Theory of fractals. The amazing world of fractals

Municipal budgetary educational institution

"Siverskaya secondary school No. 3"

Research work

mathematics.

Did the job

8th grade student

Emelin Pavel

scientific adviser

mathematic teacher

Tupitsyna Natalya Alekseevna

p. Siversky

year 2014

Mathematics is all permeated with beauty and harmony,

You just have to see this beauty.

B. Mandelbrot

Introduction

Chapter 1. The history of the emergence of fractals. _______ 5-6 pp.

Chapter 2. Classification of fractals.____________________6-10pp.

geometric fractals

Algebraic fractals

Stochastic fractals

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

Chapter 4. Application of fractals _______________13-15pp.

Chapter 5 Practical work __________________ 16-24pp.

Conclusion_________________________________25.page

List of literature and Internet resources _______ 26 p.

Introduction

Maths,

if you look at it right,

reflects not only the truth,

but also incomparable beauty.

Bertrand Russell

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

Mathematics is the oldest science. It seemed to most people that the geometry in nature was limited to such simple shapes as a line, a circle, a polygon, a sphere, and so on. As it turned out, many natural systems are so complex that using only familiar objects of ordinary geometry to model them seems hopeless. How, for example, to build a model of a mountain range or tree crown in terms of geometry? How to describe the diversity of biological diversity that we observe in the world of plants and animals? How to imagine the whole complexity of the circulatory system, consisting of many capillaries and vessels and delivering blood to every cell of the human body? Imagine the structure of the lungs and kidneys, resembling trees with a branchy crown in structure?

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

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

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

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

Objective: creating objects that are very similar to nature.

Research methods Keywords: comparative analysis, synthesis, modeling.

Tasks:

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

G. Koch, V. Sierpinsky and others;

familiarity with various types of fractal sets;

study of popular science literature on this issue, acquaintance with

scientific hypotheses;

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

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

conducting an experiment to create your own fractal images.

Core question of the job:

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

Subject of study: Fractal geometry.

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

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

Research methods: analytical, search.

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

Expected results: In the course of work, I will be able to expand my knowledge in the field of mathematics, see the beauty of fractal geometry, and start working on creating my own fractals.

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

Chapter 1

B Enua Mandelbrot

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

Fractal (lat. fractus - crushed, broken, broken) - a term meaning a complex geometric figure with the property of self-similarity, that is, composed of several parts, each of which is similar to the entire figure as a whole.

The mathematical objects to which it refers are characterized by extremely interesting properties. In ordinary geometry, a line has one dimension, a surface has two dimensions, and a spatial figure is three-dimensional. Fractals, on the other hand, are not lines or surfaces, but, if you can imagine it, something in between. With an increase in size, the volume of the fractal also increases, but its dimension (exponent) is not an integer, but a fractional value, and therefore the border of the fractal figure is not a line: at high magnification, it becomes clear that it is blurred and consists of spirals and curls, repeating in small the scale of the figure itself. Such geometric regularity is called scale invariance or self-similarity. It is she who determines the fractional dimension of fractal figures.

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

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

Liquid, gas, solid are the three usual physical states of matter that exists in the three-dimensional world. But what is the dimension of the puff of smoke, clouds, or rather, their boundaries, continuously blurred by turbulent air movement?

Basically, fractals are classified into three groups:

Algebraic fractals

Stochastic fractals

geometric fractals

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

Chapter 2. Classification of fractals

geometric fractals

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

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

Fractals of this class are the most visual, because they are immediately visible self-similarity at any scale of observation. In the two-dimensional case, such fractals can be obtained by specifying some broken line, called a generator. In one step of the algorithm, each of the segments that make up the broken line is replaced by a broken line-generator, in the appropriate scale. As a result of the endless repetition of this procedure (or, more precisely, when passing to the limit), a fractal curve is obtained. With the apparent complexity of the resulting curve, its general form is given only by the shape of the generator. Examples of such curves are: Koch curve (Fig.7), Peano curve (Fig.8), Minkowski curve.

At the beginning of the 20th century, mathematicians were looking for curves that did not have a tangent at any point. This meant that the curve abruptly changed its direction, and, moreover, at an enormously high speed (the derivative is equal to infinity). The search for these curves was caused not just by the idle interest of mathematicians. The fact is that at the beginning of the 20th century, quantum mechanics developed very rapidly. Researcher M. Brown sketched the trajectory of suspended particles in water and explained this phenomenon as follows: randomly moving liquid atoms hit suspended particles and thereby set them in motion. After such an explanation of Brownian motion, scientists were faced with the task of finding a curve that would best show the motion of Brownian particles. To do this, the curve had to meet the following properties: not have a tangent at any point. The mathematician Koch proposed one such curve.

To the Koch curve is a typical geometric fractal. The process of its construction is as follows: we take a single segment, divide it into three equal parts and replace the middle interval with an equilateral triangle without this segment. As a result, a broken line is formed, consisting of four links of length 1/3. At the next step, we repeat the operation for each of the four resulting links, and so on ...

The limit curve is Koch curve.

Snowflake Koch. By performing a similar transformation on the sides of an equilateral triangle, you can get a fractal image of a Koch snowflake.

T
Another simple representative of a geometric fractal is Sierpinski square. It is built quite simply: The square is divided by straight lines parallel to its sides into 9 equal squares. The central square is removed from the square. It turns out a set consisting of 8 remaining squares of the "first rank". Doing the same with each of the squares of the first rank, we get a set consisting of 64 squares of the second rank. Continuing this process indefinitely, we obtain an infinite sequence or Sierpinski square.

Algebraic fractals

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

They are obtained using non-linear processes in n-dimensional spaces. It is known that nonlinear dynamical systems have several stable states. The state in which the dynamical system finds itself after a certain number of iterations depends on its initial state. Therefore, each stable state (or, as they say, an attractor) has a certain area of ​​initial states, from which the system will necessarily fall into the considered final states. 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 attraction regions 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 fancy multicolor patterns. A surprise for mathematicians was the ability to generate very complex structures using primitive algorithms.

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

Part of the boundary of the Mandelbrot set, magnified 200 times.

The Mandelbrot set contains points that duringendless the number of iterations does not go to infinity (points that are black). Points belonging to the boundary of the set(this is where complex structures arise) go to infinity in a finite number of iterations, and points lying outside the set go to infinity after several iterations (white background).

P



An example of another algebraic fractal is the Julia set. There are 2 varieties of this fractal. Surprisingly, the Julia sets are formed according to the same formula as the Mandelbrot set. The Julia set was invented by the French mathematician Gaston Julia, after whom the set was named.

And
interesting fact, some algebraic fractals strikingly resemble images of animals, plants and other biological objects, as a result of which they are called biomorphs.

Stochastic fractals

Another well-known class of fractals are stochastic fractals, which are obtained if any of its parameters are randomly changed in an iterative process. This results in objects very similar to natural ones - asymmetrical trees, indented coastlines, etc.

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

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

E
If we look at this fractal in a section, then we will see this fractal is voluminous, and has a “roughness”, just because of this “roughness” there is a very important application of this fractal.

Let's say you want to describe the shape of a mountain. Ordinary figures from Euclidean geometry will not help here, because they do not take into account the surface topography. But when combining conventional geometry with fractal geometry, you can get the very “roughness” of the mountain. Plasma must be applied to an ordinary cone and we will get the relief of the mountain. Such operations can be performed with many other objects in nature, thanks to stochastic fractals, nature itself can be described.

Now let's talk about geometric fractals.

.

Chapter 3 "The Fractal Geometry of Nature"

Why is geometry often referred to as "cold" and "dry"? One reason is its inability to describe the shape of a cloud, mountain, coastline or tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, tree bark is not smooth; but complexity of a completely different level. The number of different length scales of natural objects for all practical purposes is infinite. "

(Benoit Mandelbrot "The Fractal Geometry of Nature" ).

To The beauty of fractals is twofold: it delights the eye, as evidenced by at least the world-wide exhibition of fractal images, organized by a group of Bremen mathematicians under the leadership of Peitgen and Richter. Later, the exhibits of this grandiose exhibition were captured in illustrations for the book "The Beauty of Fractals" by the same authors. But there is another, more abstract or sublime, aspect of the beauty of fractals, open, according to R. Feynman, only to the mental gaze of the theorist, in this sense, fractals are beautiful with the beauty of a difficult mathematical problem. Benoit Mandelbrot pointed out to his contemporaries (and, presumably, to his descendants) an unfortunate gap in Euclid's Elements, according to which, not noticing the omission, for almost two millennia mankind comprehended the geometry of the surrounding world and learned the mathematical rigor of presentation. Of course, both aspects of the beauty of fractals are closely interconnected and do not exclude, but mutually complement each other, although each of them is self-sufficient.

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

As for the correspondence to the real world, fractal geometry describes a very wide class of natural processes and phenomena, and therefore we can, following B. Mandelbrot, rightfully speak about the fractal geometry of nature. New - fractal objects have unusual properties. The lengths, areas and volumes of some fractals are equal to zero, others turn to infinity.

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

sea ​​shells

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

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

P fern is also a good example of a fractal among flora.

Peacocks everyone is known for their colorful plumage, in which solid fractals are hidden.

Ice, frost patterns on the windows, these are also fractals

O
t enlarged image leaflet, before tree branches- you can find fractals in everything

Fractals are everywhere and everywhere in the nature around us. The entire universe is built according to surprisingly harmonious laws with mathematical precision. Is it possible after that to think that our planet is a random clutch of particles? Hardly.

Chapter 4

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

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

The advantages of fractal image compression algorithms are the very small size of the packed file and the short image recovery time. Fractally packed images can be scaled without the appearance of pixelization (poor image quality - large squares). 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 drawback.

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

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

T
Fractal geometry is also used to design of antenna devices. This was first used by American engineer Nathan Cohen, who then lived in the center of Boston, where the installation of external antennas on buildings was prohibited. Cohen cut out a Koch curve shape from aluminum foil and then pasted it onto a piece of paper before attaching it to a receiver. It turned out that such an antenna works no worse than a conventional one. And although the physical principles of such an antenna have not been studied so far, this did not prevent Cohen from establishing his own company and setting up their serial production. At the moment, the American company “Fractal Antenna System” has developed a new type of antenna. Now you can stop using protruding external antennas in mobile phones. The so-called fractal antenna is located directly on the main board inside the device.

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

Chapter 5. Practical work.

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

First - "Necklace"(Fig. 7). The circle is the initiator of this fractal. This circle consists of a certain number of the same circles, but of smaller sizes, and it itself is one of several circles that are the same, but of larger sizes. So the process of education is endless and it can be carried out both in one direction and in the opposite direction. Those. the figure can be enlarged by taking only one small arc, or it can be reduced by considering its construction from smaller ones.

rice. 7.

Fractal "Necklace"

The second fractal is "Victory"(Fig. 8). He got this name because it outwardly resembles the Latin letter “V”, that is, “victory”-victory. This fractal consists of a certain number of small “v”, which make up one large “V”, and in the left half, in which the small ones are placed so that their left halves form one straight line, the right part is built in the same way. Each of these "v" is built in the same way and continues this to infinity.

Fig.8. Fractal "Victory"

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

Fig. 9. Fractal "Square"

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

Rice. 11. Fractal "Rose *"

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

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

R
is. 12. Fractal "Pentagram".

Rice. 13. Fractal "Pentagram *"

Rice. 14 fractal "Black hole"

Experiment No. 1 "Tree"

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

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

FROM The next step is to write the script.

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

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

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

As a final step, I copied this image and pasted it downsized and rotated. Here is the end result.

Conclusion

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

Fractal graphics is not just a set of self-repeating images, it is a model of the structure and principle of any being. Our whole life is represented by fractals. All nature around us consists of them. It should be noted that fractals are widely used in computer games, where terrains are often fractal images based on three-dimensional models of complex sets. Fractals greatly facilitate the drawing of computer graphics; with the help of fractals, many special effects, various fabulous and incredible pictures, etc. are created. Also, with the help of fractal geometry, trees, clouds, coasts and all other nature are drawn. Fractal graphics are needed everywhere, and the development of "fractal technologies" is one of the most important tasks today.

In the future, I plan to learn how to build algebraic fractals when I study complex numbers in more detail. I also want to try to build my fractal image in the Pascal programming language using cycles.

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

1. Compress images and information

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

3. Data encryption using fractal algorithms

4. Creating fractal music

5. System modeling

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

Thus, the concept of fractals becomes not only a part of "pure" science, but also an element of human culture. Fractal science is still very young and has a great future ahead of it. The beauty of fractals is far from being exhausted and will still give us many masterpieces - those that delight the eye, and those that bring true pleasure to the mind.

10. References

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

Vitolin D. The use of fractals in computer graphics. // Computerworld-Russia.-1995

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

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

Morozov A.D. Introduction to the theory of fractals. Nizhny Novgorod: Nizhegorod Publishing House. university 1999

Paytgen H.-O., Richter P. H. The beauty of fractals. - M.: "Mir", 1993.

Internet resources

http://www.ghcube.com/fractals/determin.html

http://fractals.nsu.ru/fractals.chat.ru/

http://fractals.nsu.ru/animations.htm

http://www.cootey.com/fractals/index.html

http://fraktals.ucoz.ru/publ

http://sakva.narod.ru

http://rusnauka.narod.ru/lib/author/kosinov_n/12/

http://www.cnam.fr/fractals/

http://www.softlab.ntua.gr/mandel/

http://subscribe.ru/archive/job.education.maths/201005/06210524.html

Fractals have been known for almost a century, are well studied and have numerous applications in life. However, 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.

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 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. 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, 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 the 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 generated 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 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 we continue this process indefinitely, then in the limit we get a fractal.

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. Over complex numbers, the basic mathematical operations 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 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 drawing 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, one can obtain 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.

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

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

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

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

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

⇡ Benoit Mandelbrot: the father of fractal geometry

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

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

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

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

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

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

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

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

Gaston Julia (always masked - WWI injury)

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

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

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

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

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

⇡ Lauren Carpenter: art created by nature

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

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

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

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

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

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

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

One of the first 3D renderings using the fractal algorithm

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

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

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

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

⇡ Fractal antennas: less is better, but better

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

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

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

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

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

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

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

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

⇡ Fractal dimensions: the mind does not understand

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

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

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

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

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

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

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

⇡ Fractal in the browser

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

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

⇡ XaoS: fractals for every taste

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

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

⇡ Fractal Zoomer: compact fractal generator

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

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

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

⇡ Mandelbulb3D: 3D fractal editor

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

And this fractal can be eaten

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

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

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

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

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

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

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

⇡ Aural: how fractals sing

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

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

Fractals: musical pause

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

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

⇡ Conclusion

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

We have already written about how the abstract mathematical theory of chaos has found applications in a variety of sciences - from physics to economics and political science. Now we will give another similar example - the theory of fractals. There is no strict definition of the concept of "fractal" even in mathematics. They say something like that, of course. But the “ordinary person” does not understand this. How about, for example, this phrase: "A fractal is a set with a fractional Hausdorff dimension, which is greater than the topological one." Nevertheless, they, fractals, surround us and help to understand many phenomena from different spheres of life.

How it all started

For a long time, no one except professional mathematicians was interested in fractals. Before the advent of computers and related software. Everything changed in 1982, when Benoit Mandelbrot's book "The Fractal Geometry of Nature" was published. This book has become a bestseller, not so much because of the simple and understandable presentation of the material (although this statement is very relative - a person who does not have a professional mathematical education will not understand anything in it), but because of the computer illustrations of fractals given, which are really mesmerizing. Let's look at these pictures. They are really worth it.

And there are many such pictures. But what does all this splendor have to do with our real life and what surrounds us in nature and the everyday world? It turns out the most direct.

But first, let's say a few words about the fractals themselves, as geometric objects.

What is a fractal, in simple terms

First. How they, fractals, are built. This is a rather complicated procedure that uses special transformations on the complex plane (you don’t need to know what it is). The only important thing is that these transformations are repetitive (occur, as they say in mathematics, iterations). It is as a result of this repetition that fractals arise (the ones you saw above).

Second. A fractal is a self-similar (exactly or approximately) structure. This means the following. If you bring a microscope that magnifies the image, for example, 100 times, to any of the presented pictures, and look at a fragment of a fractal piece that has fallen into the eyepiece, you will find that it is identical to the original image. If you take a stronger microscope that magnifies the image 1000 times, you will find that a piece of the fragment of the previous image that fell into the eyepiece has the same or very similar structure.

This leads to a very important conclusion for what follows. A fractal has an extremely complex structure that repeats itself on different scales. But the more we get deeper into its device, the more complex it becomes in general. And the quantitative estimates of the properties of the original picture may begin to change.

Now we will leave abstract mathematics and move on to the things around us - so, it would seem, simple and understandable.

Fractal objects in nature

Coastline

Imagine that you are photographing an island, such as Britain, from Earth orbit. You will get the same image as on the geographic map. The smooth outline of the coast, from all sides - the sea.

Finding the length of the coastline is very simple. Take an ordinary thread and carefully lay it along the borders of the island. Then, measure its length in centimeters and multiply the resulting number by the scale of the map - there are some kilometers in one centimeter. Here is the result.

And now the next experiment. You fly in an airplane at a bird's eye view and photograph the coastline. It turns out a picture similar to photographs from a satellite. But this coastline is indented. Small bays, gulfs, fragments of land protruding into the sea appear on your pictures. All this is true, but could not be seen from the satellite. The structure of the coastline is becoming more complex.

Let's say, having arrived home, you made a detailed map of the coastline based on your pictures. And we decided to measure its length with the help of the same thread, laying it out strictly according to the new data you received. The new coastline length value will exceed the old one. And significant. This is intuitively clear. After all, now your thread should go around the shores of all bays and bays, and not just go along the coast.

Note. We zoomed out and things got a lot more complex and confusing. Like fractals.

And now for another iteration. You are walking along the same coast. And fix the relief of the coastline. It turns out that the shores of the bays and bays that you shot from the plane are not at all as smooth and simple as you thought in your pictures. They have a complex structure. And so, if you map this "pedestrian" coastline, it will grow even longer.

Yes, there are no infinities in nature. But it is quite clear that the coastline is a typical fractal. It remains the same, but its structure becomes more and more complex as you look closer (think of the microscope example).

This is truly an amazing phenomenon. We are accustomed to the fact that any geometric object limited in size on a plane (square, triangle, circle) has a fixed and finite length of its boundaries. But here everything is different. The length of the coastline in the limit turns out to be infinite.

Wood

Let's imagine a tree. Ordinary tree. Some kind of loose linden. Let's look at her trunk. around the root. It is a slightly deformed cylinder. Those. has a very simple form.

Let's lift our eyes up. Branches begin to emerge from the trunk. Each branch, at its beginning, has the same structure as the trunk - cylindrical, in terms of geometry. But the structure of the whole tree has changed. It has become much more complex.

Now let's look at these branches. Smaller branches extend from them. At their base they have the same slightly deformed cylindrical shape. Like the same trunk. And then much smaller branches depart from them. And so on.

The tree reproduces itself, at every level. At the same time, its structure is constantly becoming more complex, but remains similar to itself. Isn't it a fractal?

Circulation

Here is the human circulatory system. It also has a fractal structure. There are arteries and veins. According to one of them, blood comes to the heart (veins), according to others it comes from it (arteries). And then, the circulatory system begins to resemble the same tree that we talked about above. Vessels, while maintaining their structure, become thinner and more branched. They penetrate into the most remote areas of our body, bring oxygen and other vital components to every cell. This is a typical fractal structure that reproduces itself on smaller and smaller scales.

River drains

"From afar, the Volga River flows for a long time." On a geographical map, this is such a blue winding line. Well, the major tributaries are marked. Oka, Kama. What if we zoom out? It turns out that these tributaries are much larger. Not only near the Volga itself, but also near the Oka and Kama. And they have their own tributaries, only smaller ones. And those have theirs. A structure emerges that is surprisingly similar to the human circulatory system. And again the question arises. What is the extent of this entire water system? If you measure the length of only the main channel, everything is clear. You can read it in any textbook. What if everything is measured? Again, in the limit, infinity is obtained.

Our Universe

Of course, on the scale of billions of light years, it, the Universe, is arranged uniformly. But let's take a closer look at it. And then we will see that there is no homogeneity in it. Somewhere there are galaxies (star clusters), somewhere there is emptiness. Why? Why the distribution of matter obeys irregular hierarchical laws. And what happens inside galaxies (another zoom out). Somewhere there are more stars, somewhere less. Somewhere there are planetary systems, as in our solar system, but somewhere not.

Doesn't the fractal essence of the world manifest itself here? Now, of course, there is a huge gap between the general theory of relativity, which explains the emergence of our universe and its structure, and fractal mathematics. But who knows? Perhaps all this will someday be brought to a "common denominator", and we will look at the space around us with completely different eyes.

To practical matters

Many such examples can be cited. But let's get back to more prosaic things. Take, for example, economics. It would seem, and here fractals. It turns out, very much so. An example of this is the stock markets.

Practice shows that economic processes are often chaotic and unpredictable. Mathematical models that existed until today that tried to describe these processes did not take into account one very important factor - the ability of the market to self-organize.

This is where the theory of fractals comes to the rescue, which have the properties of "self-organization", reproducing themselves at the level of different scales. Of course, a fractal is a purely mathematical object. And in nature, and in the economy, they do not exist. But there is a concept of fractal phenomena. They are fractals only in a statistical sense. Nevertheless, the symbiosis of fractal mathematics and statistics makes it possible to obtain sufficiently accurate and adequate forecasts. This approach is especially effective in the analysis of stock markets. And these are not "notions" of mathematicians. Expert data shows that many participants in the stock markets spend a lot of money to pay specialists in the field of fractal mathematics.

What does the theory of fractals give? It postulates a general, global dependence of pricing on what happened in the past. Of course, locally the pricing process is random. But random jumps and falls in prices, which can occur momentarily, have the peculiarity of gathering in clusters. Which are reproduced on a large scale of time. Therefore, by analyzing what was once, we can predict how long this or that market development trend (growth or fall) will last.

Thus, on a global scale, this or that market "reproduces" itself. Assuming random fluctuations caused by a mass of external factors at each particular moment in time. But global trends persist.

Conclusion

Why is the world arranged according to the fractal principle? The answer, perhaps, is that fractals, as a mathematical model, have the property of self-organization and self-similarity. At the same time, each of their forms (see the pictures given at the beginning of the article) is arbitrarily complex, but lives its own life, developing similar forms to itself. Isn't that how our world works?

And here is the society. Some idea comes up. Quite abstract at first. And then "penetrates the masses." Yes, it does change somehow. But in general it is preserved. And it turns at the level of most people into a goal designation of the life path. Here is the same USSR. The next congress of the CPSU adopted the next landmark decisions, and it all went downhill. On a smaller scale. City committees, party committees. And so on for each person. repeating structure.

Of course, fractal theory does not allow us to predict future events. And this is hardly possible. But much that surrounds us, and what happens in our daily life, allows us to look with completely different eyes. Conscious.

I came across a mention of the "Theory of Fractals" in the TV series "Jeremiah" and became interested in this rather elegant theory, which modern metaphysicians use to prove the existence of God. The theory of fractals has a very young age. It appeared in the late sixties at the intersection of mathematics, computer science, linguistics and biology. At that time, computers were increasingly penetrating people's lives, scientists began to use them in their research, and the number of computer users was growing. For the mass use of computers, it became necessary to facilitate the process of communication between a person and a machine. If at the very beginning of the computer era, a few user programmers selflessly entered commands in machine codes and received results in the form of endless ribbons of paper, then with the massive and loaded mode of using computers, it became necessary to invent a programming language that would be understandable to a machine, and in at the same time, would be easy to learn and use. That is, the user would need to enter only one command, and the computer would decompose it into simpler ones, and would already execute them. To facilitate the writing of translators, the theory of fractals arose at the intersection of computer science and linguistics, which allows one to strictly specify the relationship between algorithmic languages. And the Danish mathematician and biologist A. Lindenmeer came up with one such grammar in 1968, which he called the L-system, which, he believed, also models the growth of living organisms, in particular the formation of bushes and branches in plants.

A fractal (lat. fractus - crushed, broken, broken) is a complex geometric figure that has the property of self-similarity, that is, it is composed of several parts, each of which is similar to the entire figure as a whole. In a broader sense, fractals are understood as sets of points in Euclidean space that have a fractional metric dimension (in the sense of Minkowski or Hausdorff), or a metric dimension strictly greater than the topological one. A fractal form of the cauliflower subspecies (Brassica cauliflora). A fractal is an infinitely self-similar geometric figure, each fragment of which is repeated when zoomed out.

Benoit Mandelbrot can rightfully be considered the father of fractals. Mandelbrot is the inventor of the term "fractal". Mandelbrot
wrote: “I coined the word “fractal”, taking as a basis the Latin adjective “fractus”, meaning irregular, recursive,
fragmentary. The first definition of fractals was also given by B. Mandelbrot. The figure shows just the classic fractal model - the Mandelbrot Set.

Speaking primitively, the fractal theory is the ability of chaotic structures to self-organize into a system. An attractor is a set of states (more precisely, points of the phase space) of a dynamic system, to which it tends over time. The simplest variants of the attractor are an attractive fixed point (for example, in the problem of a pendulum with friction) and a periodic trajectory (an example is self-excited oscillations in a positive feedback loop), but there are also much more complex examples. Some dynamical systems are always chaotic, but in most cases chaotic behavior is observed only when the parameters of the dynamical system belong to some special subspace.

The most interesting are the cases of chaotic behavior, when a large set of initial conditions leads to a change in the orbits of the attractor. An easy way to demonstrate a chaotic attractor is to start from a point in the attractor's region of attraction and then plot its subsequent orbit. Due to the state of topological transitivity, this is similar to mapping the picture of a complete finite attractor. For example, in a system describing a pendulum, the space is two-dimensional and consists of position and velocity data. You can plot the positions of the pendulum and its speed. The position of the pendulum at rest will be a point, and one period of oscillation will look like a simple closed curve on the graph. A graph in the form of a closed curve is called an orbit. The pendulum has an infinite number of such orbits, forming in appearance a collection of nested ellipses.

Most types of motion are described by simple attractors, which are bounded cycles. Chaotic motion is described by strange attractors, which are very complex and have many parameters. For example, a simple three-dimensional weather system is described by the famous Lorenz attractor, one of the most famous diagrams of chaotic systems, not only because it was one of the first, but also because it is one of the most complex. Another such attractor is the Rössler map, which has a double period, similar to the logistic map. Strange attractors appear in both systems, both in continuous dynamical systems (such as the Lorentz system) and in some discrete ones (eg Hénon maps). Some discrete dynamical systems are called Julia systems by origin. Both strange attractors and Julia systems have a typical recursive, fractal structure. The Poincare-Bendixson theorem proves that a strange attractor can arise in a continuous dynamical system only if it has three or more dimensions. However, this limitation does not work for discrete dynamical systems. Discrete two- and even one-dimensional systems can have strange attractors. The movement of three or more bodies experiencing gravitational attraction under certain initial conditions may turn out to be a chaotic movement.

So, the property of chaotic systems to self-organize with the help of irregular attractors, according to some mathematicians, is an unprovable proof of the existence of God and His energy of creation of all things. Mystery!