Application of fractal theory. Space Research Laboratory

To present the whole variety of fractals, it is convenient to resort to their generally accepted classification.

2.1 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 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.

Fig 1. Construction of the Koch triad curve.

Let's consider one of these fractal objects - the triadic Koch curve. The construction of the curve begins with a segment of unit length (Fig. 1) - this is the 0th generation of the Koch curve. Next, each link (one segment in the zero generation) is replaced by formative element, designated in Fig. 1 by n=1. As a result of this replacement, the next generation of the Koch curve is obtained. In the 1st generation, this is a curve of four straight links, each length 1/3 . To obtain the 3rd generation, the same actions are performed - each link is replaced with a reduced forming element. So, to obtain each subsequent generation, all links of the previous generation must be replaced with a reduced forming element. Curve n-th generation for any finite n called prefractal. Figure 1 shows five generations of the curve. At n As the Koch curve approaches infinity, it becomes a fractal object.

Figure 2. Construction of the Harter-Haithway "dragon".

To obtain another fractal object, you need to change the construction rules. Let the forming element be two equal segments connected at right angles. In the zeroth generation, we replace the unit segment with this generating element so that the angle is on top. We can say that with such a replacement there is a displacement of the middle of the link. When constructing subsequent generations, the rule is followed: the very first link on the left is replaced with a forming element so that the middle of the link is shifted to the left of the direction of movement, and when replacing subsequent links, the directions of displacement of the middles of the segments must alternate. Figure 2 shows the first few generations and the 11th generation of the curve built according to the principle described above. Limit fractal curve (at n tending to infinity) is called Harter-Haithway's dragon .

In computer graphics, the use of geometric fractals is necessary when obtaining images of trees, bushes, and coastlines. Two-dimensional geometric fractals are used to create three-dimensional textures (patterns on the surface of an object).

2.2 Algebraic fractals

This is the most large group fractals. They are obtained using nonlinear processes in n-dimensional spaces. Two-dimensional processes are the most studied. Interpreting a nonlinear iterative process as a discrete dynamic system, one can use the terminology of the theory of these systems: phase portrait, steady 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 initial states, from which the system will necessarily fall into the final states under consideration. Thus, the phase space of the system is divided into areas of attraction attractors. If 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. A surprise for mathematicians was the ability to generate very complex non-trivial structures using primitive algorithms.

Fig 3. Mandelbrot set.

As an example, consider the Mandelbrot set (see Fig. 3 and Fig. 4). The algorithm for its construction is quite simple and is based on a simple iterative expression:

Z = Z[i] * Z[i] + C,

Where Z i and C- complex variables. Iterations are performed for each starting point C rectangular or square region - a subset of the complex plane. The iterative process continues until Z[i] will not go beyond the circle of radius 2, the center of which lies at the point (0,0), (this means that the attractor of the dynamical system is at infinity), or after a sufficiently large number of iterations (for example, 200-500) Z[i] will converge to some point on the circle. Depending on the number of iterations during which Z[i] remained inside the circle, you can set the color of the point C(If Z[i] remains inside the circle for a sufficiently long time large quantity iterations, the iteration process stops and this raster point is painted black).

Fig. 4. A section of the boundary of the Mandelbrot set, enlarged by 200 times.

The above algorithm gives an approximation to the so-called Mandelbrot set. The Mandelbrot set contains points that, during infinite the number of iterations does not go to infinity (the points 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).

2.3 Stochastic fractals

There are other classifications of fractals, for example, dividing fractals into deterministic (algebraic and geometric) and non-deterministic (stochastic).

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 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 "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 it 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 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 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

Beginning with 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

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.

Julia set

Another Julia set

Fractals arise naturally when studying nonlinear dynamical systems. The most studied case is when a dynamical system is specified 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 of this kind is Newton's pools.

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;

trajectory boundary Brownian motion on surface. 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.

Computer science

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 some close one) 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 these algorithms were not widely used.

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.

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 is derived from the Latin fractus and means consisting of fragments. It was proposed 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.
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. Actually found easy way representations of 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 very simple case a small part of a fractal contains information about the entire fractal. Mandelbrot's definition of a fractal is: "A fractal is a structure consisting of parts that are in some sense similar to the whole."

There are a large number of mathematical objects called fractals (Sierpinski triangle, Koch snowflake, Peano curve, Mandelbrot set and Lorentz attractors). Fractals are described with great accuracy by many physical phenomena and formations of the real world: mountains, clouds, turbulent (vortex) currents, roots, branches and leaves of trees, blood vessels, which is far from corresponding to simple geometric figures. For the first time, Benoit Mandelbrot spoke about the fractal nature of our world in his seminal work “Fractal Geometry of Nature”.
The term fractal was introduced by Benoit Mandelbrot in 1977 in his fundamental work Fractals, Form, Chaos and Dimension. According to Mandelbrot, the word fractal comes from Latin words fractus - fractional and frangere - to break, which reflects the essence of a fractal as a “broken”, irregular set.

Classification of fractals.

In order to present the whole variety of fractals, it is convenient to resort to their generally accepted classification. There are three classes of fractals.

1. Geometric fractals.

Fractals of this class are the most visual. In the two-dimensional case, they are obtained using a 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.

Let's consider an example of one of these fractal objects - the triadic Koch curve.

Construction of the triadic Koch curve.

Let's take a straight segment of length 1. Let's call it seed. Let's divide the seed into three equal parts 1/3 long, discard the middle part and replace it with a broken line of two links 1/3 long.

We will get a broken line consisting of 4 links with a total length of 4/3 - the so-called first generation.

In order to move to the next generation of the Koch curve, it is necessary to discard and replace the middle part of each link. Accordingly, the length of the second generation will be 16/9, the third - 64/27. if we continue this process ad infinitum, the result is a triadic Koch curve.

Let us now consider the properties of the triadic Koch curve and find out why fractals were called “monsters”.

Firstly, this curve has no length - as we have seen, with the number of generations its length tends to infinity.

Secondly, it is impossible to construct a tangent to this curve - each of its points is an inflection point at which the derivative does not exist - this curve is not smooth.

Length and smoothness are the fundamental properties of curves, which are studied both by Euclidean geometry and by the geometry of Lobachevsky and Riemann. Traditional methods of geometric analysis turned out to be inapplicable to the triadic Koch curve, so the Koch curve turned out to be a monster - a “monster” among the smooth inhabitants of traditional geometries.

Construction of the Harter-Haithaway "dragon".

To obtain another fractal object, you need to change the construction rules. Let the forming element be two equal segments connected at right angles. In the zeroth generation, we replace the unit segment with this generating element so that the angle is on top. We can say that with such a replacement there is a displacement of the middle of the link. When constructing subsequent generations, the rule is followed: the very first link on the left is replaced with a forming element so that the middle of the link is shifted to the left of the direction of movement, and when replacing subsequent links, the directions of displacement of the middles of the segments must alternate. The figure shows the first few generations and the 11th generation of the curve built according to the principle described above. A curve with n tending to infinity is called the Harter-Haithaway dragon.
In computer graphics, the use of geometric fractals is necessary when obtaining images of trees and bushes. Two-dimensional geometric fractals are used to create three-dimensional textures (patterns on the surface of an object).

2.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.

Mandelbrot set.

As an example, consider the Mandelbrot set. The algorithm for its construction is quite simple and is based on a simple iterative expression: Z = Z[i] * Z[i] + C, Where Zi And C- complex variables. Iterations are performed for each starting point from a rectangular or square region - a subset of the complex plane. The iterative process continues until Z[i] will not go beyond the circle of radius 2, the center of which lies at the point (0,0), (this means that the attractor of the dynamic system is at infinity), or after a sufficiently large number of iterations (for example, 200-500) Z[i] will converge to some point on the circle. Depending on the number of iterations during which Z[i] remained inside the circle, you can set the color of the point C(If Z[i] remains inside the circle for a sufficiently large number of iterations, the iteration process stops and this raster point is painted black).

3. 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. Two-dimensional stochastic fractals are used in modeling terrain and sea surfaces.
There are other classifications of fractals, for example, dividing fractals into deterministic (algebraic and geometric) and non-deterministic (stochastic).

About the use of fractals

First of all, fractals are a field of amazing mathematical art, when with the help of the simplest formulas and algorithms, pictures of extraordinary beauty and complexity are obtained! Leaves, trees and flowers are often visible in the contours of the constructed images.

Some of the most powerful applications of fractals lie in computer graphics. Firstly, this is fractal compression of images, and secondly, the construction of landscapes, trees, plants and the generation of fractal textures. Modern physics and mechanics are just beginning to study the behavior of fractal objects. And, of course, fractals are used directly in mathematics itself.
The advantages of fractal image compression algorithms are the very small size of the packed file and short image recovery time. Fractal packed images can be scaled without causing pixelation. 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.
The tendency of fractals to resemble mountains, flowers and trees is exploited by some graphic editors, for example, fractal clouds from 3D studio MAX, fractal mountains in World Builder. Fractal trees, mountains and entire landscapes are defined by simple formulas, are easy to program and do not break up into separate triangles and cubes when approached.
One cannot ignore the use of fractals in mathematics itself. In set theory, the Cantor set proves the existence of perfect nowhere dense sets; in measure theory, the self-affine function "Cantor's Ladder" is a good example of a distribution function of a singular measure.
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.
With a fractal approach, chaos ceases to be blue disorder and acquires a fine structure. 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.

About constructing fractals

Successive approximation method

Looking at this picture, it is not difficult to understand how you can build a self-similar fractal (in this case, the Sierpinski pyramid). We need to take a regular pyramid (tetrahedron), then cut out its middle (octahedron), resulting in four small pyramids. With each of them we perform the same operation, etc. This is a somewhat naive but clear explanation.

Let us consider the essence of the method more strictly. Let there be some IFS system, i.e. compression mapping system S=(S 1 ,...,S m ) S i:R n ->R n (for example, for our pyramid the mappings have the form S i (x)=1/2*x+o i , where o i are the vertices of the tetrahedron, i=1,..,4). Then we choose some compact set A 1 in R n (in our case we choose a tetrahedron). And we define by induction the sequence of sets A k:A k+1 =S 1 (A k) U...U S m (A k). It is known that sets A k with increasing k approximate the desired attractor of the system better and better S.

Note that each of these iterations is an attractor recurrent system of iterated functions(English term Digraph IFS, RIFS and also Graph-directed IFS) and therefore they are easy to build using our program.

Point-by-point or probabilistic method

This is the easiest method to implement on a computer. For simplicity, we consider the case of a flat self-affine set. So let (S

) - some system of affine contractions. Display S

representable as: S

Fixed matrix size 2x2 and o

Two-dimensional vector column.

Let us take the fixed point of the first mapping S 1 as starting point:
x:= o1;
Here we take advantage of the fact that all fixed points of compression S 1 ,..,S m belong to the fractal. You can select an arbitrary point as the starting point and the sequence of points generated by it will be drawn to a fractal, but then several extra points will appear on the screen.
Let's mark the current point x=(x 1 ,x 2) on the screen:
putpixel(x 1 ,x 2 ,15);
Let's randomly choose a number j from 1 to m and recalculate the coordinates of point x:
j:=Random(m)+1;
x:=S j (x);
We go to step 2, or, if we have done a sufficiently large number of iterations, we stop.

Note. If the compression ratios of the mappings S i are different, then the fractal will be filled with points unevenly. If the mappings S i are similar, this can be avoided by slightly complicating the algorithm. To do this, at the 3rd step of the algorithm, the number j from 1 to m must be chosen with probabilities p 1 =r 1 s,..,p m =r m s, where r i denote the compression coefficients of the mappings Si, and the number s (called the similarity dimension) is found from the equation r 1 s +...+r m s =1. The solution to this equation can be found, for example, by Newton's method.

About fractals and their algorithms

Fractal comes from the Latin adjective "fractus", and in translation means consisting of fragments, and the corresponding Latin verb "frangere" means to break, that is, to create irregular fragments. 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 term 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).

Adjustments

Let me make some adjustments to the algorithms proposed in the book by H.-O. Peitgen and P.H. Richter “The Beauty of Fractals” M. 1993 purely to eradicate typos and facilitate understanding of the processes since after studying them much remained a mystery to me. Unfortunately, these “understandable” and “simple” algorithms lead a rocking lifestyle.

The construction of fractals is based on a certain nonlinear function of a complex process with feedback z => z 2 +c since z and c are complex numbers, then z = x + iy, c = p + iq it is necessary to decompose it into x and y to go into a plane more realistic for the common man:

x(k+1)=x(k) 2 -y(k) 2 + p,
y(k+1)=2*x(k)*y(k) + q.

A plane consisting of all pairs (x,y) can be considered as if for fixed values p and q, and with dynamic ones. In the first case, by going through all the points (x, y) of the plane according to the law and coloring them depending on the number of repetitions of the function necessary to exit the iterative process or not coloring them (black color) when the permissible maximum of repetitions is exceeded, we will obtain a display of the Julia set. If, on the contrary, we determine the initial pair of values (x,y) and trace its coloristic fate with dynamically changing values of the parameters p and q, then we obtain images called Mandelbrot sets.

On the question of algorithms for coloring fractals.

Usually the body of a set is represented as a black field, although it is obvious that black color can be replaced by any other, but this is also a little interesting result. Getting an image of a set colored in all colors is a task that cannot be solved using cyclic operations because the number of iterations of the sets forming the body is equal to the maximum possible and is always the same. It is possible to color a set in different colors by using the result of checking the loop exit condition (z_magnitude) or something similar to it, but with other mathematical operations, as a color number.

Application of a "fractal microscope"

to demonstrate boundary phenomena.

Attractors are centers leading the struggle for dominance on the plane. A boundary appears between the attractors, representing a florid pattern. By increasing the scale of consideration within the boundaries of the set, one can obtain non-trivial patterns that reflect the state of deterministic chaos - a common phenomenon in the natural world.

The objects studied by geographers form a system with very complexly organized boundaries, and therefore their identification becomes not a simple practical task. Natural complexes have cores of typicality that act as attractors that lose their influence on the territory as it moves away.

Using a fractal microscope for the Mandelbrot and Julia sets, one can form an idea of boundary processes and phenomena that are equally complex regardless of the scale of consideration and thus prepare the specialist’s perception for an encounter with a dynamic and seemingly chaotic natural object in space and time, for an understanding of fractal geometry nature. The multicolored colors and fractal music will definitely leave a deep imprint in the minds of students.

Thousands of publications and vast Internet resources are devoted to fractals, but for many specialists far from computer science, this term seems completely new. Fractals, as objects of interest to specialists in various fields of knowledge, should receive a proper place in computer science courses.

Examples

SIEPINSKI GRID

This is one of the fractals that Mandelbrot experimented with when developing the concepts of fractal dimensions and iterations. Triangles formed by connecting the midpoints of a larger triangle are cut from the main triangle, forming a triangle, with big amount holes. In this case, the initiator is the large triangle and the template is the operation of cutting out triangles similar to the larger one. You can also get a three-dimensional version of a triangle by using an ordinary tetrahedron and cutting out small tetrahedrons. The dimension of such a fractal is ln3/ln2 = 1.584962501.

To obtain Sierpinski carpet, take a square, divide it into nine squares, and cut out the middle one. We will do the same with the rest, smaller squares. Eventually, a flat fractal grid is formed, having no area but with infinite connections. In its spatial form, the Sierpinski sponge is transformed into a system of end-to-end forms, in which each end-to-end element is constantly replaced by its own kind. This structure is very similar to a section of bone tissue. Someday such repeating structures will become an element of building structures. Their statics and dynamics, Mandelbrot believes, deserve close study.

KOCH CURVE

The Koch curve is one of the most typical deterministic fractals. It was invented in the nineteenth century by a German mathematician named Helge von Koch, who, while studying the work of Georg Kontor and Karl Weierstrasse, came across descriptions of some strange curves with unusual behavior. The initiator is a straight line. The generator is an equilateral triangle, the sides of which are equal to a third of the length of the larger segment. These triangles are added to the middle of each segment over and over again. In his research, Mandelbrot experimented extensively with Koch curves, and produced figures such as Koch Islands, Koch Crosses, Koch Snowflakes, and even three-dimensional representations of the Koch curve by using a tetrahedron and adding smaller tetrahedrons to each of its faces. The Koch curve has dimension ln4/ln3 = 1.261859507.

MANDELBROT FRACTAL

This is NOT the Mandelbrot set, which you see quite often. The Mandelbrot set is based on nonlinear equations and is a complex fractal. This is also a variant of the Koch curve, although this object is not similar to it. The initiator and generator are also different from those used to create fractals based on the Koch curve principle, but the idea remains the same. Instead of joining equilateral triangles to a curve segment, squares are joined to a square. Due to the fact that this fractal occupies exactly half of the allotted space at each iteration, it has a simple fractal dimension of 3/2 = 1.5.

DARER 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 in which the ratio of the larger side to the smaller side 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.

A variant of this fractal can be obtained by using a hexagon as an initiator. This fractal is called the Star of David and it is quite similar to a hexagonal version of the Koch Snowflake. The fractal dimension of the Darer pentagon is ln6/ln(1+g), where g is the ratio of the length of the larger side of the triangle to the length of the smaller one. In this case, g is the Golden Ratio, so the fractal dimension is approximately 1.86171596. Fractal dimension of the Star of David ln6/ln3 or 1.630929754.

Complex fractals

In fact, if you magnify a small area of any complex fractal and then do the same with a small area of that area, the two magnifications will be significantly different from each other. The two images will be very similar in detail, but they will not be completely identical.

Figure 1. Mandelbrot set approximation

Compare, for example, the pictures of the Mandelbrot set shown here, one of which was obtained by enlarging a certain area of the other. As you can see, they are absolutely not identical, although on both we see a black circle, from which different sides flaming tentacles are coming. These elements are repeated indefinitely in the Mandelbrot set in decreasing proportions.

Deterministic fractals are linear, whereas complex fractals are not. Being nonlinear, these fractals are generated by what Mandelbrot called nonlinear algebraic equations. Good example is the process Zn+1=ZnІ + C, which is the equation used to construct the Mandelbrot and Julia set of the second degree. Solving these mathematical equations involves complex and imaginary numbers. When the equation is interpreted graphically in the complex plane, the result is a strange figure in which straight lines turn into curves and self-similarity effects appear, albeit not without deformations, at various scale levels. At the same time, the whole picture as a whole is unpredictable and very chaotic.

As you can see by looking at the pictures, complex fractals are indeed very complex and cannot be created without the help of a computer. To obtain colorful results, this computer must have a powerful mathematical coprocessor and a monitor with high resolution. Unlike deterministic fractals, complex fractals are not calculated in 5-10 iterations. Almost every point on a computer screen is like a separate fractal. During mathematical processing, each point is treated as a separate drawing. Each point corresponds to a specific value. The equation is built in for each point and is performed, for example, 1000 iterations. To obtain a relatively undistorted image in a time period acceptable for home computers, it is possible to carry out 250 iterations for one point.

Most of the fractals we see today are beautifully colored. Perhaps fractal images gain such great aesthetic significance precisely because of their color schemes. After the equation is calculated, the computer analyzes the results. If the results remain stable, or fluctuate around a certain value, the dot usually turns black. If the value at one step or another tends to infinity, the point is painted in a different color, maybe blue or red. During this process, the computer assigns colors to all motion speeds.

Typically, fast moving dots are colored red, while slower ones are colored yellow, and so on. Dark spots are probably the most stable.

Complex fractals differ from deterministic fractals in the sense that they are infinitely complex, but can still be generated by a very simple formula. Deterministic fractals do not require formulas or equations. Just take some drawing paper and you can build a Sierpinski sieve up to 3 or 4 iterations without any difficulty. Try this with lots of Julia! It's easier to go measure the length of England's coastline!

MANDELBROT SET

Fig 2. Mandelbrot set

Mandelbrot and Julia sets are probably the two most common among complex fractals. They can be found in many scientific journals, book covers, postcards, and computer screen savers. The Mandelbrot set, which was constructed by Benoit Mandelbrot, is probably the first association that people have when they hear the word fractal. This fractal, which resembles a carding machine with flaming tree-like and circular areas attached to it, is generated by the simple formula Zn+1=Zna+C, where Z and C are complex numbers and a is a positive number.

The Mandelbrot set, which can most often be seen, is the Mandelbrot set of the 2nd degree, that is, a = 2. The fact that the Mandelbrot set is not only Zn+1=ZnІ+C, but a fractal, the indicator in the formula of which can be any positive number, has misled many. On this page you see an example of the Mandelbrot set for different meanings indicator a.
Figure 3. The appearance of bubbles at a=3.5

The process Z=Z*tg(Z+C) is also popular. By including the tangent function, the result is a Mandelbrot set surrounded by an area resembling an apple. When using the cosine function, air bubble effects are obtained. In short, there are an infinite number of ways to configure the Mandelbrot set to produce different beautiful pictures.

A LOT OF JULIA

Surprisingly, Julia sets are formed according to the same formula as the Mandelbrot set. The Julia set was invented by the French mathematician Gaston Julia, after whom the set was named. The first question that arises after a visual acquaintance with the Mandelbrot and Julia sets is “if both fractals are generated according to the same formula, why are they so different?” First look at the pictures of the Julia set. Oddly enough, there are different types of Julia sets. When drawing a fractal using different starting points (to begin the iteration process), different images are generated. This only applies to the Julia set.

Figure 4. Julia set

Although it can't be seen in the picture, a Mandelbrot fractal is actually many Julia fractals connected together. Each point (or coordinate) of the Mandelbrot set corresponds to a Julia fractal. Julia sets can be generated using these points as initial values in the equation Z=ZI+C. But this does not mean that if you select a point on the Mandelbrot fractal and enlarge it, you can get the Julia fractal. These two points are identical, but only in a mathematical sense. If you take this point and calculate it using this formula, you can get a Julia fractal corresponding to a certain point of the Mandelbrot fractal.

Self-similar sets with unusual properties in mathematics

Since the end of the 19th century, examples of self-similar objects with properties that are pathological from the point of view of classical analysis have appeared 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 by 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 - 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

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: Ψ : K ↦ ∪ i = 1 n ψ i (K) (\displaystyle \Psi \colon K\mapsto \cup _(i=1)^(n)\psi _(i)(K))

It can be shown that the mapping Ψ (\displaystyle \Psi ) is a contraction map 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. It contains all the displays ψ i , i = 1 , … , n (\displaystyle \psi _(i),\,i=1,\dots ,n)- similarity displays, and n (\displaystyle n)- number of generator links.

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 (\displaystyle z_(n)) to infinity (defined, say, as the smallest number n (\displaystyle n), at which | z n | (\displaystyle |z_(n)|)).

will exceed a fixed large value

A (\displaystyle A)

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

Stochastic fractals
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. 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.

Natural objects with fractal properties

Natural objects ( quasi-fractals) differ from ideal abstract fractals in the incompleteness and inaccuracy of repetitions of the structure. Most fractal-like structures found in nature (cloud boundaries, shorelines, trees, plant leaves, corals, ...) are quasi-fractals, since at some small scale the fractal structure disappears. Natural structures cannot be perfect fractals due to limitations imposed by the size of a living cell and, ultimately, by the size of molecules.

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.

Application

Natural Sciences

In physics, fractals naturally arise when modeling nonlinear processes, such as turbulent fluid flow, complex diffusion-adsorption processes, flames, clouds, and the like. Fractals are used in modeling porous materials, for example, in petrochemicals. In biology, they are used to model populations and to describe systems. internal organs(system of blood vessels). After the creation of the Koch curve, it was proposed to use it when calculating the length of the coastline.

Radio engineering

Fractal antennas

Using fractal geometry in design

What do a tree, a seashore, a cloud, or the blood vessels in our hand have in common? At first glance, it may seem that all these objects have nothing in common. However, in fact, there is one property of structure that is inherent in all of the listed objects: they are self-similar. From a branch, as from a tree trunk, smaller shoots extend, from them even smaller ones, etc., that is, a branch is similar to the whole tree. It is arranged in a similar way circulatory system: arterioles depart from the arteries, and from them - the smallest capillaries, through which oxygen enters the organs and tissues. Let's look at satellite images of the sea coast: we will see bays and peninsulas; Let's look at it, but from a bird's eye view: we will see bays and capes; Now imagine that we are standing on the beach and looking at our feet: there will always be pebbles that protrude further into the water than the rest. That is coastline when the scale increases, it remains similar to itself. The American (although he grew up in France) mathematician Benoit Mandelbrot called this property of objects fractality, and such objects themselves - fractals (from the Latin fractus - broken).

This concept does not have a strict definition. Therefore, the word "fractal" is not a mathematical term. Typically, a fractal is a geometric figure that satisfies one or more of the following properties: It has a complex structure at any increase in scale (unlike, for example, a straight line, any part of which is the simplest geometric figure - a segment). Is (approximately) self-similar. It has a fractional Hausdorff (fractal) dimension, which is larger than the topological one. Can be constructed using recursive procedures.

Geometry and algebra

The study of fractals at the turn of the 19th and 20th centuries 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 of a continuous function that 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 ideas of self-similarity of figures were 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 - the Levy 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 began at 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 memoir on iterations of complex rational functions, which described 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 turned to it 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.

Fractal dimensions

As you know, the dimension (number of dimensions) of a geometric figure is the number of coordinates necessary to determine the position of a point lying on this figure.
For example, the position of a point on a curve is determined by one coordinate, on a surface (not necessarily a plane) by two coordinates, and in three-dimensional space by three coordinates.
From a more general mathematical point of view, one can define the dimension in this way: an increase in linear dimensions, say, by a factor of two, for one-dimensional (from a topological point of view) objects (segment) leads to an increase in size (length) by a factor of two, for two-dimensional ones (a square ) the same increase in linear dimensions leads to an increase in size (area) by 4 times, for three-dimensional (cube) - by 8 times. That is, the “real” (so-called Hausdorff) dimension can be calculated as the ratio of the logarithm of the increase in the “size” of an object to the logarithm of the increase in its linear size. That is, for a segment D=log (2)/log (2)=1, for a plane D=log (4)/log (2)=2, for a volume D=log (8)/log (2)=3.
Let us now calculate the dimension of the Koch curve, to construct which a unit segment is divided into three equal parts and the middle interval is replaced by an equilateral triangle without this segment. When the linear dimensions of the minimum segment increase three times, the length of the Koch curve increases by log (4)/log (3) ~ 1.26. That is, the dimension of the Koch curve is fractional!

Science and art

In 1982, Mandelbrot’s book “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 designs and formulas that even a high school student can understand, the resulting images are amazing in complexity and beauty. When personal computers became quite powerful, 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.

Scheme for obtaining the Koch curve

War and Peace

As noted above, one of the natural objects that have fractal properties is the coastline. There is one thing connected with it, or more precisely, with the attempt to measure its length. interesting story, which formed the basis of Mandelbrot’s scientific article, and is also described in his book “Fractal Geometry of Nature”. We are talking about an experiment carried out by Lewis Richardson, a very talented and eccentric mathematician, physicist and meteorologist. One of the directions of his research was an attempt to find a mathematical description of the causes and likelihood of an armed conflict between two countries. Among the parameters that he took into account was the length of the common border of the two warring countries. When he collected data for numerical experiments, he discovered that data on common border Spain and Portugal are very different. This led him to the following discovery: the length of a country's borders depends on the ruler with which we measure them. The smaller the scale, the longer the border. This is due to the fact that with greater magnification it becomes possible to take into account more and more new bends of the coast, which were previously ignored due to the coarseness of the measurements. And if, with each increase in scale, previously unaccounted for bends of lines are revealed, then it turns out that the length of the boundaries is infinite! True, this does not actually happen - the accuracy of our measurements has a finite limit. This paradox is called the Richardson effect.

Constructive (geometric) fractals

The algorithm for constructing a constructive fractal in the general case is as follows. First of all, we need two suitable geometric shapes, let's call them the base and the fragment. At the first stage, the basis of the future fractal is depicted. Then some of its parts are replaced with a fragment taken at a suitable scale - this is the first iteration of the construction. Then the resulting figure again changes some parts to figures similar to the fragment, etc. If we continue this process ad infinitum, then in the limit we will get a fractal.

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

In the language of mathematics: dynamic (algebraic) fractals

Fractals of this type arise when studying nonlinear dynamic systems (hence the name). The behavior of such a system can be described by a complex nonlinear function (polynomial) f (z). Let's take some initial point z0 on the complex plane (see sidebar). Now consider such an infinite sequence of numbers on the complex plane, each next of which is obtained from the previous one: z0, z1=f (z0), z2=f (z1), ... zn+1=f (zn). Depending on the initial point z0, such a sequence can behave differently: tend to infinity as n -> ∞; converge to some end point; cyclically take a series of fixed values; More complex options are also possible.

Complex numbers

A complex number is a number consisting of two parts - real and imaginary, that is, the formal sum x + iy (x and y here are real numbers). i is the so-called imaginary unit, that is, that is, a number that satisfies the equation i^ 2 = -1. The basic mathematical operations on complex numbers are defined: addition, multiplication, division, subtraction (only the comparison operation is not defined). To display complex numbers, a geometric representation is often used - on the plane (it is called complex), the real part is plotted along the abscissa axis, and the imaginary part is plotted along the ordinate axis, and the complex number will correspond to a point with Cartesian coordinates x and y.

Thus, any point z of the complex plane has its own behavior during iterations of the function f (z), and the entire plane is divided into parts. Moreover, the points lying on the boundaries of these parts have the following property: with an arbitrarily small displacement, the nature of their behavior changes sharply (such points are called bifurcation points). So, it turns out that sets of points that have one specific type of behavior, as well as sets of bifurcation points, often have fractal properties. These are the Julia sets for the function f (z).

Dragon family

By varying the base and fragment, you can get a stunning variety of constructive fractals.
Moreover, similar operations can be performed in three-dimensional space. Examples of volumetric fractals include the “Menger sponge”, “Sierpinski pyramid” and others.
The dragon family is also considered to be a constructive fractal. Sometimes they are called by the name of their discoverers “Heavey-Harter dragons” (in their shape they resemble Chinese dragons). There are several ways to construct this curve. The simplest and most visual of them is this: you need to take a fairly long strip of paper (the thinner the paper, the better), and bend it in half. Then bend it in half again in the same direction as the first time. After several repetitions (usually after five or six folds the strip becomes too thick to be gently bent further), you need to bend the strip back, and try to create 90˚ angles at the folds. Then in profile you will get the curve of a dragon. Of course, this will only be an approximation, like all our attempts to depict fractal objects. The computer allows many more steps of this process to be depicted, and the result is a very beautiful figure.

The Mandelbrot set is constructed somewhat differently. Consider the function fc (z) = z 2 +c, where c is a complex number. Let's construct a sequence of this function with z0=0; depending on the parameter c, it can diverge to infinity or remain limited. Moreover, all values of c for which this sequence is limited form the Mandelbrot set. It was studied in detail by Mandelbrot himself and other mathematicians, who discovered many interesting properties of this set.

It can be seen that the definitions of the Julia and Mandelbrot sets are similar to each other. In fact, these two sets are closely related. Namely, the Mandelbrot set is all values of the complex parameter c for which the Julia set fc (z) is connected (a set is called connected if it cannot be divided into two disjoint parts, with some additional conditions).

Fractals and life

Nowadays, the theory of fractals is widely used in various areas of human activity. In addition to a purely scientific object for research and the already mentioned fractal painting, fractals are used in information theory to compress graphic data (the self-similarity property of fractals is mainly used here - after all, to remember a small fragment of a picture and the transformations with which you can obtain the remaining parts, much less is required memory than for storing the entire file). By adding random disturbances to the formulas that define a fractal, you can obtain stochastic fractals that very plausibly convey some real objects - relief elements, the surface of reservoirs, some plants, which is successfully used in physics, geography and computer graphics to achieve greater similarity of simulated objects with real. In radio electronics, in the last decade, antennas with a fractal shape began to be produced. Taking up little space, they provide high-quality signal reception. Economists use fractals to describe currency fluctuation curves (this property was discovered by Mandelbrot more than 30 years ago). This concludes this short excursion into the amazingly beautiful and diverse world of fractals.