotional appeal and rational utility. He publishes several monographs, including The Fractal Geometry of Nature (1982).

FRACTALS IN NATURE AND ART. Mandelbrot outlined the contours of a fractal geometry other than Euclidean. The difference did not apply to the axiom of parallelism, as in the geometries of Lobachevsky or Riemann. The difference was the rejection of Euclid's default requirement for smoothness. Some objects are inherently rough, porous or fragmented, and many of them have these properties "to the same extent at any scale." In nature, there is no shortage of such forms: sunflowers and broccoli, sea shells, ferns, snowflakes, mountain crevices, coastlines, fjords, stalagmites and stalactites, lightning.

People who are attentive and observant have long noticed that some forms show a repetitive structure when viewed "up close or from afar." Approaching such objects, we notice that only minor details change, but the shape as a whole remains almost unchanged. Based on this, the fractal is easiest to define as a geometric shape that contains repeating elements at any scale.

MYTHS AND MYSTIFICATIONS. The new layer of forms discovered by Mandelbrot became a gold mine for designers, architects, and engineers. An uncountable number of fractals are built according to the same principles of multiple repetition. From here, the fractal is easiest to define as a geometric shape that contains repeating elements at any scale. This geometric form is locally unchanging (invariant), self-similar on a scale and integral in its limitations, a true singularity, the complexity of which is revealed as it approaches, and triviality itself at a distance.

DEVIL'S LADDER. Extremely strong electrical signals are used to transfer data between computers. Such a signal is discrete. Interference or noise randomly occurs in electrical networks due to many reasons and leads to data loss when information is transmitted between computers. To eliminate the influence of noise on data transmission in the early sixties of the last century was entrusted to a group of IBM engineers, in which Mandelbrot took part.

A rough analysis showed the presence of periods during which no errors were recorded. Having singled out periods lasting an hour, the engineers noticed that between them the periods of signal passage without errors are also intermittent; there are shorter pauses lasting about twenty minutes. Thus, data transmission without errors is characterized by data packets of different lengths and pauses in noise, during which the signal is transmitted without errors. In packages of a higher rank, as it were, packages of a lower rank are built in. Such a description implies the existence of such a thing as the relative position of lower-ranking packets in a higher-ranking packet. Experience has shown that the probability distribution of these relative locations of packages is independent of their rank. This invariance indicates the self-similarity of the process of data distortion under the action of electrical noise. The very procedure for cutting out error-free pauses in a signal during data transmission could not occur to electrical engineers for the reason that this was new to them.

But Mandelbrot, who studied pure mathematics, was well aware of the Cantor set, described back in 1883 and representing dust from points obtained according to a strict algorithm. The essence of the algorithm for constructing "Cantor's dust" is as follows. Take a straight line. Remove the middle third of the segment from it, keeping the two end ones. Now we repeat the same operation with the end segments and so on. Mandelbrot discovered that this is precisely the geometry of packets and pauses in the transmission of signals between computers. The error is cumulative. Its accumulation can be modeled as follows. At the first step, we assign the value 1/2 to all points from the interval, at the second step from the interval the value 1/4, the value 3/4 to the points from the interval, etc. Step-by-step summation of these quantities makes it possible to construct the so-called "devil's ladder" (Fig. 1). The measure of "Cantor's dust" is an irrational number equal to 0.618 ..., known as the "golden ratio" or "Divine proportion".

Part II. FRACTALS ARE THE MATTER

SMILE WITHOUT A CAT: FRACTAL DIMENSION. Dimension is one of the fundamental concepts that goes far beyond mathematics. Euclid in the first book of the "Beginnings" defined the basic concepts of geometry point, line, plane. Based on these definitions, the concept of three-dimensional Euclidean space remained unchanged for almost two and a half thousand years. Numerous flirting with spaces of four, five and more dimensions essentially add nothing, but collide with what to represent them human imagination can not. With the discovery of fractal geometry, a radical revolution took place in the concept of dimension. A great variety of dimensions appeared, and among them are not only integers, but also fractional, and even irrational ones. And these dimensions are available for visual and sensual representation. In fact, we can easily imagine cheese with holes as a model of a medium whose dimension is greater than two, but falls short of three due to cheese holes lowering the dimension of the cheese mass.

To understand fractional or fractal dimension, let's turn to Richardson's paradox, which claimed that the length of Britain's rugged coastline is infinite! Louis Fry Richardson wondered about the effect of the scale of measurement on the magnitude of the measured length of the British coastline. When moving from the scale of contour maps to the scale of "coastal pebbles", he came to a strange and unexpected conclusion: the length of the coastline increases indefinitely, and this increase has no limit. Smooth curved lines don't behave like this. Richardson's empirical data, obtained on maps of increasingly larger scales, testified to a power-law increase in the length of the coastline with a decrease in the measurement step:

In this simple Richardson formula L is the measured length of the coast, ε is the value of the measurement step, and β ≈ 3/2 is the degree of growth of the coast length found by him with a decrease in the measurement step. Unlike the circumference, the length of the UK coastline increases without having a 55 limit. She is endless! One has to come to terms with the fact that the curves are broken, non-smooth, do not have a limiting length.

However, Richardson's studies suggested that they have some characteristic measure of the degree of growth in length with decreasing measurement scale. It turned out that it is this value that mystically identifies a broken line as a fingerprint of a person's personality. Mandelbrot interpreted the coastline as a fractal object - an object whose dimension coincides with the exponent β.

For example, the dimensions of the coastal boundary curves for the western coast of Norway are 1.52; for the UK - 1.25; for Germany - 1.15; for Australia - 1.13; for a relatively smooth coast of South Africa - 1.02 and, finally, for a perfectly smooth circle - 1.0.

Looking at a fragment of a fractal, you will not be able to tell what its dimension is. And the reason is not in the geometric complexity of the fragment, the fragment can be very simple, but in the fact that the fractal dimension reflects not only the shape of the fragment, but also the format of the fragment transformation in the process of constructing the fractal. The fractal dimension is, as it were, removed from the form. And thanks to this, the value of the fractal dimension remains invariant; it is the same for any fragment of the fractal at any viewing scale. It cannot be “grabbed with fingers”, but it can be calculated.

FRACTAL REPEAT. Repetition can be modeled with non-linear equations. Linear equations are characterized by a one-to-one correspondence of variables: each value X matches one and only one value at and vice versa. For example, the equation x + y = 1 is linear. The behavior of linear functions is completely determined, uniquely determined by the initial conditions. The behavior of non-linear functions is not so unambiguous, because two different initial conditions can lead to the same result. On this basis, the iteration of the repetition of the operation appears in two different formats. It can have the character of a linear reference, when at each step of the calculations there is a return to the initial condition. This is a kind of "pattern iteration". Serial production on the assembly line is "pattern iteration". Iteration in the format of linear reference does not depend on the intermediate states of the evolution of the system. Here, each new iteration starts "from the stove." It is quite a different matter when the iteration has a recursion format, i.e. the result of the previous iteration step becomes the initial condition for the next one.

The recursion can be illustrated with a Fibonacci series, represented in the form of a Girard sequence:

u n +2 = u n +1 + u n

The result is the Fibonacci numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55…

In this example, it is quite clear that the function is applied to itself without referencing the initial value. It sort of slides along the Fibonacci series, and each result of the previous iteration becomes the starting value for the next one. It is this repetition that is realized in the construction of fractal forms.

Let us show how the fractal repetition is implemented in the algorithms for constructing the “Sierpinski napkin” (using the cutting method and the CIF method).

cutting method. Take an equilateral triangle with a side r. At the first step, we cut out in the center of it an equilateral triangle turned upside down with a side length r 1 = r 0/2. As a result of this step, we get three equilateral triangles with side lengths r 1 = r 0 /2 located at the vertices of the original triangle (Fig. 2).

At the second step, in each of the three triangles formed, we cut out inverted inscribed triangles with a side length r 2 = r 1 /2 = r 0 /4. Result - 9 triangles with side length r 2 = r 0 /4. As a result, the shape of the "Sierpinski napkin" gradually becomes more and more defined. Fixation occurs at every step. All previous fixations are sort of "erased".

Method SIF, or Barnsley's Method of Systems of Iterated Functions. Given: an equilateral triangle with the coordinates of the angles A (0.0), B (1.0), C (1/2, √3/2). Z 0 is an arbitrary point inside this triangle (Fig. 3). We take a dice, on the sides of which there are two letters A, B and C.

Step 1. Throw the bone. The probability of getting each letter is 2/6 = 1/3.

• If the letter A falls out, we build a segment z 0 -A, in the middle of which we put a point z 1
• If the letter B falls out, we build a segment z 0 -B, in the middle of which we put a point z 1
• If the letter C falls out, we build a segment z 0 -C, in the middle of which we put a point z 1

Step 2. Throw the bone again.

• If the letter A falls out, we build a segment z 1 -A, in the middle of which we put a point z 2
• If the letter B falls out, we build a segment z 1 -B, in the middle of which we put a point z 2
• If the letter C falls out, we build a segment z 1 -C, in the middle of which we put a point z 2

Repeating the operation many times, we will get points z 3 , z 4 , …, z n . The peculiarity of each of them is that the point is exactly halfway from the previous one to an arbitrarily chosen vertex. Now, if we discard the initial points, for example, from z 0 to z 100 , then the rest, with a sufficiently large number of them, form the “Sierpinski napkin” structure. The more points, the more iterations, the more clearly the Sierpinski fractal appears to the observer. And this despite the fact that the process goes, it would seem, in a random way (thanks to the dice). The “Sierpinski napkin” is a kind of process attractor, that is, a figure to which all trajectories built in this process with a sufficiently large number of iterations tend. Fixing the image in this case is a cumulative, accumulative process. Each individual point, perhaps, will never coincide with the point of the Sierpinski fractal, but each subsequent point of this process organized “by chance” is attracted closer and closer to the points of the “Sierpinski napkin”.

FEEDBACK LOOP. The founder of cybernetics, Norbert Wiener, cited the helmsman on a boat as an example to describe the feedback loop. The helmsman must stay on course and is constantly assessing how well the boat is keeping to it. If the helmsman sees that the boat is deviating, he turns the helm to return it to a given course. After a while, he again evaluates and again corrects the direction of movement using the steering wheel. Thus, navigation is carried out using iterations, repetition and successive approximation of the movement of the boat to a given course.

A typical feedback loop diagram is shown in fig. 4 It comes down to changing the variable parameters (the direction of the boat) and the controlled parameter C (the course of the boat).

Consider the mapping "Bernoulli shift". Let some number belonging to the interval from 0 to 1 be chosen as the initial state. Let's write this number in the binary number system:

x 0 \u003d 0.01011010001010011001010 ...

Now one step of evolution in time is that the sequence of zeros and ones is shifted to the left by one position, and the digit that happened to be on the left side of the decimal point is discarded:

x 1 \u003d 0.1011010001010011001010 ...

x 2 \u003d 0.011010001010011001010 ...

x 3 \u003d 0.11010001010011001010 ...

Note that if the original numbers x 0 rational, then in the process of iteration the values Xn go into a periodic orbit. For example, for the initial number 11/24, in the process of iteration, we get a series of values:

11/24 -> 11/12 -> 5/6 -> 2/3 -> 1/3 -> 2/3 -> 1/3 -> …

If the original values x0 are irrational, the mapping will never reach the periodic mode. The interval of initial values ​​x 0 ∈ contains infinitely many rational points and infinitely many irrational points. Thus, the density of periodic orbits is equal to the density of orbits that never reach the periodic regime. In any neighborhood of rational value x0 there is an irrational value of the initial parameter x' 0 In this state of affairs, a subtle sensitivity to initial conditions inevitably arises. This is a characteristic sign that the system is in a state of dynamic chaos.

ELEMENTARY FEEDBACK LOOP. The reverse is a necessary condition and a consequence of any lateral glance that takes itself by surprise. The icon of the reverse loop can be the Möbius strip, in which its lower side passes into the upper one with each circle, the inner becomes outer and vice versa. The accumulation of differences during the reverse process first leads the image away from the original, and then returns to it. In logic, the reversal loop is illustrated by Epimenides' paradox: "all Cretans are liars." But Epimenides himself is a Cretan.

STRANGE LOOP. The dynamic essence of the phenomenon of a strange loop is that the image, being transformed and more and more different from the original one, returns to the original image in the process of numerous deformations, but never repeats it exactly. Describing this phenomenon, Hofstadter introduces the term "strange loop" in the book. He concludes that both Escher, Bach, and Gödel discovered or, more precisely, used strange loops in their work and creativity in the visual arts, music, and mathematics, respectively. Escher, in Metamorphoses, discovered the strange coherence of the various planes of reality. The forms of one of the artistic perspectives are plastically transformed into the forms of another artistic perspective (Fig. 5).

Rice. 5. Maurits Escher. Drawing hands. 1948

Such strangeness manifested itself in a bizarre way in music. One of the canons of Bach's Musical Offering ( Canon per Tonos- Tonal canon) is constructed in such a way that its apparent ending unexpectedly smoothly passes into the beginning, but with a shift in tone. These successive modulations take the listener higher and higher from the original pitch. However, miraculously, after six modulations we are almost back. All voices now sound exactly one octave higher than at the beginning. The only strange thing is that as we rise through the levels of a certain hierarchy, we suddenly find ourselves in almost the same place where we started our journey - return without repeat.

Kurt Gödel discovered strange loops in one of the most ancient and mastered areas of mathematics - in number theory. Gödel's theorem first saw the light as Theorem VI in his 1931 paper "On formally undecidable propositions" in Principle Mathematica. The theorem states the following: all consistent axiomatic formulations of number theory contain undecidable propositions. The judgments of number theory say nothing about the judgments of number theory; they are nothing more than judgments of number theory. There is a loop here, but no weirdness. A strange loop is hidden in the proof.

STRANGE ATTRACTOR. Attractor (from English. attract attract) a point or a closed line that attracts to itself all possible trajectories of the system's behavior. The attractor is stable, that is, in the long run, the only possible behavior is the attractor, everything else is temporary. An attractor is a spatio-temporal object covering the entire process, being neither its cause nor its effect. It is formed only by systems with a limited number of degrees of freedom. Attractors can be a point, a circle, a torus, and a fractal. In the latter case, the attractor is called "strange" (Fig. 6).

A point attractor describes any stable state of the system. In phase space, it is a point around which local trajectories of a "node", "focus" or "saddle" are formed. This is how the pendulum behaves: at any initial speed and any initial position, after a sufficient time, under the action of friction, the pendulum stops and comes to a state of stable equilibrium. A circular (cyclic) attractor is a movement back and forth, like an ideal pendulum (without friction), in a circle.

Strange attractors ( strange attractors) seem strange only from the outside, but the term "strange attractor" spread immediately after the appearance in 1971 of the article "The Nature of Turbulence" by David Ruel and the Dutchman Floris Takens (see also). Ruelle and Takens wondered if any attractor has the right set of characteristics: stability, a limited number of degrees of freedom, and non-periodicity. From a geometric point of view, the question seemed like a pure puzzle. What form should an infinitely extended trajectory, drawn in a limited space, have in order to never repeat or intersect itself? To reproduce each rhythm, the orbit must be an infinitely long line on limited area in other words, be self-swallowing (Figure 7).

By 1971 in scientific literature there was already one sketch of such an attractor. Eduard Lorentz made it an appendix to his 1963 paper on deterministic chaos. This attractor was stable, non-periodic, had a small number of degrees of freedom, and never crossed itself. If this happened, and he returned to a point that he had already passed, the movement would be repeated in the future, forming a toroidal attractor, but this did not happen.

The strangeness of the attractor lies, as Ruel believed, in three non-equivalent, but in practice signs that exist together:

• fractality (nesting, similarity, consistency);
• determinism (dependence on initial conditions);
• singularities (a finite number of defining parameters).

Part III. IMAGINARY LIGHTNESS OF FRACTAL FORMS

IMAGINARY NUMBERS, PHASE PORTRAITS AND PROBABILITY. Fractal geometry is based on the theory of imaginary numbers, dynamical phase portraits and probability theory. The theory of imaginary numbers assumes that there is a square root of minus one. Gerolamo Cardano in his work "The Great Art" ("Ars Magna", 1545) presented the general solution of the cubic equation z 3 + pz + q = 0. Cardano uses imaginary numbers as a means of technical formalism to express the roots of the equation. He notices a strangeness, which he illustrates with a simple equation x 3 = 15x + 4. This equation has one obvious solution: x = 4. However, the generalizing formula gives a strange result. It contains the root of a negative number:

Rafael Bombelli, in his book on algebra (L'Algebra, 1560), pointed out that = 2 ± i, and this immediately allowed him to obtain a real root x = 4. In such cases, when complex numbers are conjugate, a real root is obtained , and complex numbers serve as a technical aid in the process of obtaining a solution to a cubic equation.

Newton believed that solutions containing a root of minus one should be considered "without physical meaning" and discarded. In the XVII-XVIII centuries, an understanding was formed that something imaginary, spiritual, imaginary is no less real than everything real taken together. We can even give the exact date of November 10, 1619, when Descartes formulated the manifesto of the new thinking "cogito ergo sum". From this moment on, thought is an absolute and undoubted reality: “if I think, then it means that I exist”! More precisely thought is now perceived as reality. Descartes' idea of ​​an orthogonal coordinate system, thanks to imaginary numbers, finds its completion. Now it is possible to fill these imaginary numbers with meanings.

In the 19th century, the works of Euler, Argan, Cauchy, Hamilton developed an arithmetic apparatus for working with complex numbers. Any complex number can be represented as the sum of X + iY, where X and Y are real numbers familiar to us, and i imaginary unit (essentially it is √–1). Each complex number corresponds to a point with coordinates (X, Y) on the so-called complex plane.

The second important concept, the phase portrait of a dynamical system, was formed in the 20th century. After Einstein showed that everything moves at the same speed with respect to light, the idea of ​​being able to express the dynamic behavior of a system in the form of frozen geometric lines, the so-called phase portrait of a dynamic system, acquired a clear physical meaning.

Let's illustrate it on the example of a pendulum. The first experiments with a pendulum Jean Foucault conducted in 1851 in the cellar, then in the Paris Observatory, then under the dome of the Pantheon. Finally, in 1855, Foucault's pendulum was hung under the dome of the Saint-Martin-des-Champs church in Paris. The length of the rope of the Foucault pendulum is 67 m, the weight of the kettlebell is 28 kg. From a great distance, the pendulum looks like a point. The point is always stationary. Approaching, we distinguish a system with three typical trajectories: a harmonic oscillator (sinϕ ≈ ϕ), a pendulum (oscillations back and forth), a propeller (rotation).

Where a local observer sees one of three possible configurations of the ball's motion, an analyst detached from the process can assume that the ball makes one of three typical motions. This can be shown on one plane. It is necessary to agree that we will move the "ball on a thread" to an abstract phase space that has as many coordinates as the number of degrees of freedom the system under consideration has. In this case we are talking about two degrees of freedom speed v and the angle of inclination of the thread with the ball to the vertical ϕ. In the coordinates ϕ and v, the trajectory of the harmonic oscillator is a system of concentric circles; as the angle ϕ increases, these circles become oval, and when ϕ = ± π the closure of the oval is lost. This means that the pendulum has switched to propeller mode: v = const(Fig. 8).

Rice. 8. Pendulum: a) trajectory in the phase space of an ideal pendulum; b) the trajectory in the phase space of a pendulum swinging with damping; c) phase portrait

There may be no lengths, durations, or movements in the phase space. Here every action is pre-given, but not every action is real. From geometry, only topology remains, instead of measures, parameters, instead of dimensions, dimensions. Here, any dynamical system has its own unique imprint of the phase portrait. And among them there are rather strange phase portraits: being complex, they are determined by a single parameter; being commensurate, they are disproportionate; being continuous, they are discrete. Such strange phase portraits are characteristic of systems with a fractal configuration of attractors. The discreteness of the centers of attraction (attractors) creates the effect of a quantum of action, the effect of a gap or a jump, while the trajectories remain continuous and produce a single bound form of a strange attractor.

CLASSIFICATION OF FRACTALS. The fractal has three hypostases: formal, operational and symbolic, which are orthogonal to each other. And this means that the same form of a fractal can be obtained using different algorithms, and the same number of fractal dimensions can appear in completely different fractals. Taking into account these remarks, we classify fractals according to symbolic, formal and operational features:

• symbolically, the dimension characteristic of a fractal can be integer or fractional;
• on a formal basis, fractals can be connected, like a leaf or a cloud, and disconnected, like dust;
• On the operational basis, fractals can be divided into regular and stochastic.

Regular fractals are built according to a strictly defined algorithm. The construction process is reversible. You can repeat all the operations in reverse order, erasing any image created in the process of the deterministic algorithm, point by point. A deterministic algorithm can be linear or non-linear.

Stochastic fractals, similar in a stochastic sense, arise when in the algorithm for their construction, in the process of iterations, some parameters change randomly. The term "stochastic" comes from the Greek word stochasis- conjecture, conjecture. A stochastic process is a process whose nature of change cannot be accurately predicted. Fractals are produced according to the whim of nature (fault surfaces of rocks, clouds, turbulent flows, foam, gels, soot particle contours, changes in stock prices and river levels, etc.), they are devoid of geometric similarity, but stubbornly reproduce in each fragment the statistical properties of the whole on average. The computer allows you to generate sequences of pseudo-random numbers and immediately simulate stochastic algorithms and forms.

LINEAR FRACTALS. Linear fractals are named so for the reason that they are all built according to a certain linear algorithm. These fractals are self-similar, are not distorted by any change in scale, and are not differentiable at any of their points. To construct such fractals, it is sufficient to specify a base and a fragment. These elements will be repeated many times, zooming out to infinity.

Dust of Kantor. In the 19th century, the German mathematician Georg Ferdinand Ludwig Philipp Kantor (1845–1918) proposed to the mathematical community a strange set of numbers between 0 and 1. The set contained an infinite number of elements in the specified interval and, moreover, had zero dimension. An arrow fired at random would hardly have hit at least one element of this set.

First you need to choose a segment of unit length (first step: n = 0), then divide it into three parts and remove the middle third (n = 1). Further, we will do exactly the same with each of the formed segments. As a result of an infinite number of repetitions of the operation, we obtain the desired set of "Cantor's dust". Now there is no opposition between the discontinuous and the infinitely divisible. “Cantor's dust” is both (see Fig. 1). "Cantor Dust" is a fractal. Its fractal dimension is 0.6304…

One of the two-dimensional analogues of the one-dimensional Cantor set was described by the Polish mathematician Vaclav Sierpinski. It is called "cantor carpet" or more often "Sierpinski carpet". He is strictly self-similar. We can calculate its fractal dimension as ln8/lnЗ = 1.89… (Fig. 9).

LINES FILLING THE PLANE. Consider a whole family of regular fractals, which are curves capable of filling a plane. Leibniz also stated: “If we assume that someone puts many dots on paper by chance,<… >I say that it is possible to reveal a constant and complete, subject to a certain rule, a geometric line that will pass through all points. This statement by Leibniz contradicted Euclidean understanding of dimension as the smallest number of parameters by which the position of a point in space is uniquely determined. In the absence of a rigorous proof, these ideas of Leibniz remained on the periphery of mathematical thought.

Peano curve. But in 1890, the Italian mathematician Giuseppe Peano constructed a line that completely covers a flat surface, passing through all its points. The construction of the "Peano curve" is shown in fig. 10.

While the topological dimension of the Peano curve is equal to one, its fractal dimension is equal to d = ln(1/9)/ln(1/3) = 2. In the framework of fractal geometry, the paradox was resolved in the most natural way. A line, like a cobweb, can cover a plane. In this case, a one-to-one correspondence is established: each point of the line corresponds to a point on the plane. But this correspondence is not one-to-one, because each point on the plane corresponds to one or more points on the line.

Hilbert curve. A year later, in 1891, an article by the German mathematician David Hilbert (1862–1943) appeared in which he presented a curve covering a plane without intersections or tangency. The construction of the "Hilbert curve" is shown in fig. eleven.

The Hilbert curve was the first example of FASS curves (spaceFilling, selfAvoiding, Simple and selfSimilar space-filling self-avoiding, simple and self-similar lines). The fractal dimension of the Gilbert line, as well as the Peano curve, is equal to two.

Minkowski tape. Herman Minkowski, a close friend of Hilbert's from his student days, constructed a curve that does not cover the entire plane, but forms something like a ribbon. When constructing the "Minkowski tape" at each step, each segment is replaced by a broken line consisting of 8 segments. At the next stage, with each new segment, the operation is repeated on a scale of 1:4. The fractal dimension of the Minkowski strip is d = ln(l/8)/ln(1/4) = 1.5.

NONLINEAR FRACTALS. The simplest non-linear mapping of the complex plane onto itself is the Julia mapping zgz 2 + С considered in the first part. feedback loop (Fig. 13).

In the process of iterations for a fixed value of the constant C, depending on an arbitrary starting point Z 0 , the point Z n at n-> ∞ can be either finite or infinite. It all depends on the position of Z 0 relative to the origin z = 0. If the calculated value is finite, then it is included in the Julia set; if it goes to infinity, then it is cut off from the Julia set.

The form obtained after applying the Julia map to points of some surface is uniquely determined by the parameter C. For small C, these are simple connected loops; for large C, these are clusters of disconnected but strictly ordered points. By and large, all Julia forms can be divided into two large families - connected and disconnected mappings. The former resemble "Koch's snowflake", the latter "Cantor's dust".

The diversity of Julia's shapes baffled mathematicians when they were first able to observe these shapes on computer monitors. Attempts to rank this variety were of a very arbitrary nature and amounted to the fact that the basis for the classification of Julia maps was the Mandelbrot set, whose boundaries, as it turned out, are asymptotically similar to Julia maps.

With C = 0, the repetition of the Julia mapping gives a sequence of numbers z 0 , z 0 2 , z 0 4 , z 0 8 , z 0 16 ... As a result, three options are possible:

• for |z 0 |< 1 в процессе итераций числа z n по модулю будут уменьшаться, последовательно приближаясь к нулю. Иными словами, ноль есть точечный аттрактор;
• for |z 0 | > 1 in the course of iterations, the numbers z n increase in absolute value, tending to infinity. In this case, the attractor is a point at infinity, and we exclude such values ​​from the Julia set;
• for |z 0 | = 1 all points of the sequence continue to remain on this unit circle. In this case, the attractor is a circle.

Thus, at C = 0, the boundary between the attractive and repulsive starting points is a circle. In this case, the mapping has two fixed points: z = 0 and z = 1. The first of them is attractive, since the derivative of the quadratic function at zero is 0, and the second is repulsive, since the derivative of the quadratic function at the value of the parameter one is equal to two.

Consider the situation when the constant C is a real number, i.e. we seem to be moving along the axis of the Mandelbrot set (Fig. 14). At C = –0.75, the boundary of the Julia set self-crosses and the second attractor appears. The fractal at this point bears the name of the San Marco fractal, given to it by Mandelbrot in honor of the famous Venetian cathedral. Looking at the figure, it is not difficult to understand why Mandelbrot came up with the idea to name the fractal in honor of this structure: the resemblance is amazing.

Rice. 14. Changing the form of the Julia set with a decrease in the real value of C from 0 to -1

Decreasing C further to -1.25, we get a new type form with four fixed points, which persist up to C< 2. При С = 2 множество Жюлиа вырождается в отрезок, который тут же распадается в пыль, аналогичную «пыли Кантора» (рис. 15).

Rice. 15. The appearance of new forms of the Julia set with a decrease in the real value C< –1

So, even staying on the axis of the Mandelbrot fractal (constant C is a real number), we "captured" in the field of attention and in some way ranked a fairly large variety of Julia shapes from circle to dust. Now consider the sign areas of the Mandelbrot fractal and the corresponding forms of the Julia fractals. First of all, let's describe the Mandelbrot fractal in terms of "cardioid", "kidneys" and "onions" (Fig. 16).

The main cardioid and the circle adjacent to it form the basic shape of the Mandelbrot fractal. They are adjacent to an infinite number of its own copies, which are commonly called kidneys. Each of these buds is surrounded by an infinite number of smaller buds that look alike. The two largest buds above and below the main cardioid were called onions.

The Frenchman Adrien Dowdy and the American Bill Hubbard, who studied the typical fractal of this set (C = –0.12 + 0.74i), called it the “rabbit fractal” (Fig. 17).

When crossing the boundary of the Mandelbrot fractal, Julia fractals always lose their connection and turn into dust, which is usually called “Fatou dust” in honor of Pierre Fatou, who proved that for certain values ​​of C, a point at infinity attracts the entire complex plane, except for a very thin set like dust ( Fig. 18).

STOCHASTIC FRACTALS. There is a significant difference between a strictly self-similar von Koch curve and, for example, the coast of Norway. The latter, not being strictly self-similar, exhibits similarity in a statistical sense. At the same time, both curves are broken so much that you cannot draw a tangent to any of their points, or, in other words, you cannot differentiate it. Such curves are kind of "monsters" among the normal Euclidean lines. The first to construct a continuous function that does not have a tangent at any of its points was Karl Theodor Wilhelm Weierstrass. His work was presented to the Royal Prussian Academy on 18 July 1872 and published in 1875. The functions described by Weierstrass look like noise (Fig. 19).

Look at a stock bulletin chart, a summary of temperature fluctuations or air pressure fluctuations, and you will find some regular irregularity. Moreover, when the scale is increased, the nature of the irregularity is preserved. And this refers us to fractal geometry.

Brownian motion is one of the most famous examples of a stochastic process. In 1926, Jean Perrin received the Nobel Prize for his study of the nature of Brownian motion. It was he who drew attention to the self-similarity and non-differentiability of the Brownian trajectory.