Municipal budgetary educational institution

"Siverskaya secondary school №3"

Research

mathematics.

Did the job

student of grade 8-1

Emelin Pavel

supervisor

mathematic teacher

Tupitsyna Natalia Alekseevna

Siversky settlement

year 2014

Mathematics is all permeated with beauty and harmony,

Only this beauty must be seen.

B. Mandelbrot

Introduction ____________________________________ 3-4 p.

Chapter 1.History of the origin of fractals ._______ 5-6 pp.

Chapter 2. Classification of fractals ._____________ 6-10 p.

Geometric fractals

Algebraic fractals

Stochastic fractals

Chapter 3. "Fractal geometry of nature" ______ 11-13 p.

Chapter 4. Application of fractals _______________ 13-15 p.

Chapter 5 Practical work __________________ 16-24 p.

Conclusion _________________________________ 25.p

References and Internet resources ________ 26 p.

Introduction

Maths,

if you look at it correctly,

reflects not only the truth,

but also incomparable beauty.

Bertrand Russell

The word “fractal” is something that a lot of people talk about these days, from scientists to high school students. It appears on the covers of many mathematics textbooks, scientific journals, and computer software boxes. Today, colored images of fractals can be found everywhere: from postcards, T-shirts to pictures on the desktop of a personal computer. So what are these colored shapes that we see around?

Mathematics is the oldest science. It seemed to most people that geometry in nature is limited to such simple figures as a line, circle, polygon, sphere, etc. As it turned out, many natural systems are so complex that using only familiar objects of ordinary geometry to model them seems hopeless. How, for example, can you model a mountain ridge or tree crown in terms of geometry? How to describe the diversity of biological diversity that we observe in the world of plants and animals? How to imagine all the complexity of the circulatory system, consisting of many capillaries and vessels and delivering blood to every cell of the human body? Imagine the structure of the lungs and kidneys, resembling the structure of trees with a branched crown?

Fractals are suitable tools for investigating the questions posed. Often what we see in nature intrigues us with the endless repetition of the same pattern, enlarged or reduced several times. For example, a tree has branches. These branches have smaller branches, etc. In theory, the “forking” element repeats itself infinitely many times, getting smaller and smaller. The same can be seen when looking at a photograph of a mountainous relief. Try to zoom in on the mountain range a little - you will see the mountains again. This is how the self-similarity characteristic of fractals is manifested.

The study of fractals opens up wonderful possibilities, both in the study of an infinite number of applications and in the field of mathematics. The use of fractals is very extensive! After all, these objects are so beautiful that they are used by designers, artists, with the help of them many elements of trees, clouds, mountains, etc. are drawn in graphics. But fractals are even used as antennas in many cell phones.

For many chaologists (scientists who study fractals and chaos), this is not just a new field of knowledge that unites mathematics, theoretical physics, art and computer technology - this is a revolution. This is the discovery of a new type of geometry, the geometry that describes the world around us and which can be seen not only in textbooks, but also in nature and everywhere in the boundless universe..

In my work, I also decided to "touch" the world of beauty and determined for myself ...

purpose of work: Create objects that look very natural.

Research methods: comparative analysis, synthesis, modeling.

Tasks:

acquaintance with the concept, history of occurrence and research of B. Mandelbrot,

G. Koch, V. Sierpinsky and others;

acquaintance with various types of fractal sets;

study of popular science literature on this issue, acquaintance with

scientific hypotheses;

finding confirmation of the theory of fractality of the surrounding world;

study of the application of fractals in other sciences and in practice;

conducting an experiment to create your own fractal images.

Fundamental Job Question:

Show that mathematics is not a dry, soulless subject, it can express the spiritual world of a person individually and in society as a whole.

Subject of study: Fractal geometry.

Object of study: fractals in mathematics and in the real world.

Hypothesis: Everything that exists in the real world is a fractal.

Research methods: analytical, search.

Relevance the declared topic is determined, first of all, by the subject of research, which is fractal geometry.

Expected results: In the course of work, I will be able to expand my knowledge in the field of mathematics, see the beauty of fractal geometry, start working on creating my own fractals.

The result of the work will be the creation of a computer presentation, a newsletter and a booklet.

Chapter 1 history of origin

B enoie mandelbrot

The concept of "fractal" was invented by Benoit Mandelbrot. The word comes from the Latin "fractus" meaning "broken, shattered."

Fractal (Latin fractus - crushed, broken, shattered) is a term meaning a complex geometric figure with the property of self-similarity, that is, composed of several parts, each of which is similar to the entire figure as a whole.

The mathematical objects to which it refers are characterized by extremely interesting properties. In conventional geometry, a line has one dimension, a surface has two dimensions, and a spatial figure is three-dimensional. Fractals, on the other hand, are not lines or surfaces, but, if one can imagine this, something in between. With an increase in size, the volume of the fractal also increases, but its dimension (exponent) is not an integer value, but a fractional one, and therefore the border of a fractal figure is not a line: at high magnification it becomes clear that it is blurred and consists of spirals and curls, repeating in a small scale the figure itself. This geometric regularity is called scale invariance or self-similarity. It is she who determines the fractional dimension of fractal figures.

Before the advent of fractal geometry, science dealt with systems enclosed in three spatial dimensions. Thanks to Einstein, it became clear that three-dimensional space is only a model of reality, and not reality itself. In fact, our world is located in a four-dimensional space-time continuum.
Thanks to Mandelbrot, it became clear what the four-dimensional space looks like, figuratively speaking, the fractal face of Chaos. Benoit Mandelbrot discovered that the fourth dimension includes not only the first three dimensions, but also (this is very important!) The intervals between them.

Recursive (or fractal) geometry is replacing Euclidean. The new science is able to describe the true nature of bodies and phenomena. Euclidean geometry dealt only with artificial, imaginary objects belonging to three dimensions. Only the fourth dimension can turn them into reality.

Liquid, gas, solid are the three usual physical states of matter that exists in the three-dimensional world. But what is the dimension of a club of smoke, clouds, or rather, their boundaries, continuously eroded by turbulent air movement?

Basically, fractals are classified into three groups:

Algebraic fractals

Stochastic fractals

Geometric fractals

Let's take a closer look at each of them.

Chapter 2. Classification of fractals

Geometric fractals

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

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.

Fractals of this class are the most illustrative, because self-similarity is immediately visible in them at any scale of observation. In a two-dimensional case, such fractals can be obtained by specifying a certain broken line, called a generator. In one step of the algorithm, each of the segments that make up the polyline is replaced by a polyline-generator, in the appropriate scale. As a result of endless repetition of this procedure (or, more precisely, when passing to the limit), a fractal curve is obtained. With the apparent complexity of the resulting curve, its general appearance is set only by the shape of the generator. Examples of such curves are: the Koch curve (Fig. 7), the Peano curve (Fig. 8), the Minkowski curve.

At the beginning of the twentieth century, mathematicians were looking for curves that have no tangent at any point. This meant that the curve changes its direction abruptly, and, moreover, at an enormous speed (the derivative is equal to infinity). The search for these curves was motivated not simply by the idle interest of mathematicians. The fact is that at the beginning of the twentieth century, quantum mechanics developed very rapidly. Researcher M. Brown sketched the trajectory of suspended particles in water and explained this phenomenon as follows: randomly moving atoms of a liquid hit the suspended particles and thereby set them in motion. After such an explanation of Brownian motion, scientists were faced with the task of finding a curve that would best show the motion of Brownian particles. For this, the curve had to meet the following properties: not have a tangent at any point. The mathematician Koch proposed one such curve.

TO The Koch line is a typical geometric fractal. The process of its construction is as follows: we take a unit segment, divide it into three equal parts and replace the middle interval with an equilateral triangle without this segment. As a result, a polyline is formed, consisting of four links of length 1/3. At the next step, we repeat the operation for each of the four resulting links, etc.

The limiting curve is Koch curve.

Koch's snowflake. By performing similar transformations on the sides of an equilateral triangle, you can get a fractal image of a Koch snowflake.

T
Another simple representative of a geometric fractal is Sierpinski square. It is built quite simply: The square is divided by straight lines parallel to its sides into 9 equal squares. The central square is removed from the square. The result is a set consisting of 8 remaining "first rank" squares. Proceeding in the same way with each of the squares of the first rank, we get a set consisting of 64 squares of the second rank. Continuing this process infinitely, we get an infinite sequence or Sierpinski square.

Algebraic fractals

This is the largest group of fractals. Algebraic fractals get their name from the fact that they are constructed using simple algebraic formulas.

They are obtained using nonlinear processes in n-dimensional spaces. It is known that nonlinear dynamical systems have several stable states. The state in which the dynamical system finds itself after a certain number of iterations depends on its initial state. Therefore, each stable state (or, as they say, an attractor) has a certain region of initial states, from which the system will necessarily fall into the final states under consideration. Thus, the phase space of the system is divided into areas of attraction attractors. If a two-dimensional space is a phase space, then by coloring the regions of attraction with different colors, one can obtain color phase portrait this system (iterative process). By changing the color selection algorithm, you can get complex fractal paintings with bizarre multicolor patterns. A surprise for mathematicians was the ability to generate very complex structures using primitive algorithms.

As an example, consider the Mandelbrot set. It is built using complex numbers.

A section of the boundary of the Mandelbrot set, magnified 200 times.

The Mandelbrot set contains points that duringendless the number of iterations does not go to infinity (points with black color). Points belonging to the boundary of the set(this is where complex structures arise) go to infinity after a finite number of iterations, and points lying outside the set go to infinity after several iterations (white background).

NS



An example of another algebraic fractal is the Julia set. There are 2 types of this fractal. Surprisingly, Julia sets are formed according to the same formula as the Mandelbrot set. The Julia set was invented by the French mathematician Gaston Julia, after whom the set was named.

AND
interesting fact, some algebraic fractals strikingly resemble images of animals, plants and other biological objects, as a result of which they are called biomorphs.

Stochastic fractals

Another well-known class of fractals are stochastic fractals, which are obtained if any of its parameters are randomly changed in an iterative process. At the same time, objects are obtained that are very similar to natural ones - asymmetric trees, indented coastlines, etc.

Plasma is a typical representative of this group of fractals.

D
To construct it, a rectangle is taken and a color is determined for each corner. Next, the central point of the rectangle is found and painted in a color equal to the arithmetic mean of the colors at the corners of the rectangle plus some random number. The larger the random number, the more "ragged" the drawing will be. If we assume that the color of the point is the height above sea level, we will get instead of plasma - a mountain range. It is on this principle that mountains are modeled in most programs. Using an algorithm similar to plasma, a height map is built, various filters are applied to it, a texture is applied and the photorealistic mountains are ready

E
If we look at this fractal in a cut, we will see this fractal volumetric, and has “roughness”, just because of this “roughness” there is a very important application of this fractal.

Let's say you want to describe the shape of a mountain. Ordinary figures from Euclidean geometry will not help here, because they do not take into account the surface relief. But when you combine the usual geometry with the fractal, you can get the very "roughness" of the mountain. Plasma should be applied to an ordinary cone and we will get the relief of the mountain. Such operations can be performed with many other objects in nature; thanks to stochastic fractals, one can describe nature itself.

Now let's talk about geometric fractals.

.

Chapter 3 "Fractal geometry of nature"

“Why is geometry often called“ cold ”and“ dry? ”One of the reasons is its inability to describe the shape of a cloud, mountain, coastline or tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, tree bark is not smooth, the lightning does not travel in a straight line More generally, I argue that many objects in Nature are so irregular and fragmented that compared to Euclid - a term that in this work refers to all standard geometry - Nature has more than just greater complexity, but the complexity of a completely different level. The number of different scales of length of natural objects for all practical purposes is infinite. "

(Benoit Mandelbrot "Fractal geometry of nature" ).

TO The racism of fractals is twofold: it delights the eye, as evidenced by at least the world-wide exhibition of fractal images, organized by a group of mathematicians from Bremen under the leadership of Peitgen and Richter. Later, the exhibits of this grandiose exhibition were captured in illustrations for the book by the same authors "The Beauty of Fractals". But there is another, more abstract or sublime, aspect of the beauty of fractals, open, according to R. Feynman, only to the mental eye of the theorist, in this sense, fractals are beautiful with the beauty of a difficult mathematical problem. Benoit Mandelbrot pointed out to his contemporaries (and, presumably, descendants) an annoying gap in Euclid's Principles, according to which, without noticing the omission, for almost two millennia of mankind comprehended the geometry of the surrounding world and learned the mathematical rigor of presentation. Of course, both aspects of the beauty of fractals are closely interconnected and do not exclude, but mutually complement each other, although each of them is self-sufficient.

Mandelbrot's fractal geometry of nature is a real geometry that satisfies the definition of geometry proposed in the Erlangen Program by F. Klein. The fact is that before the appearance of non-Euclidean geometry N.I. Lobachevsky - L. Bolyai, there was only one geometry - the one that was presented in the "Elements", and the question of what is geometry and which of the geometries is the geometry of the real world did not arise, and could not arise. But with the advent of another geometry, the question arose of what geometry is in general, and which of the many geometries corresponds to the real world. According to F. Klein, geometry is studying such properties of objects that are invariant with respect to transformations: Euclidean - invariants of the group of motions (transformations that do not change the distance between any two points, i.e. representing a superposition of parallel translations and rotations with or without a change in orientation) , geometry of Lobachevsky-Bolyai - invariants of the Lorentz group. Fractal geometry studies the invariants of the group of self-affine transformations, i.e. properties expressed by power laws.

As for the correspondence to the real world, fractal geometry describes a very wide class of natural processes and phenomena, and therefore, following B. Mandelbrot, we can rightfully speak about the fractal geometry of nature. New - fractal objects have unusual properties. The lengths, areas and volumes of some fractals are equal to zero, while others go to infinity.

Nature often creates amazing and beautiful fractals, with perfect geometry and such harmony that you just freeze with admiration. And here are their examples:

Sea shells

Lightning admire with their beauty. Lightning fractals are neither random nor regular

Fractal form subspecies of cauliflower(Brassica cauliflora). This particular view is a particularly symmetrical fractal.

NS swamp is also a good example of a fractal among flora.

Peacocks everyone is known for their colorful plumage, in which solid fractals are hidden.

Ice, frosty patterns on the windows they are also fractals

O
t enlarged image leaflet, before tree branches- fractals can be found in everything

Fractals are everywhere and everywhere in the nature around us. The entire Universe is built according to surprisingly harmonious laws with mathematical precision. How can you then think that our planet is a random cohesion of particles? Hardly.

Chapter 4. Application of fractals

Fractals are finding more and more applications in science. The main reason for this is that they describe the real world sometimes even better than traditional physics or mathematics. Here are some examples:

O
the days of the most powerful fractal applications lie in computer graphics... This is fractal image compression. Modern physics and mechanics are just beginning to study the behavior of fractal objects.

The advantages of fractal image compression algorithms are very small packed file size and short image recovery time. Fractally packed images can be scaled without the appearance of pixelation (poor image quality - large squares). But the compression process takes a long time and sometimes takes hours. The lossy fractal packing algorithm allows you to set the compression ratio, similar to the jpeg format. The algorithm is based on finding large pieces of an image similar to some small pieces. And only which piece is similar to which is written to the output file. When compressing, they usually use a square grid (pieces - squares), which leads to a slight angularity when restoring the image, the hexagonal grid is devoid of such a drawback.

Iterated has developed a new "Sting" image format that combines fractal and waveform (such as jpeg) lossless compression. The new format allows you to create images with the possibility of subsequent high-quality scaling, and the volume of graphic files is 15-20% of the volume of uncompressed images.

In mechanics and physics fractals are used due to the unique property of repeating the outlines of many objects of nature. Fractals allow you to approximate trees, rock surfaces, and cracks with a higher accuracy than approximations with a set of lines or polygons (for the same amount of stored data). Fractal models, like natural objects, have "roughness", and this property is retained at an arbitrarily large magnification of the model. The presence of a uniform measure on fractals allows one to apply integration, potential theory, use them instead of standard objects in the equations already studied.

T
Fractal geometry is also used to antenna design... This was first applied by the American engineer Nathan Cohen, who was then living in downtown Boston, where the installation of external antennas on buildings was prohibited. Cohen cut a Koch curve out of aluminum foil and glued it to a piece of paper, then attached it to the receiver. It turned out that such an antenna works no worse than the usual one. And although the physical principles of such an antenna have not yet been studied, this did not prevent Cohen from founding his own company and starting their serial production. At the moment the American company "Fractal Antenna System" has developed a new type of antenna. Now you can stop using protruding external antennas in mobile phones. The so-called fractal antenna is located directly on the main board inside the device.

There are also many hypotheses about the use of fractals - for example, the lymphatic and circulatory systems, lungs, and much more also have fractal properties.

Chapter 5. Practical work.

First, let's dwell on the "Necklace", "Victory" and "Square" fractals.

First - "Necklace"(fig. 7). This fractal is initiated by a circle. This circle consists of a certain number of the same circles, but smaller in size, and it itself is one of several circles, which are the same, but larger in size. So the process of education is endless and it can be conducted both in one direction and in the opposite direction. Those. the figure can be enlarged by taking just one small arc, or it can be reduced by considering its construction from smaller ones.

rice. 7.

Fractal "Necklace"

The second fractal is "Victory"(fig. 8). It got this name because it outwardly resembles the Latin letter "V", that is, "victory" - victory. This fractal consists of a certain number of small “v” constituting one large “V”, moreover, in the left half, where the small ones are placed so that their left halves make up one straight line, the right side is constructed in the same way. Each of these "v" is built in the same way and this continues indefinitely.

Fig. 8. Fractal "Victory"

The third fractal is "Square" (fig. 9)... Each of its sides consists of one row of cells, in the form of squares, the sides of which also represent rows of cells, etc.

Fig.9. Fractal "Square"

The fractal was named "Rose" (Fig. 10), due to its external resemblance to this flower. The construction of a fractal is associated with the construction of a series of concentric circles, the radius of which varies in proportion to the given ratio (in this case, R m / R b = ¾ = 0.75.). After that, a regular hexagon is inscribed in each circle, the side of which is equal to the radius of the circle described around it.

Rice. 11. Fractal "Rose *"

Next, we turn to the regular pentagon, in which we draw its diagonals. Then, in the resulting pentagon at the intersection of the corresponding segments, draw diagonals again. Let's continue this process to infinity and get the "Pentagram" fractal (Fig. 12).

Let's introduce an element of creativity and our fractal will take the form of a more visual object (Fig. 13).

R
is. 12. Fractal "Pentagram".

Rice. 13. Fractal "Pentagram *"

Rice. 14 fractal "Black hole"

Experiment No. 1 "Tree"

Now that I understood what a fractal is and how to build it, I tried to create my own fractal images. In Adobe Photoshop, I created a small subroutine or action, the peculiarity of this action is that it repeats the actions that I do, and this is how I get a fractal.

To begin with, I created a background for our future fractal with a resolution of 600 by 600. Then I drew 3 lines on this background - the basis of our future fractal.

WITH the next step is to write the script.

duplicate the layer ( layer> duplicate) and change the blend type to " Screen" .

Let's call it " fr1". Let's copy this layer (" fr1") 2 more times.

Now we need to switch to the last layer. (fr3) and merge it twice with the previous one ( Ctrl + E). Decrease layer brightness ( Image> Ajustments> Brightness / Contrast , brightness set 50% ). Merge again with the previous layer and crop the edges of the entire drawing to remove invisible parts.

In the last step, I copied this image and pasted it down and rotated. This is what happened in the final result.

Conclusion

This work is an introduction to the world of fractals. We have considered only the smallest part of what fractals are, on the basis of what principles they are built.

Fractal graphics are not just a set of self-repeating images, they are a model of the structure and principle of any being. Our whole life is represented by fractals. All nature around us consists of them. It should be noted that fractals are widely used in computer games, where terrain reliefs are often fractal images based on three-dimensional models of complex sets. Fractals make it very easy to draw computer graphics; with the help of fractals, many special effects, various fabulous and incredible pictures, etc. are created. Also, with the help of fractal geometry, trees, clouds, banks and all other nature are drawn. Fractal graphics are needed everywhere, and the development of "fractal technologies" is one of the most important tasks today.

In the future, I plan to learn how to build algebraic fractals when I study complex numbers in more detail. I also want to try to build my fractal images in the Pascal programming language using loops.

It should be noted the use of fractals in computer technology, in addition to simply constructing beautiful images on a computer screen. Fractals in computer technology are used in the following areas:

1. Compression of images and information

2. Hiding information in the image, in the sound, ...

3. Data encryption using fractal algorithms

4. Creation of fractal music

5. System modeling

In our work, far from all areas of human knowledge are given where the theory of fractals has found its application. We only want to say that no more than a third of a century has passed since the theory appeared, but during this time fractals for many researchers became a sudden bright light in the night, which illuminated hitherto unknown facts and patterns in specific areas of data. With the help of the theory of fractals, they began to explain the evolution of galaxies and the development of the cell, the emergence of mountains and the formation of clouds, the movement of prices on the stock exchange and the development of society and the family. Maybe at first this fascination with fractals was even too violent and attempts to explain everything with the help of the theory of fractals were unjustified. But, without a doubt, this theory has a right to exist, and we regret that recently it has somehow been forgotten and remained the lot of the elite. When preparing this work, it was very interesting for us to find the application of THEORY in PRACTICE. Because very often there is a feeling that theoretical knowledge stands aside from the reality of life.

Thus, the concept of fractals becomes not only a part of "pure" science, but also an element of universal human culture. Fractal science is still very young and has a great future ahead of it. The beauty of fractals is far from being exhausted and will give us a lot of masterpieces - those that delight the eye, and those that bring true delight to the mind.

10. References

Bozhokin S.V., Parshin D.A. Fractals and multifractals. RHD 2001 .

Vitolin D. Application of fractals in computer graphics. // Computerworld-Russia.-1995

Mandelbrot B. Self-affine fractal sets, "Fractals in physics". M .: Mir 1988

Mandelbrot B. Fractal geometry of nature. - M .: "Institute for Computer Research", 2002.

Morozov A.D. Introduction to the theory of fractals. N. Novgorod: Publishing house of Nizhny Novgorod. University 1999

Peitgen H.-O., Richter P. H. Beauty of fractals. - M .: "Mir", 1993.

Internet resources

http://www.ghcube.com/fractals/determin.html

http://fractals.nsu.ru/fractals.chat.ru/

http://fractals.nsu.ru/animations.htm

http://www.cootey.com/fractals/index.html

http://fraktals.ucoz.ru/publ

http: // sakva .narod .ru

http://rusnauka.narod.ru/lib/author/kosinov_n/12/

http://www.cnam.fr/fractals/

http://www.softlab.ntua.gr/mandel/

http://subscribe.ru/archive/job.education.maths/201005/06210524.html

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 practically do not have external antennas of a 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 have received great attention with the development of information technology. 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.

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

What do a tree, a seashore, a cloud, or blood vessels in our hand have in common? At first glance, it may seem that all these objects have nothing in common. However, in fact, there is one property of structure inherent in all the listed objects: they are self-similar. From the branch, as well as from the trunk of the tree, there are smaller branches, from them - even smaller ones, etc., that is, the branch is like 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 the 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 let's imagine that we are standing on the beach and looking at our feet: there are always pebbles that protrude into the water further than the rest. That is, the coastline remains similar to itself when zoomed in. The American (though raised in France) mathematician Benoit Mandelbrot called this property of objects fractality, and such objects themselves - fractals (from the Latin fractus - broken).

This concept has no strict definition. Therefore, the word "fractal" is not a mathematical term. Usually a fractal is called a geometric figure that satisfies one or more of the following properties: It has a complex structure at any magnification (as opposed to, for example, a straight line, any part of which is the simplest geometric figure - a line segment). Is (approximately) self-similar. Has a fractional Hausdorff (fractal) dimension, which is greater 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 rather episodic than systematic, because earlier mathematicians mainly studied "good" objects that were amenable to research using general methods and theories. In 1872, the German mathematician Karl Weierstrass constructs an example of a continuous function that is nowhere differentiable. However, its construction was entirely abstract and difficult to perceive. Therefore, in 1904, the Swede Helge von Koch invented a continuous curve, which has no tangent anywhere, and it is quite simple to draw. It turned out that it has the properties of a fractal. One of the variants 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, he published his article "Plane and spatial curves and surfaces, consisting of parts similar to the whole", which describes another fractal - the Lévy C-curve. All of these above fractals can be conditionally attributed to one class of constructive (geometric) fractals.

Another class is dynamic (algebraic) fractals, which include the Mandelbrot set. The first studies in this direction began at the beginning of the 20th century and are associated with the names of the French mathematicians Gaston Julia and Pierre Fatou. In 1918, Julia's almost two-hundred-page memoir, devoted to iterations of complex rational functions, was published, in which Julia's sets were 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 objects discovered. Despite the fact that this work glorified Julia among the mathematicians of the time, it was quickly forgotten. It wasn't until half a century later that computers came to attention again: it was they who made the wealth and beauty of the world of fractals visible.

Fractal dimensions

As you know, the dimension (number of measurements) of a geometric figure is the number of coordinates required 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, you can define the dimension in this way: 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) twice, 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 an increase in the "size" of an object to the logarithm of an increase in its linear size. That is, for the segment D = log (2) / log (2) = 1, for the plane D = log (4) / log (2) = 2, for the 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 by three times, the length of the Koch curve increases by log (4) / log (3) ~ 1.26. That is, the dimension of the Koch curve is fractional!

Science and art

In 1982, Mandelbrot's book "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. In his presentation, Mandelbrot made the main emphasis not on cumbersome formulas and mathematical constructions, but on the geometric intuition of the readers. Thanks to computer-generated illustrations and historical tales, with which the author skillfully diluted the scientific component of the monograph, the book became a bestseller, and fractals became known to the general public. Their success among non-mathematicians is largely due to the fact that with the help of very simple constructions and formulas that 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 with 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". This is an experiment that was staged 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 the two countries. Among the parameters that he took into account was the length of the common border of the two warring countries. When he collected data for numerical experiments, he found that in different sources the data on the common border between Spain and Portugal are very different. This prompted him to discover the following: the length of a country's borders depends on the ruler with which we measure them. The smaller the scale, the longer the border is. This is due to the fact that with a higher magnification it becomes possible to take into account more and more coastal bends, which were previously ignored due to the roughness of the measurements. And if, with each increase in scale, the previously unaccounted for bends of the lines will open, then it turns out that the length of the boundaries is infinite! True, in reality 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 a base and a fragment. At the first stage, the basis of the future fractal is depicted. Then some of its parts are replaced with a fragment taken at a suitable scale - this is the first iteration of construction. Then, the resulting figure again changes some parts into figures similar to a fragment, and so on. If we continue this process indefinitely, then in the limit we get a fractal.

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

In the language of mathematics: dynamic (algebraic) fractals

Fractals of this type arise 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). Take some starting point z0 on the complex plane (see sidebar). Now consider such an infinite sequence of numbers on the complex plane, each of the following 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 in different ways: tend to infinity as n -> ∞; converge to some end point; cyclically take a number of fixed values; more complex options are also possible.

Complex numbers

A complex number is a number consisting of two parts - real and imaginary, that is, the formal sum x + iy (here x and y are real numbers). i is the so-called. imaginary unit, that is, that is, a number that satisfies the equation i ^ 2 = -1. The basic mathematical operations are defined over complex numbers - 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 laid on the abscissa, and the imaginary part on the ordinate, 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. In this case, the points lying on the boundaries of these parts have the following property: for an arbitrarily small displacement, the nature of their behavior changes sharply (such points are called bifurcation points). So, it turns out that sets of points with 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).

Family of dragons

By varying the base and fragment, you can get an amazing variety of constructive fractals.
Moreover, similar operations can be performed in three-dimensional space. Examples of volumetric fractals are Menger's sponge, Sierpinski pyramid and others.
The dragon family is also referred to as constructive fractals. Sometimes they are called by the name of the discoverers "dragons of the Highway Harter" (in their form they resemble Chinese dragons). There are several ways to plot this curve. The simplest and most intuitive of them is this: you need to take a sufficiently long strip of paper (the thinner the paper, the better), and fold it in half. Then bend it twice again in the same direction as the first time. After several repetitions (usually after five to six folds, the strip becomes too thick to be neatly bent further), you need to unbend 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 in a slightly different way. Consider the function fc (z) = z 2 + с, 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 the 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 is 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 the values ​​of the complex parameter c for which the Julia set fc (z) is connected (a set is called connected if it cannot be split into two disjoint parts, with some additional conditions).

Fractals and life

Today, 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, much less is required memory than for storing the entire file). By adding random perturbations to the formulas defining 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 with a fractal shape. Taking up little space, they provide quite high-quality signal reception. Economists use fractals to describe currency rate curves (a property discovered by Mandelbrot over 30 years ago). This concludes this small excursion into the amazingly beautiful and diverse world of fractals.

Chaos is an order that needs to be deciphered.

Jose Saramago, The Double

“The 20th century will be remembered for future generations only thanks to the creation of the theories of relativity, quantum mechanics and chaos ... the theory of relativity got rid of Newton's illusions about absolute space-time, quantum mechanics dispelled the dream of the determinism of physical events, and, finally, chaos debunked Laplace's fantasy about complete predetermination of systems development ”. These words of the famous American historian and popularizer of science, James Gleick, reflect the enormous importance of the issue, which is only briefly covered in the article brought to the attention of the reader. Our world emerged from chaos. However, if chaos did not obey its own laws, if there was no special logic in it, it would not be able to generate anything.

The new is the well-forgotten old

Let me take another quote from Gleick:

The thought of an inner similarity, that the great can be invested in the small, has long caressed the human soul ... According to Leibniz, a drop of water contains the entire world shining with colors, where water splashes sparkle and other unknown universes live. "See the world in a grain of sand" - Blake urged, and some scientists tried to follow his covenant. The first researchers of seminal fluid tended to see in each sperm a kind of homunculus, that is, a tiny, but already fully formed human.

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

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

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

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

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

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

At the origins of chaos theory

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

The first researcher of chaos is considered to be the French mathematician, physicist and philosopher Henri Poincaré. Back at the end of the 19th century. while studying the behavior of a system with three bodies interacting gravitationally, he noticed that there may be non-periodic orbits that are constantly and do not move away from a particular point, and do not approach it.

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

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

Of the thousands of publications that made up the specialized literature on the problem of chaos, hardly any was cited more often than the article "Deterministic nonperiodic flow" written by Lorentz in 1963. Although computer simulations had already transformed weather forecasting from an “art to a science” during this work, long-term forecasts were still unreliable and unreliable. The reason for this was the very butterfly effect.

In the same 1960s, mathematician Stephen Smale of the University of California gathered a research group of young like-minded people in Berkeley. He was previously awarded the Fields Medal for Outstanding Research in Topology. Smale studied dynamical systems, in particular nonlinear chaotic oscillators. To reproduce all the disorder of the van der Pol oscillator in phase space, he created a structure known as a "horseshoe" - an example of a dynamical system with chaotic dynamics.

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

In 1972, University of Maryland mathematician James Yorke read the aforementioned paper by Lorenz, which amazed him. York saw in the article a living physical model and considered it his sacred duty to convey to physicists what they did not discern in the works of Lorenz and Smale. He forwarded a copy of Lorenz's article to Smale. He was amazed to discover that an obscure meteorologist (Lorenz) ten years earlier had discovered that disorder, which he himself once considered mathematically improbable, and sent copies to all his colleagues.

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

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

May tried to draw the attention of biologists to the fact that animal populations go through more than just ordered cycles. On the way to chaos, a whole cascade of period doubling arises. It is at the points of bifurcation that a slight increase in the fecundity of individuals could lead, for example, to the replacement of the four-year cycle of the gypsy moth population by an eight-year one. American Mitchell Feigenbaum decided to start by calculating the exact values ​​of the parameter that generated such changes. His calculations showed that it didn’t matter what the initial population was — it was still steadily approaching the attractor. Then, with the first doubling of periods, the attractor, like a dividing cell, bifurcated. Then the next multiplication of periods took place, and each point of the attractor began to divide again. The number - the invariant obtained by Feigenbaum - allowed him to predict exactly when this will happen. The scientist discovered that he could predict this effect for a very complex attractor - at two, four, eight points ... Speaking in the language of ecology, he could predict the actual number that is achieved in populations during annual fluctuations. So Feigenbaum discovered in 1976 the "period doubling cascade", drawing on May's work and his studies of turbulence. His theory reflected natural law that applies to all systems undergoing a transition from orderly to chaos. York, May and Feigenbaum were the first in the West to fully understand the importance of period doubling and were able to convey this idea to the entire scientific community. May stated that chaos must be taught.

Soviet mathematicians and physicists advanced in their research independently of foreign colleagues. The study of chaos was initiated by the work of A. N. Kolmogorov in the 1950s. But the ideas of foreign colleagues did not remain without their attention. The pioneers of chaos theory are the Soviet mathematicians Andrei Nikolaevich Kolmogorov and Vladimir Igorevich Arnold and the German mathematician Jurgen Moser, who built a chaos theory called KAM (Kolmogorov-Arnold-Moser theory). Another outstanding compatriot of ours, a brilliant physicist and mathematician Yakov Grigor'evich Sinai, applied considerations similar to "Smale's horseshoe" in thermodynamics. Hardly in the 70s Western physicists became acquainted with Lorentz's work, when it gained fame in the USSR. In 1975, when York and May were still making considerable efforts to gain the attention of their colleagues, Sinai and his comrades organized a research group in Gorky to study this problem.

In the last century, when narrow specialization and dissociation between different disciplines became the norm in science, mathematicians, physicists, biologists, chemists, physiologists, economists fought over similar problems without hearing each other. Ideas that require a change in the usual worldview always struggle to make their way. However, it gradually became clear that such things as changes in animal populations, fluctuations in prices in the market, changes in weather, the distribution of celestial bodies in size and much, much more, obey the same laws. "Realization of this fact forced managers to reconsider their attitude to insurance, astronomers - from a different angle to look at the solar system, politicians - to change their minds about the causes of armed conflicts."

By the mid-1980s, the situation had changed a lot. The ideas of fractal geometry united scientists who were puzzled by their own observations and did not know how to interpret them. For chaos researchers, mathematics has become an experimental science, computers have replaced laboratories. Graphic representations have become of paramount importance. The new science gave the world a special language, new concepts: phase portrait, attractor, bifurcation, section of phase space, fractal ...

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

Fractal geometry of nature

Let's try to figure out what a fractal is and "what it is eaten with." And you can really eat some of them, like, for example, a typical representative shown in the photo.

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

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

The mathematical concept of a fractal distinguishes objects with structures of various scales, both large and small, and thus reflects the hierarchical principle of organization. Of course, the different branches of a tree, for example, cannot be exactly aligned with each other, but they can be considered similar in a statistical sense. Likewise, cloud shapes, mountain outlines, seashore line, flame patterns, vascular system, ravines, lightning, viewed at different scales, appear similar. Although this idealization may turn out to be a simplification of reality, it significantly increases the depth of the mathematical description of nature.

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

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

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

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

Interested in a wide variety of fields of science, Mandelbrot turned to mathematical linguistics, then came the turn of game theory. He also proposed his own approach to economics, pointing out the orderliness of scale in the spread of cities and towns. Studying the little-known work of the English scientist Lewis Richardson, published after the death of the author, Mandelbrot was faced with the phenomenon of the coastline. In the article "How long is the UK coastline?" he investigates in detail this question, which few people have thought about before him, and comes to unexpected conclusions: the length of the coastline is equal to ... infinity! The more accurately you try to measure it, the greater its value is!

To describe such phenomena, Mandelbrot had the idea to start from the idea of ​​dimension. The fractal dimension of an object serves as a quantitative characteristic of one of its features, namely, its filling of space.

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

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

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

Fractional measurement allows you to calculate characteristics that cannot be clearly defined in any other way: the degree of unevenness, discontinuity, roughness or instability of an object. For example, a winding coastline, despite its immeasurable length, has a roughness inherent only to it. Mandelbrot pointed out the ways of calculating fractional measurements of objects in the surrounding reality. Creating his geometry, he put forward the law about the disordered forms that occur in nature. The law said: the degree of instability is constant at different scales.

A special kind of fractals are time fractals... In 1962, Mandelbrot was faced with the task of eliminating noise on telephone lines that caused problems for computer modems. The quality of the signal transmission depends on the likelihood of errors. Engineers have struggled with noise reduction with puzzling and costly tricks, but they didn’t get impressive results. Based on the work of the founder of the theory of sets, Georg Cantor, Mandelbrot showed that the occurrence of noise - the generation of chaos - cannot be avoided in principle, therefore the proposed methods of dealing with them will not bring results. In search of the regularity of the occurrence of noise, he receives "Cantor dust" - a fractal sequence of events. It is interesting that the distribution of stars in the Galaxy obeys the same laws:

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

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

As an example, we will give several typical situations where it is useful to use the concept of a fractal structure. Columbia University professor Christopher Scholz specialized in the study of the shape and structure of the earth's solid matter, he studied earthquakes. In 1978 he read Mandelbrot's book Fractals: Form, Randomness and Dimension » and tried to apply the theory to the description, classification and measurement of geophysical objects. Scholz found that fractal geometry provided science with an efficient method of describing the specific hilly landscape of the Earth. The fractal measurement of the planet's landscapes opens the door to comprehending its most important characteristics. Metallurgists have found the same thing on a different scale - applied to surfaces of various types of steel. In particular, the fractal measurement of a metal surface often makes it possible to judge its strength. A huge number of fractal objects produce the phenomenon of crystallization. The most common type of fractals that arise during crystal growth are dendrites, which are extremely widespread in nature. Ensembles of nanoparticles often demonstrate the implementation of Levy dust. These ensembles, in combination with the absorbed solvent, form transparent compacts - Levy glasses, potentially important materials for photonics.

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

Once you have mastered the language of fractals, you can describe the shape of a cloud as clearly and simply as an architect describes a building using blueprints that use the language of traditional geometry.<...>Only a few decades have passed since Benoit Mandelbrot said: "The geometry of nature is fractal!"

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

Post Scriptum

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

So, the atlas summarized the results of interdisciplinary research and presented a comparative analysis of completely different data in the widest range of time and space. The book shows that “the processes occurring in the earthly spheres are due to a large number of interacting factors that cause different reactions in different areas (and at different times)”, which speaks of “the need for an integrated approach to the analysis of geodynamic, space, social, economic and medical observations ". It remains to express the hope that these fundamental works will be continued.

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 new weapons 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 this or that influence on our life. One of these discoveries was a 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 understand 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, it would be necessary to start debriefing from the standpoint of mathematics, however, before delving into we will philosophize a little. Each person has a natural curiosity, thanks to which he learns the world around him. Often, in his striving for knowledge, he tries to operate with logic in judgments. So, analyzing the processes that are happening around, he tries to calculate the relationship and deduce certain patterns. The greatest minds on the planet are preoccupied with these tasks. Roughly speaking, our scientists are looking for patterns where they do not exist, and indeed should not be. And yet, 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 itself, with all its branches and knots, is itself similar to 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. The list is endless. All these random shapes are easily described by a fractal algorithm. Now we come to consider what a fractal is from the point of view 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, the formulation as such does not exist. Usually it is interpreted as a self-similar set, a part of the whole, which is repeated by its structure at the micro level. This term was invented 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, on an enlarged scale, 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 background, or How it all began

At the turn of the 19th and 20th centuries, the study of the nature of fractals was of an episodic nature. This is because mathematicians preferred to study objects amenable to research 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 perceive. Then the Swede Helge von Koch went, who in 1904 built a continuous curve that has no tangent anywhere. It is fairly easy to draw and has fractal properties as it turns out. One of the variants of this curve was named after its author - "Koch 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 article "Plane and spatial curves and surfaces, consisting of parts, like a whole." In it, he described a new species - the Lévy C-curve. All of the above figures are conventionally referred to as geometric fractals.

Dynamic, or algebraic fractals

This class includes the Mandelbrot set. The first researchers of this direction were the French mathematicians Pierre Fatou and Gaston Julia. 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 made the author famous 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 who used a computer to carry out calculations (such a volume cannot be done manually), which made it possible to construct an image of these figures.

Spatial imagination

Mandelbrot began his scientific career at the IBM Research Center. Studying the possibility of transmitting data over long distances, scientists are 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 results of measurements, he drew attention to a strange pattern, namely: the graphs of the noise looked the same at 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 figurative thinking, he translated any algebraic problem into the geometric field, where the correct answer is obvious. So it comes as no surprise featured rich and became the father of fractal geometry. After all, the awareness of this figure can come only when you study the drawings and ponder the meaning of these strange swirls that form a 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 from the work of Gaston Julia and was finalized by Mandelbrot. Gaston tried to imagine what a set looks like, built on the basis of a simple formula, which is iterated over 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, use the formula to find a new value. We substitute it into the formula and find the following. The result is large. To represent such a set, it is required to perform this operation a huge number of times: hundreds, thousands, millions. This is what Benoit did. He processed the sequence and transferred the results into graphical form. Subsequently, he colored the resulting shape (each color corresponds to a certain number of iterations). This graphic is named "Mandelbrot fractal".

L. Carpenter: an art made 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 prototype aircraft, he came up with the idea to use an 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. And so Lauren came across Mandelbrot's book Fractals: Form, Randomness and Dimension. In it, Benoit gave many examples, showing that there are fractals in nature (phwa), 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 barrage of criticism from his colleagues. They argued that a fractal is just a beautiful picture with no value, which is 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 solution

The principle used by Lauren turned out to be simple. It consists in dividing the larger ones into smaller elements, and those into similar smaller ones, and so on. Carpenter, using large triangles, divided 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.

First 3D visualization based on fractal algorithm

Within a few years, Lauren applied his best practices in a large-scale project - the animated video Vol Libre, shown at the 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 a digital craftsman and artist, Beddard created a series of highly intriguing geometric shapes that he called Faberge fractals. Outwardly, they resemble decorative eggs of a Russian jeweler, they have the same brilliant intricate pattern. Beddard used a boilerplate method to create his digital renderings of 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 shapes are extraordinarily 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 present everywhere. Nature is created from self-similar figures, we just do not 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 shapes. 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. For example, musician Jonathan Colton writes music using fractal algorithms. He claims corresponds 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 others.

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, they began to use it in the Forex market. Now the fractal indicator is found on all trading platforms and is used in a trading technique called price breakouts. This technique was developed by Bill Williams. 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 examined 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, 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 on a different scale. We can say with confidence that fractals still hold many secrets that we have yet to discover.