The concepts of fractal and fractal geometry, which appeared in the late 70s, have become firmly established in the everyday life of mathematicians and programmers since the mid-80s. The word fractal is derived from the Latin fractus and in translation means consisting of fragments. It was proposed by Benoit Mandelbrot in 1975 to refer to the irregular but self-similar structures that he studied. 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, Kantor, Hausdorff But only in our time it was possible to combine their works 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 required, with the help of several coefficients, to define lines and surfaces of a very complex shape. From the point of view of computer graphics, fractal geometry is indispensable for the generation of artificial clouds, mountains, and the surface of the sea. In fact, a way has been found to easily represent complex non-Euclidean objects, the images of which are very similar to natural ones.
One of the main properties of fractals is self-similarity. In the simplest case, a small part of the fractal contains information about the entire fractal. The definition of a fractal given by Mandelbrot is as follows: "A fractal is a structure consisting of parts that are in some sense similar to the whole."

There are a large number of mathematical objects called fractals (the Sierpinski triangle, the Koch snowflake, the Peano curve, the Mandelbrot set and the Lorentz attractors). Fractals describe with great accuracy 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 shapes. For the first time, Benoit Mandelbrot spoke about the fractal nature of our world in his seminal work "The 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 the Latin words fractus - fractional and frangere - to break, which reflects the essence of the fractal as a "broken", irregular set.

Classification of fractals.

In order to represent 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 obvious. In the two-dimensional case, they are obtained using a polyline (or surface in the three-dimensional case) called a generator. In one step of the algorithm, each of the segments that make up the broken line is replaced by a broken line generator in the appropriate scale. As a result of the endless repetition of this procedure, a geometric fractal is obtained.

Consider, for example, one of such fractal objects - the Koch triadic curve.

Construction of the triadic Koch curve.

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

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

In order to move on 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 this process is continued indefinitely, the result will be a triadic Koch curve.

Let us now consider the holy triadic Koch curve and find out why fractals were called "monsters".

First, 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. The 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 "dragon" Harter-Hateway.

To obtain another fractal object, you need to change the construction rules. Let the generating element be two equal segments connected at right angles. In the zero 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, a shift in the middle of the link occurs. When constructing the next generations, the rule is followed: the very first link on the left is replaced by a generating element so that the middle of the link is shifted to the left of the direction of movement, and when replacing the next links, the directions of displacement of the midpoints 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. The curve with n tending to infinity is called the Harter-Hateway 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 non-linear processes in n-dimensional spaces. Two-dimensional processes are the most studied. Interpreting a nonlinear iterative process as a discrete dynamical system, one can use the terminology of the theory of these systems: phase portrait, steady state process, attractor, etc.
It is known that nonlinear dynamical systems have several stable states. The state in which the dynamical system finds itself after a certain number of iterations depends on its initial state. Therefore, each stable state (or, as they say, an attractor) has a certain area of ​​initial states, from which the system will necessarily fall into the considered final states. Thus, the phase space of the system is divided into areas of attraction of attractors. If the phase space is two-dimensional, then by coloring the attraction regions 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 fancy multicolor patterns. A surprise for mathematicians was the ability to generate very complex non-trivial structures using primitive algorithms.

The 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 are 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 dynamical system is at infinity), or after a sufficiently large number of iterations (for example, 200-500) Z[i] converges 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 any of its parameters are randomly changed in an iterative process. This results in objects very similar to natural ones - asymmetrical trees, indented coastlines, etc. Two-dimensional stochastic fractals are used in modeling the terrain and sea surface.
There are other classifications of fractals, for example, the division of fractals into deterministic (algebraic and geometric) and non-deterministic (stochastic).

About the use of fractals

First of all, fractals are an area of ​​amazing mathematical art, when with the help of the simplest formulas and algorithms, pictures of extraordinary beauty and complexity are obtained! In the contours of the constructed images, leaves, trees and flowers are often guessed.

One of the most powerful applications of fractals lies in computer graphics. Firstly, it 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 applied directly in mathematics itself.
The advantages of fractal image compression algorithms are the very small size of the packed file and the short image recovery time. Fractally packed pictures can be scaled without the appearance of pixelation. But the compression process takes a long time and sometimes lasts for hours. The lossy fractal packing algorithm allows you to set the compression level, similar to the jpeg format. The algorithm is based on the search for large pieces of the image 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 picture; a hexagonal grid is free from such a disadvantage.
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 look like 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 whole landscapes are given by simple formulas, are easy to program and do not fall apart into separate triangles and cubes when approached.
You can not 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 "Cantor ladder" function is a good example of a singular measure distribution function.
In mechanics and physics, fractals are used due to their unique property to repeat the outlines of many natural objects. Fractals allow you to approximate trees, mountain surfaces, and fissures with higher accuracy than approximations with line segments or polygons (with the same amount of stored data). Fractal models, like natural objects, have "roughness", and this property is preserved at an arbitrarily large increase in the model. The presence of a uniform measure on fractals makes it possible to apply integration, potential theory, to use them instead of standard objects in the equations already studied.
With the 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 building fractals

Method of successive approximations

Looking at this picture, it is not difficult to understand how a self-similar fractal can be built (in this case, the Sierpinski pyramid). We need to take an ordinary pyramid (tetrahedron), then cut out its middle (octahedron), as a result of which we get four small pyramids. With each of them we perform the same operation, and so on. This is a somewhat naive, but illustrative explanation.

Let us consider the essence of the method more strictly. Let there be some IFS system, i.e. contraction mapping system S=(S 1 ,...,S m ) S i:R n ->R n (for example, for our pyramid, the mappings look like 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 determine 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 the sets A k with increasing k approximate the required attractor of the system S.

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

Construction by points or probabilistic method

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

) is some system of affine contractions. Mappings S

representable as: S

Fixed matrix of size 2x2 and o

Two-dimensional vector column.

• Let's take a fixed point of the first mapping S 1 as a starting point:
  x:=o1;
  Here we use the fact that all fixed contraction points S 1 ,..,S m belong to the fractal. An arbitrary point can be chosen as a starting point and the sequence of points generated by it will shrink to a fractal, but then a few extra points will appear on the screen.
• Note the current point x=(x 1 ,x 2) on the screen:
  putpixel(x 1 ,x 2 ,15);
• We randomly choose a number j from 1 to m and recalculate the coordinates of the 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, then we stop.

Note. If the coefficients of compression of the mappings S i are different, then the fractal will be filled with points unevenly. If the mappings S i are similarities, 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 the probabilities p 1 =r 1 s ,..,p m =r m s , where r i denote the contraction coefficients of the mappings S i , and the number s (called the similarity dimension) is found from the equation r 1 s +...+r m s =1. The solution of 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 firmly entered the everyday life of mathematicians and programmers since the mid-80s. The term was proposed by Benoit Mandelbrot in 1975 to refer to the irregular but self-similar structures that he studied. The birth of fractal geometry is usually associated with the publication in 1977 of Mandelbrot's book "The Fractal Geometry of Nature" - "The Fractal Geometry of Nature". His works used the scientific results of other scientists who worked in the same field in the period 1875-1925 (Poincaré, Fatou, Julia, Kantor, Hausdorff).

Adjustments

Let me make some adjustments to the algorithms proposed in the book by H.-O. Paytgen and P.H. Richter "The Beauty of Fractals" M. 1993, purely to eradicate typos and make it easier to understand 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 \u003d z 2 + c since z and c are complex numbers, then z \u003d x + iy, c \u003d p + iq, it is necessary to decompose it into x and y in order to go to more real for the common man plane:

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

The plane consisting of all pairs (x, y) can be considered as with fixed values p and q, as well as for dynamic ones. In the first case, sorting 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 (black) when the allowable maximum of repetitions is increased, we get the mapping 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 get images called Mandelbrot sets.

On the question of fractal coloring algorithms.

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

Application of the "fractal microscope"

to demonstrate frontier phenomena.

Attractors are the centers fighting for dominance on the plane. Between the attractors there is a border representing a swirling pattern. By increasing the scale of consideration within the boundaries of the set, one can obtain non-trivial patterns reflecting 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, in connection with which their implementation becomes a difficult practical task. Natural complexes have cores of typicality acting as attractors that lose their power of 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 perception of a specialist for a meeting with a dynamic and seemingly chaotic in space and time natural object, for understanding fractal geometry nature. Multicolored colors and fractal music will definitely leave a deep mark on the minds of students.

Thousands of publications and huge Internet resources are devoted to fractals, however, 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 their proper place in the course of computer science.

Examples

SIERPINSKI GRID

This is one of the fractals that Mandelbrot experimented with when developing the concepts of fractal dimensions and iterations. Triangles formed by joining the midpoints of the larger triangle are cut from the main triangle to form a triangle, with more holes. In this case, the initiator is a large triangle and the template is an operation to cut triangles similar to the larger one. You can also get a 3D version of a triangle by using an ordinary tetrahedron and cutting out smaller tetrahedra. 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. In the end, a flat fractal grid is formed, which has no area, but with infinite connections. In its spatial form, the Sierpinski sponge is transformed into a system of through forms, in which each through 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. Initiator - direct 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 a lot with Koch curves, and obtained 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 tetrahedra to each of its faces. The Koch curve has dimension ln4/ln3 = 1.261859507.

Fractal Mandelbrot

This is NOT the Mandelbrot set that you see quite often. The Mandelbrot set is based on non-linear equations and is a complex fractal. This is also a variant of the Koch curve, despite the fact that this object does not look like it. The initiator and generator are also different from those used to create fractals based on the principle of the Koch curve, but the idea remains the same. Instead of attaching equilateral triangles to a curve segment, squares are attached 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'S PENTAGON

A fractal looks like a bunch of pentagons squeezed together. In fact, it is formed by using a pentagon as an initiator and isosceles triangles, the ratio of the largest side to the smallest in which is exactly equal to the so-called golden ratio (1.618033989 or 1/(2cos72)) as the 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 the initiator. This fractal is called the Star of David and is quite similar to the hexagonal version of Koch's 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 side. In this case, g is the Golden Ratio, so the fractal dimension is approximately 1.86171596. The fractal dimension of the Star of David is ln6/ln3 or 1.630929754.

Complex fractals

In fact, if you zoom in on a small area of ​​any complex fractal and then do the same on 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 won't be completely identical.

Fig 1. Approximation of the Mandelbrot set

Compare, for example, the pictures of the Mandelbrot set shown here, one of which was obtained by increasing some area of ​​the other. As you can see, they are absolutely not identical, although on both we see a black circle, from which flaming tentacles go in different directions. These elements repeat indefinitely in the Mandelbrot set in decreasing proportion.

Deterministic fractals are linear, while complex fractals are not. Being non-linear, these fractals are generated by what Mandelbrot called non-linear algebraic equations. A good example is the process Zn+1=ZnІ + C, which is the equation used to construct the Mandelbrot and Julia sets 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, self-similarity effects appear at various scale levels, although not without deformations. 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 impossible to create without the help of a computer. To get colorful results, this computer must have a powerful math coprocessor and a high-resolution monitor. Unlike deterministic fractals, complex fractals are not calculated in 5-10 iterations. Almost every dot on the computer screen is like a separate fractal. During mathematical processing, each point is treated as a separate pattern. Each point corresponds to a certain 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 interval 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 have gained such great aesthetic value 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 will usually turn 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 movement speeds.

Usually, fast moving dots are painted red, while slower ones are yellow, and so on. Dark dots are probably the most stable.

Complex fractals differ from deterministic fractals in that they are infinitely complex, yet can be generated by a very simple formula. Deterministic fractals don't need 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 to do it with lots of Julia! It's easier to go measure the length of the coastline of England!

MANDERBROT SET

Fig 2. Mandelbrot set

The 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 built by Benoit Mandelbrot, is probably the first association that people have when they hear the word fractal. This fractal, resembling a card with glowing tree and circle 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 most commonly seen Mandelbrot set is the 2nd degree Mandelbrot set, i.e. a=2. The fact that the Mandelbrot set is not only Zn+1=ZnІ+C, but a fractal whose exponent in the formula can be any positive number misled many people. On this page you see an example of the Mandelbrot set for various values ​​of the exponent a.
Figure 3. The appearance of bubbles at a=3.5

The process Z=Z*tg(Z+C) is also popular. Thanks to the inclusion of the tangent function, the Mandelbrot set is obtained, 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 tweak the Mandelbrot set to produce various beautiful pictures.

MULTIPLE JULIA

Surprisingly, the Julia sets are formed according to the same formula as the Mandelbrot set. The Julia set was invented by the French mathematician Gaston Julia, after whom the set was named. The first question that arises after a visual acquaintance with the Mandelbrot and Julia sets is "if both fractals are generated by 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 start the iteration process), different images are generated. This only applies to the Julia set.

Fig 4. Julia set

Although it can't be seen in the picture, a Mandelbrot fractal is actually a bunch of 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 increase it, you can get a Julia fractal. These two points are identical, but only in a mathematical sense. If we take this point and calculate it according to this formula, we can get the Julia fractal corresponding to a certain point of the Mandelbrot fractal.

fractal

Fractal (lat. fractus- crushed, broken, broken) - a geometric figure that has the property of self-similarity, that is, it is composed of several parts, each of which is similar to the whole figure as a whole. 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 other than topological. Fractasm is an independent exact science of studying and compiling fractals.

In other words, fractals are geometric objects with a fractional dimension. For example, the dimension of a line is 1, an area is 2, a volume is 3. A fractal has a dimension value between 1 and 2 or between 2 and 3. For example, the fractal dimension of a crumpled paper ball is approximately equal to 2.5. In mathematics, there is a special complex formula for calculating the dimension of fractals. The ramifications of the tracheal tubes, the leaves on the trees, the veins in the arm, the river are fractals. In simple terms, a fractal is a geometric figure, a certain part of which is repeated over and over again, changing in size - this is the principle of self-similarity. Fractals are similar to themselves, they are similar to themselves at all levels (ie, 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, whether it be a cloud or an oxygen molecule.

The word "chaos" suggests something unpredictable, but in fact, chaos is quite ordered and obeys certain laws. The purpose 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 a person has when he hears the word "fractal". By the way, Mandelbrot determined that the fractal dimension of the coastline of England is 1.25.

Fractals are increasingly being 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 most practical aspect of fractal geometry. 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 the price of wool using Brownian motion.

The word "fractal" can be used not only as a mathematical term. A fractal in the press and popular science literature can be called figures that have any of the following properties:

It has a non-trivial structure on all scales. This is the difference from regular figures (such as a circle, an ellipse, the 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, zooming in does not lead to a simplification of the structure, on all scales we will see an equally complex picture.

It is self-similar or approximately self-similar.

It has a fractional metric dimension or a metric dimension that is superior to the topological one.

The most useful use of fractals in computer technology is fractal data compression. At the same time, pictures are compressed much better than it is done by conventional methods - up to 600:1. Another advantage of fractal compression is that when you zoom in, there is no pixelation effect that drastically worsens the picture. Moreover, a fractally compressed image after magnification often looks even better than before. Computer scientists also know that fractals of infinite complexity and beauty can be generated with simple formulas. The film industry makes extensive use of 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 a better understanding of the dynamics of complex flows. Flames can also be modeled using fractals. Porous materials are well represented in fractal form due to the fact that they have a very complex geometry. To transmit data over distances, fractal-shaped antennas 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, such as coasts, clouds, tree crowns, snowflakes, the circulatory system and the alveolar system of humans or animals.

Fractals, especially on the plane, are popular for their combination of beauty and ease of construction with 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 introduced by Benoit Mandelbrot in 1975 and gained wide popularity with the release of his book "The Fractal Geometry of Nature" in 1977.

The figure on the left shows a Darer Pentagon fractal as a simple example, which looks like a bunch of pentagons squeezed together. In fact, it is formed by using a pentagon as an initiator and isosceles triangles, the ratio of the largest side to the smallest in which is exactly equal to the so-called golden ratio (1.618033989 or 1/(2cos72°)) as the 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 of expressing such unpredictable systems turns out to be true not in exact equalities, but in representations of the system's behavior - in graphs of strange attractors that look like fractals. Thus chaos theory, thought of by many as unpredictability, turns out to be the science of predictability even in the most unstable systems. The doctrine of dynamical systems shows that simple equations can generate such chaotic behavior in which the system never returns to a stable state and no regularity appears at the same time. 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 the minimum technical system (TS) implies the flow within the TS of two types of processes - the main and supporting ones, and this division is conditional and relative. Any process can be the main one in relation to the supporting ones, and any of the supporting processes can be considered the main one in relation to “their” supporting processes. The circles in the diagram indicate the physical effects that ensure the flow of those processes, for which it is not necessary to specially create “own” TS. These processes are the result of the interaction between substances, fields, substances and fields. To be precise, the physical effect is a vehicle, the principle of which we cannot influence, and we do not want or have no opportunity to interfere in its structure.

The flow of the main process shown in the diagram is ensured by the existence of three supporting processes that are the main ones for the TS that generate them. For the sake of fairness, 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 not as simple as shown in the diagram. A useful (needed to a person) process cannot be performed with 100% efficiency. The dissipated energy is spent on the creation of harmful processes - heating, vibration, etc. As a result, in parallel with the beneficial process, harmful ones arise. It is not always possible to replace a “bad” process with a “good” one, so new processes have to be organized to compensate for the consequences that are 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 lubricating components and parts or replacing them periodically.

In connection with the existence of the inevitable influence of a changeable Environment, a useful process may need to be controlled. Management can be carried out both with the help of automatic devices, and directly by a person. The process diagram is actually a set of special commands, i.e. algorithm. The essence (description) of each command is a combination 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 an ordinary subroutine - and here we also find a fractal. The method of R. Koller, created a quarter of a century ago, 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 the end of the 19th century, examples of self-similar objects with pathological properties from the point of view of classical analysis 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 - 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 - a non-self-intersecting continuous curve of infinite length that does not have a tangent at any point;

the Peano curve is a continuous curve passing through all points of a 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 in a plane. We define an arbitrary broken line with a finite number of links, called a generator. Next, we replace each segment in it with a generator (more precisely, a broken line similar to a generator). In the resulting broken line, we again replace each segment with a generator. Continuing to infinity, in the limit we get a fractal curve. 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),

Levy Curve,

minkowski curve,

Hilbert Curve,

Broken (curve) dragon (Fractal Harter-Hateway),

Peano curve.

Using a similar procedure, a Pythagorean tree is obtained.

Fractals as Fixed Points of Contraction Mappings

The self-similarity property can be mathematically rigorously expressed as follows. Let be contraction maps 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 compact sets 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. In it, all mappings are similarity mappings, and is the number of generator links.

For the Sierpinski triangle and the mapping , , 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 under the mapping .

In the case when 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 . So, for the Sierpinski triangle we get .

According to the same Banach theorem, starting from any compact set and applying to it iterations of the mapping , 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 set of Julia

Fractals naturally arise in the study of nonlinear dynamical systems. The most studied case is when the 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 we tend to n to infinity. This sequence can:

strive for infinity

strive for the ultimate

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

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

Sets of values z 0 , for which the sequence exhibits one specific type of behavior, as well as sets of 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 considering the set of those parameter values ​​for which the sequence ( z n) demonstrates a certain behavior for 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 well-known 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 complement the Mandelbrot set, you can color the points depending on the speed of striving ( z n) to infinity (defined, say, as the smallest number n, where | z n| exceeds a fixed large value A.

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

Stochastic fractals

Randomized fractal based on the Julia set

Natural objects often have a fractal shape. For their modeling, stochastic (random) fractals can be used. Examples of stochastic fractals:

trajectory of Brownian motion on the plane and in space;

boundary of the trajectory of Brownian motion on the plane. In 2001, Lawler, Schramm, and Werner proved Mandelbrot's conjecture 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, in 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 systems of internal organs (system of blood vessels).

Radio engineering

fractal antennas

The use of fractal geometry in the design of antenna devices was first applied by American engineer Nathan Cohen, who then lived in downtown Boston, where it was forbidden to install external antennas on buildings. Nathan cut out a figure in the form of a Koch curve from aluminum foil and pasted it on a sheet of paper, then attached it to the receiver. Cohen founded his own company and launched 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, you can store a contraction map for which this image (or some close to it) is a fixed point. One of the variants of this algorithm was used [ source unspecified 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 build 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

Netsukuku's IP address assignment system uses the principle of fractal information compression to compactly store information about network nodes. Each node on the Netsukuku network stores only 4 KB of information about the state of neighboring nodes, while any new node connects to the general 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 a completely decentralized, and therefore, the most stable operation of the entire network.

Fractal properties are not a whim and not a fruit of idle fantasy of mathematicians. By studying them, we learn to distinguish and predict the important features of the objects and phenomena around us, which before, if not completely ignored, were estimated only approximately, qualitatively, by eye. For example, by comparing the fractal dimensions of complex signals, encephalograms, or heart murmurs, doctors can diagnose some serious diseases at an early stage, when the patient can still be helped. Also, the analyst, comparing the previous behavior of prices, at the beginning of the formation of the model, can foresee its further development, thereby avoiding gross errors in forecasting.

Irregularity of fractals

The first property of fractals is their irregularity. If a fractal is described by a function, then the property of irregularity in mathematical terms will mean that such a function is not differentiable, that is, not smooth at any point. Actually, this has the most direct relation to the market. Price fluctuations are sometimes so volatile and changeable that it confuses many traders. Our task is to sort out all this chaos and bring it to order.

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Self-similarity of fractals

The second property says that a fractal is an object that has the property of self-similarity. This is a recursive model, each part of which repeats in its development the development of the entire model as a whole and is reproduced at various scales without visible changes. However, changes still occur, which can significantly affect our perception of the object.

Self-similarity means that the object does not have a characteristic scale: if it had such a scale, you would immediately distinguish the enlarged copy of the fragment from the original image. Self-similar objects have an infinite number of scales for all tastes. The essence of self-similarity can be explained by the following example. Imagine that you have a picture of a “real” geometric line, “length without width”, as Euclid defined the line, and you are playing with a friend, trying to guess whether he shows you the original picture (original) or a picture of any fragment of a straight line. No matter how hard you try, you will never be able to distinguish the original from the enlarged copy of the fragment, the straight line is arranged in the same way in all its parts, it is similar to itself, but this remarkable property of it is somewhat hidden by the uncomplicated structure of the straight line itself, its “straightness” (Fig. 7).

If you also cannot distinguish a snapshot of some object from a properly enlarged snapshot of any of its fragments, then you have a self-similar object. All fractals that have at least some symmetry are self-similar. And this means that some fragments of their structure are strictly repeated at certain spatial intervals. Obviously, these objects can be of any nature, and their appearance and shape remain unchanged regardless of the scale. An example of a self-similar fractal:

In finance, this concept is not a baseless abstraction, but a theoretical restatement of a practical market saying—namely, that the movements of a stock or currency are superficially similar, regardless of time frame and price. The observer cannot tell by the appearance of the graph whether the data is for weekly, daily, or hourly changes.

Of course, not all fractals have such a regular, endlessly repeating structure as those wonderful exhibits of the future museum of fractal art, which were born from the imagination of mathematicians and artists. Many fractals found in nature (fault surfaces of rocks and metals, clouds, currency quotes, turbulent flows, foam, gels, soot particle contours, etc.) lack geometric similarity, but stubbornly reproduce in each fragment the statistical properties of the whole. Fractals with a non-linear form of development were named by Mandelbrot as multifractals. A multifractal is a quasi-fractal object with a variable fractal dimension. Naturally, real objects and processes are much better described by multifractals.

Such statistical self-similarity, or self-similarity on average, distinguishes fractals among many natural objects.

Consider an example of self-similarity in the foreign exchange market:

In these figures, we see that they are similar, while having a different time scale, in Fig. and the 15 minute scale, in Fig. b weekly price scale. As you can see, these quotes do not have the ability to perfectly repeat each other, however, we can consider them similar.

Even the simplest of fractals - geometrically self-similar fractals - have unusual properties. For example, the von Koch snowflake has a perimeter of infinite length, although it limits a finite area (Fig. 9). In addition, it is so prickly that it is impossible to draw a tangent to it at any point of the contour (a mathematician would say that a von Koch snowflake is nowhere differentiable, that is, not smooth at any point).

Mandelbrot found that the results of fractional measurement remain constant for various degrees of enhancement of the object's irregularity. In other words, there is regularity (correctness, orderliness) for any irregularity. When we treat something as random, it indicates that we do not understand the nature of this randomness. In market terms, this means that the formation of the same typical formations must occur in different time frames. A one-minute chart will describe a fractal formation in the same way as a monthly one. This "self-similarity" found on the charts of commodity and financial markets shows all the signs that the market's actions are closer to the behavioral paradigm of "nature" than the behavior of economic, fundamental analysis.

In these figures, you can find confirmation of the above. On the left is a graph with a minute scale, on the right is a weekly one. The currency pairs Dollar/Yen (Fig. 9 (a)) and Euro/Dollar (Fig. 9 (b)) are shown here with different price scales. Even though the JPY/USD currency pair has a different volatility in relation to EUR/USD, we can observe the same price movement structure.

fractal dimension

The third property of fractals is that fractal objects have a dimension other than Euclidean (in other words, a topological dimension). The fractal dimension is a measure of the complexity of the curve. By analyzing the alternation of sections with different fractal dimensions and how the system is affected by external and internal factors, one can learn to predict the behavior of the system. And most importantly, to diagnose and predict unstable conditions.

In the arsenal of modern mathematics, Mandelbrot found a convenient quantitative measure of the imperfection of objects - the sinuosity of the contour, the wrinkling of the surface, the fracturing and porosity of the volume. It was proposed by two mathematicians - Felix Hausdorff (1868-1942) and Abram Samoylovich Besikovich (1891-1970). Now it deservedly bears the glorious names of its creators (Hausdorff-Besikovich dimension) - Hausdorff-Besikovich dimension. What is dimension and why do we need it in relation to the analysis of financial markets? Before that, we knew only one type of dimension - topological (Fig. 11). The word dimension itself indicates how many dimensions an object has. For a segment, a straight line, it is equal to 1, i.e. we have only one dimension, namely the length of a segment or a straight line. For a plane, the dimension will be 2, since we have a two-dimensional dimension, length and width. For space or solid objects, the dimension is 3: length, width, and height.

Let's take the example of computer games. If the game is made in 3D graphics, then it is spatial and voluminous, if in 2D graphics, the graphics are displayed on a plane (Fig. 10).

The most unusual (it would be more correct to say - unusual) in the Hausdorff-Besikovich dimension was that it could take not only integers, as a topological dimension, but also fractional values. Equal to one for a straight line (infinite, semi-infinite, or for a finite segment), the Hausdorff-Besicovitch dimension increases as the tortuosity increases, while the topological dimension stubbornly ignores all changes that occur with the line.

Dimension characterizes the complication of a set (for example, a straight line). If it is a curve with a topological dimension equal to 1 (a straight line), then the curve can be complicated by an infinite number of bends and branches to such an extent that its fractal dimension approaches two, i.e. will fill almost the entire plane (Fig. 12)

By increasing its value, the Hausdorff-Besikovich dimension does not change it abruptly, as the topological dimension would do "in its place", the transition from 1 immediately to 2. The Hausdorff-Besikovich dimension - and this at first glance may seem unusual and surprising, takes fractional values : equal to one for a straight line, it becomes 1.15 for a slightly sinuous line, 1.2 for a more sinuous line, 1.5 for a very sinuous line, and so on.

It was in order to emphasize the ability of the Hausdorff-Besikovich dimension to take fractional, non-integer values ​​that Mandelbrot came up with his own neologism, calling it the fractal dimension. So, the fractal dimension (not only Hausdorff-Besikovich, but also any other) is a dimension that can take not necessarily integer values, but also fractional ones.

For linear geometric fractals, the dimension characterizes their self-similarity. Consider Fig. 17(A), the line consists of N=4 segments, each of which has a length of r = 1/3. As a result, we get the ratio:

D = logN/log(1/r)

The situation is quite different when we talk about multifractals (non-linear). Here the dimension loses its meaning as a definition of the similarity of an object and is defined through various generalizations, much less natural than the unique dimension of self-similar objects.

In the foreign exchange market, the dimension can characterize the volatility of price quotes. Each currency pair has its own behavior in terms of prices. For the Pound/Dollar pair (Fig. 13(a)) it is more calm than for the Euro/Dollar (Fig. 13(b)). The most interesting thing is that these currencies move with the same structure to price levels, however, they have different dimensions, which can affect intraday trading and changes in models that elude the inexperienced look.

On fig. Figure 14 shows the dimension in relation to the mathematical model, in order for you to more deeply penetrate the meaning of this term. Note that all three figures show the same cycle. On fig. and the dimension is 1.2, in Fig. b, the dimension is 1.5, and in Fig. in 1.9. It can be seen that with an increase in the dimension, the perception of the object becomes more complicated, the amplitude of oscillations increases.

In financial markets, dimension is reflected not only as price volatility, but also as a detail of cycles (waves). Thanks to it, we will be able to distinguish whether a wave belongs to a certain time scale. On fig. 15 shows the Euro/Dollar pair on a daily price scale. Pay attention, you can clearly see the formed cycle and the beginning of a new, larger cycle. Switching to the hourly scale and zooming in on one of the cycles, we can see smaller cycles, and part of a large one located on D1 (Fig. 16). Loop detailing, i.e. their dimension allows us to determine from the initial conditions how the situation can develop in the future. We can say that: the fractal dimension reflects the scale invariance property of the set under consideration.

The concept of invariance was introduced by Mandelbrot from the word "sealant" - scalable, i.e. when an object has the property of invariance, it has different display scales.

On fig. 16 circle A highlights a mini-cycle (detailed wave), circle B - a wave of a larger cycle. It is precisely because of the dimension that we cannot always determine ALL cycles on the same price scale.

We will talk about the problems of determining and developing properties of non-periodic cycles in the “Cycles in the foreign exchange market” section, now the main thing for us was to understand how and where dimension manifests itself in financial markets.

Thus, we can say that fractals as models are used when the real object cannot be represented in the form of classical models. And this means that we are dealing with non-linear relationships and the non-deterministic (random) nature of the data. Nonlinearity in the ideological sense means the multivariance of development paths, the availability of a choice from alternative paths and a certain pace of evolution, as well as the irreversibility of evolutionary processes. Nonlinearity in the mathematical sense means a certain type of mathematical equations (nonlinear differential equations) containing the desired quantities in powers greater than one or coefficients that depend on the properties of the medium. A simple example of a non-linear dynamical system:

Johnny grows 2 inches a year. This system explains how Johnny's height changes over time. Let x(n) be Johnny's height this year. Let his growth in the next year be written as x (n + 1). Then we can write the dynamical system in the form of an equation:

x(n+1) = x(n) + 2.

See? Isn't this simple math? If we enter Johnny's height today x (n) = 38 inches, then on the right side of the equation we get Johnny's height next year, x (n+1) = 40 inches:

x(n+1) = x(n) + 2 = 38 + 2 = 40.

Moving from right to left in an equation is called iteration (repetition). We can repeat the equation again by entering Johnny's new height of 40 inches on the correct side of the equation (i.e. x(n) = 40) and we get x(n+1) = 42. If we iterate (repeat) the equation 3 times, we we get Johnny's height in 3 years, namely 44 inches, starting with a height of 38 inches.

This is a deterministic dynamic system. If we want to make it non-deterministic (stochastic), we could make a model like this: Johnny grows 2 inches per year, more or less, and write the equation as:

x(n+1) = x(n) + 2 + e

where e is a small error (small relative to 2), represents some probability distribution.

Let's go back to the original deterministic equation. The original equation, x(n+1) = x(n) + 2, is linear. Linear means you are adding variables or constants, or multiplying variables by constants. For example, the equation

z(n+l) = z(n) + 5 y(n) -2 x(n)

is linear. But if you multiply the variables, or raise them to a power greater than one, the equation (system) will become non-linear. For example, the equation

x(n+1) = x(n) 2

is non-linear because x(n) is squared. The equation

is non-linear because two variables, x and y, are multiplied.

When we apply classical models (for example, trend, regression, etc.), we say that the future of an object is uniquely determined, i.e. depends entirely on the initial conditions and is amenable to a clear forecast. You can independently perform one of these models in Excel. An example of a classical model can be represented as a constantly decreasing or increasing trend. And we can predict its behavior, knowing the past of the object (the initial data for modeling). And fractals are used in the case when the object has several options for development and the state of the system is determined by the position in which it is currently located. That is, we are trying to simulate a chaotic development. This system is the interbank foreign exchange market.

Let us now consider how one can obtain from a straight line what we call a fractal, with its inherent properties.

On fig. 17(A) shows the Koch curve. Take a line segment, its length = 1, i.e. still a topological dimension. Now we will divide it into three parts (each 1/3 of the length), and remove the middle third. But we will replace the middle third with two segments (each 1/3 of the length), which can be represented as two sides of an equilateral triangle. This is stage two (b) of the design depicted in fig. 17(A). At this point we have 4 smaller parts, each 1/3 of the length, so the whole length is 4(1/3) = 4/3. We then repeat this process for each of the 4 smaller lobes of the line. This is stage three (c). This will give us 16 even smaller line segments, each 1/9 of the length. So the whole length is now 16/9 or (4/3) 2 . As a result, we got a fractional dimension. But not only this distinguishes the resulting structure from a straight line. It has become self-similar and it is impossible to draw a tangent at any of its points (Fig. 17 (B)).

Content

What do a tree, a seashore, a cloud, or 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 the structure that is inherent in all the listed objects: they are self-similar. From the branch, as well as from the trunk of a tree, smaller processes depart, from them - even smaller ones, etc., that is, a branch is similar to the whole tree. The circulatory system is arranged in a similar way: arterioles depart from the arteries, and from them - the smallest capillaries through which oxygen enters organs and tissues. Let's look at satellite images of the sea coast: we will see bays and peninsulas; let's take a 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, the coastline remains similar to itself when zoomed in. American mathematician Benoit Mandelbrot called this property of objects fractal, 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. Usually, a fractal is a geometric figure that satisfies one or more of the following properties: It has a complex structure at any magnification (unlike, for example, a straight line, any part of which is the simplest geometric figure - a segment). It is (approximately) self-similar. It has a fractional Hausdorff (fractal) dimension, which is larger than the topological one. Can be built with recursive procedures.

Geometry and Algebra

The study of fractals at the turn of the 19th and 20th centuries was more episodic than systematic, because earlier mathematicians mainly studied “good” objects that could be studied using general methods and theories. In 1872, German mathematician Karl Weierstrass builds 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 it is quite simple to draw it. It turned out that it has the properties of a fractal. One variation 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, in which another fractal is described - the Lévy C-curve. All these fractals listed above can be conditionally attributed to 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, almost two hundred pages of Julia's memoir, devoted to iterations of complex rational functions, was published, in which Julia sets are described - a whole family of fractals closely related to the Mandelbrot set. This work was awarded the prize of the French Academy, but it did not contain a single illustration, so it was impossible to appreciate the beauty of the discovered objects. Despite the fact that this work made Julia famous among the mathematicians of the time, it was quickly forgotten. Again, attention 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 measurements) 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, in three-dimensional space by three coordinates.
From a more general mathematical point of view, the dimension can be defined as follows: an increase in linear dimensions, say, twice, 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 (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, for the construction of which a unit segment is divided into three equal parts and the middle interval is replaced by an equilateral triangle without this segment. With an increase in the linear dimensions of the minimum segment three times, the length of the Koch curve increases in log (4) / log (3) ~ 1.26. That is, the dimension of the Koch curve is fractional!

Science and art

In 1982, Mandelbrot's book "The 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 made the main emphasis in his presentation not on ponderous formulas and mathematical constructions, but on the geometric intuition of readers. Thanks to computer-assisted illustrations and historical stories, with which the author skillfully diluted the scientific component of the monograph, the book became a bestseller, and the 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 trend in art appeared - fractal painting, and almost any computer owner could do it. Now on the Internet you can easily find many sites dedicated 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. One interesting story is connected with it, or rather, with an attempt to measure its length, which formed the basis of Mandelbrot's scientific article, and is also described in his book "The Fractal Geometry of Nature". We are talking about an experiment that was set up 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 between the two warring countries. When he collected data for numerical experiments, he found that in different sources the data on the common border of Spain and Portugal differ greatly. This led him to the following discovery: the length of the country's borders depends on the ruler with which we measure them. The smaller the scale, the longer the border will be. This is due to the fact that at higher magnification it becomes possible to take into account more and more bends of the coast, which were previously ignored due to the roughness of the measurements. And if, with each zoom, previously unaccounted bends of lines are opened, then it turns out that the length of the borders is infinite! True, in fact this does not 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 by a fragment taken in a suitable scale - this is the first iteration of the construction. Then, in the resulting figure, some parts again change to figures similar to a fragment, and so on. If this process is continued indefinitely, then a fractal will be obtained in the limit.

Consider this process using the example of the Koch curve (see sidebar on the previous page). Any curve can be taken as the basis of the Koch curve (for the Koch snowflake, this is a triangle). But we confine ourselves to the simplest case - a segment. The fragment is a broken line shown on the top of 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, and so on. The figure shows the first four steps of this process.

The language of mathematics: dynamic (algebraic) fractals

Fractals of this type arise in the study of nonlinear dynamical systems (hence the name). The behavior of such a system can be described by a complex nonlinear function (polynomial) f (z). Let us take some initial point z0 on the complex plane (see sidebar). Now consider such an infinite sequence of numbers on the complex plane, each 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 number of fixed values; more complex options are 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. Basic mathematical operations are defined over complex numbers - 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 along the ordinate axis, while 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 character of 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: for an arbitrarily small displacement, the nature of their behavior changes dramatically (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 are "Menger's sponge", "Sierpinski's pyramid" and others.
The family of dragons is also referred to constructive fractals. They are sometimes referred to by the name of the discoverers as the "dragons of Heiwei-Harter" (they resemble Chinese dragons in their shape). There are several ways to construct this curve. The simplest and most obvious of them is this: you need to take a sufficiently long strip of paper (the thinner the paper, the better), and bend it in half. Then again bend it in half in the same direction as the first time. After several repetitions (usually after five or six folds, the strip becomes too thick to be carefully bent further), you need to straighten the strip back, and try to form 90˚ angles at the folds. Then the curve of the dragon will turn out in profile. Of course, this will only be an approximation, like all our attempts to depict fractal objects. The computer allows you to depict many more steps in this process, 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 us construct a sequence of this function with z0=0, depending on the parameter c, it can diverge to infinity or remain bounded. Moreover, all values ​​of c for which this sequence is bounded 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 non-intersecting parts, with some additional conditions).

fractals and life

Nowadays, the theory of fractals is widely used in various fields 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 (here, the self-similarity property of fractals is mainly used - after all, in order to remember a small fragment of a picture and transformations with which you can get the rest of the parts, it takes much less memory than to store the entire file). By adding random perturbations to the formulas that define the fractal, you can get stochastic fractals that very plausibly convey some real objects - relief elements, the surface of water bodies, 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, they began to produce antennas that have a fractal shape. Taking up little space, they provide quite high-quality signal reception. Economists use fractals to describe currency fluctuation curves (this property was discovered by Mandelbrot over 30 years ago). This concludes this short excursion into the world of fractals, amazing in its beauty and diversity.

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 figures in beauty and variety can be obtained from relatively simple structures 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 of 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 built by recursive procedures.

At the turn of the 19th and 20th centuries, the study of fractals was more episodic than systematic, because earlier 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 it is quite simple to draw it. It turned out that it has the properties of a fractal. One variation 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, in which another fractal is described - the Lévy C-curve. All of the above fractals can be conditionally attributed to one class of constructive (geometric) fractals.

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

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

In 1982, Mandelbrot's book "The 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 made the main emphasis in his presentation not on ponderous formulas and mathematical constructions, but on the geometric intuition of readers. Thanks to computer-assisted illustrations and historical stories, with which the author skillfully diluted the scientific component of the monograph, the book became a bestseller, and the 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 trend in art appeared - fractal painting, and almost any computer owner could do it. Now on the Internet you can easily find many sites dedicated to this topic.