Start in science. Research work "Journey to the world of fractals Fractals in the real world the object of study

Stavropol regional open scientific conference of schoolchildren

Section: mathematics

Job title:Study of the features of fractal models for practical application

9614524388, vkel [email protected] mail . en

Place of work : st Grigoropolisskaya

MOU secondary school No. 2, 8th grade.

Supervisor: Kuznetsova Elena

Ivanovna, teacher of mathematics and computer science

MOU secondary school No. 2

Art. Grigoropolisskaya, 2018

Introduction______________________________________________________________3-4pp.

Chapter 1. The history of the emergence of fractals.

Chapter 2. Classification of fractals._______________________________________6-10pp.

geometric fractals

Algebraic fractals

Stochastic fractals

Chapter 3. "Fractal geometry of nature" _________________________________ 11-13pp.

Chapter 4. Application of fractals

Chapter 5 Practical work ____________________________________________ 16-24p.

Conclusion ______________________________________________________________25.page

List of references and Internet resources ____________________________________ 26 p.

Introduction

Mathematics, properly looked at, reflects not only truth, but incomparable beauty.

Bertrand Russell

The word "fractal" is something that a lot of people are talking about these days, from scientists to high school students. It appears on the covers of many math textbooks, scientific journals and boxes with computer software. Color images of fractals today 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 the geometry in nature was limited to such simple shapes as a line, a circle, a polygon, a sphere, and so on. 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, to build a model of a mountain range 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 the whole 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 trees with a branchy crown in structure?

Fractals are a suitable means for exploring the questions posed. Often what we see in nature intrigues us with the endless repetition of the same pattern, enlarged or reduced by several times. For example, a tree has branches. These branches have smaller branches, and so on. Theoretically, the "fork" element repeats infinitely many times, getting smaller and smaller. The same thing can be seen when looking at a photograph of a mountainous terrain. Try zooming in a bit on the mountain range --- you will see the mountains again. This is how the property characteristic of fractals manifests itselfself-similarity.

For many chaologists (scientists who study fractals and chaos), this is not just a new field of knowledge that combines 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…

Objective: creating objects that are very similar to nature.

Research methods Keywords: comparative analysis, synthesis, modeling.

Tasks:

acquaintance with the concept, history of occurrence and research of B. Mandelbrot, G. Koch, V. Sierpinsky and others;

familiarity 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 use of fractals in other sciences and in practice;

conducting an experiment to create your own fractal images.

Core question of the job:

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

Subject of study: Fractal geometry.

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

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

Research methods: analytical, search.

Relevance of 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, and start working on creating my own fractals.

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

Chapter 1

B Enua Mandelbrot

The term "fractal" was coined by Benoit Mandelbrot. The word comes from the Latin "fractus", meaning "broken, shattered".

Fractal (lat. fractus - crushed, broken, broken) - 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 ordinary 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 you can imagine it, something in between. With an increase in size, the volume of the fractal also increases, but its dimension (exponent) is not an integer, but a fractional value, and therefore the border of the fractal figure is not a line: at high magnification, it becomes clear that it is blurred and consists of spirals and curls, repeating in small the scale of the figure itself. Such geometric regularity is called scale invariance or self-similarity. It is she who determines the fractional dimension of fractal figures.

Recursive (or fractal) geometry is replacing Euclidean. new science 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.

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 a classic 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, and people who are simply interested.

It was with them that the history of fractals began. This type of fractals is obtained by simple geometric constructions. Usually, when constructing these fractals, one proceeds as follows: a "seed" is taken - an axiom - a set of segments, on the basis of which the fractal will be built. Further, a set of rules is applied to this "seed", which transforms it into some geometric figure. Further, the same set of rules is again 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 the mind) an infinite number of transformations, we will get a geometric fractal.

Fractals of this class are the most visual, because they are immediately visible self-similarity at any scale of observation. In the two-dimensional case, such fractals can be obtained by specifying some broken line, called a generator. In one step of the algorithm, each of the segments that make up the broken line is replaced by a broken line-generator, in the appropriate scale. As a result of the 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 form is given only by the form of the generator. Examples of such curves are: Koch curve (Fig.7), Peano curve (Fig.8), Minkowski curve.

Researcher M. Brown sketched the trajectory of suspended particles in water and explained this phenomenon as follows: randomly moving liquid atoms hit 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. To do 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 curve is a typical geometric fractal. The process of its construction is as follows: we take a single segment, divide it into three equal parts and replace the middle interval with an equilateral triangle without this segment. As a result, a broken line 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, and so on ...

The limit curve is Koch curve.

Snowflake Koch. By performing a similar transformation 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. It turns out a set consisting of 8 remaining squares of the "first rank". Doing the same with each of the squares of the first rank, we get a set consisting of 64 squares of the second rank. Continuing this process indefinitely, we obtain an infinite sequence or Sierpinski square.

Algebraic fractals

E This is the largest group of fractals. Algebraic fractals got their name because they are built using simple algebraic formulas.

They are obtained using non-linear 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 area of initial states, from which the system will necessarily fall into the considered final states. A surprise for mathematicians was the ability to generate very complex structures using primitive algorithms.

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

Part 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 that are black). Points belonging to the boundary of the set(this is where complex structures arise) go to infinity beyond finite number iterations, and the points lying outside the set go to infinity after several iterations (white background).

P

An example of another algebraic fractal is the Julia set.There are 2 varieties of this fractal.Surprisingly, the 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. In this case, objects are obtained that are very similar to natural ones - asymmetrical trees, cut coastlines etc.

A typical representative of this group of fractals is "plasma".

D

To construct it, a rectangle is taken and a color is determined for each of its corners. Next, the center 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 "torn" the picture will be.If you look at this fractal in a section, then we will see this fractal is voluminous, and has a “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 topography. But when combining conventional geometry with fractal geometry, you can get the very “roughness” of the mountain.

Now let's talk about geometric fractals..

Chapter 3 "The Fractal Geometry of Nature"

"Why is geometry often referred to as 'cold' and 'dry'? One reason is its inability to describe the shape of a cloud, mountain, coastline, or tree." (Benoit Mandelbrot "The Fractal Geometry of Nature" ).

To The beauty 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 Bremen mathematicians under the leadership of Peitgen and Richter. Later, the exhibits of this grandiose exhibition were captured in illustrations for the book "The Beauty of Fractals" by the same authors.

As for the correspondence to the real world, fractal geometry describes a very wide class of natural processes and phenomena, and therefore we can, following B. Mandelbrot, 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, others turn to infinity.

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

sea shells

Lightning admiring their beauty. The fractals created by lightning are not random or regular.

fractal shape subspecies of cauliflower(Brassica cauliflora). This special kind is a particularly symmetrical fractal.

P fern 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, frost patterns on the windows, these are also fractals

O
t enlarged image leaflet, before tree branches- you can find fractals 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. Is it possible after that to think that our planet is a random clutch of particles? Hardly.

Chapter 4

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
days of the most powerful applications of fractals lie in computer graphics. This is fractal compression of images.

In mechanics and physics fractals are used due to the unique property to repeat the outlines of many natural objects. Fractals allow you to approximate trees, mountain surfaces, and fissures with higher accuracy than approximations with line segments or polygons (with the same amount of stored data).

T
Fractal geometry is also used to design of antenna devices. This was first used by American engineer Nathan Cohen, who then lived in downtown Boston, where the installation of external antennas on buildings was prohibited. There are also many hypotheses about the use of fractals - for example, the lymphatic and circulatory systems, the lungs, and much more also have fractal properties.

Chapter 5. Practical work.

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

First - "Necklace"(Fig. 7). The circle is the initiator of this fractal. This circle consists of a certain number of the same circles, but of smaller sizes, and it itself is one of several circles that are the same, but of larger sizes. So the process of education is endless and it can be carried out both in one direction and in the opposite direction.

The second fractal is "Victory"(Fig. 8). He 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”, which make up one large “V”, and in the left half, in which the small ones are placed so that their left halves form one straight line, the right part is built in the same way. Each of these "v" is built in the same way and continues this to infinity.

The third fractal is "Square" (Fig. 9). Each of its sides consists of one row of cells, shaped like squares, whose sides also represent rows of cells, and so on.

Fractal "Rose" (Fig. 10), due to the external resemblance to this flower. A regular hexagon is inscribed in each circle, the side of which is equal to the radius of the circle circumscribed around it.

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

Experiment No. 1 "Tree"

Now that I understand what a fractal is and how to build one, I tried to create my own fractal images.

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.

So, I got a full-fledged fractal! The basis of this fractal is the first three lines, which I mentioned at the beginning of the research work.

The fractal property is mini Christmas trees on the sides of the main Christmas tree, small Christmas trees also have their own small Christmas trees, and so on ad infinitum. This time we will draw arbitrary lines - the basis of our future fractal.

P
After more repetitions, you get such a pretty Christmas tree!

Experiment #2

P
building fractals using the recursion method in the PascalABC environment.

The Pythagorean tree is a type of fractal based on a figure known as "Pythagorean Pants". If in the classical Pythagorean tree the angle is 45 degrees, then it is also possible to construct a generalized Pythagorean tree using other angles. Such a tree is often called the Pythagorean tree blown by the wind.

If we represent only the segments connecting in some way the chosen "centers" of the triangles, then we get a bare Pythagorean tree. By combining the procedures described above in one program, I got a fractal landscape.

Conclusion

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

Fractal graphics is not just a set of self-repeating images, it is 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 wide application fractals in computer games, with the help of fractals a lot of special effects, various fabulous and incredible pictures, etc. are created. Also, with the help of fractal geometry, trees, clouds, coasts 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 image in the Pascal programming language using cycles.

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

1. Compress images and information

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

3. Data encryption using fractal algorithms

4. Creating fractal music

5. System modeling

In my work, not all areas of human knowledge are given, where the theory of fractals has found its application. I just want to say that no more than a third of a century has passed since the emergence of the theory, but during this time fractals have become a sudden phenomenon for many researchers. bright light in the night, which illuminated hitherto unknown facts and patterns in specific areas of data. Using 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. In preparing this work, it was very interesting for us to find applications of THEORY in PRACTICE. Because it often feels like theoretical knowledge stand apart from the realities of life.

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 still give us many masterpieces - those that delight the eye, and those that bring true pleasure to the mind.

10. References

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

Vitolin D. The use 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. Nizhny Novgorod: Nizhegorod Publishing House. university 1999

Paytgen H.-O., Richter P. H. The 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

Appendix

rice. 7. Fractal "Necklace" Fig.8. Fractal "Victory"

Fig.9. Fractal "Square" 10. Fractal "Rose"

Rice. 12. Fractal "Pentagram" fractal "Black Hole"

The most ingenious discoveries in science can radically change human life. The invented vaccine can save millions of people, the creation of weapons, on the contrary, takes these lives. More recently (at the scale of human evolution) we have learned to "tame" electricity - and now we cannot imagine life without all these convenient devices that use electricity. But there are also discoveries that few people attach importance to, although they also greatly influence our lives.

One of these “imperceptible” discoveries is fractals. You have probably heard this catchy word, but do you know what it means and how many interesting things are hidden in this term?

Every person has a natural curiosity, a desire to learn about the world around him. And in this aspiration, a person tries to adhere to logic in judgments. Analyzing the processes taking place around him, he tries to find the logic of what is happening and deduce some regularity. The biggest minds on the planet are busy with this task. Roughly speaking, scientists are looking for a pattern where it should not be. Nevertheless, even in chaos, one can find a connection between events. And this connection is a fractal.

Our little daughter, four and a half years old, is now at that wonderful age when the number of questions “Why?” many times greater than the number of answers that adults have time to give. Not so long ago, looking at a branch raised from the ground, my daughter suddenly noticed that this branch, with knots and branches, itself looked like a tree. And, of course, the usual question “Why?” followed, for which the parents had to look for a simple explanation that the child could understand.

The similarity of a single branch with a whole tree discovered by a child is a very accurate observation, which once again testifies to the principle of recursive self-similarity in nature. Very many organic and inorganic forms in nature are formed similarly. Clouds, sea shells, the "house" of a snail, the bark and crown of trees, the circulatory system, and so on - the random shapes of all these objects can be described by a fractal algorithm.

⇡ Benoit Mandelbrot: the father of fractal geometry

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

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

The mathematical basis for the emergence of the theory of fractals was laid many years before the birth of Benoit Mandelbrot, but it could only develop with the advent of computing devices. At the beginning of his scientific career, Benoit worked at the IBM research center. At that time, the center's employees were working on data transmission over a distance. In the course of research, scientists were faced with the problem of large losses arising from noise interference. Benoit faced a difficult and very important task - to understand how to predict the occurrence of noise interference in electronic circuits when the statistical method is ineffective.

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

During his lifetime, Benoit Mandelbrot repeatedly said that he did not deal with formulas, but simply played with pictures. This man thought very figuratively, and translated any algebraic problem into the field of geometry, where, according to him, the correct answer is always obvious.

It is not surprising that it was a man with such a rich spatial imagination who became the father of fractal geometry. After all, the realization of the essence of fractals comes precisely when you begin to study drawings and think about the meaning of strange swirl patterns.

A fractal pattern does not have identical elements, but has similarity at any scale. Build this image with a high degree manual detailing was previously simply impossible, it required a huge amount of calculations. For example, French mathematician Pierre Joseph Louis Fatou described this set more than seventy years before Benoit Mandelbrot's discovery. If we talk about the principles of self-similarity, then they were mentioned in the works of Leibniz and Georg Cantor.

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

Gaston Julia (always masked - WWI injury)

This French mathematician wondered what a set would look like if it were constructed from a simple formula iterated by a feedback loop. If explained “on the fingers”, this means that for a specific number we find a new value using the formula, after which we substitute it again into the formula and get another value. The result is a large sequence of numbers.

To get a complete picture of such a set, you need to do a huge amount of calculations - hundreds, thousands, millions. It was simply impossible to do it manually. But when powerful computing devices appeared at the disposal of mathematicians, they were able to take a fresh look at formulas and expressions that had long been of interest. Mandelbrot was the first to use a computer to calculate the classical fractal. Having processed a sequence consisting of a large number of values, Benoit transferred the results to a graph. Here's what he got.

Subsequently, this image was colored (for example, one way of coloring is by the number of iterations) and became one of the most popular images ever created by man.

As the ancient saying attributed to Heraclitus of Ephesus says, "You cannot enter the same river twice." It is the best suited for interpreting the geometry of fractals. No matter how detailed we examine a fractal image, we will always see a similar pattern.

Those wishing to see how an image of Mandelbrot space would look like when magnified many times over can do so by uploading an animated GIF.

⇡ Lauren Carpenter: art created by nature

The theory of fractals soon found practical application. Since it is closely related to the visualization of self-similar images, it is not surprising that the first who adopted the algorithms and principles of construction unusual shapes were artists.

The future co-founder of the legendary Pixar studio, Loren C. Carpenter, began working at Boeing Computer Services in 1967, which was one of the divisions of the well-known corporation engaged in the development of new aircraft.

In 1977, he created presentations with prototypes of flying models. Lauren was responsible for developing images of the aircraft being designed. He was supposed to create pictures of new models, showing future aircraft with different parties. At some point, the future founder of Pixar Animation Studios came up with the creative idea to use an image of mountains as a background. Today, any schoolchild can solve such a problem, but at the end of the seventies of the last century, computers could not cope with such complex calculations - there were no graphic editors, not to mention applications for 3D graphics. In 1978, Lauren accidentally saw Benoit Mandelbrot's book Fractals: Form, Randomness and Dimension in a store. In this book, his attention was drawn to the fact that Benoit gave a lot of examples of fractal forms in real life and proved that they can be described by a mathematical expression.

This analogy was chosen by the mathematician not by chance. The fact is that as soon as he published his research, he had to face a whole flurry of criticism. The main thing that his colleagues reproached him with was the uselessness of the developed theory. “Yes,” they said, “these are beautiful pictures, but nothing more. The theory of fractals has no practical value.” There were also those who generally believed that fractal patterns were simply a by-product of the work of "devil machines", which in the late seventies seemed to many to be something too complicated and unexplored to be completely trusted. Mandelbrot tried to find an obvious application of the theory of fractals, but, by and large, he did not need to do this. The followers of Benoit Mandelbrot over the next 25 years proved to be of great use to such a "mathematical curiosity", and Lauren Carpenter was one of the first to put the fractal method into practice.

Having studied the book, the future animator seriously studied the principles of fractal geometry and began to look for a way to implement it in computer graphics. In just three days of work, Lauren was able to render a realistic image. mountain system on your computer. In other words, with the help of formulas, he painted a completely recognizable mountain landscape.

The principle that Lauren used to achieve her goal was very simple. It consisted in dividing a larger geometric figure into small elements, and these, in turn, were divided into similar figures of a smaller size.

Using larger triangles, Carpenter broke them up into four smaller ones and then repeated this procedure over and over again until he had a realistic mountain landscape. Thus, he managed to become the first artist to use a fractal algorithm in computer graphics to build images. As soon as it became known about the work done, enthusiasts around the world picked up this idea and began to use the fractal algorithm to simulate realistic natural forms.

One of the first 3D renderings using the fractal algorithm

Just a few years later, Lauren Carpenter was able to apply his achievements in a much larger project. The animator based them on a two-minute demo, Vol Libre, which was shown on Siggraph in 1980. This video shocked everyone who saw it, and Lauren received an invitation from Lucasfilm.

The animation was rendered on a VAX-11/780 computer from Digital Equipment Corporation at a clock speed of five megahertz, and each frame took about half an hour to draw.

Working for Lucasfilm Limited, the animator created the same 3D landscapes for the second feature in the Star Trek saga. In The Wrath of Khan, Carpenter was able to create an entire planet using the same principle of fractal surface modeling.

Currently, all popular applications for creating 3D landscapes use the same principle of generating natural objects. Terragen, Bryce, Vue and other 3D editors rely on a fractal surface and texture modeling algorithm.

⇡ Fractal antennas: less is better, but better

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

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

Fractal drawings fascinate with their patterns. They definitely resemble images of space objects - nebulae, galaxy clusters, and so on. Therefore, it is quite natural that when Mandelbrot voiced his theory of fractals, his research aroused increased interest among those who studied astronomy. One such amateur named Nathan Cohen, after attending a lecture by Benoit Mandelbrot in Budapest, was inspired by the idea of practical application of the knowledge gained. True, he did it intuitively, and chance played an important role in his discovery. As a radio amateur, Nathan sought to create an antenna with the highest possible sensitivity.

The only way to improve the parameters of the antenna, which was known at that time, was to increase its geometric dimensions. However, the owner of Nathan's downtown Boston apartment was adamantly opposed to installing large rooftop devices. Then Nathan began to experiment with various forms of antennas, trying to get the maximum result with the minimum size. Fired up with the idea of fractal forms, Cohen, as they say, randomly made one of the most famous fractals out of wire - the “Koch snowflake”. The Swedish mathematician Helge von Koch came up with this curve back in 1904. It is obtained by dividing the segment into three parts and replacing the middle segment with an equilateral triangle without a side coinciding with this segment. The definition is a bit difficult to understand, but the figure is clear and simple.

There are also other varieties of the "Koch curve", but the approximate shape of the curve remains similar

When Nathan connected the antenna to the radio receiver, he was very surprised - the sensitivity increased dramatically. After a series of experiments, the future professor at Boston University realized that an antenna made according to a fractal pattern has a high efficiency and covers a much wider frequency range compared to classical solutions. In addition, the shape of the antenna in the form of a fractal curve can significantly reduce the geometric dimensions. Nathan Cohen even developed a theorem proving that to create a broadband antenna, it is enough to give it the shape of a self-similar fractal curve.

The author patented his discovery and founded a firm for the development and design of fractal antennas Fractal Antenna Systems, rightly believing that in the future, thanks to his discovery, cell phones will be able to get rid of bulky antennas and become more compact.

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

⇡ Fractal dimensions: the mind does not understand

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

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

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

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

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

What is the usual dimension is clear to anyone. If the dimension is equal to one, we get a straight line, if two - flat figure, three is the volume. However, such an understanding of dimension in mathematics does not work with fractal curves, where this parameter has a fractional value. The fractal dimension in mathematics can be conditionally considered as "roughness". The higher the roughness of the curve, the greater its fractal dimension. A curve that, according to Mandelbrot, has a fractal dimension higher than its topological dimension, has an approximate length that does not depend on the number of dimensions.

Currently, scientists are finding more and more areas for the application of fractal theory. With the help of fractals, you can analyze fluctuations in stock prices, explore all kinds of natural processes, such as fluctuations in the number of species, or simulate the dynamics of flows. Fractal algorithms can be used for data compression, for example for image compression. And by the way, to get a beautiful fractal on your computer screen, you don't have to have a doctoral degree.

⇡ Fractal in the browser

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

There are only two simple shapes at your disposal - a square and a circle. You can add them to the canvas, scale (to scale along one of the axes, hold down the Shift key) and rotate. Overlapping on the principle of Boolean addition operations, these simplest elements form new, less trivial forms. Further, these new forms can be added to the project, and the program will repeat the generation of these images indefinitely. At any stage of working on a fractal, you can return to any component of a complex shape and edit its position and geometry. It's a lot of fun, especially when you consider that the only tool you need to be creative is a browser. If you do not understand the principle of working with this recursive vector editor, we advise you to watch the video on the official website of the project, which shows in detail the entire process of creating a fractal.

⇡ XaoS: fractals for every taste

Many graphic editors have built-in tools for creating fractal patterns. However, these tools are usually secondary and do not allow you to fine-tune the generated fractal pattern. In cases where it is necessary to build a mathematically accurate fractal, the XaoS cross-platform editor will come to the rescue. This program makes it possible not only to build a self-similar image, but also to perform various manipulations with it. For example, in real time, you can “walk” through a fractal by changing its scale. Animated movement along a fractal can be saved as an XAF file and then played back in the program itself.

XaoS can load a random set of parameters, as well as use various image post-processing filters - add a blurred motion effect, smooth out sharp transitions between fractal points, simulate a 3D image, and so on.

⇡ Fractal Zoomer: compact fractal generator

Compared to other fractal image generators, it has several advantages. Firstly, it is quite small in size and does not require installation. Secondly, it implements the ability to define the color palette of the picture. You can choose shades in RGB, CMYK, HVS and HSL color models.

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

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

⇡ Mandelbulb3D: 3D fractal editor

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

And this fractal can be eaten

The Mandelbulb3D program can create three-dimensional objects with a similar shape. To obtain a 3D surface using the fractal algorithm, the authors of this application, Daniel White and Paul Nylander, converted the Mandelbrot set to spherical coordinates. The Mandelbulb3D program they created is a real three-dimensional editor that models fractal surfaces. different forms. Since we often observe fractal patterns in nature, an artificially created fractal three-dimensional object seems incredibly realistic and even “alive”.

It may look like a plant, it may resemble a strange animal, a planet, or something else. This effect is enhanced by an advanced rendering algorithm that makes it possible to obtain realistic reflections, calculate transparency and shadows, simulate the effect of depth of field, and so on. Mandelbulb3D has a huge amount of settings and rendering options. You can control the shades of light sources, choose the background and the level of detail of the modeled object.

The Incendia fractal editor supports double image smoothing, contains a library of fifty different three-dimensional fractals and has a separate module for editing basic shapes.

The application uses fractal scripting, with which you can independently describe new types of fractal structures. Incendia has texture and material editors, and a rendering engine that allows you to use volumetric fog effects and various shaders. The program has an option to save the buffer during long-term rendering, animation creation is supported.

Incendia allows you to export a fractal model to popular 3D graphics formats - OBJ and STL. Incendia includes a small Geometrica utility - a special tool for setting up the export of a fractal surface to a three-dimensional model. Using this utility, you can determine the resolution of a 3D surface, specify the number of fractal iterations. Exported models can be used in 3D projects when working with 3D editors such as Blender, 3ds max and others.

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

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

⇡ Aural: how fractals sing

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

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

Fractals: musical pause

In fact, fractals can help write music even without software. But this can only be done by someone who is truly imbued with the idea of natural harmony and at the same time has not turned into an unfortunate “nerd”. It makes sense to take a cue from a musician named Jonathan Coulton, who, among other things, writes compositions for Popular Science magazine. And unlike other artists, Colton publishes all of his works under a Creative Commons Attribution-Noncommercial license, which (when used for non-commercial purposes) provides for free copying, distribution, transfer of work to others, as well as its modification (creation of derivative works) in order to adapt it to your needs.

Jonathan Colton, of course, has a song about fractals.

⇡ Conclusion

In everything that surrounds us, we often see chaos, but in fact this is not an accident, but an ideal form, which fractals help us to discern. Nature is the best architect, the ideal builder and engineer. It is arranged very logically, and if somewhere we do not see patterns, this means that we need to look for it on a different scale. People understand this better and better, trying to imitate natural forms in many ways. Engineers design speaker systems in the form of a shell, create antennas with snowflake geometry, and so on. We are sure that fractals still keep a lot of secrets, and many of them have yet to be discovered by man.

EXPLORING THE WORLD OF FRACTALS

Vasilyeva Marina Vladimirovna

3rd year student, Faculty of Informatics, SSAU im. Academician S.P. Koroleva, Russian Federation, Samara

Tishin Vladimir Viktorovich

Supervisor, Associate Professor, Department of Applied Mathematics, SSAU

them. Academician S.P. Koroleva, Russian Federation, Samara

Introduction

The world of fractals is an amazing, huge and diverse world. It captivates, conquers, but sometimes it is difficult to understand it. Fractal drawings are the peak of the master's inspiration on the way to the perfect unity of mathematics, computer science and art. Recently, geometric models of natural objects have been depicted using combinations of simple shapes such as lines, triangles, circles, spheres, polyhedra. But with a set of these well-known figures, it is not easy to describe more complex natural objects, for example, porous materials, cloud shapes, tree crowns. New computer tools, without which modern science cannot do, bring mathematics to an extremely high level. When you study fractals, you understand that it is very difficult to draw a line between mathematics and computer science, because they are closely intertwined, trying to discover unique, unique models. Fractals bring us closer to understanding some natural processes and phenomena. Therefore, the topic of fractals interested me.

I have a problem: how to build a fractal using mathematical formulas.

Hypothesis: if you study the patterns of constructing fractals, then they can be modeled.

Research methods: analysis, synthesis, modeling.

Purpose: to build fractals using computer technology.

Tasks: explore fractals; study the history of the emergence and use of fractals.

Relevance: I believe that fractals are the future, they better convey our changing and complex world. Fractals help to study various processes and phenomena.

Research result: development of an algorithm for constructing fractals.

Theoretical and practical significance: using an algorithm for constructing fractals to study their properties.

The concept of "fractal"

The concepts of "fractal" and "fractal geometry" appeared in the 70-80s of the XX century. They are firmly entrenched in the use of mathematicians and programmers. The word "fractal", which in Latin means broken, divided into parts, was proposed by Benoit Mandelbrot, an American mathematician, in 1975, in order to designate irregular self-similar structures. Mandelbrot gave the following definition: "a fractal is a structure consisting of parts that are in some sense similar to the whole." It should be noted that the self-similarity property reflects main feature natural objects.

From the point of view of mathematics, a fractal is, first of all, a set of fractional dimensions. It is known that the dimension of a segment is 1, a square - 2, a cube and a parallelepiped - 3. Fractional dimension is the main property of fractals.

With the release of Mandelbrot's book "The Fractal Geometry of Nature" in 1977, the birth of fractal geometry is associated. It applied the scientific results of scientists, among them Poincaré, Fatou, Julia, Kantor, Hausdorff, who worked in the period 1875-1925. in the same area. And only in our time it was possible to combine these works into a single system.

Fractal geometry is a revolution in mathematics and the mathematical description of nature. Benoit Mandelbrot himself, the discoverer of fractal geometry, writes about it this way: “Clouds are not spheres, mountains are not cones, coastlines are not circles, and the crust is not smooth, and lightning does not propagate in a straight line. Nature shows us not just a higher degree, but a completely different level of complexity. The number of different length scales in structures is infinite.”

By viewing fractal objects at different scales, one can easily detect the same basic elements. Patterns that are repeated determine the fractional dimension of an unusual geometric figure.

Fractal classification

It is convenient to resort to their generally accepted classification in order to represent the whole variety of fractals. Fractals are divided into geometric, algebraic and stochastic.

Geometric fractals include: the Koch curve, the dragon curve, the Levy curve, the Minkowski curve, the Sierpinski triangle, the Sierpinski carpet, the Cantor set, and the Pythagorean tree.

Fractals of this class are the most visual, since self-similarity is immediately visible in them. In the two-dimensional case, they can be obtained using a broken line, which is called a generator, in the three-dimensional case - a surface. Each of the segments that make up the broken line, in one step of the algorithm, is replaced by a broken line-generator, in the appropriate scale. Thus, a fractal curve is obtained as a result of the endless repetition of this procedure. With the apparent complexity of the resulting curve, its general form is given only by the shape of the generator.

Algebraic fractals: Mandelbrot set, Julia set, Newton basins, biomorphs.

Algebraic fractals are the most numerous. To construct algebraic fractals, iterations of nonlinear mappings are used, which are given by simple algebraic formulas. Two-dimensional processes are considered the most studied. It should be noted that nonlinear dynamic systems have several stable states. The initial state determines the state in which the dynamical system finds itself after a certain number of iterations. The ability to generate very complex non-trivial structures with the help of primitive algorithms came as a surprise to mathematicians.

Stochastic fractals include plasma and randomized fractal.

The term "stochasticity" comes from the Greek word and means "assumption".

No matter how similar it is to the coastline, the Koch curve cannot be used as its model, because it is the same everywhere, self-similar and, one might say, too “correct”. All natural objects are created at the whim of nature, in this process there is always an accident. Stochastic fractals are such fractals, in the construction of which randomly in an iterative system, some parameters change. At the same time, very similar to natural objects are obtained, such as asymmetrical trees, indented coastlines. Two-dimensional stochastic fractals are used in modeling the terrain and sea surface.

Application of fractals

The main application of fractals is modern computer graphics. With their help, you can create flat sets and surfaces of a very complex shape, while changing the parameters in the given equations.

Fractal geometry is indispensable for generating artificial clouds, mountain landscapes, seas. Scientists have found a simple way to depict complex objects, in which images resemble natural forms.

The most useful use of fractals in computer science is considered fractal data compression. The basis of this type of compression is that fractal geometry describes the real world quite well. Pictures are compressed even much better than using conventional methods. When the image is enlarged, no pixelation effect is observed, this is another advantage of fractal compression. With fractal compression, after zooming, the picture often looks even better than before.

It should be noted that fractals are used in data encryption using fractal algorithms.

To transmit data over a distance, antennas are used that have fractal shapes, which greatly reduces their weight and size.

Also, with the help of fractals, you can simulate complex physical processes, for example, flames. Fractal shapes are quite good at conveying porous materials that have a very complex geometric structure. Such knowledge is used in oil science.

The theory of fractals is also used in the study of the structure of the Universe.

In biology, examples such as biosensor interactions and heartbeats, modeling of chaotic processes can be considered. Fractals are used in their works by artists, designers, and composers.

Algorithms for constructing fractals

Consider the Mandelbrot set. In mathematics, the Mandelbrot set is a fractal, which is defined as a set of points on the complex plane, the iterative sequence does not go to infinity and is given by the formulas z 0 =0, Z n +1 = Z n 2 +M. To build a given sequence of points, i.e., a fractal, let's move from complex form records using transformations to convenient formulas for construction.

If the expression Z n +1 \u003d Z n 2 + M is reformulated as an iterative sequence of coordinate values of the complex plane x and y, that is, taking Z \u003d X + iY and M \u003d p + iq (where i is the imaginary unit), then we get algorithm with formulas (1): X n +1 =X n 2 –Y n 2 +p; Y n +1 \u003d 2X n Y n + q, with parameters p \u003d - 0.5219;

First we set X n = 0; Y n = 0, and according to formulas (1) we obtain at the first step of calculations: X n +1 =0 2 –0 2 –0.5219= – 0.5219; Y n +1 \u003d 2 0 0 + 0.4999.

Now we assume X n \u003d X n +1 \u003d - 0.5219; Y n = Y n +1 = 0.4999, and according to formulas (1) we obtain at the second step: X n +1 = (–0.5219) 2 – (0.4999) 2 – 0.5219 = – 0, 4994...;

Y n +1 \u003d 2 (-0.5219) (0.4999) + 0.4999 \u003d - 0.0218 ....

Then we assume X n = X n +1 = – 0.4994...; Y n \u003d Y n +1 \u003d -0.0218, and again, according to formulas (1), we continue further. That is, at each subsequent step of calculations (iterations), the previous values of X n +1 and Y n +1 must be substituted into formulas (1) as new values of X n and Y n .

In the Microsoft Excel program, you can make 32,000 such "steps" - calculations, and then build ("points") a graph of the function Y n +1 \u003d f (X n +1), which will look like a "blazing sun". Moreover, by changing the numerical values of the parameters p and q, other objects can be seen on the same graph; for example, at p = – 0.5; q \u003d 0.4999 instead of the "sun" you get " spiral galaxy».

I will present the algorithm that I have compiled for constructing the Mandelbrot fractals "blazing sun" and "spiral galaxy" in the Microsoft Excel program. In practice, 100 iterations are sufficient to achieve acceptable accuracy.

Table1 .

Algorithm for constructing the "blazing sun" Mandelbrot fractal in Microsoft Excel (for 100 iterations)

6. Write in cell H1 the variable Y n +1. 7. Enter the value 0 in cell A2.

8. Enter the value 0 in cell B2.

11.Enter the value -0.5219 in cell D2.

Insert->Charts->Scatter->Scatter with smooth curves

Table2 .

Algorithm for constructing the Mandelbrot fractal "spiral galaxy" in the program "Microsoft Excel" (for 100 iterations)

1. Write in cell A1 the variable X n

2. Write down the variable Y n in cell B1.

3. Write down the p parameter in cell D1.

4. Write down the q parameter in cell E1.

5.Write the variable X n +1 into cell G1.

6. Write in cell H1 the variable Y n +1.

7. Enter the value 0 in cell A2.

8. Enter the value 0 in cell B2. .

9. Enter the formula =G2 in cell A3.

10. Enter the formula =H2 in cell B3.

11.Enter the value -0.5 in cell D2.

12. Enter the value 0.4999 in cell E2.

13. Enter in cell G2 the formula =A2^2-B2^2+$D$2

14. Enter the formula =2*A2*B2+$E$2 in cell H2

15. Stretch cell A3 by the lower right corner to A101.

16. Stretch cell B3 by the lower right corner to B101.

17. Stretch cell G2 by the lower right corner to G101.

18. Stretch cell H2 by the lower right corner to H101.

19.Select the range from G2 to H101.

20. To build a figure, do the following:

Insert->Charts->Scatter->Scatter with smooth curves

Consider the "Hilbert curve" fractal given by formula (2):

y(x) = (cos 0.5 x⋅ cos 200 x + |x| 0,5 − 0,7)(4 − x 2) 0.01 . Let's find the range of allowable values of this expression. Under arithmetic square root the function cos(x) is found, so cos(x) ≥ 0.

I will present the algorithm that I have compiled for constructing the “Hilbert curve” fractal in the “Microsoft Excel” program according to this formula (2) in the allowable range of values, choosing a step equal to 0.01.

Table3 .

Algorithm for constructing the "Hilbert curve" fractal in the program "Microsoft Excel"

1. Write down the variable x in cell A1.

2. Write down the variable y in cell B1.

3. Write in cell A2 the value -π/2, according to the range of allowable values XЄ[-π/2; π/2],

4. Enter the formula =A2+0.01 into cell A3.

5. Stretch cell A3 by the lower right corner to cell A316 (up to a value of 1.57).

6. Enter the formula in cell B2

=((SQRT(COS(A2)))*COS(200*A2)+SQRT(ABS(A2))-0.7)*(4-A2*A2)^0.01

7. Stretch cell B2 by the lower right corner to cell B316.

8. Select the range of values from A2 to B316.

9. To build a figure, do the following:

Insert->Charts->Scatter->Scatter with smooth curves

Consider the Mandelbrot fractal "Dragon curve" given by the systems of equations (3) and (4), respectively:

First we set X n = 0; Y n = 0. We randomly set the parameter m, which varies from 0 to 1. If m > 0.5, then we apply the system of equations (3) to construct a fractal, otherwise - (4). Each new value is obtained from the previous one depending on a random number.

I will present the algorithm that I compiled for constructing the Mandelbrot fractal “Dragon Curve” in the Microsoft Excel program.

Table4 .

Algorithm for constructing the Mandelbrot fractal "Dragon Curve" in the program "Microsoft Excel"

1. Write the number n in cell A1.

2. Write to cell B1 random variable m.

3. In cell C1 write x.

4. In cell D1 write y.

5. In cell A2, write 1.

6. In cell A3 enter the formula =A2+1

7. Stretch A3 to cell A 11363

8. In cell B2, write the random number function =RAND()

9. Stretch cell B2 to B 11363

10. Enter the value 0 in cell C2

11. Enter in cell C3 the formula =IF(B3>0.5;-0.4*C2-1;0.76*C2-0.4*D2)

12. Stretch cell C3 to cell C 11363

13. Enter the value 0 in cell D2.

14. Enter in cell D3 the formula =IF(B3>0.5;-0.4*D2+0.1;0.4*C2+0.76*D2)

15. Stretch cell D3 to cell D11363

16. Select cells from C2 to D11363

17. To build a figure, do the following:

Insert->Charts->Scatter

Conclusion

The computer can be characterized as a new means of cognition. Thanks to him, you can see connections and meanings that have been hidden from us until now.

Performing research work, I was convinced that the scope of fractals is extremely large. Their help is needed, for example, when it is required to define lines and surfaces of a very complex shape using several coefficients.

We can say that actually found lung way, a convenient representation of complex non-Euclidean objects, the images of which are similar to natural ones.

Fractals allow us to look at mathematics from a completely different perspective, open our eyes. It would seem that ordinary calculations are made with ordinary numbers, but this gives us in its own way unique, inimitable results that allow us to feel like a creator of nature. Fractals make it clear that mathematics is also the science of beauty.

Bibliography:

1.Benoit Mandelbrot. "The Fractal Geometry of Nature", 1977.

2. Mandelbrot B. Fractal geometry of nature. M.: Institute of Computer Research, 2002. - 656 p.

3.Morozov A.D. Introduction to the theory of fractals. Moscow-Izhevsk: Institute for Computer Research, 2002. - 160 p.

4. About fractals. [Electronic resource] - Access mode. - URL: http://elementy.ru/posters/fractals

5. Pererva L.M., Yudin V.V. P 27 Fractal modeling: tutorial/ under total ed. V.N. Gryanik. Vladivostok: Publishing House of VGUES, 2007. - 186 p.

Martynov Daniil

Project Manager:

Martynova Ludmila Yurievna

Institution:

MOU "Kriushinskaya secondary school"

During research work in mathematics "Fractals around us" a 8th grade student set a goal to show that mathematics is not a soulless subject, it can express the spiritual world of a person and society by creating its own geometric fractal " Star».

In the research work on mathematics "Fractals around us", the author builds a geometric fractal "Star" within the framework of the project and gives recommendations on the practical application of the created fractal, tries to find a connection between fractals and Pascal's triangles in the process of mathematical research.

In the proposed Mathematics project "Fractals around us" the author comes to the conclusion that the new ideas of fractal geometry will help to study many mysterious phenomena surrounding nature. Image processing and pattern recognition methods using new concepts enable researchers to apply this mathematical apparatus to a quantitative description of a huge number of natural objects and structures.

Introduction
1. Justification and construction of the geometric fractal "Star".
2. Finding a connection between fractals and Pascal's triangles.
3. Recommendations for the practical application of the created fractal.
Conclusion

Introduction

Many of my classmates think that mathematics is an exact and boring science, problems, equations, graphs, formulas…. What can be interesting here? Geometry of the 21st century. Cold, complex, uninteresting...

"Why is it called that? One of the reasons is its inability to describe the shape of a cloud, a mountain, a tree, or a seashore. Clouds are not spheres, mountains are not cones, coastlines are not circles, and the crust is not smooth, and lightning does not spread in a straight line. Nature shows us not just a higher degree, but a completely different level of complexity" Benoit Mandelbrot.

In my research work, I tried to refute the above. This became possible after the discovery of fractals - self-similar figures with a number of interesting properties, which made it possible to compare fractals with objects of nature.

Hypothesis – « Everything that exists in the real world is a fractal».

Target - to show that mathematics is not a soulless subject, it can express the spiritual world of a person and society by creating its own geometric fractal " Star».

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

Analyze and review the literature on the research topic.
Consider and study different types of fractals.
Establish the relationship between Pascal's triangle, literary works.
Invent and create your own fractal, create a program for constructing a graphic image of a geometric fractal " Star».
Consider the possibilities of practical application of the created fractal.

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

Structure of the research work includes an introduction, two chapters, a conclusion, a list of references, applications.

In the introduction the relevance and novelty of the research topic are substantiated, the problem, subject, goal, tasks, stages of work, theoretical and practical significance of the work are identified.

In the first chapter the question of the history of the concept of a fractal, the classification of fractals, the use of fractals is revealed.

In the second chapter it is investigated and proved that the geometric figure created by us " Star"is a fractal, by changing the parameters of the created fractal, we got a whole gallery of beautiful ornaments that can be used for practical applications: in the production of fabrics, finishing materials, in valeology.

Christolubova Angelina

The most ingenious discoveries in science can radically change human life. The invented vaccine can save millions of people, the creation of weapons, on the contrary, takes these lives. More recently (on the scale of human evolution) we have learned to "tame" electricity - and now we can not imagine life without all these convenient devices that use electricity. But there are also discoveries that few people attach importance to, although they also greatly influence our lives.

Download:

Preview:

Municipal budgetary educational institution

gymnasium №2 in Salska

"Department of Natural and Mathematical Disciplines"

Research

subject: " Fractals in our life».

Christolubova Angelina Mikhailovna,

student of 8 "B" class.

Supervisor:

Kuzminchuk Elena Sergeevna,

teacher of mathematics and computer science.

Salsk

2015

Introduction

Fractal classification

Application of fractals

Conclusion.

Bibliography.

Applications.

Introduction

Large fleas are bitten by fleas

Those fleas are tiny crumbs,

As they say, ad infinitum.

Jonathan Swift

The most ingenious discoveries in science can radically change human life. The invented vaccine can save millions of people, the creation of weapons, on the contrary, takes these lives. More recently (on the scale of human evolution) we have learned to "tame" electricity - and now we can not imagine life without all these convenient devices that use electricity. But there are also discoveries that few people attach importance to, although they also greatly influence our lives.

One of these "imperceptible" discoveries is fractals. You have probably heard this catchy word, but do you know what it means and how many interesting things are hidden in this term?

Every person has a natural curiosity, a desire to learn about the world around him. And in this aspiration, a person tries to adhere to logic in judgments. Analyzing the processes taking place around him, he tries to find the logic of what is happening and deduce some regularity. The biggest minds on the planet are busy with this task. Roughly speaking, scientists are looking for a pattern where it should not be. However, even in chaos, one can find a connection between events. And this connection is a fractal.

Today, one can hardly find a person involved in or interested in science who has not heard about fractals. Looking at them, it is hard to believe that these are not creations of nature and mathematical formulas are hidden behind them. Fractals strikingly resemble objects of living and inanimate nature around us. In a word, they are "real". Most likely, that is why, once seen, a person can no longer forget them.

An interesting thought is given in his book "The Fractal Geometry of Nature" by the American mathematician Benoit Mandelbrot: "Why is geometry often called cold and dry? One of the reasons is that it is unable to accurately describe the shape of a cloud, mountain, tree or seashore. Clouds - they are not spheres, shore lines are not circles, and the crust is not smooth, and lightning does not travel in a straight line.Nature shows us not just a higher degree, but a completely different level of complexity.The number of different length scales in structures is always infinite.The existence of these structures challenges us with the difficult task of studying those forms that Euclid dismissed as formless - the task of studying the morphology of the amorphous. Mathematicians, however, neglected this challenge and preferred to move further and further away from nature, inventing theories that do not correspond to anything that what can be seen or felt.

Everything that exists in the real world is a fractal - this is our hypothesis, but the goal This work shows that mathematics is not a soulless subject, it can express the spiritual world of a person individually and in society as a whole.

Object of studyfractals appear in mathematics and in the real world. In the course of our work, we identified the followingresearch objectives:

Analyze and review the literature on the research topic.
Consider and study different types of fractals.
To give an idea of the fractals encountered in our lives.

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

Structure of the research workwas determined by the logic of the study and the tasks set. It includes an introduction, two chapters, a conclusion, a list of references, and appendices.

The history of the concept of "fractal"

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

Georg Cantor (Cantor, 1845-1918) - German mathematician, logician, theologian, creator of the theory infinite sets, using a simple recursive (repeating) procedure, turned the line into a set of unconnected points. He took the line and removed the central third and then repeated the same with the remaining segments. It turned out the so-called Dust of Kantor (appendices 1, 2).

Giuseppe Peano (1858-1932) - Italian mathematician depicted a special line. He took a straight line and replaced it with 9 segments 3 times shorter than the length of the original line. Then he did the same with each segment. And so on ad infinitum. The uniqueness of such a line is that it fills the entire plane. Later, a similar construction was carried out in three-dimensional space (appendices 3, 4).

The very word "fractal" appeared thanks to the brilliant scientist Benoit Mandelbrot (Appendix 5).

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

The definition of a fractal given by Mandelbrot is as follows: "A fractal is a structure consisting of parts that are in some sense similar to the whole."

A fractal pattern does not have identical elements, but has similarity at any scale. To build such an image with a high degree of detail manually was simply impossible before, it required a huge amount of calculations.

One of the first drawings of a fractal was a graphical interpretation of the Mandelbrot set, which was born thanks to the research of Gaston Maurice Julia (Appendix 6).

Many objects in nature have fractal properties, such as coasts, clouds, tree crowns, snowflakes, the circulatory system and the alveolar system of humans or animals.

Fractal classification

Fractals are divided into groups. The largest groups are:

Geometric fractals;

Algebraic fractals;

Application of fractals

Conclusion.

In addition to the useful role that fractal geometry plays in describing the complexity of natural objects, it also offers a good opportunity to popularize mathematical knowledge. The concepts of fractal geometry are clear and intuitive. Its forms are attractive from an aesthetic point of view and have a variety of applications. Therefore, fractal geometry may help to refute the view of mathematics as a dry and inaccessible discipline and will become an additional incentive for students to master this interesting and fascinating science.

Even the scientists themselves experience almost childish delight, watching rapid development of this new language - the language of fractals.

In everything that surrounds us, we often see chaos, but in fact this is not an accident, but an ideal form, which fractals help us to discern. Nature is the best architect, ideal builder and engineer. It is arranged very logically, and if somewhere we do not see patterns, this means that we need to look for it on a different scale. People understand this better and better, trying to imitate natural forms in many ways. Engineers design speaker systems in the form of a shell, create antennas with snowflake geometry, and so on. We are sure that fractals still keep a lot of secrets, and many of them have yet to be discovered by man.

As a result of the study, it was possible to find out that 42.5% of the respondents met with fractals, 15% of the respondents know what a fractal is, 62.5% of the students and teachers of the surveyed students and teachers of the MBOU gymnasium No. 2 in Salska would like to know what a fractal is.

After the discovery of fractals, it became obvious to many that the good old forms of Euclidean geometry are much inferior to most natural objects due to the lack of some irregularity, disorder and unpredictability in them. It is possible that the new ideas of fractal geometry will help to study many mysterious phenomena of the surrounding nature.

We managed to show that everything that exists in the real world is a fractal. We are convinced that those who deal with fractals discover a beautiful, wonderful world in which mathematics, nature and art reign. We hope that after getting acquainted with our work, you, like us, will be convinced that mathematics is beautiful and amazing.

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