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 (at the scale of human evolution) we have learned to "tame" electricity - and now we cannot imagine life without all these convenient devices that use electricity. But there are also discoveries that few people attach importance to, although they also greatly influence our lives.

One of these “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 simply impossible before, it required a huge amount of calculations. 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 way of coloring is by the number of iterations) and became one of the most popular images ever created by man.

As the ancient saying attributed to Heraclitus of Ephesus says, "You cannot 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 at Boeing Computer Services in 1967, 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. In this book, his attention was drawn to the fact that Benoit gave a lot of 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 smaller figures.

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 take the advances in modern technology 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 very graphic illustration of 3D fractals in everyday life is a 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 a demo of his program, but promises that when he does, in order to work with Aural, he will not need to learn the theory of fractals - just play 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.

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. The American mathematician Benoit Mandelbrot (albeit raised in France) 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 studied 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, almost two hundred pages of Julia's memoir, devoted to iterations of complex rational functions, was published, in which Julia sets are described - 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 the middle interval is replaced by 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. One interesting story is connected with it, or rather, with an attempt to measure its length, 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 you continue this process indefinitely, then in the limit you 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.

MINISTRY OF HIGHER AND PROFESSIONAL EDUCATION

IRKUTSK STATE ACADEMY OF ECONOMY

DEPARTMENT OF INFORMATION SYSTEMS

According to economic and mathematical models and methods

FRACTAL THEORY AND ITS APPLICATIONS

Prepared by: Leader:

Pogodaeva E.A. Tolstikova T.V.

Chetverikov S.V.

IRKUTSK 1997

All images are similar and

Yet not one on the other

Goy is not like; their choirs

I will point to the secret law

Yut, to the holy riddle...

J. W. Goethe.

plant metamorphosis.

WHY ARE WE TALKING ABOUT FRACTALS?

In the second half of our century in natural science there were
fundamental changes that gave rise to the so-called theory
self-organization, or synergetics. She was born suddenly, as if on
crossing several lines of scientific research. One of the decisive
initial impulses was betrayed to her by Russian scientists at the turn of the
fifties - sixties. In the fifties the scientist
Analytical chemist B.P. Belousov discovered redox
chemical reaction. Discovery and study of self-oscillations and autowaves during
Belousov reactions

S. E. Shnolem, A. M. Zhabotinsky, V.I. Krinsky, A.N. Zaikin, G.R.
Ivanitsky - perhaps the most brilliant page of the fundamental
Russian science in the postwar period. Fast and successful learning
reaction Belousov - Zhabotinsky worked in science as a trigger
hook: they immediately remembered that processes of this kind were known before
kind and that many natural phenomena ranging from the formation of galaxies
to tornadoes, cyclones and the play of light on reflective surfaces (so
called caustics), - in fact, the processes of self-organization. They are
can be of a very different nature: chemical, mechanical,
optical, electrical, etc. Moreover, it turned out that
has long been ready and perfectly developed mathematical theory
self-organization. Its basis was laid by the works of A. Poincaré and A. A.
Lyapunov at the end of the last century. Dissertation "On sustainability
movement" was written by Lyapunov in 1892.

The mathematical theory of self-organization forces us in a new way
look at the world around us. Let us explain how it differs from
classical worldview, since we will need to know this when
the study of fractal objects.

"The classical uniquely deterministic worldview
can be symbolized by a flat, smooth surface on which
balls collide, having received a certain amount of motion.
The future fate of each such body is uniquely determined by its
"past" at the previous moment of time (momentum, charge) and
interaction with other bodies. No integrity of such a system
does not possess." (L. Belousov. Messengers of a living thunderstorm. \\ Knowledge is power. N
2. 1996. - p.32). Thus classical science believed that the future
such a system is rigidly and unambiguously determined by its past and, subject to
knowledge of the past, unlimitedly predictable.

Modern mathematics has shown that in some cases this is not
like this: for example, if the balls hit a convex wall, then the negligible
the differences in their trajectories will grow indefinitely, so that
the behavior of the system becomes unpredictable at some point.
Thus, the positions of unambiguous determinism were undermined even
in relatively simple situations.

A worldview based on the theory of self-organization,
symbolized by the image of a mountainous country with valleys through which rivers flow,
and watershed ridges. This country has powerful feedback
- both negative and positive. If the body rolls down
along the slope, then there is a positive
the feedback, if it tries to climb up, is negative.
Nonlinear (sufficiently strong) feedbacks are an indispensable condition
self-organization. Nonlinearity in the ideological sense means
multivariate paths of evolution, the presence of a choice of alternative paths
and a certain rate of evolution, as well as the irreversibility of evolutionary
processes. For example, consider the interaction of two bodies: A and B. B -
elastic tree trunk, A is a mountain stream in our country. The flow bends
trunk in the direction of water movement, but upon reaching a certain
bending the trunk under the action of an elastic force can straighten out, repelling
water particles back. That is, we see an alternative interaction
two bodies A and B. Moreover, this interaction occurs in such a way that
that the A-B relationship is positive, and the B-A relationship is negative. The condition is met
non-linearity.

Moreover, in self-organization theory, we can force our
mountainous country to "live", that is, to change in time. At the same time, it is important
select variables of different order. Such a hierarchy of variables
time is a necessary condition for ordering self-organization.
Break it, "mix" the times - chaos will come (for example, an earthquake,
when shifts in the geological order occur in a matter of minutes, and
should - for several millennia). However, as it turns out, living
systems are not so afraid of chaos: they live on its limit all the time,
sometimes even falling into it, but still they know how, when necessary, from it
get out. In this case, the most important are the slowest
time variables (they are called parameters). It is the parameter values
determine what set of sustainable solutions the system will have and,
thus, what structures can be implemented in it at all. AT
same time faster

(dynamic) variables are responsible for the specific choice of realizable
stable states among the possible ones.

The principles of non-linearity and alternatives for choosing the development of any
process, development of the system is also implemented in the construction of fractals.

As it has become clear in recent decades (due to the development of the theory
self-organization), self-similarity occurs in a variety of objects and
phenomena. For example, self-similarity can be observed in tree branches and
shrubs, when dividing a fertilized zygote, snowflakes, crystals
ice, with the development of economic systems (Kondratiev waves), the structure
mountain systems, in the structure of clouds. All of the above and others
similar to them in their structure are called fractal. That is, they
possess the properties of self-similarity, or scale invariance. And this
means that some fragments of their structure are strictly repeated through
certain spatial intervals. It is clear that these objects
can be of any nature, and their appearance and form remain unchanged
regardless of scale.

Thus, we can say that fractals as models are used in
case when the real object cannot be represented in the form of classical
models. And this means that we are dealing with nonlinear relationships and
non-deterministic nature of the data. Nonlinearity in the worldview
sense means the multivariance of development paths, the availability of a choice from
alternative paths and a certain pace of evolution, as well as the irreversibility
evolutionary processes. Non-linearity in the mathematical sense means
a certain kind of mathematical equations (nonlinear differential
equations) containing the desired quantities in powers greater than one or
coefficients depending on the properties of the medium. That is, when we use
classical models (for example, trend, regression, etc.), we
we say that the future of the object is uniquely determined. And we can
predict it, knowing the past of the object (input data for
modeling). And fractals are used when an object has
several development options and the state of the system is determined
the position it is in at the moment. That is, we
trying to simulate a chaotic development.

What gives us the use of fractals?

They allow you to greatly simplify complex processes and objects, which is very
important for modeling. Allows you to describe unstable systems and
processes and, most importantly, to predict the future of such objects.

FRACTAL THEORY

BACKGROUND OF THE APPEARANCE

The theory of fractals has a very young age. She appeared in
late sixties at the intersection of mathematics, computer science, linguistics
and biology. At that time, computers were increasingly penetrating life.
people, scientists began to apply them in their research, the number of
computer users. For mass use
computers, it became necessary to facilitate the process of communication between a person and
machine. If at the very beginning of the computer era a few
programmers-users selflessly entered commands in machine
codes and received results in the form of endless paper tapes, then with
massive and loaded mode of using computers arose
the need to invent a programming language that was
would be understandable to the machine, and at the same time, would be easy to learn and
application. That is, the user would only need to enter one
command, and the computer would decompose it into simpler ones, and execute
would already have them. To facilitate the writing of translators, at the intersection of computer science
and linguistics, a theory of fractals arose, which allows you to strictly set
relationships between algorithmic languages. And the Danish mathematician and
biologist A. Lindenmeer came up with one such grammar in 1968,
which he called the L-system, which, as he believed, also models the growth
living organisms, especially the formation of bushes and branches in plants.

Here is what his model looks like. Set alphabet - arbitrary set
characters. Allocate one, the initial word, called an axiom, - you can
consider that it corresponds to the initial state of the organism - the embryo.
And then they describe the rules for replacing each character of the alphabet with a certain
a set of symbols, that is, they set the law of development of the embryo. Operate
the rules are as follows: we read each symbol of the axiom in order and replace
it to the word specified in the substitution rule.

Thus, after reading the axiom once, we get a new line
characters, to which we again apply the same procedure. Step by step
an increasingly longer string appears - each of these steps can be
considered as one of the successive stages in the development of the "organism".
By limiting the number of steps, determine when the development is considered complete.

THE ORIGIN OF THE THEORY OF FRACTALS

Benoit Mandelbrot can rightfully be considered the father of fractals.
Mandelbrot is the inventor of the term "fractal". Mandelbrot
wrote: “I came up with the word “fractal”, based on the Latin
adjective "fractus", meaning irregular, recursive,
fragmentary. The first definition of fractals was also given by B. Mandelbrot:

A fractal is a self-similar structure whose image does not depend on
scale. This is a recursive model, each part of which repeats in its own
development development of the entire model as a whole.

To date, there are many different mathematical models
fractals. The distinguishing feature of each of them is that
they are based on some recursive function, for example: xi=f(xi-1).
With the use of computers, researchers have the opportunity to obtain
graphic images of fractals. The simplest models do not require large
calculations and they can be implemented directly in a computer science lesson, while
other models are so demanding on computer power that they
implementation is carried out using a supercomputer. Incidentally, in the US
fractal models are studied by the National Application Center
for Supercomputers (NCSA). In this work, we only want to show
several fractal models that we managed to get.

Mandelbrot model.

Benoit Mandelbrot proposed a fractal model, which has already become
classic and is often used to showcase how typical
example of the fractal itself, and to demonstrate the beauty of fractals,
which also attracts researchers, artists, just
interested people.

The mathematical description of the model is as follows: on the complex plane in
some interval for each point with the recursive function is calculated
Z=Z2+c. It would seem, what is so special about this function? But after N
repetitions of this procedure for calculating the coordinates of points, on
complex plane, a surprisingly beautiful figure appears, something
pear-like.

In the Mandelbrot model, the changing factor is the starting point
c, and the parameter z, is dependent. Therefore, to construct a fractal
Mandelbrot there is a rule: the initial value of z is zero (z=0)!
This restriction is introduced so that the first derivative of the function
z at the starting point was equal to zero. And this means that in the initial
point, the function has a minimum, and henceforth it will take only
big values.

We want to note that if the fractal recursive formula has a different
view, then you should choose another value of the starting point for
parameter Z. For example, if the formula looks like z=z2+z+c, then the initial
point will be:

2*z+1=0 ???z= -1/2.

In this work, we have the opportunity to bring images of fractals,
which were built in the NCSA. We received the image files via
Internet network.

Fig.1 Mandelbrot fractal

You already know the mathematical model of the Mandelbrot fractal. now we
Let's show how it is implemented graphically. Model starting point
equals zero. Graphically, it corresponds to the center of the pear body. Through N
steps will fill the entire body of the pear and in the place where it ended
the last iteration, the “head” of the fractal begins to form.
The "head" of the fractal will be exactly four times smaller than the body, since
the mathematical formula of a fractal is a square
polynomial. Then again, after N iterations, the "body" begins to form
"kidney" (to the right and left of the "body"). Etc. The more given
the number of iterations N, the more detailed the image of the fractal will be,
the more different processes it will have. Schematic representation
The growth stages of the Mandelbrot fractal are shown in Fig. 2:

Fig.2 Scheme of the formation of the Mandelbrot fractal

Figures 1 and 2 show that each subsequent formation on the "body"
exactly repeats in its structure the body itself. This is the distinctive
feature that this model is a fractal.

The following figures show how the position of the point will change,
corresponding to the parameter z, for different initial positions of the point
c.

A) Starting point in the "body" B) Starting point
dot in head

C) Starting point in the "kidney" D) Starting point in
"kidney" of the second level

E) Starting point in the "kidney" of the third level

From figures A - E it is clearly seen how with each step more and more
the structure of the fractal becomes more complicated and the parameter z has an increasingly complex
trajectory.

Limitations on the Mandelbrot model: there is evidence that in
the Mandelbrot model |z|

Julia model (Julia set)

The Julia fractal model has the same equation as the model
Mandelbrot: Z=Z2+c, only here the variable parameter is
not c, but z.

Accordingly, the entire structure of the fractal changes, since now on
the starting position is not subject to any restrictions. Between
models of Mandelbrot and Julia, there is such a difference: if the model
Mandelbrot is static (since initial z is always
zero), then the Julia model is a dynamic fractal model. On the
rice. 4 shows a graphical representation of the Julia fractal.

Rice. 4 Model Julia

As can be seen from the fractal drawing, it is symmetrical with respect to the central
dots shape, while the Mandelbrot fractal has a shape that is symmetrical
about the axis.

Sierpinski carpet

The Sierpinski carpet is considered another fractal pattern. It is under construction
as follows: a square is taken, divided into nine squares,
cut out the central square. Then with each of the eight remaining
squares, a similar procedure is performed. And so on ad infinitum. AT
As a result, instead of a whole square, we get a carpet with a peculiar
symmetrical pattern. This model was first proposed by the mathematician
Sierpinsky, after whom it got its name. Carpet example
Sierpinski can be seen in Fig. 4d.

Fig.4 Construction of the Sierpinski carpet

4. Koch curve

At the beginning of the 20th century, mathematicians were looking for curves that could not be found anywhere else.
points do not have a tangent. This meant that the curve abruptly changed its
direction, and, moreover, at an enormously high speed (the derivative
is equal to infinity). The search for these curves was caused not simply by
idle interest of mathematicians. The fact is that at the beginning of the twentieth century, very
quantum mechanics developed rapidly. Researcher M.Brown
sketched the trajectory of the movement of suspended particles in water and explained this
phenomenon is as follows: randomly moving atoms of a liquid collide with
suspended particles and thereby set them in motion. After such
explanation of Brownian motion, scientists were faced with the task of finding such
curve that best approximates the motion
Brownian particles. For this, the curve had to correspond to the following
properties: not have a tangent at any point. Mathematician Koch
proposed one such curve. We won't go into explanations
rules for its construction, but simply give its image, from which all
becomes clear (Fig. 5).

Fig.5 Stages of constructing the Koch curve

The Koch curve is another example of a fractal, since each of its
part is a reduced image of the entire curve.

6. Graphic images of various fractals

In this paragraph, we decided to place graphic images of various
fractals that we received from the Internet. Unfortunately we are not
were able to find a mathematical description of these fractals, but in order to
to understand their beauty, only drawings are enough.

Rice. 6 Examples of graphical representation of fractals

II SECTION

APPLICATION OF THE THEORY OF FRACTALS IN ECONOMY

TECHNICAL ANALYSIS OF FINANCIAL MARKETS

The financial market in the developed countries of the world has existed for more than one hundred
years. For centuries, people have bought and sold securities.
This type of transactions with securities brought income to market participants
because the prices of stocks and bonds fluctuated all the time,
were constantly changing. For centuries, people have bought securities at
same price and sold when they became more expensive. But sometimes
the buyer's expectations did not come true and the prices for the purchased papers started
fall, thus, he not only did not receive income, but also suffered
losses. For a very long time, no one thought about why this happens:
the price rises and then falls. People simply saw the result of the action and did not
thought about the causal mechanism that generates it.

This happened until an American financier, one of
publishers of the well-known newspaper "Financial Times", Charles Dow did not
published a number of articles in which he expounded his views on
functioning of the financial market. The Dow noticed that stock prices
subject to cyclical fluctuations: after a long period of growth,
a long fall, then another rise and fall. Thus,
Charles Dow first noticed that it is possible to predict the future
the behavior of the stock price, if its direction is known for some
last period.

Fig.1 Price behavior according to Ch.Dow

Subsequently, on the basis of the discoveries made by Ch. Dow, a whole
theory of technical analysis of the financial market, which received
called Dow Theory. This theory dates back to the nineties
nineteenth century, when C. Dow published his articles.

Technical analysis of the markets is a method of predicting the future
behavior of the price trend, based on knowledge of the history of its behavior.
Technical analysis for forecasting uses mathematical
properties of trends, not the economic performance of securities.

In the middle of the twentieth century, when the whole scientific world was only interested in
that the emerging theory of fractals, another well-known American
financier Ralph Elliot proposed his theory of the behavior of stock prices,
which was based on the use of fractal theory.

Elliot proceeded from the fact that the geometry of fractals does not exist.
only in living nature, but also in social processes. to public
He attributed the processes to trading in shares on the stock exchange.

ELLIOT WAVE THEORY

Elliot Wave Theory is one of the oldest technical theories.
analysis. Since its inception, none of the users have contributed to it
any notable changes. On the contrary, all efforts were directed to
that the principles formulated by Elliot loomed more and
more clearly. The result is obvious. With the help of Elliot's theory,
the best forecasts for the movement of the American Dow Jones index.

The basis of the theory is the so-called wave diagram. The wave is
discernible price movement. Following the rules of development of mass
psychological behavior, all price movements are divided into five waves in
direction of a stronger trend, and three waves in the opposite direction
direction. For example, in the case of a dominant trend, we will see five
waves when the price moves up and three - when moving (correcting) down.

To indicate a five-wave trend, numbers are used and for
the opposite three-wave - letters. Each of the five wave movements
called impulse, and each of the three-won - corrective. So
each of the waves 1,3,5, A and C is impulse, and 2,4, and B -
corrective.

Rice. 7 Elliott Wave Chart

Elliot was one of the first to clearly define the operation of Geometry
Fractals in nature, in this case - in the price chart. He
suggested that in each of the just shown impulse and
corrective waves is also an Elliot wave chart.
In turn, those waves can also be decomposed into components, and so
Further. Thus Elliot applied the theory of fractals to the decomposition
trend into smaller and more understandable parts. Knowledge of these parts in more
smaller scale than the largest waveform is important because
that traders (financial market participants), knowing in what part
charts they are in, can confidently sell securities when
a corrective wave starts and should buy them when it starts
impulse wave.

Fig.8 Fractal structure of the Elliott diagram

FIBONACCCI NUMBERS AND WAVES CHARACTERISTICS

Ralph Elliot first came up with the idea of ​​using a number sequence
Fibonacci for making forecasts within the framework of technical analysis. With
using Fibonacci numbers and coefficients, you can predict the length
each wave and the time of its completion. Without touching the issue of time,
Let's turn to the most commonly used rules for determining the length
Elliot waves. By length, in this case, we mean
rise or fall in prices.

impulse waves.

Wave 3 usually has a length of 1.618 of wave 1, less often - equal to
her.

Two of the impulse waves are often equal in length, usually waves 5
and 1. This usually happens if wavelength 3 is less than 1.618
wavelength 1.

Often there is a ratio in which the wavelength 5 is equal to 0.382
or 0.618 the distance traveled by the price from the beginning of wave 1 to the end
waves 3.

Corrections

The lengths of the corrective waves make up a certain coefficient
Fibonacci from the length of the previous impulse wave. In accordance with
by the alternation rule, waves 2 and 4 must alternate in percentage
ratio. The most common example is the following:
wave 2 was 61.8% of wave 1, while wave 4 could be
only 38.2% or 50% of wave 3.

CONCLUSION

In our work, not all areas of human knowledge are given,
where the theory of fractals found its application. We only want to say that
no more than a third of a century has passed since the emergence of the theory, but for this
time fractals for many researchers have become a sudden bright light
in the nights that illuminated hitherto unknown facts and patterns in
specific data areas. Using the theory of fractals began to explain
the evolution of galaxies and the development of the cell, the emergence of mountains and the formation
clouds, the movement of prices on the stock exchange and the development of society and the family. Maybe
maybe at first this passion for fractals was even too
stormy and attempts to explain everything using the theory of fractals were
unjustified. But, without a doubt, this theory has the right to
existence, and we regret that lately it has somehow been forgotten
and remained the lot of the chosen. In preparing this work, we
It is very interesting to find applications of THEORY in PRACTICE. because
very often there is a feeling that theoretical knowledge is in
away from real life.

At the end of our work, we want to bring enthusiastic words
godfather of fractal theory, Benoit Mandelbrot: "The geometry of nature
fractal! Nowadays it sounds as bold and absurd as
the famous exclamation of G. Galileo: “But still it spins!” in the XVI
century.

LIST OF USED SOURCES

Sheipak ​​I.A. Fractals, graftals, bushes… //Chemistry and Life. 1996 №6

Comprehension of chaos //Chemistry and life. 1992 №8

Erlich A. Technical analysis of commodity and stock markets, M: Infra-M, 1996

Materials from the Internet.

The Fibonacci sequence - a sequence proposed in 1202
by the medieval mathematician Leonardo Fibonacci. Refers to the species
return sequences. a1=1, a2=1, ai=ai-1+ai-2.
Fibonacci coefficients - the quotient of dividing two neighboring terms
Fibonacci sequences: K1=ai/ai-1=1.618,

K2=ai-1/ai=0.618. These coefficients are the so-called
"golden section".

share price

stock price chart

Often, brilliant discoveries made in science can radically change our lives. So, for example, the invention of a vaccine can save many people, and the creation of a new weapon leads to murder. Literally yesterday (on the scale of history) a person "tamed" electricity, and today he can no longer imagine his life without it. However, there are also such discoveries that, as they say, remain in the shadows, and despite the fact that they also have some influence on our lives. One of these discoveries was the fractal. Most people have not even heard of such a concept and will not be able to explain its meaning. In this article, we will try to deal with the question of what a fractal is, consider the meaning of this term from the standpoint of science and nature.

Order in chaos

In order to understand what a fractal is, one should start the debriefing from the position of mathematics, however, before delving into it, we philosophize a little. Each person has a natural curiosity, thanks to which he learns the world around him. Often, in his desire for knowledge, he tries to operate with logic in his judgments. So, analyzing the processes that take place around, he tries to calculate the relationships and derive certain patterns. The biggest minds on the planet are busy solving these problems. Roughly speaking, our scientists are looking for patterns where they are not, and should not be. Nevertheless, even in chaos there is a connection between certain events. This connection is the fractal. As an example, consider a broken branch lying on the road. If we look closely at it, we will see that it, with all its branches and knots, itself looks like a tree. This similarity of a separate part with a single whole testifies to the so-called principle of recursive self-similarity. Fractals in nature can be found all the time, because many inorganic and organic forms are formed in a similar way. These are clouds, and sea shells, and snail shells, and tree crowns, and even the circulatory system. This list can be continued indefinitely. All these random shapes are easily described by the fractal algorithm. Here we come to consider what a fractal is from the standpoint of the exact sciences.

Some dry facts

The word “fractal” itself is translated from Latin as “partial”, “divided”, “fragmented”, and as for the content of this term, there is no wording as such. Usually it is treated as a self-similar set, a part of the whole, which is repeated by its structure at the micro level. This term was coined in the seventies of the twentieth century by Benoit Mandelbrot, who is recognized as the father. Today, the concept of a fractal means a graphic representation of a certain structure, which, when enlarged, will be similar to itself. However, the mathematical basis for the creation of this theory was laid even before the birth of Mandelbrot himself, but it could not develop until electronic computers appeared.

Historical reference, or How it all began

At the turn of the 19th and 20th centuries, the study of the nature of fractals was episodic. This is due to the fact that mathematicians preferred to study objects that can be investigated on the basis of general theories and methods. In 1872, the German mathematician K. Weierstrass constructed an example of a continuous function that is nowhere differentiable. However, this construction turned out to be completely abstract and difficult to understand. Next came the Swede Helge von Koch, who in 1904 built a continuous curve that has no tangent anywhere. It is quite easy to draw, and, as it turned out, it is characterized by fractal properties. One of the variants of this curve was named after its author - "Koch's snowflake". Further, the idea of ​​self-similarity of figures was developed by the future mentor of B. Mandelbrot, the Frenchman Paul Levy. In 1938 he published the paper "Plane and Spatial Curves and Surfaces Consisting of Parts Like a Whole". In it, he described a new species - the Levy C-curve. All of the above figures conditionally refer to such a form as geometric fractals.

Dynamic or algebraic fractals

The Mandelbrot set belongs to this class. The French mathematicians Pierre Fatou and Gaston Julia became the first researchers in this direction. In 1918 Julia published a paper based on the study of iterations of rational complex functions. Here he described a family of fractals that are closely related to the Mandelbrot set. Despite the fact that this work glorified the author among mathematicians, it was quickly forgotten. And only half a century later, thanks to computers, Julia's work received a second life. Computers made it possible to make visible to every person the beauty and richness of the world of fractals that mathematicians could "see" by displaying them through functions. Mandelbrot was the first to use a computer to carry out calculations (it is impossible to carry out such a volume manually) that made it possible to build an image of these figures.

Man with spatial imagination

Mandelbrot began his scientific career at the IBM Research Center. Studying the possibilities of data transmission over long distances, scientists were faced with the fact of large losses that arose due to noise interference. Benoit was looking for ways to solve this problem. Looking through the measurement results, he drew attention to a strange pattern, namely: the noise graphs looked the same on different time scales.

A similar picture was observed both for a period of one day, and for seven days, or for an hour. Benoit Mandelbrot himself often repeated that he does not work with formulas, but plays with pictures. This scientist was distinguished by imaginative thinking, he translated any algebraic problem into a geometric area, where the correct answer is obvious. So it is not surprising, distinguished by the rich and became the father of fractal geometry. After all, the awareness of this figure can come only when you study the drawings and think about the meaning of these strange swirls that form the pattern. Fractal drawings do not have identical elements, but they are similar at any scale.

Julia - Mandelbrot

One of the first drawings of this figure was a graphic interpretation of the set, which was born thanks to the work of Gaston Julia and was finalized by Mandelbrot. Gaston was trying to imagine what a set looks like when it is built from a simple formula that is iterated by a feedback loop. Let's try to explain what has been said in human language, so to speak, on the fingers. For a specific numerical value, using the formula, we find a new value. We substitute it into the formula and find the following. The result is a large one. To represent such a set, you need to do this operation a huge number of times: hundreds, thousands, millions. This is what Benoit did. He processed the sequence and transferred the results to graphical form. Subsequently, he colored the resulting figure (each color corresponds to a certain number of iterations). This graphic image is called the Mandelbrot fractal.

L. Carpenter: art created by nature

The theory of fractals quickly found practical application. Since it is very closely related to the visualization of self-similar images, the first to adopt the principles and algorithms for constructing these unusual forms were artists. The first of these was the future founder of Pixar studio Lauren Carpenter. While working on the presentation of aircraft prototypes, he came up with the idea to use the image of mountains as a background. Today, almost every computer user can cope with such a task, and in the seventies of the last century, computers were not able to perform such processes, because there were no graphic editors and applications for three-dimensional graphics at that time. Loren came across Mandelbrot's Fractals: Shape, Randomness, and Dimension. In it, Benois gave many examples, showing that there are fractals in nature (fiva), he described their various forms and proved that they are easily described by mathematical expressions. The mathematician cited this analogy as an argument for the usefulness of the theory he was developing in response to a flurry of criticism from his colleagues. They argued that a fractal is just a beautiful picture of no value, a by-product of electronic machines. Carpenter decided to try this method in practice. Having carefully studied the book, the future animator began to look for a way to implement fractal geometry in computer graphics. It took him only three days to render a completely realistic image of the mountain landscape on his computer. And today this principle is widely used. As it turned out, creating fractals does not take much time and effort.

Carpenter's decision

The principle used by Lauren turned out to be simple. It consists in dividing larger ones into smaller elements, and those into similar smaller ones, and so on. Carpenter, using large triangles, crushed them into 4 small ones, and so on, until he got a realistic mountain landscape. Thus, he became the first artist to apply the fractal algorithm in computer graphics to construct the required image. Today, this principle is used to simulate various realistic natural forms.

The first 3D visualization based on the fractal algorithm

A few years later, Lauren applied his work in a large-scale project - an animated video Vol Libre, shown on Siggraph in 1980. This video shocked many, and its creator was invited to work at Lucasfilm. Here the animator was able to fully realize himself, he created three-dimensional landscapes (the whole planet) for the feature film "Star Trek". Any modern program ("Fractals") or application for creating three-dimensional graphics (Terragen, Vue, Bryce) uses the same algorithm for modeling textures and surfaces.

Tom Beddard

A former laser physicist and now digital artist and artist, Beddard created a series of highly intriguing geometric shapes that he called Faberge's fractals. Outwardly, they resemble the decorative eggs of a Russian jeweler, they have the same brilliant intricate pattern. Beddard used a template method to create his digital renderings of the models. The resulting products are striking in their beauty. Although many refuse to compare a handmade product with a computer program, it must be admitted that the resulting forms are unusually beautiful. The highlight is that anyone can build such a fractal using the WebGL software library. It allows you to explore various fractal structures in real time.

fractals in nature

Few people pay attention, but these amazing figures are everywhere. Nature is made up of self-similar figures, we just don't notice it. It is enough to look through a magnifying glass at our skin or a leaf of a tree, and we will see fractals. Or take, for example, a pineapple or even a peacock's tail - they consist of similar figures. And the Romanescu broccoli variety is generally striking in its appearance, because it can truly be called a miracle of nature.

Musical pause

It turns out that fractals are not only geometric shapes, they can also be sounds. So, the musician Jonathan Colton writes music using fractal algorithms. He claims to correspond to natural harmony. The composer publishes all his works under the CreativeCommons Attribution-Noncommercial license, which provides for free distribution, copying, transfer of works by other persons.

Fractal indicator

This technique has found a very unexpected application. On its basis, a tool for analyzing the stock exchange market was created, and, as a result, it began to be used in the Forex market. Now the fractal indicator is found on all trading platforms and is used in a trading technique called a price breakout. Bill Williams developed this technique. As the author comments on his invention, this algorithm is a combination of several "candles", in which the central one reflects the maximum or, conversely, the minimum extreme point.

Finally

So we have considered what a fractal is. It turns out that in the chaos that surrounds us, in fact, there are ideal forms. Nature is the best architect, the ideal builder and engineer. It is arranged very logically, and if we cannot find a pattern, this does not mean that it does not exist. Maybe you need to look at a different scale. We can say with confidence that fractals still keep a lot of secrets that we have yet to discover.

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

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

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.

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. There they naturally appear in the study of nonlinear dynamic systems. The most studied case is when the dynamical system is specified by iterations polynomial or holomorphic function of a complex of variables on surface. Some of the most famous fractals of this type are the Julia set, 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 simple procedure 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.