Often, brilliant discoveries made in science can radically change our lives. For example, the invention of a vaccine can save many people, but the creation of new weapons leads to murder. Literally yesterday (on the scale of history) man “tamed” electricity, and today he can no longer imagine his life without it. However, there are also discoveries that, as they say, remain in the shadows, despite the fact that they also have one or another impact on our lives. One of these discoveries was the fractal. Most people have never even heard of this concept and will not be able to explain its meaning. In this article we will try to understand the question of what a fractal is and consider the meaning of this term from the perspective of science and nature.

Order in chaos

In order to understand what a fractal is, we should begin the debriefing from the position of mathematics, but before delving into it, we will philosophize a little. Every person has a natural curiosity, thanks to which he learns about the world around him. Often, in his quest for knowledge, he tries to use logic in his judgments. Thus, by analyzing the processes that occur around him, he tries to calculate relationships and derive certain patterns. The greatest minds on the planet are busy solving these problems. Roughly speaking, our scientists are looking for patterns where there are none, and there should not be any. And yet, even in chaos there is a connection between certain events. This connection is what the fractal is. As an example, consider a broken branch lying on the road. If we look closely at it, we will see that with all its branches and twigs it itself looks like a tree. This similarity of a separate part with a single whole indicates the so-called principle of recursive self-similarity. Fractals can be found all over the place in nature, because many inorganic and organic forms are formed in a similar way. These are clouds, sea shells, snail shells, tree crowns, and even the circulatory system. This list can be continued indefinitely. All these random shapes are easily described by a fractal algorithm. Now we have come to consider what a fractal is from the perspective of exact sciences.

Some dry facts

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

Historical background, or How it all began

At the turn of the 19th and 20th centuries, the study of the nature of fractals was sporadic. This is explained by the fact that mathematicians preferred to study objects that could be researched on the basis of general theories and methods. In 1872, the German mathematician K. Weierstrass constructed an example of a continuous function that is not differentiable anywhere. However, this construction turned out to be entirely abstract and difficult to perceive. Next came the Swede Helge von Koch, who in 1904 constructed a continuous curve that had no tangent anywhere. It's fairly easy to draw and turns out to have fractal properties. One of the variants of this curve was named after its author - “Koch snowflake”. Further, the idea of ​​self-similarity of figures was developed by the future mentor of B. Mandelbrot, the Frenchman Paul Levy. In 1938, he published the article "Plane and spatial curves and surfaces consisting of parts similar to the whole." In it, he described a new type - the Lewy C-curve. All of the above figures are conventionally classified as geometric fractals.

Dynamic or algebraic fractals

The Mandelbrot set belongs to this class. The first researchers in this direction were the French mathematicians Pierre Fatou and Gaston Julia. In 1918, Julia published a paper based on the study of iterations of rational complex functions. Here he described a family of fractals that are closely related to the Mandelbrot set. Despite the fact that this work glorified the author among mathematicians, it was quickly forgotten. And only half a century later, thanks to computers, Julia’s work received a second life. Computers made it possible to make visible to every person the beauty and richness of the world of fractals that mathematicians could “see” by displaying them through functions. Mandelbrot was the first to use a computer to carry out calculations (such a volume cannot be done manually) that made it possible to construct an image of these figures.

A person with spatial imagination

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

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

Julia - Mandelbrot

One of the first drawings of this figure was a graphic interpretation of the set, which was born out of the work of Gaston Julia and was further developed by Mandelbrot. Gaston tried to imagine what a set would look like based on a simple formula that was iterated through a feedback loop. Let's try to explain what has been said in human language, so to speak, on the fingers. For a specific numerical value, we find a new value using a formula. We substitute it into the formula and find the following. The result is large. To represent such a set it is necessary to perform this operation a huge number of times: hundreds, thousands, millions. This is what Benoit did. He processed the sequence and transferred the results to graphical form. Subsequently, he colored the resulting figure (each color corresponds to a certain number of iterations). This graphic image was named “Mandelbrot fractal”.

L. Carpenter: art created by nature

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

Carpenter's solution

The principle Lauren used was simple. It consists of dividing larger ones into small elements, and those into similar smaller ones, and so on. Carpenter, using large triangles, split them into 4 small ones, and so on, until he had a realistic mountain landscape. Thus, he became the first artist to use a fractal algorithm in computer graphics to construct the required image. Today this principle is used to imitate various realistic natural forms.

The first 3D visualization using a fractal algorithm

A few years later, Lauren applied his developments in a large-scale project - the animated video Vol Libre, shown on Siggraph in 1980. This video shocked many, and its creator was invited to work at Lucasfilm. Here the animator was able to realize his full potential; he created three-dimensional landscapes (an entire planet) for the feature film "Star Trek". Any modern program (“Fractals”) or application for creating 3D graphics (Terragen, Vue, Bryce) uses the same algorithm for modeling textures and surfaces.

Tom Beddard

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

Fractals in nature

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

Musical pause

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

Fractal indicator

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

Finally

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

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• Introduction
• 1. The concept of a fractal
• 2. Classification of fractals
• 4. Application of fractals
• Conclusion
• List of used literature

Introduction

The appearance of self-similar mathematical objects a hundred or more years ago was of no interest to almost anyone; they were of interest only to the authors of these objects. Moreover, some scientists dubbed them “monsters” and did not believe that they had anything to do with the real world and science.

Attitudes towards self-similar mathematical objects changed with the advent of computers, when the first images of algebraic and stochastic fractals appeared. Immediately after this, they interested not only mathematicians, but also physicists, biologists, acousticians, and everyone who came across natural objects in their work. Mathematicians were attracted to fractals by the simplicity of the formulas that describe such complex structures, physicists by the opportunity to reconsider physics from a new position, biologists by the correspondence of fractal images with various biological objects.

Fractals have not yet exhausted themselves; fractal objects are still being found in new areas of science. They are used by physicists, biologists, sociologists, economists and many others. Fractals have not been fully studied; new applications are being found for them, changing our attitude both to fractals themselves and to nature.

The object of the work is the phenomenon of fractals.

The subject of the work is the place of fractals in modern science.

The purpose of the work is to consider fractals as both a simple and complex phenomenon.

Objectives of the work: consider the concept of fractals, types of fractals, the history of the emergence and study of fractals, the application of fractals in practice.

1. The concept of a fractal

The concepts of fractal and fractal geometry, which appeared in the late 70s, have become firmly established among mathematicians and programmers since the mid-80s of the 20th century. The word fractal is derived from the Latin fractus and translated means consisting of fragments. Mandelbrot B. Fractal geometry of nature, p. 5 - M.: Institute of Computer Research, 2002. . It was proposed by Benoit Mandelbrot in 1975 to designate irregular but self-similar structures, which he studied. Mandelbrot B. Fractal geometry of nature, p. 5 - M.: Institute for Computer Research, 2002. . The birth of fractal geometry is usually associated with the publication of Mandelbrot’s book “The Fractal Geometry of Nature” in 1977. His works used the scientific results of other scientists who worked in the period 1875-1925 in the same field (Poincaré, Fatou, Julia, Cantor, Hausdorff But only in our time has it been possible to combine their work into a single system.

The role of fractals in computer graphics today is quite large. They come to the rescue, for example, when it is necessary, using several coefficients, to define lines and surfaces of very complex shapes. From the point of view of computer graphics, fractal geometry is indispensable when generating artificial clouds, mountains, and sea surfaces. In fact, a way has been found to easily represent complex non-Euclidean objects, the images of which are very similar to natural ones.

One of the main properties of fractals is self-similarity. In the simplest case, a small part of a fractal contains information about the entire fractal.

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.” Feder E. Fractals: World 1991, p.67.

It should be noted that 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 in question has any of the following properties:

1. Has a non-trivial structure on all scales. This is in contrast to regular figures (such as a circle, ellipse, graph of a smooth function): if we consider a small fragment of a regular figure on a very large scale, it will look like a fragment of a straight line. For a fractal, increasing the scale does not lead to a simplification of the structure; on all scales we will see an equally complex picture.

2. Is self-similar or approximately self-similar.

3. Has a fractional metric dimension or a metric dimension that exceeds the topological one.

4. Can be constructed using a recursive procedure Feder E. Fractals: World 1991, p. 133.

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

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

Fractals are primarily the language of geometry. However, their main elements are not directly observable. In this respect, they are fundamentally different from the usual objects of Euclidean geometry, such as a straight line or a circle. Fractals are expressed not in primary geometric forms, but in algorithms, sets of mathematical procedures.

These algorithms are transformed into geometric shapes using a computer. The repertoire of algorithmic elements is inexhaustible. Once you master the language of fractals, you can describe the shape of a cloud as clearly and simply as an architect describes a building using drawings that use the language of traditional geometry.

2. Classification of fractals

Geometric fractals. Fractals of this class are the most visual. In the two-dimensional case, they are obtained using some broken line (or surface in the three-dimensional case), called a generator. In one step of the algorithm, each of the segments that make up the polyline is replaced with a generator polyline, on the appropriate scale. As a result of endless repetition of this procedure, a geometric fractal is obtained.

Algebraic fractals. This is the largest group of fractals. They are obtained using nonlinear processes in n-dimensional spaces. Two-dimensional processes are the most studied. When interpreting a nonlinear iterative process as a discrete dynamic system, one can use the terminology of the theory of these systems: phase portrait, steady-state process, attractor, etc.

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

Scholastic fractals. Natural objects that arise as a result of complex random processes often have a fractal shape. Stochastic (random) fractals can be used to model them. Examples of stochastic fractals:

1. trajectory of Brownian motion on the plane and in space;

2. boundary of the trajectory of Brownian motion on a plane. In 2001, Lawler, Schramm and Werner proved Mandelbort's conjecture that its dimension is 4/3.

3. Schramm-Löwner evolutions - conformally invariant fractal curves arising in critical two-dimensional models of statistical mechanics, for example in the Ising model and percolation.

4. various types of randomized fractals, that is, fractals obtained using a recursive procedure into which a random parameter is introduced at each step. Plasma is an example of the use of such a fractal in computer graphics.

Fractal monotype, or stochatypy, are trends in the visual arts that consist in obtaining an image of a random fractal. Schroeder M. Fractals, chaos, power laws. Miniatures from endless paradise. - Izhevsk: RHD, 2001, p.26.

3. The history of fractals

It is noteworthy that the appearance of fractals (not yet given this name) in the mathematical literature about a hundred years ago was met with regrettable hostility, as has happened in the history of the development of many other mathematical ideas. One famous mathematician, Charles Hermite, even dubbed them monsters. At least the general consensus recognized them as a pathology of interest only to researchers who abuse mathematical fads, and not to real scientists.

As a result of the efforts of Benoit Mandelbrot, this attitude changed, and fractal geometry became a respected applied science. Mandelbrot coined the term fractal based on Hausdorff's theory of fractal (fractional) dimension, proposed in 1919. Many years before the appearance of his first book on fractal geometry, Mandelbrot began researching the appearance of monsters and other pathologies in nature. He found a niche for the disreputable Cantor sets, Peano curves, Weierstrass functions and their many variations, which were considered nonsense. He and his students discovered many new fractals, such as fractal Brownian motion for modeling forest and mountain landscapes, river level fluctuations and heartbeats. With the publication of his books, applications of fractal geometry began to appear like mushrooms after rain. This affected both many applied sciences and pure mathematics. Even the film industry was not left out. Millions of people admired the mountain landscape in the film “Star Migration II: The Wrath of Khan,” constructed using fractals Peitgen H.-O., Richter P. H. The beauty of fractals. - M.: Mir 1993, p.45.

French mathematician Henri Poincaré initiated research into nonlinear dynamics around 1890, leading to modern chaos theory. Interest in the subject increased markedly when Edward Lorenz, a nonlinear weather modeler, discovered in 1963 that long-term weather forecasts were impossible. Lorenz noted that even small errors in measuring the current state of weather conditions can lead to completely incorrect predictions about future weather conditions. This essential dependence on initial conditions underlies the mathematical theory of chaos.

Particle trajectories of Brownian motion, which were studied by Robert Brown as early as 1828 and Albert Einstein in 1905, are an example of fractal curves, although their mathematical description was not given until 1923 by Norbert Wiener. In 1890, Peano constructed his famous curve - a continuous mapping that transforms a segment into a square and, therefore, increases its dimension from one to two. The Koch snowflake boundary (1904), whose dimension d » 1.2618, is another well-known dimension-increasing curve.

The fractal, in no way resembling a curve, which Mandelbrot called dust, is the classical Cantor set (1875 or earlier). This set is so sparse that it does not contain intervals, but nevertheless has the same number of points as the interval. Mandelbrot used such “dust” to model stationary noise in telephony. Fractal dust of one kind or another appears in numerous situations. In fact, it is a universal fractal in the sense that any fractal - an attractor of a system of iterated functions - is either fractal dust or its projection onto a space with a lower dimension Peitgen H.-O., Richter P., p. 22.

Various tree-like fractals were used not only to model tree-plants, but also the bronchial tree (air-bearing branches in the lungs), the functioning of the kidneys, the circulatory system, etc. It is interesting to note Leonardo da Vinci’s assumption that all branches of a tree at a given height, added together , equal in thickness to the trunk (below their level). This implies a fractal model for the tree crown in the form of a fractal surface.

Many remarkable properties of fractals and chaos are revealed by studying iterated mappings. In this case, they start with some function y = f(x) and consider the behavior of the sequence f(x), f(f(x)), f(f(f(x))),... In the complex plane, work of this kind ascends , apparently, to the name of Cayley, who investigated Newton's method of finding the root as applied to complex, and not just real, functions (1879). Remarkable progress in the study of iterated complex mappings was made by Gaston Julia and Pierre Fatou (1919). Naturally, everything was done without the help of computer graphics. These days, many have already seen colorful posters depicting *Julia sets and the Mandelbrot set, which is closely related to them. It is natural to start mastering the mathematical theory of chaos with iterated mappings.

The study of fractals and chaos opens up wonderful possibilities, both in the study of an infinite number of applications and in the field of pure mathematics. But at the same time, as often happens in the so-called new mathematics, the discoveries are based on the pioneering work of the great mathematicians of the past. Sir Isaac Newton understood this when he said, “If I have seen further than others, it is because I have stood on the shoulders of giants.”

4. Application of fractals

Computer graphics

Fractals are widely used in computer graphics to construct images of natural objects, such as trees, bushes, mountain landscapes, sea surfaces, etc.

Physics and other natural sciences

In physics, fractals naturally arise when modeling nonlinear processes, such as turbulent fluid flow, complex random diffusion-adsorption processes, flames, clouds, etc. Fractals are also used when modeling porous materials, for example, in petrochemistry. In biology, they are used to model populations and to describe internal organ systems (the blood vessel system).

Literature

Among literary works there are those that have a textual, structural or semantic fractal nature. In text fractals, elements of text are potentially endlessly repeated:

1. A non-branching infinite tree, identical to themselves from any iteration (“The priest had a dog...”, “The parable of a philosopher who dreams that he is a butterfly who dreams that she is a philosopher who dreams...”, “It is a false statement that the statement is true, the statement is false...").

2. Non-branching endless texts with variations (“Peggy had a funny goose…”) and texts with extensions (“The House That Jack Built”).

3. In structural fractals, the text layout is potentially fractal

4. Wreath of sonnets (15 poems), wreath of sonnets (211 poems), wreath of wreaths of sonnets (2455 poems).

5. “Stories within a story” (“The Book of One Thousand and One Nights”, J. Pototsky “Manuscript Found in Saragossa”).

6. Prefaces that hide the authorship (U. Eco “The Name of the Rose”).

In semantic and narrative fractals, the author talks about the infinite similarity of a part to the whole

H. L. Borges “In the circle of ruins”

J. Cortazar “Yellow Flower”

J. Perek “Kunstkamera”

Fractal antennas.

The use of fractal geometry in the design of antenna devices was first used by American engineer Nathan Cohen, who then lived in downtown Boston, where the installation of external antennas on buildings was prohibited. Nathan cut out a Koch curve shape from aluminum foil and glued it onto a piece of paper, then attached it to the receiver. It turned out that such an antenna works no worse than a regular one. And although the physical principles of operation of such an antenna have not yet been studied, this did not stop Cohen from founding his own company and launching their serial production.

Image compression.

There are algorithms for compressing images using fractals. They are based on the idea that instead of an image, one can store a compression mapping for which the image is a fixed point.

Decentralized networks.

The IP address assignment system in the Netsukuku network uses the principle of fractal information compression to compactly store information about network nodes. Each node in the Netsukuku network stores only 4 KB of information about the state of neighboring nodes, while any new node connects to the common network without the need for central regulation of the distribution of IP addresses, which, for example, is typical for the Internet. Thus, the principle of fractal information compression guarantees completely decentralized, and therefore, the most stable operation of the entire network.

Conclusion

Most people believe that fractals are just beautiful pictures that please the eye. Fortunately, this is not the case, and fractals are used in many areas of human activity. There is already a theoretical basis for creating new areas of their application, such as diagnosing diseases, predicting damage during dynamic impact, and many others. But, despite the theoretical inexhaustibility of the use of fractals, it can be assumed that over time the main directions of their application will emerge.

Only a few decades have passed since Benoit Mandelbrot declared: “The geometry of nature is fractal!” Today we can already assume much more, namely, that fractality is the primary principle of construction of all natural objects without exception.

Conclusions:

1. The nature of fractals is carefully studied by scientists

2. In the future, many problems in medicine, the computer industry, science, etc. will be solved with the help of fractals.

List of used literature

fractal natural graphics

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

2. Peitgen H.-O., Richter P. H. The beauty of fractals. - M.: Mir, 1993.

3. Feder E. Fractals-M.: Mir, 1991.

4. Schroeder M. Fractals, chaos, power laws. Miniatures from endless paradise. - Izhevsk: RHD, 2001.

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Recently I learned about such interesting objects of the mathematical world as fractals. But they exist not only in mathematics. They surround us everywhere. Fractals are natural. I will talk about what fractals are, about the types of fractals, about examples of these objects and their applications in this article. To begin with, I’ll briefly tell you what a fractal is.

A fractal (Latin fractus - crushed, broken, broken) is a complex geometric figure that has the property of self-similarity, that is, composed of several parts, each of which is similar to the entire figure. In a broader sense, fractals are understood as sets of points in Euclidean space that have a fractional metric dimension (in the sense of Minkowski or Hausdorff), or a metric dimension different from the topological one. As an example, I will insert a picture depicting four different fractals.

I'll tell you a little about the history of fractals. The concepts of fractal and fractal geometry, which appeared in the late 70s, have become firmly established among mathematicians and programmers since the mid-80s. The word "fractal" was coined by Benoit Mandelbrot in 1975 to refer to the irregular but self-similar structures with which he was concerned. The birth of fractal geometry is usually associated with the publication of Mandelbrot’s book The Fractal Geometry of Nature in 1977. His works used the scientific results of other scientists who worked in the period 1875-1925 in the same field (Poincaré, Fatou, Julia, Cantor, Hausdorff). But only in our time has it been possible to combine their work into a single system.

There are a lot of examples of fractals, because, as I said, they surround us everywhere. In my opinion, even our entire Universe is one huge fractal. After all, everything in it, from the structure of the atom to the structure of the Universe itself, exactly repeats each other. But there are, of course, more specific examples of fractals from different areas. Fractals, for example, are present in complex dynamics. They are there appear naturally when studying nonlinear dynamic systems. The most studied case is when the dynamical system is specified by iterations of a polynomial or holomorphic function of a complex of variables on surface. Some of the most famous fractals of this type are the Julia set, Mandelbrot set and Newton pools. Below, in order, the pictures depict each of the above fractals.

Another example of fractals is fractal curves. It is best to explain how to construct a fractal using the example of fractal curves. One of these curves is the so-called Koch Snowflake. There is a simpleprocedure for obtaining fractal curves on a plane. Let us define an arbitrary broken line with a finite number of links, called a generator. Next, we replace each segment in it with a generator (more precisely, a broken line similar to a generator). In the resulting broken line, we again replace each segment with a generator. Continuing to infinity, in the limit we get a fractal curve. Below is the Koch Snowflake (or Curve).

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

The third example or type of fractals are stochastic fractals. Such fractals include the trajectory of Brownian motion on the plane and in space, Schramm-Löwner evolutions, various types of randomized fractals, that is, fractals obtained using a recursive procedure into which a random parameter is introduced at each step.

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

But perhaps the most interesting fractals are natural ones. Natural fractals are objects in nature that have fractal properties. And here the list is already big. I won’t list everything, because it’s probably impossible to list them all, but I’ll tell you about some. For example, in living nature, such fractals include our circulatory system and lungs. And also the crowns and leaves of trees. This also includes starfish, sea urchins, corals, sea shells, and some plants such as cabbage or broccoli. Several such natural fractals from living nature are clearly shown below.

If we consider inanimate nature, then there are much more interesting examples there than in living nature. Lightning, snowflakes, clouds, well-known to everyone, patterns on windows on frosty days, crystals, mountain ranges - all these are examples of natural fractals from inanimate nature.

We looked at examples and types of fractals. As for the use of fractals, they are used in a variety of fields of knowledge. In physics, fractals naturally arise when modeling nonlinear processes, such as turbulent fluid flow, complex diffusion-adsorption processes, flames, clouds, etc. Fractals are used when modeling porous materials, for example, in petrochemistry. In biology, they are used to model populations and to describe internal organ systems (the blood vessel system). After the creation of the Koch curve, it was proposed to use it in calculating the length of the coastline. Fractals are also actively used in radio engineering, information science and computer technology, telecommunications and even economics. And, of course, fractal vision is actively used in modern art and architecture. Here is one example of fractal patterns:

And so, with this I think to complete my story about such an unusual mathematical phenomenon as a fractal. Today we learned about what a fractal is, how it appeared, about the types and examples of fractals. I also talked about their application and demonstrated some of the fractals visually. I hope you enjoyed this little excursion into the world of amazing and fascinating fractal objects.

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Authors:
Bekbulatova Alina,
Getmanova Sofia

Leaders:
Mogutova Tatyana Mikhailovna,
Deryushkina Oksana Valerievna

Introduction.

Theoretical part of the project:

• History of the development of fractal geometry.
• The concept of a fractal.
• Types of fractals:

a) geometric fractals, examples of geometric fractals;
b) algebraic fractals, examples of algebraic fractals;
c) stochastic fractals, examples.

• Natural fractals.
• Practical application of fractals:
• in literature;
• in telecommunications;
• in medicine;
• in architecture;
• in design;
• in economics;
• in games, cinema, music
• in natural sciences
• in physics;
• in biology
• fractals for housewives
• modern paintings – fractal graphics.
• Fractal graphics.
• The role of fractal geometry in life is a hymn to fractals!

The practical part of the project

• Creation of a scientific work “Journey to the world of fractals”
• Posting on the Internet.
• Participation in Olympiads and competitions.
• Create your own fractals.
• Creation of the brochure “The Amazing World of Fractals”
• Carrying out the festival “The Amazing World of Fractals.

Introduction

The geometry is often described as cold and dry. One reason is its inability to describe everything that surrounds us: the shape of a cloud, mountain, tree or seashore. Clouds are not spheres, mountains are not cones, coastlines are not circles, and the crust is not smooth, and lightning does not travel in a straight line. With great joy for us, we learned that in the modern world there is a new geometry - the geometry of fractals.

The discovery of fractals revolutionized not only geometry, but also physics, chemistry, biology, and all areas of our lives.

Relevance of the project:

• The role of fractals in the modern world is quite large
• Convincing arguments in favor of the relevance of studying fractals is the breadth of their application

Research hypothesis:

Fractal geometry is a modern, very interesting area of ​​human knowledge. The emergence of fractal geometry is evidence of the ongoing evolution of man and the expansion of his ways of understanding the world.

Objective of the project:

Study the theory of fractals to create a scientific work “The Amazing World of Fractals” and develop and implement on a computer algorithms for drawing fractals on a plane.

Project objectives:

• Get acquainted with the history of the emergence and development of fractal geometry;
• Study the types of fractals and their application in the modern world.
• Execute fractal creation programs in Pascal and Logo programming languages
• Create a scientific work about fractals and publish it on the Internet.
• Create a brochure “The Amazing World of Fractals”
• Hold a festival “The Amazing World of Fractals” to familiarize school students with the results of our work.

We worked on the project for 4 months.

The main stages of our work:

• Gathering the necessary information: using the Internet, books, publications on this topic. (2 weeks)
• Sorting information by topic: systematizing and determining the order of writing the work. The work took 2 weeks.
• Preparation of text work: writing the text, partial preparation of systematized information. It took one month.
• Creation of the presentation: compression of systematized information, determination of the structure of the presentation, its creation and design, and took place over the course of a month.
• Learning a fractal creation program and creating your own fractals in the Pascal and Logo programming languages ​​(until today)

Theoretical part of the project

We studied the history of the creation of fractal geometry.

Interest in fractal objects was revived in the mid-70s of the 20th century.

The birth of fractal geometry is usually associated with the publication of Mandelbrot’s book “The Fractal Geometry of Nature” in 1977. His works used the scientific results of other scientists who worked in the period 1875-1925 in the same field (Poincaré, Fatou, Julia, Cantor, Hausdorff But only in our time has it been possible to combine their work into a single system.

So what is a fractal?

Fractal - a geometric figure composed of several parts, each of which is similar to the entire figure.

A small part of a fractal contains information about the entire fractal. Today, the word “fractal” most often means a graphic representation of a structure that is similar to itself on a larger scale.

Fractals are divided into geometric, geometric and stochastic.

Geometric fractals are also called classical. They are the most visual, since they have so-called rigid self-similarity, which does not change when the scale changes. This means that no matter how close you zoom in on the fractal, you still see the same pattern.

Let us give the most famous examples of geometric fractals.

Snowflake Koch.

Invented in 1904 by German mathematician Helge von Koch.

To construct it, a single segment is taken, divided into three equal parts, and the middle link is replaced by an equilateral triangle without this link. In the next step, we repeat the operation for each of the four resulting segments. As a result of endless repetition of this procedure, a fractal curve is obtained.

Durer's pentagon.

A fractal looks like a bunch of pentagons squeezed together. In fact, it is formed by using a pentagon as an initiator and isosceles triangles, the ratio of the larger side to the smaller side is exactly equal to the so-called golden ratio. These triangles are cut out from the middle of each pentagon, resulting in a figure similar to 5 small pentagons glued to one big.

Sierpinski's napkin.

In 1915, Polish mathematician Waclaw Sierpinski came up with an interesting object.

To construct it, take a solid equilateral triangle. In the first step, an inverted equilateral triangle is removed from the center. The second step removes three inverted triangles from the remaining three triangles, and so on.

Dragon Curve.

Invented by Italian mathematician Giuseppe Peano.

Sierpinski carpet.

A square is taken, divided into nine equal squares, the middle one is thrown away, and the same operation is repeated ad infinitum with the rest.

The second type of fractals is algebraic fractals.

They got their name because they are built on the basis of algebraic formulas. As a result of mathematical processing of this formula, a point of a certain color is displayed on the screen. The result is a strange figure in which straight lines turn into curves and self-similarity effects appear at various scale levels. Almost every point on a computer screen is like a separate fractal.

Examples of the most famous algebraic fractals.

Mandelbrot set.

Mandelbrot sets are the most common among algebraic fractals. It can be found in many scientific journals, book covers, postcards, and computer screen savers. This fractal resembles a carding machine with flaming tree-like and circular areas attached to it.

Lots of Julia.

The Julia set was invented by the French mathematician Gaston Julia. An equally famous algebraic fractal.

Newton Pools.

Stochastic fractals.

Fractals, during the construction of which in an iterative system some parameters change randomly, are called stochastic. The term "stochasticity" comes from the Greek word meaning "assumption".

In this case, the resulting objects are very similar to natural ones - asymmetrical trees, rugged coastlines, etc. Two-dimensional stochastic fractals are used in modeling terrain and sea surfaces.

These fractals are used in modeling terrain and sea surfaces, and the electrolysis process. This group of fractals has become widespread thanks to the work of Michael Barnsley from the Georgia Institute of Technology.
A typical representative of this class of fractals is "Plasma".

The most understandable for us are the so-called natural fractals.

“The Great Book of Nature is written in the language of geometry” (Galileo Galilei).

Natural fractals.

• In wildlife:
• Starfish and urchins
• Flowers and plants (broccoli, cabbage)
• Tree crowns and plant leaves
• Fruit (pineapple)
• Circulatory system and bronchi of humans and animals
• In inanimate nature:
• Borders of geographical objects (countries, regions, cities)
• Frosty patterns on window glass
• Stalactites, stalagmites, helictites.

Almost all natural formations: tree crowns, clouds, mountains, coastlines have a fractal structure.
What does it mean?

If you look at a fractal object as a whole, then at a part of it on an enlarged scale, then at a part of this part, it is not difficult to see that they look the same.

Marine fractals.

An octopus is a bottom-dwelling sea animal from the order of cephalopods.

Its bodies and suckers on all eight tentacles of this animal have a fractal structure.

Another typical representative of the fractal underwater world is coral.

There are over 3,500 species of corals known in nature.

Green fractal – fern leaves.

Fern leaves have the shape of a fractal figure - they are self-similar.

Onion is a fractal that makes you cry. Of course, it is a simple fractal: ordinary circles of different diameters, one might even say a primitive fractal.

A striking example of a fractal in nature is “Romanescu", also known as "Romanesque broccoli" or "coral cauliflower".

Cauliflower- typical fractal.

Let's look at the structure of cauliflower.

If you cut one of the flowers, it is obvious that the same cauliflower remains in your hands, only smaller in size. We can keep cutting again and again, even under a microscope - but all we get are tiny copies of the cauliflower.

Matryoshka - souvenir toy- a typical fractal. The principle of fractality is obvious when all the figures of a wooden toy are lined up in a row and not nested inside each other.

Man is a fractal.

A child is born and grows, and this process is accompanied by the principle of “self-similarity”, fractality.

The scope of fractals is wide.

Fractals in literature

Among literary works there are those that have a textual, structural or fractal nature. In literary fractals, elements of the text are endlessly repeated:

The priest had a dog
he loved her.
She ate a piece of meat
he killed her.
Buried in the ground
Caption wrote:
The priest had a dog...

“Here is the house.
Which Jack built.
And here is the wheat.

In the house,
Which Jack Built
And here is a cheerful tit bird,
Which cleverly steals wheat,
Which is stored in a dark closet
In the house,
Which Jack built..." .

Fractals in telecommunications.

To transmit data over distances, antennas with fractal shapes are used, which greatly reduces their size and weight.

Fractals in medicine.

Currently, fractals are widely used in medicine. The human body itself consists of many fractal structures: the circulatory system, muscles, bronchi, bronchial tracts in the lungs, arteries.

The theory of fractals is used to analyze electrocardiograms.

Assessing the magnitude and rhythms of the fractal dimension allows one to judge at an earlier stage and with greater accuracy and information about disturbances of homeostasis and the development of specific heart diseases.

X-ray images processed using fractal algorithms provide a higher-quality picture, and, accordingly, better diagnostics!!

Another area of ​​active use of fractals is gastroenterology.

A new research method in medicine, electrogastroenterography is a research method that allows you to evaluate the bioelectrical activity of the stomach, duodenum and other parts of the gastrointestinal tract.

Fractals in architecture.

The fractal principle of the development of natural and geometric objects penetrates deep into architecture both as an image of the external solution of the object, and as an internal principle of architectural form formation.

Designers from all over the world started use in your work remarkable fractal structures, only recently described by prominent mathematicians.

The use of fractals has brought almost all areas of modern design to a new level.

The introduction of fractal structures has increased both the visual and functional components of the design in many cases.

Designer Takeshi Miyakawa dreamed of becoming a mathematician as a child.

How else can we explain this piece of furniture: the Fractal 23 bedside table contains 23 drawers of various sizes and proportions, which somehow manage to coexist with each other inside the cubic body, filling almost all the space available to them.

Fractals in economics.

Recently, fractals have become popular among economists for analyzing stock exchange rates, currency and trading markets.
Fractals appear on the market quite often.

Fractals in games.

Today, many games (perhaps the most striking example of Minecraft), where various kinds of natural landscapes are present, use fractal algorithms in one way or another. A large number of programs have been created for generating landscapes and landscapes based on fractal algorithms.

Fractals in cinema.

In cinema, a fractal algorithm is used to create various fantastic landscapes. Fractal geometry allows special effects artists to easily create objects such as clouds, smoke, flames, starry skies, etc. What then can we say about fractal animation, it is truly an amazing sight.

Electonic music.

The spectacle of fractal animation is successfully used by VJs. Such video installations are especially often used at concerts of electronic music performers.

Natural Sciences.

Fractals are often used in geology and geophysics. It is no secret that the coasts of islands and continents have a certain fractal dimension, knowing which one can very accurately calculate the lengths of the coasts.

The study of fault tectonics and seismicity is sometimes also studied using fractal algorithms.

Geophysics uses fractals and fractal analysis to study magnetic field anomalies, to study the propagation of waves and oscillations in elastic media, to study climate and many other things.

Fractals in physics.

In physics, fractals are used very widely. In solid state physics, fractal algorithms make it possible to accurately describe and predict the properties of solid, porous, spongy bodies, and aerogels. This helps in the creation of new materials with unusual and useful properties.
An example of a solid is crystals.

The study of turbulence in flows adapts very well to fractals.

The transition to a fractal representation makes the work of engineers and physicists easier, allowing them to better understand the dynamics of complex systems.
Using fractals you can also simulate flames.

Fractals in biology.

In biology, they are used to model populations and to describe internal organ systems (the blood vessel system). After the creation of the Koch curve, it was proposed to use it when calculating the length of the coastline.

Fractals for housewives.

It is easy to transfer the theory of fractals to the home, including the kitchen.

The result of application can be anything: fractal earrings, fractal tasty liver and much more. You only need to use knowledge and ingenuity!

Fractal graphics are widely used in the modern world. The paintings are popular - the result of fractal graphics.

And this is no coincidence. Admire the beauty of fractal graphics!

Practical part of the project

• Created a scientific work “Journey to the World of Fractals”
• We studied programs for creating fractals in the Pascal and Logo programming languages.
• Created your own fractals.
• We made our own “Sierpinski Napkin” and “Sierpinski Carpet”
• Made “Fractal earrings”
• Created a series of paintings “Miracles of Fractal Graphics”
• Published the work “Journey to the World of Fractals” on the Internet.
• We took part with the work “Journey to the World of Fractals” in the VII All-Russian Olympiad for schoolchildren and students “Science 2.0” in the academic subject “Mathematics”. We took first place.
• We took part in the All-Russian competition “Great Discoveries and Inventions” with the work “Journey to the World of Fractals”. We took first place.
• We took part with the work “Journey to the World of Fractals” in the VIII All-Russian Olympiad for schoolchildren and students “I am a researcher” in the academic subject of Mathematics. We took first place.
• Created a presentation “The Amazing World of Fractals”
• Created brochures “Using Fractals” and “Fractals Around Us”
• We held the festival “The Amazing World of Fractals” for students in grades 8-11.”

So, we can say with complete confidence about the enormous practical application of fractals and fractal algorithms today.

The range of areas where fractals are used is very extensive and diverse.

And for sure, in the near future, fractals, fractal geometry, will become close and understandable to each of us. We cannot live without them in our lives!

Let's hope that the emergence of fractal geometry is evidence of the ongoing evolution of man and the expansion of his ways of knowing and understanding the world. Perhaps our children will also easily and meaningfully operate with the concepts of fractals and nonlinear dynamics, as we operate with the concepts of classical physics and Euclidean geometry.

Results of the project

• We got acquainted with the history of the emergence and development of fractal geometry;
• We studied the types of fractals and their application in the modern world.
• Created our own fractals in the Pascal and Logo programming languages
• Created a scientific paper on fractals.
• Created brochures “Fractals around us” and “Use of fractals”
• We held the festival “The Amazing World of Fractals” for students in grades 8-11.

Nature is a perfect creation, scientists are convinced, who discover the proportions of the golden section in the structure of the human body, and fractal figures in the head of a cauliflower.

“The study and observation of nature gave birth to science,” wrote Cicero in the first century BC. In later times, with the development of science and its distance from the study of nature, scientists are surprised to discover what was known to our ancestors, but was not confirmed by scientific methods.

It is interesting to find similar formations in the micro- and macrocosm; it can also be inspiring that science can describe the geometry of these formations. The circulatory system, a river, lightning, tree branches... all of these are similar systems, consisting of different particles and different in scale.

Proportions of the “golden ratio”

Even the ancient Greeks, and possibly the Egyptians, knew the proportion of the “golden section”. Luca Pacioli, a Renaissance mathematician, called this ratio the “divine proportion.” Later, scientists discovered that the golden ratio, which is so pleasing to the human eye and which is often found in classical architecture, art and even poetry, can be found everywhere in nature.

The golden ratio is a division of a segment into two unequal parts, in which the short part is related to the long one as the long part is to the entire segment. The ratio of the long part to the entire segment is an infinite number, an irrational fraction 0.618..., the ratio of the short part is 0.382...

If you build a rectangle with sides whose ratio is equal to the proportion of the “golden ratio”, and inscribe another “golden rectangle” into it, another one into that one, and so on ad infinitum inwards and outwards, then a spiral can be drawn along the corner points of the rectangles. It is interesting that such a spiral will coincide with a cut of a nautilus shell, as well as other spirals found in nature.

Illustration: Homk/wikipedia.org

Nautilus fossil.
Photo: Studio-Annika/Photos.com

Nautilus shell.
Photo: Chris 73/en.wikipedia.org

The proportion of the golden ratio is perceived by the human eye as beautiful and harmonious. And the proportion 0.618... is equal to the ratio of the previous to the next number in the Fibonacci series. Fibonacci numbers appear everywhere in nature: this is the spiral along which the branches of a plant adjoin the stem, the spiral along which the scales on a pine cone grow or the grains on a sunflower. Interestingly, the number of rows spinning counterclockwise and clockwise are adjacent numbers in the Fibonacci series.

The head of a broccoli cabbage and a ram's horn twists in a spiral... And in the human body itself, of course, healthy and of normal proportions, the golden ratio ratios are found.

Vitruvian Man. Drawing by Leonardo da Vinci.

1, 1, 2, 3, 5, 8, 13, 21, ... are numbers in the Fibonacci series, in which each subsequent term is obtained from the sum of the previous two. Distant spiral galaxies photographed by satellites also spin in Fibonacci spirals.

Spiral galaxy.
Photo: NASA

Three tropical cyclones.
Photo: NASA

The DNA molecule is twisted in a double helix.

Twisted human DNA.
Illustration: Zephyris/en.wikipedia.org

The hurricane twists in a spiral, the spider weaves its web in a spiral.

Web of a cross spider.
Photo: Vincent de Groot/videgro.net

The “golden proportion” can also be seen in the body structure of the butterfly, in relation to the thoracic and abdominal parts of its body, as well as in the dragonfly. And most eggs fit, if not into the rectangle of the golden ratio, then into a derivative of it.

Illustration: Adolphe Millot

Fractals

Other interesting shapes that we can see everywhere in nature are fractals. Fractals are figures made up of parts, each of which is similar to the whole figure - doesn't this remind you of the principle of the golden ratio?

Trees, lightning, bronchi and the human circulatory system have a fractal shape; ferns and broccoli are also called ideal natural illustrations of fractals. “Everything is so complicated, everything is so simple” is how nature works, people notice, listening to it with respect.

“Nature has endowed man with the desire to discover the truth,” wrote Cicero, with whose words I would like to end the first part of the article on geometry in nature.

Broccoli is a perfect natural illustration of a fractal.
Photo: pdphoto.org

Fern leaves have the shape of a fractal figure - they are self-similar.
Photo: Stockbyte/Photos.com

Green fractals: fern leaves.
Photo: John Foxx/Photos.com

Veins on a yellowed leaf, shaped like a fractal.
Photo: Diego Barucco/Photos.com

Cracks on a stone: fractal in macro.
Photo: Bob Beale/Photos.com

Branches of the circulatory system on the ears of a rabbit.
Photo: Lusoimages/Photos.com

Lightning strike - fractal branch.
Photo: John R. Southern/flickr.com

Branch of arteries in the human body.

Winding river and its branches.
Photo: Jupiterimages/Photos.com

Ice frozen on glass has a self-similar pattern.
Photo: Schnobby/en.wikipedia.org

An ivy leaf with branching veins - fractal in shape.
Photo: Wojciech Plonka/Photos.com