Fractals in the real world. Research object. Research "Travel to World Fractals

How was the fractal

Mathematical forms known as fractals belong to the genius of an outstanding scientist Benoit Mandelbrot. He taught Mathematics most of his life at the US University of Yale. In 1977 - 1982, Mandelbrot published scientific worksdevoted to the study of "fractal geometry" or "geometry of nature", in which at first glance, random mathematical forms on the component elements, which appended at the closest review by repeating, - which also proved the presence of a certain sample for copying. The opening of the Mandelbroke had significant consequences in the development of physics, astronomy and biology.

Fractals in nature

In nature, many objects have fractal properties, for example: crowns of trees, cauliflower, clouds, a blood and alveolar system of humans and animals, crystals, snowflakes, whose elements are built into one complex structure, coast (fractal concept allowed scientists to measure the British Islands coastline and Other, previously immeasurable, objects).

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

Fractals in digital technology

Fractal geometry made an invaluable contribution to the development of new technologies in the field of digital music, as well as made possible compression of digital images. Existing fractal image compression algorithms are based on the storage principle of the compressive image instead of the digital picture itself. For compressing image, the main picture remains a fixed point. Microsoft used one of the variants of this algorithm when eating his encyclopedia, but for one reason or another, this idea did not receive a wide dissemination.

The mathematical basis fractal graphics lies fractal geometry, where the methods of inheritance from the initial "parent objects" are based on the basis of the methods of building "heir images". The concepts of fractal geometry and fractal graphics appeared only about 30 years ago, but it was already firmly included in the use of computer designers and mathematicians.

The basic concepts of fractal computer graphics are:

Fractal triangle - fractal figure - fractal object (hierarchy in descending order)
Fractal straight
Fractal composition
"Parent Object" and "Object Heir"

As in the vector and three-dimensional graphics, the creation of fractal images mathematically calculated. The main difference from the first two types of graphics is that the fractal image is built by equation or system of equations - nothing but the formula in the memory of the computer is not necessary to store all calculations - and such compactness of the mathematical apparatus allowed the use of this idea in computer graphics. Simply changing the coefficients of the equation, you can easily get a completely different fractal image - with the help of several mathematical coefficients, the surfaces and lines are specified very complex formThat allows you to implement such a composition of compositions as horizontally and vertical, symmetry and asymmetry, diagonal directions and much more.

How to build a fractal?

The creator of fractals acts as an artist, photographer, sculptor, and an inventor scientist at the same time. What are the stages of the work of the creation of the drawing "from scratch"?

set the pattern of the mathematical formula
explore the convergence of the process and vary its parameters
select image image
choose a palette of flowers

Among fractal graphic editors and other graphic programs can be allocated:

"Art Dabbler"
"Painter" (Without a computer, not a single artist will never reach the programmers of possibilities only by using a pencil and pen brushes)
"Adobe Photoshop" (but here the image "from scratch" is not created, and, as a rule, only processed)

Consider the device of an arbitrary fractal geometric shape. In its center there is a simplest element - an equilateral triangle, which received the same name: "fractal". On the average segment of the parties, we will construct equilateral triangles with a side of one third of the side of the initial fractal triangle. In the same principle, even smaller triangles-heirs of the second generation are being built - and so indefinitely. The object, which as a result turned out, is called a "fractal figure", from the sequences of which we get a "fractal composition".

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

Fractals and ancient mandalas

This is a mandala to attract money. Approve that the red color works like money magnet. And the vessels do not remind you of anything? They seemed to me very familiar and I was engaged in the study of the mandala as a fractal.

In principle, the mandala is a geometric symbol of a complex structure, which is interpreted as a model of the universe, "Cosmos Map." Here is the first sign of fractality!

They are embroidered on the tissue, drawn on the sand, perform with non-ferrous powders and made from metal, stone, wood. A bright and fascinating look, makes it a beautiful decoration of floors, walls and ceilings of temples in India. On ancient indian language Mandala denotes a mystical circle of the relationship of the spiritual and material energies of the Universe or a different flower of life.

I wanted to write an overview of fractal mandalas very small, with a minimum of paragraphs, showing that the relationship clearly exists. However, trying to find aware and associate information about fractals and mandalas in a single whole, I had a feeling of a quantum jump in the space unknown to me.

We demonstrate the immensity of this topic quote: "Such fractal compositions or mandalas can be used as in the form of paintings, elements of the design of residential and working space, wearable amulets, in the form of video tapes, computer programs... "In general, the topic for the research of fractals is just a huge.

One thing I can say exactly, the world is much more diverse and richer than the wretched ideas of our mind about him.

Fractal marine animals

My guesses about fractal marine animals were not groundless. Here are the first representatives. Octopus - Sea Donnaya Animal from Charton Fit.

Looking at this photo, I became an obvious fractal structure of his body and suckers on all eight tentacles of this animal. Suction cups on adult octopus tentacles reaches up to 2000.

It is interesting that the octopus is three hearts: one (most importantly) drives blue blood throughout the body, and two other - gill - pushing the blood through the gills. Some types of these deep-water fractals of poisonous.

Adapting and masking under environmentThe octopus has a very useful ability to change the color.

Octopresses are considered the most "smart" among all invertebrates. Learn people, get used to those who feed them. It would be interesting to look at octopus, which are easy to train, have good memory and even distinguish geometric shapes. But the age of these fractal animals is a non-national - a maximum of 4 years.

A person uses the ink of this living fractal and other charts. They are in demand from artists for their durability and a beautiful brown tone. In Mediterranean cuisine, the octopus is a source of vitamins B3, B12, potassium, phosphorus and selenium. But I think that these nautical fractals need to be able to prepare to enjoy their food consumption.

By the way, it should be noted that octopuses are predators. They hold their fractal tentacles to the victim in the form of mollusks, crustaceans and fish. It is a pity if the food of these marine fractals becomes such a beautiful mollusk. In my opinion, also a typical representative of the fractals of the marine kingdom.

This is a snail relative, a burglar-legged Glazk Glavk, he is glaucus, he is Glaucus Atlanticus, he is Glaucilla Marginata. This fractal is also unusual in that it lives and moves under the surface of the water, while holding off due to surface tension. Because Mollusk is a hermaphrodite, then after mating both "partners" put eggs. This fractal is found in all the oceans of the tropical belt.

Fractals of the Marine Kingdom

Each of us at least once in his life kept in his hands and with genuine child interest he looked at sea shell.

Usually shells are a beautiful souvenir resembling a trip to the sea. When you look at this spiral formation of invertebrate mollusks, there is no doubt in its fractal nature.

We, people, with something we remind these soft mollusks, upholstered in well-maintained concrete houses fractals, placing and moving their body in rapid cars.

Another typical representative of the fractal underwater world is coral.
In nature, more than 3500 varieties of corals are known, in the palette of which are distinguished by up to 350 color shades.

The coral is the skeleton material of the coral polyps colony, also from the invertebrate family. Their huge accumulations form whole coral reefs, the fractal method of formation of which is obvious.

Coral with complete confidence can be called fractal from the sea kingdom.

It is also used by a person in the form of souvenir or raw materials for jewelry and jewelry. But repeat the beauty and perfection of fractal nature is very difficult.

For some reason, I have no doubt that many fractal animals are also deepened in the underwater world.

Once again, fulfilling a ritual in the kitchen with a knife and a cutting board, and then, lowering the knife in cold waterI once again came up with tears in tears, how to deal with a tear fractal, which almost daily appears in my eyes.

The principle of fractality is the same as the famous Matryoshka - nesting. That is why fractality notice not immediately. In addition, the bright homogeneous color and its natural ability to cause unpleasant sensations do not contribute to steady observation over the universe and identification of fractal mathematical patterns.

But the salad bowl of lilac color due to its colors and the lack of tear phytoncides brought on reflections on the natural fractality of this vegetable. Of course, it fractal it is a simple, ordinary circumference of different diameters, you can even say a primitive fractal. But it would not hurt to remember that the ball is considered an ideal geometric figure within our universe.

A lot of articles published on the useful properties of Luke on the Internet, but somehow no one tried to study this natural copy from the point of view of fractality. I can only state the benefit of the use of fractal in the form of a bow at its kitchen.

P.S. And I have already acquired vegetable cutters for grinding fractal. Now you have to reflect how fractable is such a useful vegetable, like an ordinary white cabbage. The same principle of nesting.

Fractals in folk art

My attention attracted the story of the world-famous toy "Matryoshka". Looking at carefully, with confidence it can be said that this souvenir toy is a typical fractal.

The principle of fractality is obvious when all the figures of the wooden toy are built into a row, and not invested in each other.

My minor studies of the history of this toy fractal in the world market have shown that the roots of this beauty are Japanese. Matryoshka was always considered an invarious Russian souvenir. But it turned out that she was the prototype of the Japanese figure of the old man-sage Fukurum, brought once to Moscow from Japan.

But it was the Russian toy fishing that brought world fame to this Japanese figure. Where was the idea of \u200b\u200bfractal nesting toys, personally for me and remained a mystery. Most likely the author of this toy used the principle of the nesting of figures in each other. And the easiest way of investment is such figures of different sizes, and this is already a fractal.

An equally interesting object of the study is the painting of fractal toys. This is a decorative painting - Khokhloma. The traditional elements of Khokhloma are herbal patterns of flowers, berries and branches.

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

And if the new-fashioned painting of computer mice, the covers of laptops and phones no one will no longer surprise, then the fractal tuning of the car in a folk style is something new in autodizain. It remains only to be surprised in the manifestation of the world of fractals in our life in such an unusual way in such ordinary things for us.

Fractals in the kitchen

Each time, disassembled cauliflower into small inflorescences for blanching in boiling water, I have never paid attention to the explicit signs of fractality, while I did not have this instance in my hands.

Typical fractal representative from vegetable world On my kitchenette.

With all my love for cauliflower, it all the time came across instances with a homogeneous surface without visible signs of fractality, and even a large number of infloresions embedded in each other did not give me a reason to see the fractal vegetable in this useful vegetable.

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

Another trip to the hypermarket only confirmed the fractal cabbage status. Among the huge number of exotic vegetables, a whole box was blocked with fractals. It was a romance, or Romanesque broccoli, colored coral cabbage.

It turns out, designers and 3D artists are enthusiastic with its exotic forms similar to fractals.

Cabbage kidneys are growing on the logarithmic spiral. The first references to Cabesto Romanentic came from Italy of the 16th century.

And the cabbage broccoli is not a completely frequent guest in my diet, although the content of the useful substances and trace elements it exceeds a cauliflower at times. But its surface and the form are so homogeneous that I never occurred to see the vegetable fractal in it.

Fractals in Qilling

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

After seeing the next master class in Qulation, there was no doubt about the fractality of the queen. After all, for the manufacture of various elements for crafts from queening, a special line with circles of different diameter is used. With all the beauty and uniqueness of products, it is an incredibly simple technique.

Almost all the basic elements for crafts in quilling are made of paper. To stock paper for queening for free, spend home revision of your bookshelves. Surely, there you will find a couple of bright glossy magazines.

Qwill tools are simple and inexpensive. All you need to fulfill amateur-style quilling, you can find among your home stationery.

And the history of the queen begins in the 18th century in Europe. In the era of the Renaissance, the monks of French and Italian monasteries with the help of queening were decorated with book covers and did not even suspect a fractality invented paperwork invented. Girls from the highest society even passed a course on queen in special schools. This technique began to spread through countries and continents.

This master class video quilling for the manufacture of luxury plumage can even be called "fractals with their own hands." With the help of paper fractals, wonderful exclusive cards - Valentines and many different other interesting things are obtained. After all, fantasy, like the nature of inexhaustible.

It's no secret that the Japanese in life is strongly limited in space, in connection with which they have to be sophisticated in effectively use. Miyakava Takeshi shows how it can be done simultaneously and aesthetically. Its fractal closet confirmation that the use of fractals in design is not only a tribute to fashion, but also harmonious design solution in conditions of limited space.

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

And it seems that the principle of fractality Nature uses everywhere. Just need to look at it attentive, and it will show themselves in all its great abundance and infinity of being.

Municipal budget general education - Average secondary school

from. Doggy

Scientific and practical conference "Amazing World of Mathematics"

Research "Journey to the world of fractals"

Performed: Student 10 Class

Allahverdieva Naila

Leader: Davydova E. V.

Introduction.
Main part:

a) the concept of fractal;

b) the history of the creation of fractals;

c) the classification of fractals;

d) the use of fractals;

e) fractals in nature;

e) Fractal colors.

3. Conclusion.

Introduction.

What is hiding behind the mysterious concept of "fractal"? Probably, for many, this term is associated with beautiful images, intricate patterns and bright images created using computer graphics. But fractals are not easy picture. These are special structures that underlie everything surrounds. Brooring B. scientific world Just a few decades ago, fractals managed to produce a real revolution in the perception of surrounding reality. Using fractals, a person can create high-precision mathematical models of natural objects, systems, processes and phenomena.

Main part
The concept of fractal.

Fractal(from lat. fractus. - crushed, broken, broken) - a complex geometric figure, which has the property of self-similarity, that is, composed of several parts, each of which is similar to the whole figure. Many objects in nature have fractal properties, such as coast, clouds, trees crowns, circulatory system and the human or animal alveoli system.

Fractals, especially on the plane, are popular due to the combination of beauty with ease of construction using a computer.

History of creation.
To bring the science of fractals to a new level, the French mathematician Benoit Mandelbrot was managed - the scientist who today is recognized as the father of fractal geometry. Mandelbroid for the first time gave the definition of the term "fractal":

Quote

"Fractal is called a structure consisting of parts, which in some sense are like a whole"
In the 70s, Benoit Mandelbrot worked as a mathematical analyst at IBM. The scientist first thought about fractals in the process of studying noise in electronic networks. At first glance, there was absolutely chaotic interference during data transmission. Mandelbrot built a schedule of errors and was surprised to find that in any time scale, all fragments looked likewise. On the scale of the week noise appeared in the same sequence as on the scale of one day, an hour or minute. Mandelbrot understood that the frequency of errors when data transfer is distributed over time on the principle set forth by the Cantor in late XIX. century. Then Benooy Mandelbrot was seriously carried away by the study of fractals.
Unlike its predecessors, it was not geometric constructions for the creation of Mandelbrot Fractals, but algebraic transformations Various complexity. The mathematician used the method of reverse iterations, which implies the multiple calculation of the same function. Using the use of computers, the mathematician performed a huge amount of successive computing, the results of which displayed graphically on the complex plane. So many mandelbroke appeared - a complex algebraic fractal, which today is considered a classic of science on fractals. In some cases, the same subject can be considered simultaneously smooth and fractal. To explain why this happens, Mandelbroth brings an interesting visual example. A tangle of woolen threads, removed at a certain distance, looks like a point with dimension 1. The tangle, located nearby, looks like a two-dimensional disk. Taking it in hand, you can clearly feel the volume of the ball - now it is perceived as three-dimensional. And the fractal of the tangle can be considered only from the point of view of the observer using a magnifying device, or flies, which served on the surface of an uneven woolen thread. Therefore, the true fractality of the object depends on the point of view of the observer and on the resolution of the instrument used.
Mandelbrot noted an interesting pattern - the closer to consider the measured object, the more extended its border will be. This property can be clearly demonstrated on the example of measuring the length of one of the natural fractals - coastline. Conducting measurements by geographic map., It is possible to obtain an approximate value of length, because all irregularities and bends will not be taken into account. If you measure the measurement, taking into account all the irregularities of the relief visible from the height of human growth, the result will be somewhat different - the length of the coastline will increase significantly. And if theoretically imagine that the measuring instrument will ribbon the irregularity of each pebble, then in this case the length of the coastline will be almost infinite.
Fractal classification.

Fractals are divided into:

geometric: fractals of this class are the most visual, they immediately visible self-similarity. The history of fractals began with the geometric fractals, which were studied by mathematicians in the XIX century.

algebraic: This Fractal group received such a name because fractals are formed using simple algebraic formulas.

stochastic: are formed in case of accidental change in the iteration process of fractal parameters. Two-dimensional stochastic fractals are used in modeling the terrain and sea surface.

Geometric fractals

It was from them that the history of fractals began. This type of fractal is obtained by simple geometric constructions. Usually, when constructing these fractals, they do this: the "seed" is taken - axiom - a set of segments, on the basis of which fractal will be built. Next to this "seed" apply a set of rules that converts it to any geometric shape. Next, the same set of rules apply to each part of this figure. With each step, the figure will become more complicated and more difficult, and if we feed (at least in the mind), the infinite number of transformations - we get a geometric fractal. Classic examples Geometric fractals: Koch snowflake, leaf, triangle of Serpinsky, Dragonov broken (Appendix 1).

Algebraic fractals

The second large Fractal group is algebraic (Appendix 2). They obtained their name for ensuring that they are built on the basis of algebraic formulas are sometimes very simple. Methods for obtaining algebraic fractals are several.

Unfortunately, many terms levels of 10-11 class associated with complex numbers needed to explain the fractal construction are unknown to me and are still difficult to understand, therefore it is not possible to describe in detail the construction of fractals of this kind for me.

Initially fractal nature black and white, but if you add a little fantasy and paints, you can get a real work of the arts.

Stochastic fractals

A typical representative of this class of Fractals "Plasma" (Appendix 3). To build it, take a rectangle and for each of its angle will determine the color. Next, we find the central point of the rectangle and paint it into color equal to the average arithmetic colors at the corners of the rectangle plus a random number. The more random number - the more "torn" will be a drawing. If we now say that the color of the point is a height above sea level - we get instead of plasma - a mountain range. It is on this principle that mountains are simulated in most programs. With the help of an algorithm similar to the plasma, a map of heights is built, various filters apply to it, we apply to the texture and, please, photorealistic mountains are ready!

Application fractals

Already today, fractals are widely used in a wide variety of areas. The direction of fractal archiving of graphic information is actively developing. Theoretically, fractal archiving can compress images to the size of the point without loss of quality. With an increase in the pictures compressed according to the fractal principle, the smallest details are clearly displayed, and the grain effect is completely absent.

The principles of the theory of fractals are used in medicine for analyzing electrocardiograms, since the rhythm of heart abbreviations is also a fractal. The direction of studies of the circulatory system and other internal systems of the human body is actively developing. In biology, fractals are used to model the processes occurring within populations.
Meteorologists use fractal dependences for analyzing the intensity of the air masses, thereby appearing the possibility of more accurate prediction of weather changes. Physics of fractal media with great success solves the task of studying the dynamics of complex turbulent flows, adsorption and diffusion processes. In the petrochemical industry, fractals are used to simulate porous materials. The theory of fractals is effectively used in the financial markets. Fractal geometry is used to create powerful antenna devices.
Today, fractal theory is an independent area of \u200b\u200bscience, on the basis of which all new and new directions are being created in various fields. The significance of fractals is devoted by many scientific papers.

But these unusual objects are not only extremely helpful, but also incredibly beautiful. That is why fractals are gradually finding their place in art. Their amazing aesthetic appeal inspires many artists to create fractal paintings. Modern composers create musical works using electronic tools with various fractal characteristics. Writers apply a fractal structure to form their literary works, and designers create fractal furniture and interior items.

Fractality in nature

In 1977, the book of Mandelbrot "Fractals: Form, Accident and Dimension" was published, and in 1982 another monograph was published - "Fractal Geometry of Nature", on the pages of which the author demonstrated visual examples various fractal sets and led evidence of the existence of fractals in nature. The main idea of \u200b\u200bthe theory of Fractal Mandelbrot expressed in the following words:

"Why is the geometry often called cold and dry? One of the reasons is that it is unable to accurately describe the clouds, mountains, wood or seashore. Clouds are not spheres, the lines of the shore is not a circle, and the bark is not smooth. and zipper does not apply in a straight line. Nature demonstrates us not just more high degree, and a completely different level of complexity. The number of different lengths of lengths in the structures is always infinite. The existence of these structures gives us a challenge in the form of a difficult task of studying those forms that Euclidean dropped as formless - the tasks of the study of the morphology of amorphous. Mathematics, however, neglected by this challenge and preferred increasingly and more from nature, inventing theories that do not correspond to anything that you can see or feel. "

Many natural objects are possessed by the properties of the fractal set (Appendix 4).

Are fractals really universal structures that were taken as a basis when creating absolutely everything in this world? The form of many natural objects is as close as possible to fractals. But not all the world's existing fractals have so correct and infinitely repeated structure as the sets created by mathematicians. Mountain ridges, metal fault surfaces, turbulent flows, clouds, foam and many-many other natural fractals are deprived of perfectly accurate self-similarity. And it would be absolutely mistaken to believe that fractals are a universal key to all the secrets of the universe. With all its apparent complexity, fractals are only a simplified model of reality. But among all the fractal theories available today are the most accurate means of describing the surrounding world.

The beauty of fractals adds their bright and catchy color. Complex color schemes make fractals with beautiful and memorable. From a mathematical point of view, fractals are black and white objects, each point of which either belongs to the set, or does not belong. But the possibilities of modern computers allow you to make fractals with color and bright. And this is not a simple coloring of the neighboring areas of many random order.

Analyzing the value of each point, the program automatically determines the shade of one or another fragment. Black shows the points in which the function takes constant value. If the value of the function tends to infinity, then the point is painted in another color. The intensity of staining depends on the rate of approximation to infinity. The more repetitions are required to approach the point to a stable value, the lighter becomes its shade. And on the contrary - the points, quickly rushing to infinity, painted in bright and rich colors.
Conclusion

For the first time he heard fractals, ask the question, what is it?

On the one hand, this is a complex geometric figure, which has the characteristics of self-similarity, that is, composed of several parts, each of which is similar to the whole figure.

This concept fascinates with its beauty and mystery, manifested in the most unexpected areas: meteorology, philosophy, geography, biology, mechanics and even stories.

It is almost impossible not to see the fractal in nature, because almost every object (clouds, mountains, coastline, etc.) have a fractal structure. Most web designers, programmers have their own fractal gallery (extremely beautiful).

In essence, fractals open our eyes and allow you to look at mathematics on the other hand. It would seem that ordinary calculations are made with conventional "dry" figures, but this gives us in their own way unique results, allowing you to feel the Creator of Nature. Fractals make it clear that mathematics is also a science of beautiful.

His design work I wanted to tell about a fairly new concept in mathematics "Fractal". What is it, what are the species where they extend. I really hope that fractals are interested in you. After all, as it turned out, fractals are quite interesting and they are almost at every step.

Bibliography

http://ru.wikipedia.org/wiki.
http://www.metaphor.ru/er/misc/fractal_gallery.xml
http://fractals.narod.ru/
http://rusproject.narod.ru/article/fractals.htm.
Bondarenko V.A., Dolnikov V.L. Fractal image compression on Barncel Sloan. // Automation and telemechanics. - 1994.-N5.-C.12-20.
Watoline D. The use of fractals in machine graph. // Computerworld-Russia.-1995.-N15.-C.11.
Feder E. Fractals. Per. From English: Mir, 1991.-254c. (Jens Feder, Plenum Press, Newyork, 1988)
Application of Fractals and Chaos. 1993, Springer-Verlag, Berlin.

Attachment 1

Appendix 2.

Appendix 3.

Appendix 4.

Ministry of Education, Science and Youth of the Republic of Crimea

Municipal budgetary educational institution "Shop Educational Complex" Municipal Education Krasnoperekopsky district of the Republic of Crimea

Direction: Mathematics

Study of features of fractal models

For practical application

I've done the work:

student of grade 8 of the municipal budgetary general education institution "Shop Educational Complex" Municipal Education Krasnoperekopsky district of the Republic of Crimea

Scientific adviser:

mathematics teacher of the municipal budget educational institution "Shop Educational Complex" of the Municipal Education Krasnoperekopsky district of the Republic of Crimea

Krasnoperekopsky district - 2016

The science was made by many ingenious discoveries and inventions, thoroughly changing the lives of humanity: electricity, atomic energy, vaccine and much more. However, there are such discoveries that give little values, but they are also able to influence and affect our life. One of these discoveries are fractals that help to establish a link between events even in chaos.

American mathematician Benoit Mandelbrot in his book "Fractal Geometry of Nature" wrote: "Why is the geometry often called cold and dry? One of the reasons is that it is unable to accurately describe the shape of the cloud, mountains, wood or sea shores. The clouds are not spheres, the railway lines is not a circle, and the bark is not smooth, but the lightning does not apply in a straight line. Nature demonstrates us not just a higher degree, but a completely different level of complexity. The number of different lengths of lengths in the structures is always infinite. The existence of these structures gives us a challenge in the form of a difficult task of studying those forms that Euclidean dropped as formless - the tasks of the study of the morphology of amorphous. Mathematics, however, neglected by this challenge and chose more and more and more from nature, inventing theories that do not correspond to anything that you can see or feel. "

Hypothesis:all that exists in the world around us is a fractal.

Purpose of work:creating objects whose images are similar to natural.

Object of study:fractals in various fields of science and the real world.

Subject of study:fractal geometry.

Research tasks:

1. Acquaintance with the concept of fractal, history of its occurrence and research by B. Mandelbrot, Koch, V. Serpinsky et al.;

3. Find confirmation of the theory of fractality of the surrounding world;

4. Studying the use of fractals in other sciences and in practice;

5. Conduct an experiment to create its own fractal images.

Research methods:analytical, search, experimental.

The history of the appearance of the concept of "fractal"

Fractal geometry, as a new direction in mathematics, appeared in 1975. The concept of "fractal" first introduced into mathematics American scientist Benoit Mandelbrot. Fractal (from the English. "Fraction") - a fraction divided into parts. The definition of the fractal given by Mandelbrom, it sounds like this: "Fractal is called a structure consisting of parts, which in some sense are like a whole".

Working in the IBM Research Center, whose employees worked on the transfer of data to the distance, the complex and very important task faced Benouua - to understand how to predict the occurrence of noise interference in electronic circuits. Mandelbrot drew attention to one strange pattern - noise charts on a different scale looked equally. The same picture was observed regardless of whether it was a chart of noise in one day, a week or an hour. It was worth changing the scale of the chart, and the picture was repeated each time. Thinking into the meaning of strange patterns, the essence of fractals came to Benua.

However, the first ideas of fractal geometry arose in the 19th century.

So Georg Cantor (Cantor, 1845-1918) is a German mathematician, logic, a theologian, the creator of the theory of infinite sets, with the help of a simple repeating procedure turned the line into a set of unrelated points. He took the line and removed the central third and after that he repeated the same with the remaining segments. What happened, called the dust of the Cantor (Figure 1).

And the Italian mathematician Juseppe Peano (Giuseppe Peano; 1858-1932) took the line and replaced it by 9 segments long 3 times lower than the length of the original line. Next, he did the same with each segment. And so indefinitely. Later, similar construction was carried out in three-dimensional space (Figure 2).

One of the first Fractal drawings was a graphical interpretation of a set of mandelbroke, which was born thanks to the research of Gaston Maurice Julia (Figure 3).

All fractals can be divided into groups, but the biggest ones are:

Geometric fractals;

Algebraic fractals;

Stochastic fractals.

Geometric fractals

Geometric fractals are the most visual and they are obtained by simple geometric constructions. Take some broken (or surface in a three-dimensional case), called the generator. Then each of the segments constituting the broken, is replaced by a broken generator, on an appropriate scale. As a result of an infinite repetition of this procedure, a geometric fractal is obtained. Examples of geometric fractals can be:

1) Koch curve. At the beginning of the twentieth century, with the rapid development of quantum mechanics before scientists, the task of finding such a curve, which would best show the movement of Brownian particles. For this, the curve should have had the following property: not to have a tangential at any point. Mathematics Koh suggested one such curve: take a single segment, we divide into three equal parts and replace the average interval with an equilateral triangle without this segment. As a result, a broken form is formed, consisting of four lines of 1/3. In the next step, we repeat the operation for each of the four of the following links, etc.

Limit curve and there is a Koch curve (Figure 4) . After performing similar conversion on the sides of the equilateral triangle, you can get a fractal image of koche snowflakes.

2) Levi curve . Half of the square is taken and each side is replaced by the same fragment. The operation is repeated many times and ultimately it turns out the levion curve (Figure 5).

3) Minkowski curve. The foundation is a segment, and the generator is a broken out of eight links (two equal links continue each other) (Figure 6).

4) Peno curve (Figure 2).

5) Dragon curve (Figure 7).

6) Pythagore tree. Built on a figure known as "Pythagora pants", where on the sides rectangular triangle There are squares. For the first time, Pythagore tree built using a conventional drawing line (Figure 8).

7) Serpinsky's square. Known as a "lattice" or "Napkin" of Serpinsky (Figure 9). The square is divided by straight, parallel to its parties, on 9 equal squares. From the square removed the central square. A set consisting of 8 remaining squares "first rank" is obtained. By doing the same as each of the first rank squares, we obtain a set consisting of 64 squares of the second rank. Continuing this process infinitely, we get an infinite sequence or square of Serpinsky.

Algebraic fractals

Fractals, based on algebraic formulas, belong to algebraic fractals. This is the largest group of fractals. These include the fractal of the Mandelbrot (Figure 3) , newton Fractal (Figure 10), Many Julia (Figure 11) and many others.

Some algebraic fractals are strikingly resemble images of animals, plants and other biological objects, as a result of which the biomorphs were called.

Stochastic fractals

Stochastic fractals are another major variety of fractals that are formed by repeated repetitions of random changes to any parameters. At the same time, objects are obtained very similar to natural - asymmetrical trees, rugged coastal lines, etc.

So if you take a rectangle and to determine each of its corner. Then take it a central point and paint it into color equal to the average arithmetic colors at the corners of the rectangle plus a random number. The more random number - the more "torn" will be a drawing. Thus, it will be fractal "plasma" (Figure 12). And if we assume that the color of the point is a height above sea level - we get instead of plasma - a mountain range. It is on this principle that mountains are simulated in most programs. With the help of the algorithm, a map of the height is built, various filters are applied to it, the texture and photorealistic mountains are superimposed.

Application fractals

Fractal painting.Popular among digital artists the direction of modern art. Fractal patterns are unusual and fascinatingly acting on a person, giving birth to bright flaming images. Fabulous abstractions are created by boring mathematical formulas, but the imagination perceives them alive (Figure 13). Anyone can exercise with fractal programs and generate their fractals. Genuine art is in the ability to find a unique combination of color and form.

Fractals in the literature. Among the literary works are found, which possess fractal nature, i.e. nested by the structure of self-similarity:

1. "Here is the house.

Which built Jack.

But wheat.

Which built Jack

But the messenger bird-tit,

Which deftly steals wheat,

Which is in the dark chulana storing

Which built Jack ... ".

Samuel Marshak

2. fleas big biting flew

Flea tech - baby-crumbs,

As they say, ad infinitum.

Jonathan Swift

Fractals in medicine.The human body consists of a variety of fractal-like structures: blood, lymphotic and nervous systems, muscles, bronchi, etc. (Figure 14, 15).

Fractals in physics and mechanics.Fractal models of natural objects allow you to simulate various physical phenomena and make forecasts.

American engineer Nathan Cohen, who lived in the center of Boston, where the installation of external antennas was forbidden, cut out a figure in the form of a koch curve from an aluminum foil, stuck it on a sheet of paper and attached to the receiver. It turned out that such an antenna works no worse than usual. And although the physical principles of such an antenna have not yet been studied, this did not prevent Cohen to justify his own company and establish their serial release. IN this moment The American firm "Fractal Antenna System" produces fractal antenna for mobile phones.

Fractals in nature.Nature often creates amazing and excellent fractals, with perfect geometry and such harmony, which simply die from admiration. And here are their examples:

- sea shells;

The subspecies of cauliflower (Brassica Cauliflora), fern;

Peacock plumage;

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Tree from leaf to root.

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Fractals are everywhere and everywhere in nature around us. The whole universe is constructed by surprisingly harmonious laws with mathematical accuracy. Is it possible to think after that that our planet is a random grip of particles?

Practical work

Fractal tree.With the help of the "Drawing" toolbar of the Microsoft Word program and unacceptable conversions of the grouping, copying and insertion, I built my fractal tree. Five segments located in a certain way became the meter of my fractal.
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Figure 8. Pythagore tree

Figure 9. Serpinsky Square

Figure 10. Newton Fractal

Figure 11. Many Julia

Figure 12. Fractal "Plasma"

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Figure 14. Human blood system

Figure 15. Cluster of nerve cells

Fractals are already known for almost a century, well studied and have numerous applications in life. However, the basis of this phenomenon is a very simple idea: infinite beauty and variety of many figures can be obtained from relatively simple designs using only two operations - copying and scaling.

Evgeny Epifanov

What is common with the tree, the shores of the sea, clouds or blood vessels in our hand? At first glance, it may seem that all these objects are not united. However, in fact, there is one property of the structure inherent in all listed subjects: they are self-like. From the branch, as from the trunk of a tree, the progestion is smaller, from them are even smaller, etc., that is, the branch is similar to the whole tree. The blood system is similar in the same way: arterioles are departed from the arteries, and they are the smallest capillaries, according to which oxygen enters the organs and tissues. Let's look at the space shots of the sea coast: we will see the bays and peninsula; take a look at him, but from a bird's eye view: we will be visible bays and capes; Now imagine that we stand on the beach and look at your feet: there will always be pebbles that are further outpassing than the rest. That is, the coastline with an increase in scale remains similar to itself. This property of the objects is American (though, giving out in France) Mathematics Benoit Mandelbrot called fractality, and such objects themselves - fractals (from Latin Fractus - broken).

This concept has no strict definition. Therefore, the word "fractal" is not a mathematical term. Typically, the fractal is called a geometric shape that satisfies one or more of the following properties: has complex structure With any increase in scale (in contrast, for example, a straight line, any part of which is the simplest geometric figure - segment). Is (approximately) self-like. It has a fractional Hausdorf (fractal) dimension that is more topological. Can be built by recursive procedures.

Geometry and algebra

The study of fractals at the turn of the XIX and XX centuries was rather an episodic, rather than a systematic character, because previously mathematics were mainly studied "good" objects that were led by research using general methods and theories. In 1872, the German mathematician Karl Weiershtrass builds an example of a continuous function that is not differentiated anywhere. However, its construction was entirely abstract and difficult to perceive. Therefore, in 1904, Swede Helge von Koh came up with a continuous curve, which nowhere has no tangent, and it is quite simple to draw it. It turned out that it possesses the properties of fractal. One of the options for this curve is the name "Snowflake Koch".

The ideas of the self-similarity of the figures picked up the Frenchman Paul Pierre Levi, the future mentor Benoian Mandelbrot. In 1938, his article "Flat and spatial curves and surfaces consisting of parts like a whole" came out, in which another fractal is described - the C-curve of Levi. All of these the above fractals can be conditionally attributed to one class of structural (geometric) fractals.

Another class is dynamic (algebraic) fractals to which the set of mandelbroke. The first studies in this direction began at the beginning of the 20th century and are associated with the names of the French mathematicians of Gaston Zhulia and Pierre Fata. In 1918, almost two-hundred-status memoir Juulia came out, dedicated to the iterations of complex rational functions, which describes the sets of Julia - a whole family of fractals, closely related to a multitude of mandelbrot. This work was awarded prize French academyHowever, it did not contain any illustration, so it was impossible to evaluate the beauty of open objects. Despite the fact that this work glorified Zhulia among the mathematicians of that time, they had quite quickly forgotten about her. Again, attention to it appealed only half a century later with the advent of computers: it was they who made visible wealth and beauty of the world of fractals.

Fractal dimension

As is known, dimension (number of measurements) of the geometric shape is the number of coordinates necessary to determine the position of the point lying on this figure.
For example, the position of the point on the curve is determined by one coordinate, on the surface (not necessarily plane) with two coordinates, in three-dimensional space with three coordinates.
With a more general mathematical point of view, it is possible to determine the dimension in this way: an increase in linear dimensions, let's say, twice, for one-dimensional (from a topological point of view) of 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) 4 times, for three-dimensional (cubic) - 8 times. That is, the "real" (so-called hausdorfov) dimension can be calculated in the form of the relationship of the logarithm increase in the "size" of the object to the logarithm for an increase in its linear size. That is, for the segment d \u003d log (2) / log (2) \u003d 1, for the plane d \u003d log (4) / log (2) \u003d 2, for the volume d \u003d log (8) / log (2) \u003d 3.
We now calculate the dimension of the koch curve, to construct which a single segment is divided into three equal parts and replace the average interval with an equilateral triangle without this segment. With an increase in the linear dimensions of the minimum segment, three times the length of the koch curve increases in Log (4) / log (3) ~ 1.26. That is, the dimension of the koch curve - fractional!

Science and Art

In 1982, the Book of Mandelbrot "Fractal Geometry of Nature" was published, in which the author gathered and systematized almost all the information on fractals and in an easy and accessible manner outlined at that time. The main emphasis in its presentation of Mandelbrot did not on heavy formulas and mathematical structures, but on the geometric intuition of readers. Thanks to the illustrations obtained using a computer, and the historical bikes, 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 nonmatics is largely due to the fact that with the help of very simple designs and formulas that are able to understand the high school student, the amazing complexity and beauty of the image are obtained. When personal computers have become powerful enough, even a whole direction in art appeared - fractal painting, and almost any owner of the computer could do it. Now on the Internet you can easily find many sites dedicated to this topic.

The scheme for obtaining a koch curve

War and Peace

As noted above, one of the natural objects having fractal properties is the coastline. With him, or rather, with an attempt to measure its length, is connected interesting storythat fell back scientific article Mandelbrot, and also described in his book "Fractal Geometry of Nature". We are talking about the experiment, which put 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 reasons and the likelihood of armed conflict between the two countries. Among the parameters that he took into account was the length of the overall border of two warring countries. When he collected data for numerical experiments, it discovered that in different sources data on the total border of Spain and Portugal differ very much. It came across it on the next discovery: the length of the country's borders depends on the ruler, which we measure them. The smaller the scale, the longer the border is obtained. This is due to the fact that with a larger increase it becomes possible to take into account all new and new bends of the coast, which were previously ignored due to the rudeness of measurements. And if, with each increase in the scale, previously not taken into account lines will be opened, it turns out that the length of the boundaries of the endless! True, in fact, this does not happen - the accuracy of our measurements has a final limit. This paradox is called Richardson's Effect.

Structural (geometric) fractals

The algorithm for building a constructive fractal in the general case. First of all, we need two suitable geometric shapes, let's call them the basis and fragment. At the first stage, the basis of the future fractal is depicted. Then some of its parts are replaced by a fragment taken on a suitable scale - this is the first iteration of the construction. Then, the resulting figure again some parts change to the figures similar to the fragment, etc. If you continue this process to infinity, then the limit will be fractal.

Consider this process on the example of the koch curve (see the insert on the previous page). As the basis of the koch curve, you can take any curve (for the "Koch snowflakes" is a triangle). But we will limit ourselves to the simplest case - segment. Fragment - the broken depicted on top in the picture. After the first iteration of the algorithm in this case, the initial segment coincides with the fragment, then each of the components of its segments itself will be replaced by a broken, similar to the fragment, and so on. The figure shows the first four steps of this process.

Mathematics language: dynamic (algebraic) fractals

Fractals of this type arise in the study of nonlinear dynamic systems (hence and name). The behavior of such a system can be described by a complex nonlinear function (polynomial) F (z). Take some kind of starting point z0 on the complex plane (see the insert). Now consider such an infinite sequence of numbers on the complex plane, each of which is obtained from the previous one: z0, z1 \u003d f (z0), z2 \u003d f (z1), ... zn + 1 \u003d f (zn). Depending on the starting point Z0, this sequence can behave in different ways: to strive for infinity at n -\u003e ∞; converge to some end point; cyclically take a number of fixed values; More complex options are possible.

Complex numbers

A complex number is a number consisting of two parts - valid and imaginary, that is, the formal sum X + IY (X and Y here are real numbers). i is the so-called. imaginary unit, that is, that is, the number satisfying the equation i ^.2 \u003d -1. Over complex numbers, basic mathematical operations are defined - addition, multiplication, division, subtraction (only the comparison operation is not defined). A geometric representation is often used to display complex numbers - on the plane (it is called complex) along the abscissa axis, the actual part is shrinkled, and along the axis of the ordinate - imaginary, with the complex number it will correspond to the point with Cartesian coordinates X and Y.

Thus, any point z of the complex plane has its own character of behavior in the iterations of the function f (z), and the entire plane is divided into parts. At the same time, the points lying on the boundaries of these parts possess such a property: with an arbitrarily low displacement, the nature of their behavior changes dramatically (such points are called bifurcation points). So, it turns out that many points having one particular type of behavior, as well as many bifurcation points often have fractal properties. This is the sets of zhulia for the function f (z).

Dragon Family

Variating the basis and fragment, you can get a stunning variety of structural fractals.
Moreover, such operations can be performed in three-dimensional space. Examples of volumetric fractals can serve as "Menger", "Pyramid of Serpinsky" and others.
The structural fractals include the Dragon Family. Sometimes they are called by the name of the discovers of the "Dragons of Hayweea-Harter" (they resemble Chinese Dragons). There are several ways to build this curve. The easiest and most visual one of them is: you need to take a fairly long strip of paper (the thinner the paper, the better), and bend it in half. Then bend it twice again in the same direction as the first time. After several repetitions (usually after five or six, the folding strip becomes too thick so that it can be carefully bend further) you need to break the strip back, and try to be in place in the sections of the folds 90˚. Then the profile turns out the dragon curve. Of course, it will only approach, like all of our attempts to portray fractal objects. The computer allows you to portray much more steps of this process, and as a result, it turns out a very beautiful figure.

Mandelbrot many are somewhat different. Consider the function fc (z) \u003d z 2 + C, where C is a complex number. We construct the sequence of this function with z0 \u003d 0, depending on the parameter with it can be dispersed to infinity or remain limited. In this case, all values \u200b\u200bwith, in which this sequence is limited, just form a set of mandelbrot. It was studied in detail by Mandelbrom and other mathematicians, which opened a lot of interesting properties of this set.

It can be seen that the definitions of the sets of Julia and Mandelbrot are similar to each other. In fact, these two sets are closely connected. Namely, the set of mandelbrot is all values \u200b\u200bof the complex parameter C, in which the set of Julia Fc (z) is connected (a set is called connected if it cannot be broken into two non-cycle parts, with some additional conditions).

Fractals and life

Nowadays, fractal theory is widely used in various fields of human activity. In addition to the purely scientific facility for research and already mentioned fractal painting, fractals are used in the theory of information for compression of graphic data (here the fractal self-similarity property is used here - after all, to remember a small fragment of the drawing and transformations, with which other parts can be obtained, much less Memory than to store the entire file). Adding to the formulas specifying a fractal, random perturbations, can be obtained by stochastic fractals, which are very plausible to transmit some real objects - elements of the relief, the surface of the reservoirs, some plants that successfully applies in physics, geography and computer graphs to achieve greater similarities of simulated items with real. In Radioelectronics, the antennas, having a fractal form, began to produce antennas in the last decade. Taking a little space, they provide quite high-quality signal reception. Economists use fractals to describe curves curve curvature currency (this property was opened by Mandelbrotom more than 30 years ago). On this we will complete this little excursion to the amazing beauty and variety of the world of fractals.

Fractals in the world around us.

Performed: 9th grade student

MBOU Kirovskaya Sosh

Lithuanko Ekaterina Nikolaevna.
Leader: Mathematics Teacher

MBOU Kirovskaya Sosh

Kacoon Natalia Nikolaevna.

Introduction ........................................................................ 3.

Object of study.

Research objects.

Hypotheses.

Goals, objectives and research methods.

Research part. ................................................. 7.

Finding a connection between fractals and a triangle of Pascal.

Finding the connection between fractals and a golden section.

Finding a connection between fractals and figured numbers.

Finding communication between fractals and literary works.

3. Practical application of fractals ................................. .. 13

4. Conclusion .................................................................. .. 15

4.1 Research results.

5. Bibliography ............................................................................. .. 16

Introduction

Research Object: Fractals .

When most people seemed that geometry in nature is limited to such simple figures as a line, a circle, a conical section, a polygon, sphere, a quadratic surface, as well as their combinations. For example, what could be more beautiful than the claim that the planets in our solar system move around the sun on elliptical orbits?

However, many natural systems are so complex and irregular that the use of only familiar objects of classical geometry for their modeling seems hopeless. How for example, build a model of a mountain range or a tree crown in the terms of geometry? How to describe the variety of biological configurations that we observe the world of plants and animals? Imagine the complexity of the circulatory system consisting of a variety of capillaries and blood vessels and blood delivering to each cell of the human body. Imagine how hitrophically arranged light and kidneys, resembling trees with a branched crown.

The dynamics of real natural systems can be as complex and irregular. How to approach the modeling of cascade waterfalls or turbulent processes that determine the weather?

Fractals and mathematical chaos are suitable tools for the study of issues. Term fractalrefers to some static geometric configuration, such as an instant image of a waterfall. Chaos - The term dynamics used to describe phenomena similar to the turbulent weather behavior. Often, what we observe in nature, intrigues us to the infinite repetition of the same pattern, enlarged or reduced at how much time. For example, the tree has branches. On these branches there are smaller branches, etc. Theoretically, the element "branching" repeats infinitely many times, becoming less and less. The same can be seen, looking at the photo of the mountain relief. Try a little closer image of a mountain ridge - you will see the mountains again. So the property characteristic of fractals is manifested self-similar.

In many works on fractals, self-similarity is used as a defining property. Following Benoit Madelbrot, we accept the point of view, according to which fractals should be determined in terms of fractal (fractional) dimension. Hence the origin of the word fractal (from lat. fractus. - fractional).

The concept of fractional dimension is a complex concept that is set out in several stages. Direct is a one-dimensional object, and the plane is two-dimensional. If it is pretty twisting direct and plane, you can enhance the dimension of the resulting configuration; At the same time, the new dimension will usually be fractional in a sense, which we have to clarify. The combination of fractional dimension and self-similarity is that with the help of self-similarity, many fractional dimensions can be constructed in the most simple way. Even in the case of much more complex fractals, such as the boundary of a set of MandelbroNes, when there is no pure self-similarity, there is almost a complete repetition of the base form in an increasingly reduced form.

The word "fractal" is not a mathematical term and does not have a generally accepted strict mathematical definition. It can be used when the figure under consideration, has some of the properties listed below:

Theoretical multidimensionality (can be continued in any number of measurements).

If you consider a small fragment of a regular figure on a very large scale, it will be similar to a straight fragment. Fractal fragment on a large scale will be the same as in any other scale. For fractal, an increase in scale does not lead to a simplification of the structure, on all scales we will see the same complex picture.

Is self-like or approximately self-like, each level is similar to a whole

Length, squares and volumes of one fractals are zero, others - contact infinity.

It has a fractional dimension.

Types of fractals: algebraic, geometric, stochastic.

Algebraic Fractals are the largest group of fractals. They receive them using nonlinear processes in n-dimensional spaces, such as mandelbrot and Julia.

Second Fractal Group - geometric Fractals. The history of fractals began with geometric fractals, which were studied by mathematicians in the XIX century. Fractals of this class are the most visual, because they are immediately visible to self-similarity. This type of fractal is obtained by simple geometric constructions. When building these fractals, a set of segments are usually taken, on the basis of which fractal will be built. Next, this set is used by a set of rules that converts them into any geometric shape. Next, the same set of rules apply to each part of this figure. With each step, the figure will become more complicated and more difficult, and if you present an infinite number of such operations, a geometric fractal is obtained.

The figure right shows the triangle of Serpinsky - a geometric fractal, which is formed as follows: In the first step, we see the usual triangle, in the next step, the middle of the parties are connected, forming 4 triangles, one of which is inverted. Next, we repeat the operation done with all triangles, except inverted, and so indefinitely.

Examples of geometric fractals:

1.1 Star Koch

At the beginning of the twentieth century of mathematics, such curves were looking for such curves that are not tangent at any point. This meant that the curve changes dramatically its direction, and moreover with a tremendous high speed (the derivative is equal to infinity). The search for these curves was caused not just idle interest mathematicians. The fact is that at the beginning of the twentieth century, quantum mechanics developed very violently. The researcher M. Browon drew the trajectory of the suspended particles in water and explained this phenomenon as: the randomly moving fluid atoms are struck on suspended particles and thereby lead them into motion. After such an explanation of the Brownian movement before scientists, the task of finding such a curve, which would best approximate the movement of Brownian particles. For this, the curve was supposed to meet the following properties: Do not have a tangential at any point. Mathematics Koh offered one such curve. We will not go into explanation of the rules for its construction, but simply give its image from which everything becomes clear. One important property that the border of the snowflake of Koch is possessed ... Her endless length. It may seem amazing because we are accustomed to dealing with curves from the course of mathematical analysis. Usually smooth or at least piecewise smooth curves always have a finite length (which you can make sure integration). Mandelbrot in this regard published a number of fascinating works, in which the issue of measuring the length of the British coastline is investigated. As a model, he used a fractal curve resembling the border of the snowflakes in the exception that it introduced an element of chance that takes into account the accident in nature. As a result, it turned out that the curve describing the coastline has an infinite length.

Sponge Menger

Another famous fractal class is stochastic Fractals that are obtained if in the iterative process randomly change any parameters. At the same time, objects are very similar to natural - asymmetrical trees, rugged coastal lines, etc. .

Research objects

Triangle Pascal.

W.
the construction of the Pascal triangle is the side sides of the unit, each number is equal to the sum of two above it. The triangle can be continued indefinitely.

Pascal's triangle serves to calculate the shielding coefficients of the appearance of the form (x + 1) n. Starting from the triangle from units, calculate the values \u200b\u200bon each sequential level by the addition of adjacent numbers; The latter put a unit. Thus, it is possible to determine, for example, that (x + 1) 4 \u003d 1x 4 + 4x 3 + 6x 2 + 4x + 1x 0.

Figured numbers.

Pythagoras for the first time, in VI BC, drew attention to the fact that, helping himself with the score of the pebbles, people sometimes build stones into the right figures. You can just put the pebbles in a row: one, two, three. If you put them in two rows so that rectangles are obtained, we will find that all even numbers are obtained. You can lay out stones in three rows: the numbers divided by three are obtained. Any number that is divided into something can be represented by a rectangle, and only simple numbers cannot be "rectangles".

Linear numbers are numbers that do not decompose into factors, that is, their row coincides with a number of prime numbers, a supplemented unit: (1,2,3,5,7,11,13,17,19,23, ...). These are simple numbers.

Flat numbers - numbers representable in the form of a work of two factors (4,6,8,9,10,12,14,15, ...)

Clause numbers - numbers expressed by the work of three facilities (8,12,18,20,24,27,28, ...), etc.

Polygonal numbers:

Triangular numbers: (1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...)

Square numbers are a product of two identical numbers, that is, are complete squares: (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ..., N2, ...)

Pentagonal numbers: (1, 5, 12, 22, 35, 51, 70, 92, 117, 145, ...)

Hexagonal numbers (1, 6, 15, 28, 45, ...)

Golden section ..

Golden cross section (golden proportion, division in extreme and middle scope, harmonic division, number of fidiy) - division of continuous magnitude into parts in such respects, in which most of the way it relates to a smaller way as the entire value is greater. In the picture on the left, the point C produces the Golden section of the segment AB, if: A C: AB \u003d SV: AU.

This proportion is made to designate the Greek letter. . It is equal 1,618. From this proportion, it can be seen that with a gold section, the length of the larger segment is the average geometric lengths of the entire segment and its smaller part. Parts of the golden section are approximately 62% and 38% of the total segment. With the number associated with the sequence of integers Fibonacci : 1, 1, 2, 3, 5, 8, 13, 21, ... often found in nature. It is generated by recurrent ratio F. n + 2. \u003d F. n + 1. + F. n. from initial conditions F. 1 \u003d F. 2 = 1.

An ancient literary monument in which the division of a segment in relation to the golden section is found is the "beginning" Euclidea. Already in the second book "Beginning" Euclidea builds a golden cross section, and in the future it applies it to build some of the correct polygons and polyhedra.

Hypotheses:

Is there a connection between fractals and

triangle Pascal.

golden cross section.

figure numbers.

literary works

1.4 Objective:

1. Acquaint the listeners with the new branch of mathematics - fractals.

2. Disprove or prove the hypotheses set in work.

Research tasks:

Work and analyze literature on research.

Consider various types of fractals.

Collect the collection of fractal images for primary familiarization with the world of fractals.

Install the relationship between the triangle of Pascal, literary works, figured numbers and a golden cross section.

Research methods:

Theoretical (study and theoretical analysis of scientific and special literature; summarizing experience);

Practical (preparation of calculations, generalization of results).

Research part.

2.1 Finding a connection between fractals and a triangle of Pascal.

Triangle Pascal Triangle Serpinsky

When highlight odd numbers in the Pascal triangle, Serpin's triangle is obtained. The pattern demonstrates the properties of the coefficients used in the "arithmeticization" of computer programs that converts them into algebraic equations.

2.1 Finding a connection between fractals and a golden cross section.

The dimension of fractals.

If you look from a mathematical point of view, dimension is defined as follows.

For one-dimensional objects - an increase in 2 times of linear dimensions leads to an increase in dimensions (in this case of length) by 2 times, i.e. at 21 .

For two-dimensional objects, an increase in 2 times linear dimensions leads to an increase in size (area) 4 times, i.e. at 2 2. Let us give an example. Dan range of R radius, then S \u003d π R 2 .

If you increase by 2 times the radius, then: s1 \u003d π (2 r) 2 ; S 1 \u003d 4π r. 2 .

For three-dimensional objects, an increase in 2 times of linear dimensions leads to an increase in the volume of 8 times, i.e. 2 3.

If we take a cube, then v \u003d a 3, v "\u003d (2a) 3 \u003d 8a; v" / v \u003d 8.

However, nature does not always obey these laws. Let's try to consider the dimension of fractal objects on a simple example.

Imagine that the fly wants to sit on the tangle of wool. When she looks at it from afar, he sees only a point, the dimension of which is 0. Flied closer, she sees the circle first, its dimension 2, and then the ball - dimension 3. When the fly is sitting on the tangle, she will not see the ball, but will look for VILLIN , threads, emptiness, i.e. Object with fractional dimension.

The dimension of the object (indicator of the degree) shows how its internal area is growing. Similarly, the growth of fractal increases with increasing size. Scientists came to the conclusion that fractal is a set with fractional dimension.

Fractals as mathematical objects arose due to needs scientific knowledge in the adequate theoretical description of increasingly complex natural systems (such as mountain range, coastline, crown of wood, cascade waterfall, turbulent air flow in the atmosphere, etc.) and, ultimately, in mathematical modeling of nature in Overall. And the golden cross section is known, is one of the most vibrant and sustainable manifestations of nature harmony. Therefore, it is quite possible to identify the relationship of the aforementioned objects, i.e. Detect a golden cross section in the theory of fractals.

Recall that the gold cross section is determined by the expression
(*) And is the only positive root square equation
.

The numbers of Fibonacci 1,1,2,3,5,8,13,21 are closely related to the gold section, ... each of which is the sum of the two previous ones. Indeed, the value is the rim of a number composed of the relationship of neighboring Fibonacci numbers:
,

and the value - the rim of a row composed of the relations of Fibonacci numbers taken through one thing:

The fractal is called a structure consisting of parts like a whole. According to another definition, the fractal is a geometric object with fractional (non-mechanical) dimension. In addition, the fractal always arises as a result of an infinite sequence of the same type of geometric operations by its construction, i.e. It is a consequence of the limit transition, which relates it to the golden section, which also represents the limit of an infinite numerical series. Finally, the dimension of the fractal is usually an irrational number (like a golden cross section).

In the light of the foregoing, the detection of the fact that the dimension of many classic fractals with one degree of accuracy can be expressed through a gold cross section is not surprising. So, for example, the ratios for the dimensions of the snowflakes Koh d. SC. \u003d 1,2618595 ... and Menger Sponges d. Gm \u003d 2.7268330 ..., taking into account (*) can be recorded in the form
and
.

Moreover, the first expression error is only 0.004%, and the second expression is 0.1%, and taking into account the elementary ratio 10 \u003d 2 · 5 it follows that the values d. SC. and d. Gm There are combinations of the golden section and Fibonacci numbers.

The dimension of the carpet of Serpinsky d. Ks. \u003d 1,5849625 ... and dust of the Cantor d. PC \u003d 0.6309297 ... also can be considered close by the value to the golden section:
and
. The error of these expressions is 2%.

The dimension of the theory of fractals widely used in physical applications (for example, in the study of thermal convection) of an uneven (two-scale) set of cantor (the length of the forming segments of which -
and
- belong to each other as numbers of Fibonacci:
) , but d. MK \u003d 0.6110 ... differs from the size
Only by 1%.

Thus, the gold cross section and fractals are interrelated.

2.2 Finding the connection between fractals and figure numbers .

Consider each group of numbers.

The first number is 1. The next number is 3. It is obtained by adding to the previous number, 1, two points so that the desired figure becomes a triangle. In the third step, we add three points, keeping a triangle figure. On subsequent steps, n points are added, where N is the ordinal number of the triangular number. Each number is obtained by adding to the previous number of points. A recurrent formula for triangular numbers was obtained from this property: T n \u003d n + t n -1.

The first number is 1. The following number is 4. It is obtained by adding 3 points to the previous number in the form of a direct angle to make the square. The formula for square numbers is very simple, it comes out of the name of this group of numbers: G n \u003d n 2. But also, in addition to this formula, it is possible to derive the recurrent formula for square numbers. To do this, consider the first five square numbers:

g n \u003d g n-1 + 2n-1

2 \u003d 4 \u003d 1 + 3 \u003d 1 + 2 · 2-1

g 3 \u003d 9 \u003d 4 + 5 \u003d 4 + 2 · 3-1

g 4 \u003d 16 \u003d 9 + 7 \u003d 9 + 2 · 4-1

g 5 \u003d 25 \u003d 16 + 9 \u003d 16 + 2 · 5-1

The first number is 1. The next number is 5. It is obtained by adding four points, thus, the resulting figure takes the form of a pentagon. One side of such a pentagon contains 2 points. In the next step on one side there will be 3 points, the total number of points - 12. Let's try to output the formula for calculating pentagonal numbers. The first five pentagonal numbers: 1, 5, 12, 22, 35. They are formed as follows:

f 2 \u003d 5 \u003d 1 + 4 \u003d 1 + 3 · 2-2

f n \u003d f n-1 + 3N-2

3 \u003d 12 \u003d 5 + 7 \u003d 5 + 3 · 3-2

f 4 \u003d 22 \u003d 12 + 10 \u003d 12 + 3 · 4-2

f 5 \u003d 35 \u003d 22 + 13 \u003d 22 + 3 · 5-2

The first number is 1. Second - 6. The figure looks like a hexagon with a side of 2 points. In the third step, 15 points are already built in the form of a hexagon with a side of 3 points. Withdraw the recurrent formula:

u n \u003d u n-1 + 4n-3

2 \u003d 6 \u003d 1 + 4 · 2-3

u 3 \u003d 15 \u003d 6 + 4 · 3-3

u 4 \u003d 28 \u003d 15 + 4 · 4-3

u 5 \u003d 45 \u003d 28 + 4 · 5-3

If you look more attentive, then you can notice the connection between all recurrent formulas.

For triangular numbers: T n \u003d t n -1 + n \u003d t. n. -1 +1 n. -0

For square numbers: G n \u003d g. n. -1 +2 n. -1

For pentagonal numbers: F n \u003d f. n. -1 +3 n. -2

For hexagonal numbers: U n \u003d u. n. -1 +4 n. -3

We see that curly numbers are built on repeatability: it is clearly visible on recurrent formulas. We can safely argue that curly numbers are based on a fractal structure.

2.3 Finding a connection between fractals and literary works.

Consider the fractal precisely as a work of art, and characterized by two main characteristics: 1) part of it is somehow similar to a whole (ideally, this sequence of similarity applies to infinity, although no one has ever seen a really infinite sequence of iterations building a snowflake; 2) His perception Comes on the sequence of nested levels. Note that the fractal charm just occurs on the path of this fascinating and dizzying system levels, the return with which is not guaranteed.

How can I create endless text? This issue was asked by the hero of the story of X.-L. Burhhes "Garden of Diverging Trail": "... I asked myself how the book could be infinite. In addition, nothing comes in mind, except for the cyclical, walking around Tom, volume, in which the last page repeats the first thing that allows him to continue as much as you like. "

Let's see what other solutions can exist.

The simplest infinite text will be text from an infinite number of duplicate elements, or bobs that repeated part of which is its "tail" - the same text with any number of initial verses discarded. Schematically, such text can be depicted in the form of a unbreakable tree or periodic sequence of repetitive versions. The unit of text - the phrase, stanza or story begins, develops and ends, returning to the starting point, the transition point to the next unit of text repeating the original one. Such text can be likened to infinite periodic fraci: 0,33333 ..., it can still be written as 0, (3). It can be seen that the cutting off the "head" - any number of initial units, will not change anything, and the "tail" will accurately coincide with the whole text.

Unbranched endless tree identifier to itself from any couple.

Among such infinite works - poems for children or folk songs, like, for example, the poem about the Pop and his dog from Russian folk poetry, or the poem of M. Syasnova "Scarecrow-meadow", telling about the kitten who sings about a kitten who sings about Kitten. Or, the shortest: "The priest was the courtyard, there was a stake on the courtyard, it was urinated on Coke - not to start a fairy tale first? ... Pop was the yard ..."

I'm going and I see the bridge, under the crow brown,
I took the crows behind the tail, put it on the bridge, let the crow drown.
I'm going and I see the bridge, drown on the bridge,
I took the crows for the tail, put it under the bridge, let the crow flies ...

Unlike infinite versions, fragments of fractals of the Mandelbrot are still not identical, but are similar to each other, and this quality and gives it the fascinating charm. Therefore, in the study of literary fractals, the task of searching for suchness, similarities (and not identity) of text elements is faced.

In the case of endless intervals, the replacement of identity on the similarity was carried out in various ways. You can give at least two possibilities: 1) Creating poems with variations, 2) texts with increments.

Poems with variations are, for example, launched in the turnover of S.Nikitin and who has become a folk song "Peggy lived a cheerful goose", in which the Peggin Surrounding and their habits vary.

Peggy lived a cheerful goose,

He knew all the songs by heart.

Ah, what is a cheerful goose!

Wear, Peggy, Wear!

Peggy lived a funny puppy,

He could dance under his drawing.

Ah, to what a funny puppy!

Wear, Peggy, Wear!

Peggy has a slim giraffe,

He was Elegance, like a wardrobe,

That's a slim giraffe!

Wear, Peggy, Wear!

Peggy lived a funny penguin,

He distinguished all wines,

Ah, to what funny penguin!

Wear, Peggy, Wear!

Peggy lived a cheerful elephant,

He fought syncrophasotron,

Well, what is a cheerful elephant,

Unscrew, Peggy, Wear! ..

Already, if not infinite, then a rather large number of bakers: they argue that the cassette "Songs of our century" came out with two variations of songs, and probably the number continues to grow. The infinity of the identical verses here is trying to overcome due to the coachable, childish, naive and funny.

Another opportunity lies in texts with "increments". Such are those who are known to us since childhood a tazzle of a repka or a kolobkin, in every episode of which the number of characters increases:

"Teremok"

Muha-fuel.
Muha-Gully, Komar-Piskun.
Muha-Torjukha, Komar-Piskun, Mouse-Norushka.
Muha-Gulf, Komar-Piskun, Mouse-Norushka, Kubashka Frog.
Muha-Torry, Mosquito Piskun, Mouse-Norushka, Cubean Frog, Bunny-Pumpchair.
Muha-Gulf, Mosquito-Piskun, Mouse-Norushka, Kubashka Frog, Bunny - Pumpcharchik, Fox-sister.
Mukha-Torry, Mosquito-Piskun, Mouse-Nomushka, Cubean Frog, Bunny - Pumpcharchik, Fox-Sisters, Wristband-Gray Tail.
Muha-Torry, Komar-Piskun, Mouse-Norushka, Frog-Kubashka, Bunny - Pumpcharchik, Fox-sister, Wildren-Gray Tail, Bear, you give everyone.

Such texts have the structure of the "Christmas tree" or "matryoshki", in which each level repeats the previous one with increasing image size.

The poetic work in which each vehicle can be read independently, like a separate "floor" of the Christmas tree, as well as together, making up text that develops from one to another, and further to nature, peace and the universe, created by T.Wasilee:

Now, I think we can conclude that there are literary works with a fractal structure.

3. Practical application of fractals

Fractals are increasingly used 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:

COMPUTER SYSTEMS

The most useful use of fractals in computer science is fractal data compression. The basis of this type of compression is the fact that the real world is well described by fractal geometry. At the same time, the pictures are compressed much better than this is done by conventional methods (such as JPEG or GIF). Another advantage of fractal compression is that with an increase in the picture, the effect of pixelization is not observed (increase the size of points to sizes distorting the image). With fractal compression, after an increase, the picture often looks even better than before it.

Mechanics of liquids

1. The study of turbulence in flows is very well adjusted for fractals. Turbulent streams are chaotic and therefore they are difficult to simply simulate. And here it helps the transition to from fractal representation. What greatly facilitates the work of engineers and physicists, allowing them to better understand the dynamics of complex flows.

2. Using fractals, you can also simulate flame languages.

3. Porous materials are well represented in fractal form due to the fact that they have a very complex geometry. It is used in oil science.

Telecommunications

Antennas are used to transmit data at distances, having fractal forms, which greatly reduces their size and weight.

Physics surfaces

Fractals are used to describe the curvature of surfaces. The uneven surface is characterized by a combination of two different fractals.

MEDICINE

1. Chicoensory interactions.

2. Hearts

BIOLOGY

Simulation of chaotic processes, in particular when describing population models.

4. Conclusion

4.1 Research results

In my work, not all areas of human knowledge are given, where they found their use of fractal theory. I just want to say that no more than a third of the century passed since the emergence of the theory, but during this time, fractals for many researchers have become a sudden bright light in the night, which illuminated unknown decisilities and patterns in specific areas of data. Using the theory of fractals began to explain the evolution of galaxies and the development of the cell, the emergence of mountains and the formation of clouds, the movement of prices on the stock exchange and the development of society and family. Maybe at the first time, this fractal passion was even too violent and attempts to explain everything with the help of the theory of fractals were unjustified. But, no doubt, this theory has the right to exist.

In my work I collected interesting information On fractals, their types, dimension and properties, on their use, as well as the triangle of Pascal, figured numbers, gold section, on fractal literary works and many other things.

The following work was done during the study:

Analyzed and worked out literature on the topic of research.

Various types of fractals were considered and studied.

A collection of fractal images is collected for primary familiarization with the world of fractals.

The relationships between fractals and the triangle of Pascal, literary works, figured numbers and a golden cross section are established.

I made sure that those who are engaged in fractals, the beautiful, amazing world, in which mathematics, nature and art reign. I think after acquaintance with my work, you, like me, make sure that mathematics is beautiful and amazing.

5. Like:

1. Bogkin S.V., Parshin D.A. Fractals and multifractals. Izhevsk: NIC "Regular and chaotic dynamics", 2001. - 128С.

2. Voloshinov A. V. Mathematics and Art: KN. For those who do not only love mathematics and art, but also wishes to think about the nature of the beautiful and beauty of science. 2nd ed., Dorap. and add. - M.: Enlightenment, 2000. - 399С.

3. Gardner M. A. Neskual mathematics. Kaleidoscope puzzles. M.: Ast: Astrel, 2008. - 288C.: IL.

4. GRINCHENKO V. T., Matshapura V.T., Snairsky A.A. Introduction to nonlinear dynamics. Chaos and Fractal
. Publisher: Lki, 2007 264 pp.

5. Litinsky G.I. Functions and graphics. 2nd publication. - M.: Aslan, 1996. - 208c.: Il.

6. Morozov A. D. Introduction to the theory of fractals. Publisher: Publisher Nizhny Novgorod University, 2004

7. Richard M. Croith Fractals and Chaos in the dynamic systems Introduction to Fractals and Chaos.
Publisher: Technosphere, 2006. 488 pp.

8. surrounding usmira As solid bodies with clearly designated ... Find the formation and viewing program fractals, explore and build several fractals. Literature 1.A.I. Azevich "Twenty ...