Mathematics,
if you look at it correctly,
reflects not only the truth,
but also incomparable beauty.
Bertrand Russell.

You've certainly heard of fractals. You have certainly seen these breathtaking images from Bryce3d that are more real than reality itself. Mountains, clouds, tree bark - all this goes beyond the usual Euclidean geometry. We cannot describe the stone or the boundaries of the island with lines, circles and triangles. And here fractals come to the rescue. What are these familiar strangers? When did they appear?

History of appearance.

The first ideas of fractal geometry emerged in the 19th century. Cantor, using a simple recursive (repetitive) procedure, turned the line into a set of unconnected points (the so-called Cantor's Dust). He took a line and removed the center third and then repeated the same with the remaining segments. Peano drew a special kind of line (picture # 1). To draw it, Peano used the following algorithm.

At the first step, he took a straight line and replaced it with 9 segments 3 times shorter than the length of the original line (Part 1 and 2 of Figure 1). Then he did the same with each segment of the resulting line. And so on ad infinitum. Its uniqueness is that it fills the entire plane. It is proved that for each point on the plane, one can find a point belonging to the Peano line. Peano's Curve and Cantor's Dust went beyond ordinary geometric objects. They did not have a clear dimension. Cantor's dust was built on the basis of a one-dimensional straight line, but it consisted of points (dimension 0). And the Peano curve was built on the basis of a one-dimensional line, and the result was a plane. In many other areas of science, problems appeared, the solution of which led to strange results, like those described above (Brownian motion, stock prices).

Father of fractals

Until the 20th century, there was an accumulation of data on such strange objects, without any attempt to systematize them. That was until Benoit Mandelbrot, the father of modern fractal geometry and the word fractal, took on them. While working for IBM as a mathematical analyst, he studied noise in electronic circuits that could not be described using statistics. Gradually comparing the facts, he came to the discovery of a new direction in mathematics - fractal geometry.

What is a fractal. Mandelbrot himself derived the word fractal from the Latin word fractus, which means broken (divided into parts). And one of the definitions of a fractal is a geometric figure made up of parts and which can be divided into parts, each of which will represent a reduced copy of the whole (at least approximately).

To imagine a fractal more closely, consider the example given in the book by B. Mandelbrot "The Fractal Geometry of Nature", which has become a classic - "How long is the coast of Britain?". The answer to this question is not as simple as it seems. It all depends on the length of the tool we will be using. Having measured the coast with a kilometer ruler, we get some length. However, we will skip many small coves and peninsulas that are much smaller than our ruler. By reducing the size of the ruler to, say, 1 meter, we will take into account these details of the landscape, and, accordingly, the length of the coast will increase. Let's go ahead and measure the length of the coast using a millimeter ruler, here we will take into account details that are more than a millimeter, the length will be even greater. As a result, the answer to such a seemingly simple question can baffle anyone - the length of the coast of Britain is infinite.

A little about dimensions.

In our daily life, we constantly encounter dimensions. We estimate the length of the road (250 m), find out the area of ​​the apartment (78 m2) and look for the volume of a beer bottle (0.33 dm3) on the sticker. This concept is quite intuitively clear and, it would seem, does not require clarification. The line has dimension 1. This means that, having chosen a reference point, we can define any point on this line using 1 number - positive or negative. And this applies to all lines - a circle, a square, a parabola, etc.

Dimension 2 means that we can uniquely define any point with two numbers. Don't think that two-dimensional means flat. The surface of a sphere is also two-dimensional (it can be defined using two values ​​- angles like width and longitude).

From a mathematical point of view, the dimension is determined as follows: for one-dimensional objects - doubling their linear size leads to an increase in size (in this case, length) two times (2 ^ 1).

For 2D objects, doubling the linear dimensions will quadruple the size (for example, the area of ​​a rectangle) (2 ^ 2).

For 3-D objects, an increase in linear dimensions by two times leads to an increase in volume eight times (2 ^ 3), and so on.

Thus, the dimension D can be calculated based on the dependence of the increase in the "size" of the object S on the increase in the linear dimensions L. D = log (S) / log (L). For line D = log (2) / log (2) = 1. For the plane D = log (4) / log (2) = 2. For volume D = log (8) / log (2) = 3. It can be a little confusing, but in general it is not difficult and understandable.

Why am I telling all this? And in order to understand how to separate fractals from, say, sausage. Let's try to calculate the dimension for the Peano curve. So, we have the original line, consisting of three segments of length X, is replaced by 9 segments three times shorter. Thus, with an increase in the minimum segment by 3 times, the length of the entire line increases by 9 times and D = log (9) / log (3) = 2 - a two-dimensional object !!!

So, when the dimension of the figure obtained from some of the simplest objects (segments) is greater than the dimension of these objects, we are dealing with a fractal.

Fractals are divided into groups. The largest groups are:

Geometric fractals.

It was with them that the history of fractals began. This type of fractal is obtained by simple geometric constructions. Usually, when constructing these fractals, one does the following: a "seed" is taken - an axiom - a set of segments, on the basis of which the fractal will be constructed. Then a set of rules is applied to this "seed", which transforms it into some kind of geometric figure. Next, the same set of rules is applied to each part of this figure. With each step, the figure will become more and more complex, and if we carry out (at least in our mind) an infinite number of transformations, we will get a geometric fractal.

The Peano curve discussed above is a geometric fractal. The figure below shows other examples of geometric fractals (from left to right Koch Snowflake, Liszt, Sierpinski Triangle).

Koch Snowflake

Sheet

Sierpinski triangle

Of these geometric fractals, the first, the Koch snowflake, is very interesting and rather famous. It is built on the basis of an equilateral triangle. Each line of which ___ is replaced by 4 lines each 1/3 long of the original _ / \ _. Thus, with each iteration, the length of the curve increases by a third. And if we do an infinite number of iterations, we get a fractal - a Koch snowflake of infinite length. It turns out that our infinite curve covers a limited area. Try to do the same using methods and shapes from Euclidean geometry.

The dimension of a Koch snowflake (when a snowflake grows 3 times, its length increases 4 times) D = log (4) / log (3) = 1.2619 ...

The so-called L-Systems are well suited for constructing geometric fractals. The essence of these systems is that there is a specific set of system symbols, each of which denotes a specific action and a set of rules for character conversion. For example, describing a Koch snowflake using L-Systems in the Fractint program

; Adrian Mariano from The Fractal Geometry of Nature by Mandelbrot Koch1 ( ; set the angle of rotation 360/6 = 60 degrees Angle 6 ; Initial drawing for construction Axiom F - F - F ; Character conversion rule F = F + F - F + F)

In this description, the geometric meanings of the symbols are as follows:

F stands for draw line + turn clockwise - turn counterclockwise

The second property of fractals is self-similarity. Take the Sierpinski triangle, for example. To construct it from the center of an equilateral triangle, "cut out" a triangle. We repeat the same procedure for the three formed triangles (except for the central one) and so on ad infinitum. If we now take any of the formed triangles and enlarge it, we will get an exact copy of the whole. In this case, we are dealing with complete self-similarity.

I'll make a reservation right away that most of the fractal drawings in this article were obtained using the Fractint program. If you are interested in fractals, then this is the program must have For you. With its help, you can build hundreds of different fractals, get comprehensive information on them, and even listen to how fractals sound;).

To say that the program is good is to say nothing. It's great except for one thing - the latest version 20.0 is only available for DOS :(. You can find this program (latest version 20.0) at http://spanky.fractint.org/www/fractint/fractint.html.

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Comments (1)

Well, for a snack, an interesting example Microsoft Excel Cells A2 and B2 have the same values ​​between 0 and 1. At a value of 0.5, there is no effect.

Hello everyone who managed to make a prog on the picture of the Fratal. Who can tell me which cycle method is better for me to use to build a clearing of fern fractals with a substrate of 3d max with the number of dt iteration 100,000 on a stone with 2800 mH

There is a source code with a program for drawing the Dragon curve, also a fractal.

The article is awesome. And the ex-fur-tree is probably a coprocessor error (at the last low-order bits)

How the fractal was discovered

The mathematical forms known as fractals belong to the genius of the eminent scientist Benoit Mandelbrot. For most of his life he taught mathematics at Yale University in the USA. In 1977 - 1982, Mandelbrot published scientific works devoted to the study of "fractal geometry" or "geometry of nature", in which he broke seemingly random mathematical forms into constituent elements that, upon closer examination, were repetitive, which proved the existence of a certain pattern for copying ... Mandelbrot's discovery had significant consequences in the development of physics, astronomy and biology.

Fractals in nature

In nature, many objects have fractal properties, for example: tree crowns, cauliflower, clouds, circulatory and alveolar systems of humans and animals, crystals, snowflakes, the elements of which are arranged in one complex structure, coasts (the fractal concept allowed scientists to measure the coastline of the British Isles and other, previously immeasurable, objects).

Consider the structure of cauliflower. If you cut one of the flowers, it is obvious that the same cauliflower remains in your hands, only of a smaller size. You can keep cutting again and again, even under a microscope - however, all we get are tiny copies of the cauliflower. In this simplest case, even a small part of the fractal contains information about the entire final structure.

Fractals in digital technology

Fractal geometry has made an invaluable contribution to the development of new technologies in the field of digital music, as well as made possible the compression of digital images. Existing fractal image compression algorithms are based on the principle of storing a compressing image instead of the digital image itself. For a squeezing image, the main picture remains a fixed point. Microsoft used one of the variants of this algorithm when publishing its encyclopedia, but for one reason or another this idea was not widely disseminated.

The mathematical basis of fractal graphics is fractal geometry, where the principle of inheritance from the original "parent objects" is placed at the basis of the methods for constructing "images-heirs". The very concepts of fractal geometry and fractal graphics appeared only about 30 years ago, but have already become firmly established by computer designers and mathematicians.

The basic concepts of fractal computer graphics are:

• Fractal triangle - fractal figure - fractal object (hierarchy in descending order)
• Fractal line
• Fractal composition
• "Parent object" and "Successor object"

Just like in vector and 3D graphics, creating fractal images is mathematically calculated. The main difference from the first two types of graphics is that a fractal image is built according to an equation or a system of equations - nothing but a formula in the memory of a computer needs to be stored to perform all calculations - and such a compactness of the mathematical apparatus made it possible to use this idea in computer graphics. By simply changing the coefficients of the equation, you can easily get a completely different fractal image - using several mathematical coefficients, surfaces and lines of very complex shapes are set, which allows you to implement such composition techniques as horizontal and vertical, symmetry and asymmetry, diagonal directions and much more.

How to build a fractal?

The fractal creator plays the role of an artist, photographer, sculptor, and scientist-inventor at the same time. What are the stages of the work of creating a picture "from scratch"?

• set the shape of the picture by a mathematical formula
• investigate the convergence of the process and vary its parameters
• select the type of image
• choose a palette of colors

Among fractal graphics editors and others graphics programs can be distinguished:

• "Art Dabbler"
• "Painter" (without a computer, no artist will ever reach the possibilities laid down by programmers only with the help of a pencil and a brush pen)
• "Adobe Photoshop" (but here the image is not created "from scratch", but, as a rule, only processed)

Consider the device of an arbitrary fractal geometric figure. In its center is the simplest element - an equilateral triangle, which received the same name: "fractal". On the middle segment of the sides, construct equilateral triangles with a side equal to one third of the side of the original fractal triangle. Even smaller triangles-heirs of the second generation are built on the same principle - and so on ad infinitum. The resulting object is called a "fractal figure", from the sequences of which we get a "fractal composition".

Source: http://www.iknowit.ru/

Fractals and ancient mandalas

This is a mandala for attracting money. The color red is said to work like a money magnet. Ornate patterns remind you of nothing? They seemed very familiar to me and I started researching mandalas as a fractal.

In principle, a mandala is a geometric symbol of a complex structure, which is interpreted as a model of the Universe, a “map of the cosmos”. This is the first sign of fractality!

They are embroidered on fabric, painted on sand, made with colored powders and made of metal, stone, wood. Bright and mesmerizing appearance makes her beautiful decoration floors, walls and ceilings of temples in India. In the ancient Indian language, "mandala" means the mystical circle of interconnection of the spiritual and material energies of the Universe or, in other words, the flower of life.

I wanted to write a review about fractal mandalas very small, with a minimum of paragraphs, showing that the relationship clearly exists. However, trying to find an awareness and connect information about fractals and mandalas into a single whole, I had a feeling of a quantum leap into a space unknown to me.

I demonstrate the immensity of this topic with a quote: "Such fractal compositions or mandalas can be used both in the form of paintings, design elements for living and working premises, wearable amulets, in the form of video tapes, computer programs ..." In general, the topic for the study of fractals is simply enormous.

One thing I can say for sure, the world is much more diverse and richer than the poor ideas of our minds about it.

Fractal marine animals

My guesses about fractal marine animals were not groundless. Here are the first representatives. The octopus is a marine benthic animal from the order of cephalopods.

Looking at this photo, the fractal structure of his body and the suckers on all eight tentacles of this animal became obvious to me. The suction cups on the tentacles of an adult octopus reach up to 2,000.

An interesting fact is that the octopus has three hearts: one (main) drives blue blood throughout the body, and the other two - gill - push blood through the gills. Some of these deep sea fractals are poisonous.

By adapting and disguising itself to its environment, the octopus has a very useful ability to change color.

Octopuses are considered the smartest of all invertebrates. They get to know people, get used to those who feed them. It would be interesting to look at octopuses, which are easy to train, have a good memory and even distinguish geometric shapes. But the age of these fractal animals is short-lived - a maximum of 4 years.

Man uses the ink of this living fractal and other cephalopods. They are sought after by artists for their durability and beautiful brown tone. In Mediterranean cuisine, octopus is a source of vitamins B3, B12, potassium, phosphorus and selenium. But I think you need to be able to cook these marine fractals in order to enjoy eating them.

By the way, it should be noted that octopuses are predators. With their fractal tentacles, they hold their prey in the form of molluscs, crustaceans and fish. It's a pity if such a beautiful mollusk becomes the food of these sea fractals. In my opinion, also a typical representative of the fractals of the sea kingdom.

This is a relative of snails, the gastropod nudibranch mollusk Glaucus, aka Glaucus, aka Glaucus atlanticus, aka Glaucilla marginata. This fractal is also unusual in that it lives and moves under the surface of the water, being held by surface tension. Because the mollusk is hermaphrodite, then after mating both "partners" lay eggs. This fractal is found in all oceans in the tropical zone.

Fractals of the sea kingdom

Each of us at least once in his life held in his hands and examined a sea shell with genuine childish interest.

Usually shells are a beautiful souvenir reminiscent of a trip to the sea. When you look at this spiraling formation of invertebrate molluscs, there is no doubt about its fractal nature.

We humans are somewhat reminiscent of these soft-bodied molluscs, living in comfortable concrete fractal houses, placing and moving our bodies in fast cars.

Another typical representative of the fractal underwater world is coral.
More than 3500 coral species are known in nature, in the palette of which up to 350 color shades are distinguished.

Coral is the skeleton material of a colony of coral polyps, also from the invertebrate family. Their huge accumulations form entire coral reefs, the fractal way of formation of which is obvious.

Coral can be confidently called a fractal from the sea kingdom.

It is also used by humans as a souvenir or raw material for jewelry and ornaments. But it is very difficult to repeat the beauty and perfection of fractal nature.

For some reason, I have no doubt that many fractal animals will also be found in the underwater world.

Once again, performing a ritual in the kitchen with a knife and a cutting board, and then, lowering the knife into cold water, I was in tears once again figured out how to deal with the tear fractal, which appears almost daily in my eyes.

The principle of fractality is the same as that of the famous matryoshka - nesting. That is why fractality is not immediately noticed. In addition, the light uniform color and its natural ability to cause discomfort do not contribute to close observation of the universe and the identification of fractal mathematical laws.

But the lilac green onion, due to its color and the absence of tear phytoncides, led to reflections on the natural fractality of this vegetable. Of course, it is a simple fractal, ordinary circles of different diameters, one might even say the most primitive fractal. But it would not hurt to remember that the ball is considered an ideal geometric figure within our universe.

O useful properties onion, many articles have been published on the Internet, but somehow no one tried to study this natural specimen from the point of view of fractality. I can only state the fact of the usefulness of using a fractal in the form of an onion in my kitchen.

P.S. And I have already purchased a vegetable cutter for grinding a fractal. Now you have to think about how fractal such a healthy vegetable as ordinary white cabbage is. The same nesting principle.

Fractals in folk art

My attention was attracted by the history of the world famous toy "Matryoshka". Taking a closer look, we can confidently say that this souvenir toy is a typical fractal.

The principle of fractality is obvious when all the figures of a wooden toy are lined up, and not nested into each other.

My small studies of the history of the appearance of this toy fractal on the world market have shown that this beauty has Japanese roots. Matryoshka has always been considered a primordially Russian souvenir. But it turned out that she is the prototype of the Japanese figurine of the old sage Fukurum, once brought to Moscow from Japan.

But it was the Russian toy craft that brought world fame to this Japanese figurine. Where the idea of ​​fractal nesting of a toy came from, for me personally, remained a mystery. Most likely, the author of this toy used the principle of nesting figures into each other. And the easiest way to attach is similar figures of different sizes, and this is already a fractal.

An equally interesting object of research is the painting of a fractal toy. This is a decorative painting - khokhloma. Traditional elements of Khokhloma are herbal patterns of flowers, berries and branches.

Again, all the signs of fractality. After all, the same element can be repeated several times in different versions and proportions. The result is a folk fractal painting.

And if you won't surprise anyone with the newfangled painting of computer mice, laptop covers and phones, then fractal car tuning in the folk style is something new in auto design. It remains only to be surprised at the manifestation of the world of fractals in our life in such an unusual way in such ordinary things for us.

Fractals in the kitchen

Every time I took the cauliflower into small inflorescences for blanching in boiling water, I never once paid attention to the obvious signs of fractality until I had this specimen in my hands.

A typical plant fractal was on my kitchen table.

With all my love for cauliflower, I always came across specimens with a uniform surface without visible signs of fractality, and even a large number of inflorescences nested within each other did not give me a reason to see a fractal in this useful vegetable.

But the surface of this particular specimen with a pronounced fractal geometry did not leave the slightest doubt about the fractal origin of this type of cabbage.

Another trip to the hypermarket only confirmed the fractal status of cabbage. Among the huge number of exotic vegetables was a whole box of fractals. It was Romanescu, or Romanesque broccoli, cauliflower.

It turns out that designers and 3D artists admire its exotic, fractal-like shapes.

Cabbage buds grow in a logarithmic spiral. The first mentions of Romanescu cabbage came from Italy in the 16th century.

And broccoli cabbage is not at all a frequent guest in my diet, although in terms of content nutrients and micronutrients, it surpasses cauliflower at times. But its surface and shape are so uniform that it never occurred to me to see a vegetable fractal in it.

Fractals in quilling

Seeing the openwork crafts using the quilling technique, I never left the feeling that they remind me of something. Repetition of the same elements in different sizes - of course, this is the principle of fractality.

After watching the next master class on quilling, there was not even a doubt about the fractality of quilling. After all, for making various elements for quilling crafts, a special ruler with circles of different diameters is used. For all the beauty and uniqueness of the products, this is an incredibly simple technique.

Almost all the basic elements for quilling crafts are made from paper. To stock up on quilling paper for free, do an audit of your bookshelves at home. Surely, there you will find a couple of bright glossy magazines.

Quilling tools are simple and inexpensive. You can find everything you need for amateur quilling work among your home office supplies.

And the history of quilling begins in the 18th century in Europe. During the Renaissance, monks from French and Italian monasteries used quilling to decorate book covers and did not even suspect that the paper-rolling technique they had invented was fractal. Girls from high society even took a quilling course in special schools. This is how this technique began to spread across countries and continents.

This master class video quilling for making luxurious plumage can even be called "do-it-yourself fractals". With the help of paper fractals, wonderful exclusive Valentine cards and many different other interesting things are obtained. After all, fantasy, like nature, is inexhaustible.

It's not a secret for anyone that the Japanese are very limited in space in life, and therefore, they have to do their best to use it effectively. Takeshi Miyakawa shows how this can be done both efficiently and aesthetically. His fractal wardrobe confirms that the use of fractals in design is not only a tribute to fashion, but also a harmonious design solution in a limited space.

This example of the use of fractals in real life, applied to furniture design, showed me that fractals are real not only on paper in mathematical formulas and computer programs.

And it seems that nature uses the principle of fractality everywhere. You just need to take a closer look at it, and it will manifest itself in all its magnificent abundance and infinity of being.

So, a fractal is a mathematical set consisting of objects similar to this set. In other words, if we look at a small fragment of a fractal figure under magnification, it will look like a larger-scale part of this figure, or even the figure as a whole. For a fractal, moreover, an increase in scale does not mean a simplification of the structure. Therefore, at all levels, we will see an equally complex picture.

Fractal properties

Based on the above definition, a fractal is usually represented as a geometric figure that satisfies one or more of the following properties:

Has a complex structure at any magnification;

Approximately self-similar (the parts are similar to the whole);

Has a fractional dimension that is more topological;

Can be constructed using a recursive method.

Fractals in the outside world

Despite the fact that the concept of "fractal" seems to be extremely abstract, in life you can come across many real-life and even practical examples of this phenomenon. Moreover, from the surrounding world must certainly be considered, because they will give a better understanding of the fractal and its features.

For example, antennas for various devices, designs of which are executed by the fractal method, show their efficiency 20% higher than antennas of traditional design. In addition, the fractal antenna can operate with excellent performance simultaneously on a wide variety of frequencies. That is why modern mobile phones already practically do not have external antennas of the classical device in their design - the latter are replaced by internal fractal ones, which are mounted directly on the printed circuit board of the phone.

Fractals received great attention with the development information technologies... At present, algorithms have been developed for compressing various images using fractals, there are methods for constructing computer graphics objects (trees, mountain and sea surfaces) in a fractal way, as well as a fractal system for assigning IP addresses in some networks.

In economics, there is a way to use fractals when analyzing stock and currency quotes. Perhaps a reader trading in the Forex market has seen fractal analysis in action in a trading terminal, or even applied it in practice.

Also, in addition to objects artificially created by man with fractal properties, in natural nature there are also many such objects. Good examples of a fractal are corals, sea shells, some flowers and plants (broccoli, cauliflower), the circulatory system and bronchi of humans and animals, patterns formed on glass, natural crystals. These and many other objects have a pronounced fractal shape.

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 particularly important (at least for me). If the topic comes up again, for the third time, I will have new chances to better understand it. Fractals are among such themes. I first learned about them from the book of Nassim Taleb, and then in more detail from the book of Benoit Mandelbrot. Today, on request "fractal", you can get 20 notes on the site.

Part I. JOURNEY TO THE SOURCES

TO NAME MEANS TO FIND OUT. At 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, a French mathematician of Polish origin Benoit Mandelbrot put together the Word. From Latin words frangere(break) and fractus(discontinuous, discrete, fractional) fractal formed. Mandelbrot artfully promoted and publicized fractal as a brand with an emphasis on emotional appeal and rational utility. He publishes several monographs, including, Fractal Geometry of Nature (1982).

FRACTALS IN NATURE AND ART. Mandelbrot outlined the contours of 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 in the abandonment of Euclid's default smoothness requirement. Some objects are inherent in roughness, porosity or fragmentation, and many of them have the specified properties "to the same extent at any scale." There is no shortage of such forms in nature: sunflower and broccoli, seashells, ferns, snowflakes, mountain crevasses, coastlines, fjords, stalagmites and stalactites, lightning.

People who are attentive and observant have long noticed that some forms exhibit a repetitive pattern when viewed "near or far." Approaching such objects, we notice that only minor details change, but the shape as a whole remains almost unchanged. Based on this, a fractal is easiest to define as a geometric shape containing 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, a 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 at a distance it is triviality itself.

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

A rough analysis showed the presence of periods during which not a single error was recorded. Highlighting periods of one hour, the engineers noticed that between them the signal transmission periods without errors are also intermittent, here there are shorter pauses lasting about twenty minutes. Thus, error-free data transmission is characterized by data packets different lengths and pauses in noise, during which the signal is transmitted without errors. The packages of a higher rank are, as it were, built-in packages of a lower one. Such a description assumes the existence of such a thing as the relative position of the lowest-ranked packets in the higher-ranked packet. Experience has shown that the probability distribution of these relative locations of packets is independent of their rank. This invariance indicates the self-similarity of the data distortion process under the influence of electrical noise. The very procedure of cutting out error-free pauses in the signal during data transmission could not have occurred 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 rigorous algorithm. The essence of the algorithm for constructing "Cantor's dust" is as follows. Take a straight line segment. Remove the middle third of the segment from it, keeping the two end ones. Now we will repeat the same operation with the end segments and so on. Mandelbrot discovered that this is exactly the geometry of the packets and pauses in the transmission of signals between computers. The error is accumulating. Its accumulation can be modeled as follows. At the first step, we will assign the value 1/2 to all points from the interval, at the second step from the interval to 1/4, the value 3/4 to the points from the interval, etc. The step-by-step summation of these values ​​allows us to build 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 THE ESSENCE

SMILE WITHOUT A CAT: FRACTAL DIMENSION. Dimension is one of the fundamental concepts that goes far beyond mathematics. Euclid in the first book of "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 or more dimensions essentially add nothing, but they are faced with what the human imagination cannot imagine. With the discovery of fractal geometry, a radical revolution took place in the concept of dimension. A great variety of dimensions have appeared, and among them there are not only whole ones, but also fractional ones, and even irrational ones. And these dimensions are available for visual and sensory presentation. Indeed, we can easily imagine cheese with holes as a model of the environment, the dimension of which is more than two, but does not reach three because of the cheese holes, which reduces the dimension of the cheese mass.

To understand fractional or fractal dimensions, we turn to Richardson's paradox, which argued that Britain's rugged coastline was infinite in length! Louis Fry Richardson wondered about the effect of scale on the measured length of Britain's coastline. When passing 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. Empirical data of Richardson, obtained on maps of ever larger scales, indicated a power-law increase in the length of the coastline with a decrease in the measurement step:

In this simple Richardson formula L there is a measured length of the coast, ε Is the magnitude of the measurement step, and β ≈ 3/2 is the degree of increase in the coast length with a decrease in the measurement step found by him. Unlike the circumference, the length of the UK's coastline is increasing beyond the 55 limit. It is endless! We have to come to terms with the fact that the curves are broken, nonsmooth, do not have a limiting length.

However, Richardson's studies suggested that they have some characteristic measure of the degree to which length increases with decreasing scale. It turned out that it is this value that mystically identifies the 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 west coast of Norway are 1.52; for the UK - 1.25; for Germany - 1.15; for Australia - 1.13; for the 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 cannot 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 transformation format of the fragment in the process of constructing the fractal. Fractal dimension is, as it were, removed from the form. And due to this, the value of the fractal dimension remains invariant; it is the same for any fragment of the fractal at any scale of the survey. It cannot be “grasped with your fingers”, but it can be calculated.

FRACTAL REPEAT. Repeat can be modeled using nonlinear 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 deterministic, uniquely determined by the initial conditions. The behavior of nonlinear 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 a conveyor is “pattern iteration”. Iteration in the linear reference format does not depend on the intermediate states of the evolution of the system. Here, each new iteration starts from the stove. It is quite another 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.

Recursion can be illustrated by the Fibonacci series, represented in the form of a Girard sequence:

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

The result is Fibonacci numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55…

In this example, it is quite obvious that the function is applied to itself without referring to the initial value. It slides along the Fibonacci series, as it were, and each result of the previous iteration becomes the initial value for the next. It is this repetition that is realized when constructing fractal shapes.

Let us show how 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 with the side length turned upside down 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 formed triangles, 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 is gradually becoming more and more definite. Fixation occurs at every step. All previous commitments are "erased", as it were.

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

Step 1. Roll the bone. The probability of each letter falling out is 2/6 = 1/3.

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

Step 2. Roll the bone again.

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

Repeating the operation many times, we 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 structure of a “Sierpinski napkin”. The more points, the more iterations, the more clearly the Sierpinski fractal is to the observer. And this despite the fact that the process goes on, it would seem, in a random way (thanks to the dice). The “Sierpinski napkin” is a kind of attractor of the process, that is, the figure towards which all the trajectories constructed in this process with a sufficiently large number of iterations tend. In this case, the fixation of the image 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, used a boat helmsman as an example to describe the feedback loop. The helmsman must stay on course and continually assess how well the boat is on course. If the helmsman sees that the boat is deviating, he turns the rudder back on the set course. After a while, he evaluates again and again corrects the direction of travel with the help of the rudder. Thus, navigation is carried out with the help of iterations, repetition and sequential approach of the boat movement to a given course.

A typical feedback loop is shown in Fig. 4 It comes down to changing variable parameters (boat direction) and controlled parameter С (boat heading).

Consider the Bernoulli Shift mapping. Let some number be selected as the initial state, which belongs to the interval from 0 to 1. Let's write this number in the binary number system:

x 0 = 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 on the left side of the decimal point is discarded:

x 1 = 0.1011010001010011001010 ...

x 2 = 0.011010001010011001010 ...

x 3 = 0.11010001010011001010 ...

Note that if the original numbers x 0 rational, then during iteration the values Xn go into periodic orbit. For example, for a seed of 11/24, we will get a number of values ​​during iteration:

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

If the original values x 0 irrational, the display will never go to periodic mode. The range 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 enter the periodic regime. In any neighborhood of the rational value x 0 there is an irrational value of the original 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 HINGES. The reverse is necessary condition and the consequence of any lateral glance that catches itself by surprise. The icon of the reversing loop can be a Mobius strip, in which its lower side with each circle turns into the upper one, the inner becomes outer and vice versa. The accumulation of differences in the reverse process first removes the image from the original one, and then returns to it. In logic, the reverse loop is illustrated by Epimenides' paradox: "All Cretans are liars." But Epimenides himself was a Cretan.

STRANGE LOOP. The dynamic essence of the strange loop phenomenon boils down to the fact that an image, transforming and becoming more and more different from the original, in the process of numerous deformations returns to the original image, but never repeats it exactly. In 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 works and creativity in visual arts, music and mathematics, respectively. In Metamorphoses, Escher discovered the strange coherence of the various planes of reality. The forms of one of the artistic perspectives are plasticly transformed into the forms of another artistic perspective (Fig. 5).

Rice. 5. Maurits Escher. Drawing hands. 1948

This oddity manifested itself in a bizarre way in the music. One of the canons of Bach's "Musical Offering" ( Canon per Tonos- The tonal canon) is designed in such a way that its apparent finale unexpectedly smoothly transitions to the beginning, but with a shift in key. These successive modulations take the listener higher and higher away from the initial key. However, miraculously, after six modulations, we're almost back. All voices now sound exactly one octave higher than at the beginning. The only strangeness is that, as we climb the levels of a certain hierarchy, we suddenly find ourselves in almost the same place where we started our journey - return without replay.

Kurt Gödel discovered strange loops in one of the most ancient and mastered areas of mathematics - number theory. Gödel's theorem first saw the light of day as Theorem VI in his 1931 article "On Formally Unsolvable Judgments" in Principle Mathematica. The theorem states the following: all consistent axiomatic formulations of number theory contain undecidable propositions. Number theory judgments say nothing about number theory judgments; they are no 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 the English. attract attract) a point or a closed line that attracts all possible trajectories of the system's behavior. The attractor is stable, that is, in the long term, the only possible model of the attractor's behavior, everything else is temporary. The attractor is a spatio-temporal object that encompasses 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 a suitable set of characteristics: stability, a limited number of degrees of freedom, and non-periodicity. WITH geometric point view the question seemed to be a pure puzzle. What form should an infinitely long trajectory depicted in limited space to never repeat or cross itself? To reproduce each rhythm, the orbit must be an infinitely long line over a limited area, in other words, be self-swallowing (Fig. 7).

By 1971, there was already one sketch of such an attractor in the scientific literature. Edward Lorenz made it an appendix to his 1963 article on deterministic chaos. This attractor was stable, non-periodic, had a small number of degrees of freedom and never crossed itself. If something like that happened, and he returned to the 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 Ruelle believed, in three nonequivalent, but in practice, existing together features:

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

Part III. IMPRESSIVE LIGHTNESS OF FRACTAL FORMS

IMAGINARY NUMBERS, PHASE PORTRAITS AND PROBABILITY. Fractal geometry rests on the theory of imaginary numbers, dynamic phase portraits, and probability theory. Imaginary number theory assumes that there is a square root of minus one. Gerolamo Cardano, in his work "Great Art" ("Ars Magna", 1545), presented a general solution to the cubic equation z 3 + pz + q = 0. Cardano uses imaginary numbers as a technical formalism for expressing the roots of the equation. He notices an oddity, which he illustrates with the 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:

Raphael 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 similar cases, when complex numbers are conjugate, we get a real root , 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 “not physically meaningful” and discarded. In the XVII-XVIII centuries, the understanding was formed that something imaginary, spiritual, imaginary is no less real than everything real taken together. We can even give the exact date November 10, 1619, when Descartes formulated the manifesto of 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, acquires its completeness. 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 X + iY, where X and Y are real numbers we are used to, and i imaginary unit (in fact 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 XX century. After Einstein showed that everything moves with the same speed with respect to light, the idea of ​​the possibility of expressing the dynamic behavior of a system in the format of frozen geometric lines, the so-called phase portrait of a dynamical system, acquired a clear physical meaning.

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

Where the local observer sees one of three possible configurations of the ball's motion, the analyst removed from the process can assume that the ball performs one of three typical movements. This can be depicted on one plan. It is necessary to agree that we will move the "ball on a thread" into an abstract phase space, which has as many coordinates as the degrees of freedom of the system under consideration. 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 at 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) 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 any action is pre-given, but not all are valid. All that remains of the geometry is the topology, instead of measures, parameters, instead of dimensions, dimensions. Here, any dynamic system has its own unique imprint, a 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 maintain continuity and produce a single connected 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 fractal shape can be obtained using different algorithms, and the same fractal dimension number can appear for completely different fractals in shape. Taking these remarks into account, we classify fractals according to symbolic, formal and operational features:

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

Regular fractals are built according to a strictly defined algorithm. In this case, 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. The deterministic algorithm can be linear or non-linear.

Stochastic fractals, similar in the stochastic sense, arise when in the algorithm of their construction, in the course of iterations, any parameters change randomly. The term "stochasticity" comes from the Greek word stochasis- guess, guess. A stochastic process is a process whose nature of change cannot be accurately predicted. Fractals are produced at the whim of nature (fracture surfaces of rocks, clouds, turbulent flows, foam, gels, contours of soot particles, changes in stock prices and river levels, etc.), are devoid of geometric similarity, but persistently 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 shapes.

LINEAR FRACTALS. Linear fractals are so named because they are all built according to a certain linear algorithm. These fractals are self-similar, do not distort at any change in scale, and are not differentiable at any point. To build such fractals, it is enough to set a base and a fragment. These elements will be repeated many times with decreasing scale to infinity.

Dust of Cantor. In the 19th century, the German mathematician Georg Ferdinand Ludwig Philip Cantor (1845-1918) proposed to the mathematical community a strange set of numbers in the range from 0 to 1. The set contained an infinite number of elements in the indicated interval and, moreover, had zero dimension. An arrow fired at random would hardly have hit even one element of this multitude.

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

One of the two-dimensional analogs 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 that can fill a plane. Even Leibniz argued: “If we assume that someone puts a lot of dots on paper by chance,<… >I say that it is possible to identify a constant and integral geometric line that obeys a certain rule, which will pass through all points. " This statement by Leibniz contradicted the 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, a mathematician from Italy, 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 d = ln (1/9) / ln (1/3) = 2. Within the framework of fractal geometry, the paradox was resolved in the most natural way. A line, like a spider web, 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 appeared by the German mathematician David Hilbert (1862–1943), in which he presented a curve covering a plane without intersections or tangencies. 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 of space-filling self-avoiding, simple and self-similar lines). The fractal dimension of the Gilbert line, like the Peano curve, is two.

Minkowski's tape. Hermann Minkowski, a close friend of Hilbert from his student days, has constructed a curve that does not cover the entire plane, but forms something like a ribbon. When constructing the "Minkowski strip" 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 nonlinear mapping of the complex plane onto itself is the Julia mapping zgz 2 + C considered in the first part. It is a calculation over a closed cycle, in which the result of the previous cycle is multiplied by itself with a constant added to it, that is, it is a quadratic feedback loop (fig. 13).

In the course of iterations at a fixed value of the constant C, depending on an arbitrary initial 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 shape that is obtained after applying the Julia map to the points of a certain 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 split into two large families - connected and disconnected maps. The former are reminiscent of Koch's snowflake, the latter are Kantor's dust.

Julia's variety of forms discouraged mathematicians when they were first able to observe these forms on computer monitors. Attempts to rank this manifold were very conditional and boiled down to the fact that the Mandelbrot set was taken as the basis for the classification of Julia mappings, the boundaries of which, as it turned out, are asymptotically similar to Julia mappings.

With C = 0, repetition of the Julia map 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 the 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 attracting and repulsive initial 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 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 С = –0.75, the boundary of the Julia set self-intersects and a second attractor appears. The fractal at this point bears the name of the San Marco fractal, given to him by Mandelbrot in honor of the famous Venetian cathedral. Looking at the drawing, it is not difficult to understand why Mandelbrot had the idea to name the fractal precisely after this structure: the similarity is amazing.

Rice. 14. Change in the shape of the Julia set when the real value C decreases from 0 to -1

Decreasing further С to –1.25, we get a new typical shape with four fixed points, which remain up to values ​​С< 2. При С = 2 множество Жюлиа вырождается в отрезок, который тут же распадается в пыль, аналогичную «пыли Кантора» (рис. 15).

Rice. 15. The emergence 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's forms from circle to dust. Now let us consider the sign areas of the Mandelbrot fractal and the corresponding forms of Julia's fractals. First of all, let's describe the Mandelbrot fractal in terms of "cardioid", "kidney" and "onion" (Fig. 16).

The main cardioid and the adjacent circle form the main shape of the Mandelbrot fractal. They are adjoined by an infinite number of copies of it, which are usually called kidneys. Each of these buds is encased in an infinite number of smaller buds, similar to one another. The two largest buds above and below the main cardioid are called onions.

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

When crossing the boundary of the Mandelbrot fractal, Julia's fractals always lose connectivity and turn into dust, which is commonly called "Fatou dust" in honor of Pierre Fatou, who proved that for certain values ​​of C, an infinitely distant point attracts the entire complex plane, except for a very thin set like dust ( fig. 18).

STOCHASTIC FRACTALS. There is a significant difference between the strictly self-similar von Koch curve and, for example, the coast of Norway. The latter, while not being strictly self-similar, exhibits similarity in the statistical sense. In this case, 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 a kind of "monsters" among the normal Euclidean lines. The first to construct a continuous function that has no 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 the charts of the stock exchange, a summary of temperature or air pressure fluctuations, and you will find some kind of regular irregularity. Moreover, with an increase in scale, the character of irregularity remains. And that 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 research on the nature of Brownian motion. It was he who drew attention to the self-similarity and nondifferentiability of the Brownian trajectory.

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. I will talk about what fractals are, about the types of fractals, examples of these objects and their application in this article. To begin with, I will briefly tell you what a fractal is.

A fractal (Latin fractus - crushed, broken, broken) is a complex geometric figure with the property of self-similarity, that 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. As an example, I will insert a picture of four different fractals.

I'll tell you a little about the history of fractals. The concepts of fractal and fractal geometry, which appeared in the late 70s, have become part of the everyday life of mathematicians and programmers since the mid-80s. The word "fractal" was coined by Benoit Mandelbrot in 1975 to refer to the irregular but self-similar structures that he worked on. 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, Cantor, Hausdorff). But only in our time was it possible to combine their work into a single system.

There are a lot of examples of fractals, because, as I said, they surround us everywhere. In my opinion, even our entire Universe is one huge fractal. After all, everything in it, from the structure of the atom to the structure of the Universe itself, exactly repeats each other. But there are, of course, more specific examples of fractals from different fields. 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 a dynamical system is specified by iterations of a polynomial or a holomorphic a function of a complex of variables on surface. Some of the most famous fractals of this kind are the Julia set, the Mandelbrot set, and Newton's basins. Below, in order, the pictures show each of the above fractals.

Another example of fractals is fractal curves. It is best to explain how to build a fractal using the example of fractal curves. One of these curves is the so-called Koch Snowflake. There is a simpleprocedure for obtaining fractal curves on a plane. Let's define an arbitrary polyline 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, replace each segment with a generator again. Continuing to infinity, in the limit we get a fractal curve. The Koch Snowflake (or curve) is shown below.

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

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

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

But, perhaps, the most interesting fractals are natural ones. Natural fractals are objects in nature that have fractal properties. And here the list is already long. I will not list everything, because, probably, I cannot list all of them, but I will tell you about some. For example, in nature, such fractals include our circulatory system and lungs. And also the crowns and leaves of trees. It also includes starfish, sea ​​urchins, corals, seashells, some plants such as cabbage or broccoli. Several such natural fractals from wildlife are clearly shown below.

If we consider inanimate nature, then there are much more interesting examples than in living nature. Lightning, snowflakes, clouds, well-known to all, 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 in modeling porous materials, for example, in petrochemistry. In biology, they are used to model populations and to describe systems internal organs(blood vessel system). After the creation of the Koch curve, it was proposed to use it when calculating the length of the coastline. Fractals are also actively used in radio engineering, computer science and computer technology, telecommunications and even the economy. 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. I also talked about their application and demonstrated some of the fractals visually. I hope you enjoyed this short excursion into the world of amazing and mesmerizing fractal objects.