Fractals simple explanation. Mysterious disorder: the history of fractals and their areas of application

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

This concept does not have a strict definition. Therefore, the word "fractal" is not a mathematical term. This is usually the name given to a geometric figure that satisfies one or more of the following properties:

• has a complex structure at any magnification;
• is (approximately) self-similar;
• has a fractional Hausdorff (fractal) dimension, which is larger than the topological one;
• can be constructed by recursive procedures.

At the turn of the 19th and 20th centuries, the study of fractals was more episodic than systematic, because previously mathematicians mainly studied “good” objects that could be studied using general methods and theories. In 1872, the German mathematician Karl Weierstrass constructed an example continuous function, which is nowhere differentiable. However, its construction was entirely abstract and difficult to understand. Therefore, in 1904, the Swede Helge von Koch came up with a continuous curve that has no tangent anywhere, and is quite easy to draw. It turned out that it has the properties of a fractal. One variant of this curve is called the “Koch snowflake”.

The idea of ​​self-similarity of figures was picked up by the Frenchman Paul Pierre Levy, the future mentor of Benoit Mandelbrot. In 1938, his article “Plane and spatial curves and surfaces consisting of parts similar to the whole” was published, which described another fractal - Lévy C-curve. All of these fractals listed above can be conditionally classified as one class of constructive (geometric) fractals.

Another class is dynamic (algebraic) fractals, which include the Mandelbrot set. The first research in this direction dates back to the beginning of the 20th century and is associated with the names of the French mathematicians Gaston Julia and Pierre Fatou. In 1918, Julia published an almost two-hundred-page work devoted to iterations of complex rational functions, which describes Julia sets, a whole family of fractals closely related to the Mandelbrot set. This work was awarded a prize by the French Academy, but it did not contain a single illustration, so it was impossible to appreciate the beauty of the open objects. Despite the fact that this work made Julia famous among mathematicians of that time, it was quickly forgotten.

Attention again to the work of Julia and Fatou turned only half a century later, with the advent of computers: it was they who made visible the richness and beauty of the world of fractals. After all, Fatou could never look at the images that we now know as images of the Mandelbrot set, because required quantity calculations cannot be done manually. The first person to use a computer for this was Benoit Mandelbrot.

In 1982, Mandelbrot’s book “Fractal Geometry of Nature” was published, in which the author collected and systematized almost all the information about fractals available at that time and presented it in an easy and accessible manner. Mandelbrot placed the main emphasis in his presentation not on heavy formulas and mathematical constructions, but on the geometric intuition of readers. Thanks to illustrations obtained using a computer and historical stories, with which the author skillfully diluted the scientific component of the monograph, the book became a bestseller, and fractals became known to the general public. Their success among non-mathematicians is largely due to the fact that with the help of very simple constructions and formulas that even a high school student can understand, images of amazing complexity and beauty are obtained. When personal computers became powerful enough, even a whole direction in art appeared - fractal painting, and almost any computer owner could do it. Now on the Internet you can easily find many sites devoted to this topic.

Fractal

Fractal (lat. fractus- crushed, broken, broken) is a 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 mathematics, 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. Fractasm is an independent exact science of studying and composing fractals.

In other words, fractals are geometric objects with a fractional dimension. For example, the dimension of a line is 1, the area is 2, and the volume is 3. For a fractal, the dimension value can be between 1 and 2 or between 2 and 3. For example, the fractal dimension of a crumpled paper ball is approximately 2.5. In mathematics, there is a special complex formula for calculating the dimension of fractals. The branches of tracheal tubes, leaves on trees, veins in the hand, a river - these are fractals. In simple terms, a fractal is a geometric figure, a certain part of which is repeated again and again, changing in size - this is the principle of self-similarity. Fractals are similar to themselves, they are similar to themselves at all levels (i.e. at any scale). There are many different types of fractals. In principle, it can be argued that everything that exists in the real world is a fractal, be it a cloud or an oxygen molecule.

The word “chaos” makes one think of something unpredictable, but in fact, chaos is quite orderly and obeys certain laws. The goal of studying chaos and fractals is to predict patterns that, at first glance, may seem unpredictable and completely chaotic.

The pioneer in this field of knowledge was the French-American mathematician, Professor Benoit B. Mandelbrot. In the mid-1960s, he developed fractal geometry, the purpose of which was to analyze broken, wrinkled and fuzzy shapes. The Mandelbrot set (shown in the figure) is the first association that arises in a person when he hears the word “fractal”. By the way, Mandelbrot determined that the fractal dimension of the English coastline is 1.25.

Fractals are increasingly used in science. They describe the real world even better than traditional physics or mathematics. Brownian motion is, for example, the random and chaotic movement of dust particles suspended in water. This type of movement is perhaps the aspect of fractal geometry that has the most practical use. Random Brownian motion has a frequency response that can be used to predict phenomena involving large amounts of data and statistics. For example, Mandelbrot predicted changes in wool prices using Brownian motion.

The word "fractal" can be used not only as a mathematical term. In the press and popular science literature, a fractal can be called a figure that has any of the following properties:

It has a non-trivial structure at 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.

Is self-similar or approximately self-similar.

It has a fractional metric dimension or a metric dimension that exceeds the topological one.

The most useful use of fractals in computer technology is fractal data compression. At the same time, images are compressed much better than is done with conventional methods - up to 600:1. Another advantage of fractal compression is that when enlarged, there is no pixelation effect, which dramatically worsens the image. Moreover, a fractally compressed image often looks even better after enlargement than before. Computer scientists also know that fractals of infinite complexity and beauty can be generated by simple formulas. The film industry widely uses fractal graphics technology to create realistic landscape elements (clouds, rocks and shadows).

The study of turbulence in flows adapts very well to fractals. This allows us to better understand the dynamics of complex flows. Using fractals you can also simulate flames. Porous materials are well represented in fractal form due to the fact that they have a very complex geometry. To transmit data over distances, antennas with fractal shapes are used, which greatly reduces their size and weight. Fractals are used to describe the curvature of surfaces. An uneven surface is characterized by a combination of two different fractals.

Many objects in nature have fractal properties, for example, coasts, clouds, tree crowns, snowflakes, 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.

The first examples of self-similar sets with unusual properties appeared in the 19th century (for example, the Bolzano function, the Weierstrass function, the Cantor set). The term "fractal" was coined by Benoit Mandelbrot in 1975 and gained widespread popularity with the publication of his book "The Fractal Geometry of Nature" in 1977.

The picture on the left shows a simple example of the Darer Pentagon fractal, which looks like a bunch of pentagons squashed together. In fact, it is formed by using a pentagon as an initiator and isosceles triangles, in which the ratio of the larger side to the smaller one is exactly equal to the so-called golden ratio (1.618033989 or 1/(2cos72°)) as a generator. These triangles are cut from the middle of each pentagon, resulting in a shape that looks like 5 small pentagons glued to one large one.

Chaos theory says that complex nonlinear systems are hereditarily unpredictable, but at the same time claims that the way to express such unpredictable systems turns out to be correct not in exact equalities, but in representations of the behavior of the system - in graphs of strange attractors, which have the form of fractals. Thus, chaos theory, which many people think of as unpredictability, turns out to be the science of predictability even in the most unstable systems. The study of dynamic systems shows that simple equations can give rise to chaotic behavior in which the system never returns to a stable state and no pattern appears. Often such systems behave quite normally up to a certain value of a key parameter, then experience a transition in which there are two possibilities for further development, then four, and finally a chaotic set of possibilities.

Schemes of processes occurring in technical objects have a clearly defined fractal structure. The structure of a minimal technical system (TS) implies the occurrence within the TS of two types of processes - the main one and the supporting ones, and this division is conditional and relative. Any process can be the main one in relation to the supporting processes, and any of the supporting processes can be considered the main one in relation to “its” supporting processes. The circles in the diagram indicate physical effects that ensure the occurrence of those processes for which it is not necessary to specially create “your own” vehicles. These processes are the result of interactions between substances, fields, substances and fields. To be precise, a physical effect is a vehicle whose operating principle we cannot influence, and we do not want or do not have the opportunity to interfere with its design.

The flow of the main process shown in the diagram is ensured by the existence of three supporting processes, which are the main ones for the TS that generate them. To be fair, we note that for the functioning of even a minimal TS, three processes are clearly not enough, i.e. The scheme is very, very exaggerated.

Everything is far from being as simple as shown in the diagram. Useful ( necessary for a person) the process cannot be performed with 100% efficiency. The dissipated energy is spent on creating harmful processes - heating, vibration, etc. As a result, harmful ones arise in parallel with the beneficial process. It is not always possible to replace a “bad” process with a “good” one, so it is necessary to organize new processes aimed at compensating for the consequences harmful to the system. A typical example is the need to combat friction, which forces one to organize ingenious lubrication schemes, use expensive anti-friction materials, or spend time on lubrication of components and parts or its periodic replacement.

Due to the inevitable influence of a changeable Environment, a useful process may need to be managed. Control can be carried out either using automatic devices or directly by a person. The process diagram is actually a set of special commands, i.e. algorithm. The essence (description) of each command is the totality of a single useful process, harmful processes accompanying it, and a set of necessary control processes. In such an algorithm, the set of supporting processes is a regular subroutine - and here we also discover a fractal. Created a quarter of a century ago, R. Koller's method makes it possible to create systems with a fairly limited set of only 12 pairs of functions (processes).

Self-similar sets with unusual properties in mathematics

Starting from late XIX century, examples of self-similar objects with properties that are pathological from the point of view of classical analysis appear in mathematics. These include the following:

The Cantor set is a nowhere dense uncountable perfect set. By modifying the procedure, one can also obtain a nowhere dense set of positive length.

the Sierpinski triangle (“tablecloth”) and the Sierpinski carpet are analogues of the Cantor set on the plane.

Menger's sponge is an analogue of the Cantor set in three-dimensional space;

examples of Weierstrass and Van der Waerden of a nowhere differentiable continuous function.

Koch curve is a non-self-intersecting continuous curve of infinite length that does not have a tangent at any point;

Peano curve is a continuous curve passing through all points of the square.

the trajectory of a Brownian particle is also nowhere differentiable with probability 1. Its Hausdorff dimension is two

Recursive procedure for obtaining fractal curves

Construction of the Koch curve

There is a simple recursive procedure for obtaining fractal curves on a plane. Let us define an arbitrary broken line with a finite number of links, called a generator. Next, let’s 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. The figure on the right shows the first four steps of this procedure for the Koch curve.

Examples of such curves are:

dragon Curve,

Koch curve (Koch snowflake),

Lewy Curve,

Minkowski curve,

Hilbert curve,

Broken (curve) of a dragon (Harter-Haithway Fractal),

Peano curve.

Using a similar procedure, the Pythagorean tree is obtained.

Fractals as fixed points of compression mappings

The self-similarity property can be expressed mathematically strictly as follows. Let be contractive mappings of the plane. Consider the following mapping on the set of all compact (closed and bounded) subsets of the plane:

It can be shown that the mapping is a contraction mapping on the set of compacta with the Hausdorff metric. Therefore, by Banach's theorem, this mapping has a unique fixed point. This fixed point will be our fractal.

The recursive procedure for obtaining fractal curves described above is a special case of this construction. All mappings in it are similarity mappings, and - the number of generator links.

For the Sierpinski triangle and the map , , are homotheties with centers at the vertices of a regular triangle and coefficient 1/2. It is easy to see that the Sierpinski triangle transforms into itself when displayed.

In the case where the mappings are similarity transformations with coefficients, the dimension of the fractal (under some additional technical conditions) can be calculated as a solution to the equation. Thus, for the Sierpinski triangle we obtain .

By the same Banach theorem, starting with any compact set and applying iterations of the map to it, we obtain a sequence of compact sets converging (in the sense of the Hausdorff metric) to our fractal.

Fractals in complex dynamics

Julia set

Another Julia set

Fractals arise naturally when studying nonlinear dynamical systems. The most studied case is when a dynamical system is defined by iterations of a polynomial or a holomorphic function of a complex variable on the plane. The first studies in this area date back to the beginning of the 20th century and are associated with the names of Fatou and Julia.

Let F(z) - polynomial, z 0 is a complex number. Consider the following sequence: z 0 , z 1 =F(z 0), z 2 =F(F(z 0)) = F(z 1),z 3 =F(F(F(z 0)))=F(z 2), …

We are interested in the behavior of this sequence as it tends n to infinity. This sequence can:

strive towards infinity,

strive for the ultimate limit

exhibit cyclic behavior in the limit, for example: z 1 , z 2 , z 3 , z 1 , z 2 , z 3 , …

behave chaotically, that is, do not demonstrate any of the three types of behavior mentioned.

Sets of values z 0, for which the sequence exhibits one particular type of behavior, as well as multiple bifurcation points between different types, often have fractal properties.

Thus, the Julia set is the set of bifurcation points for the polynomial F(z)=z 2 +c(or other similar function), that is, those values z 0 for which the behavior of the sequence ( z n) can change dramatically with arbitrarily small changes z 0 .

Another option for obtaining fractal sets is to introduce a parameter into the polynomial F(z) and consideration of the set of those parameter values ​​for which the sequence ( z n) exhibits a certain behavior at a fixed z 0 . Thus, the Mandelbrot set is the set of all , for which ( z n) For F(z)=z 2 +c And z 0 does not go to infinity.

Another famous example Newton's pools are of this kind.

It is popular to create beautiful graphic images based on complex dynamics by coloring plane points depending on the behavior of the corresponding dynamic systems. For example, to complete the Mandelbrot set, you can color the points depending on the speed of aspiration ( z n) to infinity (defined, say, as the smallest number n, at which | z n| will exceed a fixed large value A.

Biomorphs are fractals built on the basis of complex dynamics and reminiscent of living organisms.

Stochastic fractals

Randomized fractal based on Julia set

Natural objects often have a fractal shape. Stochastic (random) fractals can be used to model them. Examples of stochastic fractals:

trajectory of Brownian motion on the plane and in space;

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

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

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.

In nature

Front view of the trachea and bronchi

Bronchial tree

Network of blood vessels

Application

Natural sciences

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).

Radio engineering

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 pasted it onto a piece of paper, then attached it to the receiver. Cohen founded his own company and started their serial production.

Informatics

Image compression

Main article: Fractal compression algorithm

Fractal tree

There are image compression algorithms using fractals. They are based on the idea that instead of the image itself, one can store a compression map for which this image (or something close to it) is a fixed point. One of the variants of this algorithm was used [ source not specified 895 days] by Microsoft when publishing its encyclopedia, but widespread these algorithms were not received.

Computer graphics

Another fractal tree

Fractals are widely used in computer graphics to construct images of natural objects, such as trees, bushes, mountain landscapes, sea surfaces, and so on. There are many programs used to generate fractal images, see Fractal Generator (program).

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.

So, a fractal is a mathematical set consisting of objects similar to this set. In other words, if we look at a small fragment of a fractal figure under magnification, it will look like a larger part of this figure or even the figure as a whole. For a fractal, however, increasing the scale does not mean simplifying the structure. Therefore, at all levels we will see an equally complex picture.

Properties of a fractal

Based on the definition stated above, a fractal is usually represented as a geometric figure that satisfies one or more of the following properties:

It has a complex structure at any magnification;

Approximately self-similar (the parts are similar to the whole);

It has a fractional dimension, which is larger than the topological one;

Can be built recursively.

Fractals in the world around us

Despite the fact that the concept of “fractal” seems extremely abstract, in life you can encounter many real-life and even practical examples of this phenomenon. Moreover, from the surrounding world must certainly be considered, because they will give a better understanding of the fractal and its features.

For example, antennas for various devices, the designs of which are made using the fractal method, show their operating efficiency is 20% greater than antennas of a traditional design. In addition, the fractal antenna can operate with excellent performance at a wide variety of frequencies simultaneously. That is why modern mobile phones practically no longer have external antennas of the classical device in their design - the latter have been replaced with internal fractal ones, which are mounted directly on the phone’s printed circuit board.

Fractals have received much attention with the development of information technology. Currently, compression algorithms have been developed various images using fractals, there are ways to construct objects computer graphics(trees, mountain and sea surfaces) in a fractal way, as well as a fractal system for assigning IP addresses in some networks.

In economics, there is a way to use fractals when analyzing stock and currency quotes. Perhaps the reader who trades on the Forex market has seen fractal analysis in action in a trading terminal or even used it in practice.

Also, in addition to artificially created objects with fractal properties, in natural nature There can also be many similar objects. Good examples of fractals are corals, sea shells, some flowers and plants (broccoli, cauliflower), the circulatory system and bronchi of humans and animals, patterns formed on glass, and natural crystals. These and many other objects have a pronounced fractal shape.

The most brilliant discoveries in science can radically change human life. The invented vaccine can save millions of people; the creation of weapons, on the contrary, takes away these lives. More recently (to scale 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 “inconspicuous” discoveries is fractals. You've probably heard this catchy word before, but do you know what it means and how much interesting information is hidden in this term?

Every person has a natural curiosity, a desire to understand the world around him. And in this endeavor, 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 derive some pattern. The most great minds on the planet are busy with this task. Roughly speaking, scientists are looking for a pattern where there shouldn't be one. Nevertheless, even in chaos it is possible to find connections 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 exceeds the number of answers that adults manage to give. Not long ago, while examining a branch raised from the ground, my daughter suddenly noticed that this branch, with its twigs and branches, itself looked like a tree. And, of course, what followed was the usual question “Why?”, to which 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. Many organic and inorganic forms in nature are formed in a similar way. Clouds, sea shells, a snail’s “house,” 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: father of fractal geometry

The word “fractal” itself appeared thanks to the brilliant scientist Benoit B. Mandelbrot.

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

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 in research center IBM company. At that time, the center's employees were working on transmitting data over a distance. During the research, scientists were faced with the problem of large losses arising from noise interference. Benoit faced a difficult and very important task— understand how to predict the occurrence of noise interference in electronic circuits when statistical method turns out to be ineffective.

While looking through the results of noise measurements, Mandelbrot noticed one strange pattern - the noise graphs at different scales looked the same. An identical pattern was observed regardless of whether it was a noise graph for one day, a week, or an hour. It was necessary to change the scale of the graph, and the picture was repeated every time.

During his lifetime, Benoit Mandelbrot repeatedly said that he did not study 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, awareness of the essence of fractals comes precisely when you begin to study the drawings and think about the meaning of strange swirl patterns.

A fractal pattern does not have identical elements, but is similar on any scale. Construct such an image with high degree manual detailing was previously simply impossible; it required a huge amount of calculations. For example, the 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, they were mentioned in the works of Leibniz and Georg Cantor.

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

Gaston Julia (always wearing a mask - injury from World War I)

This French mathematician wondered what a set would look like if it were built from a simple formula iterated through a loop feedback. If we explain it “on our 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 number of calculations - hundreds, thousands, millions. It was simply impossible to do this manually. But when powerful computing devices became available to 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 a classical fractal. After processing a sequence consisting of a large number of values, Benoit plotted the results on a graph. That's what he got.

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

As it says ancient saying, attributed to Heraclitus of Ephesus, “You cannot step into the same river twice.” It is perfectly suited for interpreting the geometry of fractals. No matter how detailed we look at a fractal image, we will always see a similar pattern.

Those wishing to see what an image of Mandelbrot space would look like when zoomed in many times can do so by downloading the animated GIF.

⇡ Lauren Carpenter: art created by nature

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

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

In 1977, he created presentations with prototype flying models. Loren's responsibilities included developing images of the aircraft being designed. He had to create pictures of new models, showing future aircraft from different angles. At some point, the future founder of Pixar Animation Studios came up with the creative idea of ​​using an image of mountains as a background. Today, any schoolchild can solve such a problem, but in the late seventies of the last century, computers could not cope with such a task. 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, Chance and Dimension in a store. What caught his attention in this book was that Benoit gave a lot of examples of fractal shapes in real life and argued that they can be described by a mathematical expression.

This analogy was not chosen by the mathematician by chance. The fact is that as soon as he published his research, he had to face a whole barrage of criticism. The main thing that his colleagues reproached him for was the uselessness of the theory being developed. “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 “devilish machines”, which in the late seventies seemed to many to be something too complex and unexplored to be completely trusted. Mandelbrot tried to find obvious applications for fractal theory, but in the grand scheme of things he didn't need to. Over the next 25 years, the followers of Benoit Mandelbrot proved the enormous benefits of such a “mathematical curiosity,” and Lauren Carpenter was one of the first to try the fractal method in practice.

After studying 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, he used formulas to paint a completely recognizable mountain landscape.

The principle Lauren used to achieve her goal was very simple. It consisted of 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 split them into four smaller ones and then repeated this process over and over again until he had a realistic mountain landscape. Thus, he managed to become the first artist to use a fractal algorithm for constructing images in computer graphics. As soon as word of the work became known, enthusiasts around the world took up the idea and began using the fractal algorithm to imitate realistic natural shapes.

One of the first 3D visualizations using a fractal algorithm

Just a few years later, Lauren Carpenter was able to apply his developments to much more large-scale project. The animator created a two-minute demo of Vol Libre from them, 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 with a clock speed of five megahertz, and each frame took about half an hour to render.

Working for Lucasfilm Limited, the animator created 3D landscapes using the same scheme for the second full-length film 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 a similar principle for generating natural objects. Terragen, Bryce, Vue and other 3D editors rely on a fractal algorithm for modeling surfaces and textures.

⇡ Fractal antennas: less is more

Over the past half century, life has rapidly begun to change. Most of us accept achievements modern technologies for granted. You get used to everything that makes life more comfortable very quickly. Rarely does anyone ask the questions “Where did this come from?” and “How does it work?” A microwave heats up breakfast - great, a smartphone gives you the opportunity to talk to another person - great. This seems like an obvious possibility to us.

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

Fractal patterns fascinate with their patterns. They definitely resemble the images space objects- nebulae, galaxy clusters, and so on. It is therefore quite natural that when Mandelbrot voiced his theory of fractals, his research aroused increased interest among those who studied astronomy. One of these amateurs named Nathan Cohen, after attending a lecture by Benoit Mandelbrot in Budapest, was inspired by the idea of ​​​​practical application of the acquired knowledge. True, he did this 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 the property in downtown Boston that Nathan rented was categorically against installing large devices on the roof. Then Nathan began experimenting with various forms antennas, trying to get maximum result with minimal dimensions. Inspired by the idea of ​​fractal forms, Cohen, as they say, randomly made one of the most famous fractals from wire - the “Koch snowflake”. Swedish mathematician Helge von Koch came up with this curve back in 1904. It is obtained by dividing a segment into three parts and replacing the middle segment with an equilateral triangle without a side coinciding with this segment. The definition is a little difficult to understand, but in the figure everything is clear and simple.

There are also other variations of the Koch curve, but approximate form 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 high efficiency and covers a much wider frequency range compared to classic solutions. In addition, the shape of the antenna in the form of a fractal curve makes it possible to significantly reduce the geometric dimensions. Nathan Cohen even came up with 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 company 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.

In principle, this is what happened. True, to this day Nathan is engaged in a legal battle with large corporations that are illegally using his discovery to produce compact communication devices. Some famous manufacturers mobile devices, such as Motorola, have already reached a peace agreement with the inventor of the fractal antenna.

⇡ Fractal dimensions: you can’t understand it with your mind

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

The last one, like many others famous mathematicians, loved to communicate with children, asking them questions and receiving unexpected answers. Sometimes this led to surprising consequences. For example, the nine-year-old nephew of Edward Kasner came up with the now well-known word “googol,” meaning one followed by one hundred zeros. But let's return to fractals. The American mathematician liked to ask the question how long is the US coastline. After listening to the opinion of his interlocutor, Edward himself spoke the correct answer. If you measure the length on a map using broken segments, the result will be inaccurate, because the coastline has a large number of irregularities. What happens if we measure as accurately as possible? You will have to take into account the length of every unevenness - you will need to measure every cape, every 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 when measuring each new irregularity.

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

Interestingly, following Edward's prompts, the children spoke much faster than the adults. the right decision, while the latter had trouble accepting such an incredible answer.

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

What a regular dimension is is clear to anyone. If the dimension is equal to one, we get a straight line, if two - flat figure, three - volume. However, this understanding of dimension in mathematics does not work with fractal curves, where this parameter has a fractional value. Fractal dimension in mathematics can be conventionally considered as a “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 to apply the theory of fractals. Using fractals, you can analyze fluctuations in stock exchange prices, study all sorts 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, such as image compression. And by the way, to get a beautiful fractal on your computer screen, you don’t have to have a doctorate.

⇡ Fractal in the browser

Perhaps one of the easiest ways to get a fractal pattern is to use an online vector editor from the young talented programmer Toby Schachman. The tools of this simple graphic editor are based on the same principle of self-similarity.

At your disposal there are only two simplest shapes - a quadrangle and a circle. You can add them to the canvas, scale them (to scale along one of the axes, hold down the Shift key) and rotate them. Overlapping according to the principle of Boolean addition operations, these simplest elements form new, less trivial forms. These new shapes can then be added to the project, and the program will repeat generating these images ad infinitum. At any stage of working on a fractal, you can return to any component complex shape and edit its position and geometry. A fun activity, especially when you consider that the only tool you need to create 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 fine tuning of the generated fractal pattern. In cases where it is necessary to construct a mathematically accurate fractal, the cross-platform XaoS editor will come to the rescue. This program makes it possible not only to construct a self-similar image, but also to perform various manipulations with it. For example, in real time you can take a “walk” along a fractal by changing its scale. Animated movement along a fractal can be saved as an XAF file and then reproduced in the program itself.

XaoS can load a random set of parameters, and also 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 very small in size and does not require installation. Secondly, it implements the ability to determine the color palette of a picture. You can choose shades in color models RGB, CMYK, HVS and HSL.

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

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

⇡ Mandelbulb3D: 3D fractal editor

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

You can also eat this fractal

The Mandelbulb3D program can create three-dimensional objects with a similar shape. To obtain a 3D surface using a fractal algorithm, the authors 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 resemble a plant, it may resemble a strange animal, a planet, or something else. This effect is enhanced by an advanced rendering algorithm, which 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 number of settings and rendering options. You can control the shades of light sources, select the background and level of detail of the simulated 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 designs. Incendia has texture and material editors, and the rendering engine allows you to use volumetric fog effects and various shaders. The program implements the option of saving a buffer during long-term rendering, and supports the creation of animation.

Incendia allows you to export a fractal model to popular 3D graphics formats - OBJ and STL. Incendia includes a small utility called Geometrica - special tool to configure the export of a fractal surface to a 3D model. Using this utility, you can determine the resolution of a 3D surface and 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.

IN lately work on the Incendia project has slowed down somewhat. On at the moment the author is looking for sponsors to help him develop the program.

If you don’t have enough imagination to draw a beautiful three-dimensional fractal in this program, it doesn’t matter. Use the parameters library, which is located in the INCENDIA_EX\parameters folder. Using PAR files, you can quickly find the most unusual fractal shapes, including animated ones.

⇡ Aural: how fractals sing

We usually don’t talk about projects that are just underway, but in in this case we must make an exception, this is a very unusual application. The project, called Aural, was invented by the same person who created Incendia. However, this time the program does not visualize the fractal set, but sounds it, turning it into electronic music. The idea is very interesting, especially considering the unusual properties of fractals. Aural is an audio editor that generates melodies using fractal algorithms, that is, in essence, it is an audio synthesizer-sequencer.

The sequence of sounds produced by this program is unusual and... beautiful. It may well be useful for writing modern rhythms and, it seems to us, is especially well suited for creating soundtracks for screensavers 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, you will not need to study fractal theory - you will just need to play with the parameters of the algorithm for generating a sequence of notes. Listen to how fractals sound, and.

Fractals: musical break

In fact, fractals can help you 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 follow the example of a musician named Jonathan Coulton, who, among other things, writes compositions for Popular Science magazine. And unlike other performers, 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 the work to others, as well as its modification (creation of derivative works) so that adapt it to your tasks.

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 perfect shape, which fractals help us discern. Nature is the best architect, ideal builder and engineer. It is structured very logically, and if we don’t see a pattern somewhere, 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 shell-shaped speaker systems, create snowflake-shaped antennas, and so on. We are sure that fractals still hold many secrets, and many of them have yet to be discovered by humans.

Municipal budgetary educational institution

"Siverskaya average secondary school No. 3"

Research work

in mathematics.

Done the job

8th-1st grade student

Emelin Pavel

Scientific supervisor

math teacher

Tupitsyna Natalya Alekseevna

Siversky village

2014

Mathematics is all permeated with beauty and harmony,

You just need to see this beauty.

B. Mandelbrot

Introduction__________________________________________3-4pp.

Chapter 1.history of the emergence of fractals._______5-6pp.

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_______________13-15pp.

Chapter 5 Practical work__________________16-24pp.

Conclusion_________________________________25.page

List of references and Internet resources________26 pages.

Introduction

Mathematics,

if you look at it correctly,

reflects not only the truth,

but also incomparable beauty.

Bertrand Russell

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

Mathematics – ancient science. It seemed to most people that geometry in nature was limited to such simple figures, like line, circle, polygon, sphere, etc. As it turns out, many natural systems are so complex that using only familiar objects of ordinary geometry to model them seems hopeless. How, for example, can you build a model of a mountain range or a 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 complexity of the circulatory system, consisting of many capillaries and vessels and delivering blood to every cell human body? Imagine the structure of the lungs and kidneys, reminiscent in structure of trees with a branched crown?

Fractals are suitable tools for exploring these questions. Often what we see in nature intrigues us with the endless repetition of the same pattern, increased or decreased by several times. For example, a tree has branches. On these branches there are smaller branches, etc. Theoretically, the branching element is repeated infinitely many times, becoming smaller and smaller. The same thing can be seen when looking at a photograph of mountainous terrain. Try to zoom in a little closer to the mountain range --- you will see the mountains again. This is how the property of self-similarity characteristic of fractals manifests itself.

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

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

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

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

Research methods: comparative analysis, synthesis, modeling.

Tasks:

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

G. Koch, V. Sierpinsky and others;

acquaintance with various types of fractal sets;

studying popular science literature on this issue, acquaintance with

scientific hypotheses;

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

studying the use of fractals in other sciences and in practice;

conducting an experiment to create your own fractal images.

The fundamental question of the work:

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

Subject of research: Fractal geometry.

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

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

Research methods: analytical, search.

Relevance The stated 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 begin work on creating my own fractals.

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

Chapter 1. History

B whena Mandelbrot

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

Fractal (lat. fractus - crushed, broken, broken) is a term meaning 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.

For mathematical objects, to which it belongs, are characterized by extremely interesting properties. In ordinary geometry, a line has one dimension, a surface has two dimensions, and a spatial figure has three dimensions. Fractals are not lines or surfaces, but, if you can imagine it, something in between. As the size increases, the volume of the fractal also increases, but its dimension (exponent) is not a whole value, but a fractional one, and therefore the boundary 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 at low magnification scale of the figure itself. This geometric regularity is called scale invariance or self-similarity. This is what determines the fractional dimension of fractal figures.

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

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

Liquid, gas, solid - three familiar physical condition substance that exists in the three-dimensional world. But what is the dimension of a cloud of smoke, a cloud, or more precisely, their boundaries, continuously eroded by turbulent air movement?

Basically, fractals are classified into three groups:

Algebraic fractals

Stochastic fractals

Geometric fractals

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

Chapter 2. Classification of fractals

Geometric fractals

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

This is where the history of fractals began. This type of fractal is obtained through simple geometric constructions. Usually, when constructing these fractals, they do this: they take a “seed” - an axiom - a set of segments on the basis of which the fractal will be built. Next, a set of rules is applied to this “seed”, which transforms it into some kind of geometric figure. Next, the same set of rules is applied again to each part of this figure. With each step the figure will become more and more complex, and if we carry out (at least in our minds) infinite number transformations - we get a geometric fractal.

Fractals of this class are the most visual, because self-similarity is immediately visible in them 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 polyline is replaced with a generator polyline, on the appropriate scale. As a result of endless repetition of this procedure (or, more precisely, when going to the limit), a fractal curve is obtained. Despite the apparent complexity of the resulting curve, its general appearance is determined only by the shape of the generator. Examples of such curves are: Koch curve (Fig. 7), Peano curve (Fig. 8), Minkowski curve.

At the beginning of the twentieth century, mathematicians were looking for curves that do not have a tangent at any point. This meant that the curve abruptly changed its direction, and, moreover, with a colossal high speed(the derivative is equal to infinity). The search for these curves was not just caused by the idle interest of mathematicians. The fact is that at the beginning of the twentieth century there was very rapid development quantum mechanics. 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 this explanation of Brownian motion, scientists were faced with the task of finding a curve that would in the best possible way showed the movement 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 average interval an equilateral triangle without this segment. As a result, a broken line is formed, consisting of four links of length 1/3. In the next step, we repeat the operation for each of the four resulting links, etc...

The limit curve is Koch curve.

Snowflake Koch. By performing similar transformations on the sides 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 constructed quite simply: The square is divided by straight lines parallel to its sides into 9 equal squares. The central square is removed from the square. The result is a set consisting of the 8 remaining “first rank” squares. Doing exactly the same with each of the squares of the first rank, we obtain a set consisting of 64 squares of the second rank. Continuing this process indefinitely, we obtain an infinite sequence or Sierpinski square.

Algebraic fractals

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

They are obtained using nonlinear processes in n-dimensional spaces. It is known that nonlinear dynamic systems have several stable states. The state in which I found myself dynamic system 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, phase space system is divided into areas of attraction attractors. If the phase space is two-dimensional, then by coloring the areas of attraction with different colors, one can obtain color phase portrait this system (iterative process). By changing the color selection algorithm, you can get complex fractal patterns with bizarre multicolor patterns. What came as a surprise to mathematicians was the ability, using primitive algorithms, to generate very complex structures.

As an example, consider the Mandelbrot set. They build it using complex numbers.

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

The Mandelbrot set contains points that, duringinfinite 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 in a finite number of iterations, and 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 French mathematician Gaston Julia, after whom many were 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 some of its parameters are randomly changed in an iterative process. In this case, the resulting objects are very similar to natural ones - asymmetrical trees, rugged coastlines, etc.

A typical representative of this group of fractals is “plasma”.

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

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

Let's say you need 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 a mountain. We need to apply plasma to a regular cone and we will get the relief of a mountain. Such operations can be performed with many other objects in nature; thanks to stochastic fractals, nature itself can be described.

Now let's talk about geometric fractals.

.

Chapter 3 "Fractal geometry of nature"

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

(Benoit Mandelbrot "Fractal geometry of nature" ).

TO The beauty of fractals is twofold: it delights the eye, as evidenced by the worldwide 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 by the same authors, “The Beauty of Fractals.” But there is another, more abstract or sublime, aspect of the beauty of fractals, open, according to R. Feynman, only to the mental gaze of a theorist; in this sense, fractals are beautiful because of the beauty of a difficult mathematical problem. Benoit Mandelbrot pointed out to his contemporaries (and, presumably, his descendants) an annoying gap in Euclid’s Elements, through which, without noticing the omission, almost two millennia of humanity comprehended the geometry of the surrounding world and learned the mathematical rigor of presentation. Of course, both aspects of the beauty of fractals are closely interrelated and do not exclude, but complement each other, although each of them is self-sufficient.

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

As for the correspondence to the real world, fractal geometry describes a very wide class of natural processes and phenomena, and therefore 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 zero, while others turn to infinity.

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

Sea shells

Lightning admire with their beauty. Fractals created by lightning are not arbitrary 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, frosty patterns on the windows these are also fractals

ABOUT
t enlarged image leaf, up to 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 amazingly harmonious laws with mathematical precision. Is it possible after this to think that our planet is a random concatenation of particles? Hardly.

Chapter 4. Application of fractals

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

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

The advantages of fractal image compression algorithms are very small size packed file and short picture recovery time. Fractal packed images can be scaled without the appearance of pixelation (poor image quality - large squares). But the compression process takes a long time and sometimes lasts for hours. The fractal lossy packaging algorithm allows you to set the compression level, similar to the jpeg format. The algorithm is based on searching for large pieces of the image that are similar to some small pieces. And only which piece is similar to which is written to the output file. When compressing, a square grid is usually used (pieces are squares), which leads to a slight angularity when restoring the image; a hexagonal grid does not have this drawback.

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

In mechanics and physics Fractals are used due to their unique property of repeating the outlines of many natural objects. Fractals allow you to approximate trees, mountain surfaces and cracks with higher accuracy than approximations using sets of segments or polygons (with the same amount of stored data). Fractal models, like natural objects, have a “roughness”, and this property is preserved no matter how large the magnification of the model is. The presence of a uniform measure on fractals allows one to apply integration, potential theory, and use them instead of standard objects in already studied equations.

T
Fractal geometry is also used for designing antenna devices. This was first used by the American engineer Nathan Cohen, who then lived in the center of Boston, where the installation of external antennas on buildings was prohibited. Cohen cut out a Koch curve shape from aluminum foil and then glued it onto a piece of paper and 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 such an antenna have not yet been studied, this did not stop Cohen from establishing his own company and launching their serial production. Currently, the American company “Fractal Antenna System” has developed a new type of antenna. You can now stop using mobile phones protruding external antennas. The so-called fractal antenna is located directly on the main board inside the device.

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

Chapter 5. Practical work.

First, let's look at the fractals “Necklace”, “Victory” and “Square”.

First - "Necklace"(Fig. 7). The initiator of this fractal is a circle. 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. Those. the figure can be enlarged by taking just one small arc, or it can be reduced by considering its construction from smaller ones.

rice. 7.

Fractal “Necklace”

The second fractal is "Victory"(Fig. 8). It received this name because it looks like the Latin letter “V”, that is, “victory”. This fractal consists of a certain number of small “vs” that 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, right side is built the same way. Each of these “v” is built in the same way and continues this ad infinitum.

Fig.8. Fractal "Victory"

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

Fig. 9. Fractal “Square”

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

Rice. 11. Fractal “Rose *”

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

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

R
is. 12. Fractal “Pentagram”.

Rice. 13. Fractal “Pentagram *”

Rice. 14 fractal “Black hole”

Experiment No. 1 “Tree”

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

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

WITH The next step is to write the script.

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

Let's call him " fr1". Copy this layer (" fr1") 2 more times.

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

The last step was to copy this image and paste it smaller and rotated. This is the final result.

Conclusion

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

Fractal graphics are not just a set of self-repeating images, it is a model of the structure and principle of any existing thing. Our whole life is represented by fractals. All the nature around us consists of them. It is impossible not to note the widespread use of fractals in computer games, where terrain reliefs are often fractal images based on three-dimensional models complex sets. Fractals greatly facilitate drawing computer graphics; with the help of fractals, many special effects, various fabulous and incredible pictures, etc. are created. Also, trees, clouds, shores and all other nature are drawn using fractal geometry. Fractal graphics are needed everywhere, and the development of “fractal technologies” is one of the important tasks today.

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

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

1. Compressing images and information

2. Hiding information in the image, sound,…

3. Data encryption using fractal algorithms

4. Making fractal music

5. System modeling

Our work does not list all areas of human knowledge where the theory of fractals has found its application. We only want to say that no more than a third of a century has passed since the theory arose, but during this time fractals became a sudden phenomenon for many researchers. bright light in the nights that illuminated hitherto unknown facts and patterns in specific areas of data. With the help of the theory of fractals, they began to explain the evolution of galaxies and the development of cells, the emergence of mountains and the formation of clouds, the movement of prices on the stock exchange and the development of society and family. Perhaps, at first, this passion for fractals was even too intense and attempts to explain everything using the theory of fractals were unjustified. But, without a doubt, this theory has a right to exist, and we regret that recently it has somehow been forgotten and remained the lot of the elite. In preparing this work, it was very interesting for us to find applications of THEORY in PRACTICE. Because very often there is a feeling that theoretical knowledge stands apart from life reality.

Thus, the concept of fractals becomes not only part of “pure” science, but also an element of universal human culture. Fractal science is still very young and has a great future ahead of it. The beauty of fractals is far from being exhausted and will 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. Application of fractals in computer graphics. // Computerworld-Russia.-1995

Mandelbrot B. Self-affine fractal sets, “Fractals in Physics.” M.: Mir 1988

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

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

Peitgen 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