The editors of NNN accidentally came across a very interesting material presented on the blog of user xtsarx, dedicated to elements of the theory fractals and its practical application. As is known, fractal theria plays an important role in the physics and chemistry of nanosystems. Having contributed to this good material, presented in a language accessible to a wide range of readers and supported by an abundance of graphic and even video material, we present it to your attention. We hope that NNN readers will find this material interesting.
Nature is so mysterious that the more you study it, the more questions appear... Night lightning - blue “jets” of branching discharges, frosty patterns on the window, snowflakes, mountains, clouds, tree bark - all this goes beyond the usual Euclidean geometry. We cannot describe a rock or the boundaries of an island using straight lines, circles, and triangles. And here they come to our aid fractals. What are these familiar strangers?
“Under a microscope, he discovered that on the flea
A flea that bites lives;
On that flea there is a tiny flea,
A tooth pierces a flea angrily
Flea, and so ad infinitum.” D. Swift.
A little bit of history
First ideas fractal geometry arose in the 19th century. Cantor, using a simple recursive (repeating) procedure, turned the line into a collection of unconnected points (the so-called Cantor Dust). He would take a line and remove the central third and then repeat the same with the remaining sections.
Rice. 1. Peano curve 1.2–5 iterations.
Peano drew a special kind of line. Peano did the following:: In the first step, he took a straight line and replaced it with 9 segments 3 times shorter than the length of the original line. Then he did the same with each segment of the resulting line. And so on ad infinitum. Its uniqueness is that it fills the entire plane. It is proved that for every point on the plane one can find a point belonging to the Peano line. Peano's curve and Cantor's dust went beyond ordinary geometric objects. They did not have a clear dimension. Cantor's dust seemed to be built on the basis of a one-dimensional straight line, but consisted of points (dimension 0). And the Peano curve was built on the basis of a one-dimensional line, and the result was a plane. In many other areas of science, problems appeared whose solution led to strange results similar to those described above (Brownian motion, stock prices). Each of us can do this procedure...
Father of Fractals
Until the 20th century, data about such strange objects was accumulated, without any attempt to systematize them. That was until I took them on Benoit Mandelbrot – father of modern fractal geometry and the word fractal.
Rice. 2. Benoit Mandelbrot.
While working as a mathematical analyst at IBM, he studied noise in electronic circuits that could not be described using statistics. Gradually comparing facts, he came to the discovery of a new direction in mathematics - fractal geometry.
The term “fractal” was introduced by B. Mandelbrot in 1975. According to Mandelbrot, fractal(from Latin “fractus” - fractional, broken, broken) is called structure consisting of parts similar to the whole. The property of self-similarity sharply distinguishes fractals from objects of classical geometry. Term self-similarity means the presence of a fine, repeating structure, both on the smallest scales of the object and on the macroscale.
Rice. 3. Towards the definition of the concept “fractal”.
Examples of self-similarity are: Koch, Levy, Minkowski curves, Sierpinski triangle, Menger sponge, Pythagorean tree, etc.
From a mathematical point of view, fractal- this is, first of all, set with fractional (intermediate, “not integer”) dimension. While a smooth Euclidean line fills exactly one-dimensional space, a fractal curve extends beyond the boundaries of one-dimensional space, intruding beyond the boundaries into two-dimensional space. Thus, the fractal dimension of a Koch curve will be between 1 and 2. This, first of all, means that For a fractal object, it is impossible to accurately measure its length! Of these geometric fractals, the first one is very interesting and quite famous - Koch's snowflake.
Rice. 4. Towards the definition of the concept “fractal”.
It is built on the basis equilateral triangle. Each line of which is replaced by 4 lines, each 1/3 of the original length. Thus, with each iteration, the length of the curve increases by a third. And if we make an infinite number of iterations, we will get a fractal - a Koch snowflake of infinite length. It turns out that our infinite curve covers a limited area. Try to do the same using methods and figures from Euclidean geometry.
Koch snowflake dimension(when a snowflake increases by 3 times, its length increases by 4 times) D=log(4)/log(3)=1.2619.
About the fractal itself
Fractals are finding more and more applications in science and technology. The main reason for this is that they describe the real world sometimes even better than traditional physics or mathematics. You can endlessly give examples of fractal objects in nature - these are clouds, and snow flakes, and mountains, and a flash of lightning, and finally, cauliflower. A fractal as a natural object is an eternal continuous movement, new formation and development.
Rice. 5. Fractals in economics.
Besides, fractals find application in decentralized computer networks And "fractal antennas" . The so-called “Brownian fractals” are very interesting and promising for modeling various stochastic (non-deterministic) “random” processes. In the case of nanotechnology, fractals also play an important role , because due to their hierarchical self-organization many nanosystems have a non-integer dimension, that is, they are fractals in their geometric, physicochemical or functional nature. For example, A striking example of chemical fractal systems are the molecules of “dendrimers” . In addition, the principle of fractality (self-similar, scaling structure) is a reflection of the hierarchical structure of the system and is therefore more general and universal than standard approaches to describing the structure and properties of nanosystems.
Rice. 6. “Dendrimer” molecules.
Rice. 7. Graphic model of communication in the architectural and construction process. The first level of interaction from the perspective of microprocesses.
Rice. 8. Graphic model of communication in the architectural and construction process. The second level of interaction from the perspective of macro processes (a fragment of the model).
Rice. 9. Graphic model of communication in the architectural and construction process. The second level of interaction from the perspective of macro processes (entire model)
Rice. 10. Planar development of the graphic model. The first homeostatic state.
Photo gallery of beautiful and unusual fractals
Rice. eleven.
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Correction and editing completed Filippov Yu.P.
Fractal
Fractal (lat. fractus- crushed, broken, broken) is a geometric figure that has the property of self-similarity, that is, composed of several parts, each of which is similar to the entire figure. In mathematics, fractals are understood as sets of points in Euclidean space that have a fractional metric dimension (in the sense of Minkowski or Hausdorff), or a metric dimension different from the topological one. Fractasm is an independent exact science of studying and composing fractals.
In other words, fractals are geometric objects with a fractional dimension. For example, the dimension of a line is 1, the area is 2, and the volume is 3. For a fractal, the dimension value can be between 1 and 2 or between 2 and 3. For example, the fractal dimension of a crumpled paper ball is approximately 2.5. In mathematics, there is a special complex formula for calculating the dimension of fractals. The branches of tracheal tubes, leaves on trees, veins in the hand, a river - these are fractals. In simple terms, a fractal is a geometric figure, a certain part of which is repeated again and again, changing in size - this is the principle of self-similarity. Fractals are similar to themselves, they are similar to themselves at all levels (i.e. at any scale). There are many different types of fractals. In principle, it can be argued that everything that exists in the real world is a fractal, be it a cloud or an oxygen molecule.
The word “chaos” makes one think of something unpredictable, but in fact, chaos is quite orderly and obeys certain laws. The goal of studying chaos and fractals is to predict patterns that, at first glance, may seem unpredictable and completely chaotic.
The pioneer in this field of knowledge was the French-American mathematician, Professor Benoit B. Mandelbrot. In the mid-1960s, he developed fractal geometry, the purpose of which was to analyze broken, wrinkled and fuzzy shapes. The Mandelbrot set (shown in the figure) is the first association that arises in a person when he hears the word “fractal”. By the way, Mandelbrot determined that the fractal dimension of the English coastline is 1.25.
Fractals are increasingly used in science. They describe the real world even better than traditional physics or mathematics. Brownian motion is, for example, the random and chaotic movement of dust particles suspended in water. This type of movement is perhaps the aspect of fractal geometry that has the most practical use. Random Brownian motion has a frequency response that can be used to predict phenomena involving large amounts of data and statistics. For example, Mandelbrot predicted changes in wool prices using Brownian motion.
The word "fractal" can be used not only as a mathematical term. In the press and popular science literature, a fractal can be called a figure that has any of the following properties:
It has a non-trivial structure at all scales. This is in contrast to regular figures (such as a circle, ellipse, graph of a smooth function): if we consider a small fragment of a regular figure on a very large scale, it will look like a fragment of a straight line. For a fractal, increasing the scale does not lead to a simplification of the structure; on all scales we will see an equally complex picture.
Is self-similar or approximately self-similar.
It has a fractional metric dimension or a metric dimension that exceeds the topological one.
The most useful use of fractals in computer technology is fractal data compression. At the same time, images are compressed much better than is done with conventional methods - up to 600:1. Another advantage of fractal compression is that when enlarged, there is no pixelation effect, which dramatically worsens the image. Moreover, a fractally compressed image often looks even better after enlargement than before. Computer scientists also know that fractals of infinite complexity and beauty can be generated by simple formulas. The film industry widely uses fractal graphics technology to create realistic landscape elements (clouds, rocks and shadows).
The study of turbulence in flows adapts very well to fractals. This allows us to better understand the dynamics of complex flows. Using fractals you can also simulate flames. Porous materials are well represented in fractal form due to the fact that they have a very complex geometry. To transmit data over distances, antennas with fractal shapes are used, which greatly reduces their size and weight. Fractals are used to describe the curvature of surfaces. An uneven surface is characterized by a combination of two different fractals.
Many objects in nature have fractal properties, for example, coasts, clouds, tree crowns, snowflakes, the circulatory system and the alveolar system of humans or animals.
Fractals, especially on a plane, are popular due to the combination of beauty with the ease of construction using a computer.
The first examples of self-similar sets with unusual properties appeared in the 19th century (for example, the Bolzano function, the Weierstrass function, the Cantor set). The term "fractal" was coined by Benoit Mandelbrot in 1975 and gained widespread popularity with the publication of his book "Fractal Geometry of Nature" in 1977.
The picture on the left shows a simple example of the Darer Pentagon fractal, which looks like a bunch of pentagons squashed together. In fact, it is formed by using a pentagon as an initiator and isosceles triangles, in which the ratio of the larger side to the smaller one is exactly equal to the so-called golden ratio (1.618033989 or 1/(2cos72°)) as a generator. These triangles are cut from the middle of each pentagon, resulting in a shape that looks like 5 small pentagons glued to one large one. 
Chaos theory says that complex nonlinear systems are hereditarily unpredictable, but at the same time claims that the way to express such unpredictable systems turns out to be correct not in exact equalities, but in representations of the behavior of the system - in graphs of strange attractors, which look like fractals. Thus, chaos theory, which many think of as unpredictability, turns out to be the science of predictability even in the most unstable systems. The study of dynamic systems shows that simple equations can give rise to chaotic behavior in which the system never returns to a stable state and no pattern appears. Often such systems behave quite normally up to a certain value of a key parameter, then experience a transition in which there are two possibilities for further development, then four, and finally a chaotic set of possibilities.
Schemes of processes occurring in technical objects have a clearly defined fractal structure. The structure of a minimal technical system (TS) implies the occurrence within the TS of two types of processes - the main one and the supporting ones, and this division is conditional and relative. Any process can be the main one in relation to the supporting processes, and any of the supporting processes can be considered the main one in relation to “its” supporting processes. The circles in the diagram indicate physical effects that ensure the occurrence of those processes for which it is not necessary to specially create “your own” vehicles. These processes are the result of interactions between substances, fields, substances and fields. To be precise, a physical effect is a vehicle whose operating principle we cannot influence, and we do not want or do not have the opportunity to interfere with its design. 
The flow of the main process shown in the diagram is ensured by the existence of three supporting processes, which are the main ones for the TS that generate them. To be fair, we note that for the functioning of even a minimal TS, three processes are clearly not enough, i.e. The scheme is very, very exaggerated.
Everything is far from being as simple as shown in the diagram. A process that is useful (needed by a person) cannot be performed with one hundred percent efficiency. The dissipated energy is spent on creating harmful processes - heating, vibration, etc. As a result, harmful ones arise in parallel with the beneficial process. It is not always possible to replace a “bad” process with a “good” one, so it is necessary to organize new processes aimed at compensating for consequences harmful to the system. A typical example is the need to combat friction, which forces one to organize ingenious lubrication schemes, use expensive anti-friction materials, or spend time on lubrication of components and parts or its periodic replacement.
Due to the inevitable influence of a changeable Environment, a useful process may need to be managed. Control can be carried out either using automatic devices or directly by a person. The process diagram is actually a set of special commands, i.e. algorithm. The essence (description) of each command is the totality of a single useful process, harmful processes accompanying it, and a set of necessary control processes. In such an algorithm, the set of supporting processes is a regular subroutine - and here we also discover a fractal. Created a quarter of a century ago, R. Koller's method makes it possible to create systems with a fairly limited set of only 12 pairs of functions (processes).
Self-similar sets with unusual properties in mathematics
Since the end of the 19th century, examples of self-similar objects with properties that are pathological from the point of view of classical analysis have appeared in mathematics. These include the following:
The Cantor set is a nowhere dense uncountable perfect set. By modifying the procedure, one can also obtain a nowhere dense set of positive length.
the Sierpinski triangle (“tablecloth”) and the Sierpinski carpet are analogues of the Cantor set on the plane.
Menger's sponge is an analogue of the Cantor set in three-dimensional space;
examples of Weierstrass and Van der Waerden of a nowhere differentiable continuous function.
Koch curve is a non-self-intersecting continuous curve of infinite length that does not have a tangent at any point;
Peano curve is a continuous curve passing through all points of the square.
the trajectory of a Brownian particle is also nowhere differentiable with probability 1. Its Hausdorff dimension is two
Recursive procedure for obtaining fractal curves
Construction of the Koch curve
There is a simple recursive procedure for obtaining fractal curves on a plane. Let us define an arbitrary broken line with a finite number of links, called a generator. Next, let’s replace each segment in it with a generator (more precisely, a broken line similar to a generator). In the resulting broken line, we again replace each segment with a generator. Continuing to infinity, in the limit we get a fractal curve. The figure on the right shows the first four steps of this procedure for the Koch curve.
Examples of such curves are:
dragon Curve,
Koch curve (Koch snowflake),
Lewy Curve,
Minkowski curve,
Hilbert curve,
Broken (curve) of a dragon (Harter-Haithway Fractal),
Peano curve.
Using a similar procedure, the Pythagorean tree is obtained.
Fractals as fixed points of compression mappings
The self-similarity property can be expressed mathematically strictly as follows. Let be contractive mappings of the plane. Consider the following mapping on the set of all compact (closed and bounded) subsets of the plane:
It can be shown that the mapping is a contraction mapping on the set of compacta with the Hausdorff metric. Therefore, by Banach's theorem, this mapping has a unique fixed point. This fixed point will be our fractal.
The recursive procedure for obtaining fractal curves described above is a special case of this construction. All mappings in it are similarity mappings, and - the number of generator links.
For the Sierpinski triangle and the map , , are homotheties with centers at the vertices of a regular triangle and coefficient 1/2. It is easy to see that the Sierpinski triangle transforms into itself when displayed.
In the case where the mappings are similarity transformations with coefficients, the dimension of the fractal (under some additional technical conditions) can be calculated as a solution to the equation. Thus, for the Sierpinski triangle we obtain 
.
By the same Banach theorem, starting with any compact set and applying iterations of the map to it, we obtain a sequence of compact sets converging (in the sense of the Hausdorff metric) to our fractal.
Fractals in complex dynamics
Julia set
Another Julia set
Fractals arise naturally when studying nonlinear dynamical systems. The most studied case is when a dynamical system is specified by iterations of a polynomial or a holomorphic function of a complex variable on the plane. The first studies in this area date back to the beginning of the 20th century and are associated with the names of Fatou and Julia.
Let F(z) - polynomial, z 0 is a complex number. Consider the following sequence: z 0 , z 1 =F(z 0), z 2 =F(F(z 0)) = F(z 1),z 3 =F(F(F(z 0)))=F(z 2), …
We are interested in the behavior of this sequence as it tends n to infinity. This sequence can:
strive towards infinity,
strive for the ultimate limit
exhibit cyclic behavior in the limit, for example: z 1 , z 2 , z 3 , z 1 , z 2 , z 3 , …
behave chaotically, that is, do not demonstrate any of the three types of behavior mentioned.
Sets of values z 0, for which the sequence exhibits one particular type of behavior, as well as multiple bifurcation points between different types, often have fractal properties.
Thus, the Julia set is the set of bifurcation points for the polynomial F(z)=z 2 +c(or other similar function), that is, those values z 0 for which the behavior of the sequence ( z n) can change dramatically with arbitrarily small changes z 0 .
Another option for obtaining fractal sets is to introduce a parameter into the polynomial F(z) and consideration of the set of those parameter values for which the sequence ( z n) exhibits a certain behavior at a fixed z 0 . Thus, the Mandelbrot set is the set of all , for which ( z n) For F(z)=z 2 +c And z 0 does not go to infinity.
Another famous example of this kind is Newton's pools.
It is popular to create beautiful graphic images based on complex dynamics by coloring plane points depending on the behavior of the corresponding dynamic systems. For example, to complete the Mandelbrot set, you can color the points depending on the speed of aspiration ( z n) to infinity (defined, say, as the smallest number n, at which | z n| will exceed a fixed large value A.
Biomorphs are fractals built on the basis of complex dynamics and reminiscent of living organisms.
Stochastic fractals
Randomized fractal based on Julia set
Natural objects often have a fractal shape. Stochastic (random) fractals can be used to model them. Examples of stochastic fractals:
trajectory of Brownian motion on the plane and in space;
boundary of the trajectory of Brownian motion on a plane. In 2001, Lawler, Schramm and Werner proved Mandelbrot's hypothesis that its dimension is 4/3.
Schramm-Löwner evolutions are conformally invariant fractal curves that arise in critical two-dimensional models of statistical mechanics, for example, in the Ising model and percolation.
various types of randomized fractals, that is, fractals obtained using a recursive procedure into which a random parameter is introduced at each step. Plasma is an example of the use of such a fractal in computer graphics.
In nature
Front view of the trachea and bronchi
Bronchial tree
Network of blood vessels
Application
Natural Sciences
In physics, fractals naturally arise when modeling nonlinear processes, such as turbulent fluid flow, complex diffusion-adsorption processes, flames, clouds, etc. Fractals are used when modeling porous materials, for example, in petrochemistry. In biology, they are used to model populations and to describe internal organ systems (the blood vessel system).
Radio engineering
Fractal antennas
The use of fractal geometry in the design of antenna devices was first used by American engineer Nathan Cohen, who then lived in downtown Boston, where the installation of external antennas on buildings was prohibited. Nathan cut out a Koch curve shape from aluminum foil and pasted it onto a piece of paper, then attached it to the receiver. Cohen founded his own company and started their serial production.
Computer science
Image compression
Main article: Fractal compression algorithm
Fractal tree
There are image compression algorithms using fractals. They are based on the idea that instead of the image itself, one can store a compression map for which this image (or some close one) is a fixed point. One of the variants of this algorithm was used [ source not specified 895 days] by Microsoft when publishing its encyclopedia, but these algorithms were not widely used.
Computer graphics
Another fractal tree
Fractals are widely used in computer graphics to construct images of natural objects, such as trees, bushes, mountain landscapes, sea surfaces, and so on. There are many programs used to generate fractal images, see Fractal Generator (program).
Decentralized networks
The IP address assignment system in the Netsukuku network uses the principle of fractal information compression to compactly store information about network nodes. Each node in the Netsukuku network stores only 4 KB of information about the state of neighboring nodes, while any new node connects to the common network without the need for central regulation of the distribution of IP addresses, which, for example, is typical for the Internet. Thus, the principle of fractal information compression guarantees completely decentralized, and therefore, the most stable operation of the entire network.
Mathematics,
if you look at it correctly,
reflects not only the truth,
but also incomparable beauty.
Bertrand Russell.
You have, of course, heard about fractals. You've certainly seen these breathtaking pictures from Bryce3d that are more real than reality itself. Mountains, clouds, tree bark - all this goes beyond the usual Euclidean geometry. We cannot describe a rock or the boundaries of an island using straight lines, circles, and triangles. And here fractals come to our aid. What are these familiar strangers? When did they appear?
History of appearance.
The first ideas of fractal geometry arose in the 19th century. Cantor, using a simple recursive (repeating) procedure, turned the line into a collection of unconnected points (the so-called Cantor Dust). He would take a line and remove the central third and then repeat the same with the remaining sections. Peano drew a special kind of line (Figure No. 1). To draw it, Peano used the following algorithm.
In the first step, he took a straight line and replaced it with 9 segments 3 times shorter than the length of the original line (Part 1 and 2 of Figure 1). Then he did the same with each segment of the resulting line. And so on ad infinitum. Its uniqueness is that it fills the entire plane. It is proved that for every point on the plane one can find a point belonging to the Peano line. Peano's curve and Cantor's dust went beyond ordinary geometric objects. They did not have a clear dimension. Cantor's dust seemed to be built on the basis of a one-dimensional straight line, but consisted of points (dimension 0). And the Peano curve was built on the basis of a one-dimensional line, and the result was a plane. In many other areas of science, problems appeared whose solution led to strange results similar to those described above (Brownian motion, stock prices).
Father of Fractals
Until the 20th century, data about such strange objects was accumulated, without any attempt to systematize them. That was until Benoit Mandelbrot, the father of modern fractal geometry and the word fractal, took up them. While working as a mathematical analyst at IBM, he studied noise in electronic circuits that could not be described using statistics. Gradually comparing the facts, he came to the discovery of a new direction in mathematics - fractal geometry.
What is a fractal? Mandelbrot himself derived the word fractal from the Latin word fractus, which means broken (divided into parts). And one of the definitions of a fractal is a geometric figure made up of parts and which can be divided into parts, each of which will represent a smaller copy of the whole (at least approximately).
To imagine a fractal more clearly, let’s consider an example given in B. Mandelbrot’s book “The Fractal Geometry of Nature”, which has become a classic - “What is the length of the coast of Britain?” The answer to this question is not as simple as it seems. It all depends on the length of the tool we will use. By measuring the shore using a kilometer ruler, we will get some length. However, we will miss many small bays and peninsulas that are much smaller in size than our line. By reducing the size of the ruler to, say, 1 meter, we will take into account these details of the landscape, and, accordingly, the length of the coast will become larger. Let's go further and measure the length of the shore using a millimeter ruler, we will take into account details that are larger than a millimeter, the length will be even greater. As a result, the answer to such a seemingly simple question can baffle anyone - the length of the coast of Britain is endless.
A little about dimensions.
In our daily lives we constantly encounter dimensions. We estimate the length of the road (250 m), find out the area of the apartment (78 m2) and look for the volume of a beer bottle on the sticker (0.33 dm3). This concept is quite intuitive and, it would seem, does not require clarification. The line has dimension 1. This means that by choosing a reference point, we can define any point on this line using 1 number - positive or negative. Moreover, this applies to all lines - circle, square, parabola, etc.
Dimension 2 means that we can uniquely define any point by two numbers. Don't think that two-dimensional means flat. The surface of a sphere is also two-dimensional (it can be defined using two values - angles like width and longitude).
If we look at it from a mathematical point of view, then the dimension is determined as follows: for one-dimensional objects, doubling their linear size leads to an increase in size (in this case, length) by a factor of two (2^1).
For two-dimensional objects, doubling linear dimensions results in an increase in size (for example, the area of a rectangle) by four times (2^2).
For 3-dimensional objects, doubling the linear dimensions leads to an eightfold increase in volume (2^3) and so on.
Thus, the dimension D can be calculated based on the dependence of the increase in the “size” of the object S on the increase in the linear dimensions L. D=log(S)/log(L). For line D=log(2)/log(2)=1. For the plane D=log(4)/log(2)=2. For volume D=log(8)/log(2)=3. It can be a little confusing, but in general it is not complicated and understandable.
Why am I telling all this? And in order to understand how to separate fractals from, say, sausage. Let's try to calculate the dimension for the Peano curve. So, we have the original line, consisting of three segments of length X, replaced by 9 segments three times shorter. Thus, when the minimum segment increases by 3 times, the length of the entire line increases by 9 times and D=log(9)/log(3)=2 is a two-dimensional object!!!
So, when the dimension of a figure obtained from some simple objects (segments) is greater than the dimension of these objects, we are dealing with a fractal.
Fractals are divided into groups. The largest groups are:
Geometric fractals.
This is where the history of fractals began. This type of fractal is obtained through simple geometric constructions. Usually, when constructing these fractals, they do this: they take a “seed” - an axiom - a set of segments on the basis of which the fractal will be built. Next, a set of rules is applied to this “seed”, which transforms it into some kind of geometric figure. Next, the same set of rules is applied again to each part of this figure. With each step, the figure will become more and more complex, and if we carry out (at least in our minds) an infinite number of transformations, we will get a geometric fractal.
The Peano curve discussed above is a geometric fractal. The figure below shows other examples of geometric fractals (from left to right Koch's Snowflake, Liszt, Sierpinski Triangle).
Snowflake Koch
Sheet
Sierpinski triangle
Of these geometric fractals, the first one, the Koch snowflake, is very interesting and quite famous. It is built on the basis of an equilateral triangle. Each line of which ___ is replaced by 4 lines each 1/3 the length of the original _/\_. Thus, with each iteration, the length of the curve increases by a third. And if we make an infinite number of iterations, we will get a fractal - a Koch snowflake of infinite length. It turns out that our infinite curve covers a limited area. Try to do the same using methods and figures from Euclidean geometry.
The dimension of a Koch snowflake (when a snowflake increases by 3 times, its length increases by 4 times) D=log(4)/log(3)=1.2619...
The so-called L-Systems are well suited for constructing geometric fractals. The essence of these systems is that there is a certain set of system symbols, each of which denotes a specific action and a set of symbol conversion rules. For example, the description of Koch's snowflake using L-Systems in the Fractint program
; Adrian Mariano from The Fractal Geometry of Nature by Mandelbrot Koch1 ( ;set the rotation angle to 360/6=60 degrees Angle 6 ; Initial drawing for construction Axiom F--F--F ; Character Conversion Rule F=F+F--F+F )In this description, the geometric meanings of the symbols are as follows:
F means draw a line + turn clockwise - turn counterclockwise
The second property of fractals is self-similarity. Take, for example, the Sierpinski triangle. To construct it, we “cut out” a triangle from the center of an equilateral triangle. Let's repeat the same procedure for the three triangles formed (except for the central one) and so on ad infinitum. If we now take any of the resulting triangles and enlarge it, we will get an exact copy of the whole. In this case we are dealing with complete self-similarity.
Let me make a reservation right away that most of the fractal drawings in this article were obtained using the Fractint program. If you are interested in fractals, then this is a must have program for you. With its help, you can build hundreds of different fractals, get comprehensive information on them, and even listen to how fractals sound;).
To say that the program is good is to say nothing. It's great, except for one thing - the latest version 20.0 is only available in the DOS version:(. You can find this program (latest version 20.0) at http://spanky.fractint.org/www/fractint/fractint.html.
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Comments
Well, for starters, an interesting example from Microsoft Excel. Cells A2 and B2 have the same values between 0 and 1. with a value of 0.5 there is no effect.
Hello to everyone who managed to make a program using a fratal picture. Who can tell me which cycle method is best for me to use to build a clearing of fractal ferns with a 3d max backing with a dt iteration of 100,000 on a stone with 2800 mH
There is a source code with a program for drawing the Dragon curve, also a fractal.
The article is awesome. And Excel is probably a coprocessor error (on the last low-order digits)
Hello everybody! My name is, Ribenek Valeria, Ulyanovsk and today I will post several of my scientific articles on the LCI website.
My first scientific article in this blog will be devoted to fractals. I will say right away that my articles are designed for almost any audience. Those. I hope they will be of interest to both schoolchildren and students.
Recently I learned about such interesting objects of the mathematical world as fractals. But they exist not only in mathematics. They surround us everywhere. Fractals are natural. I will talk about what fractals are, about the types of fractals, about examples of these objects and their applications in this article. To begin with, I’ll briefly tell you what a fractal is.
Fractal(Latin fractus - crushed, broken, broken) is a complex geometric figure that has the property of self-similarity, that is, composed of several parts, each of which is similar to the entire figure. In a broader sense, fractals are understood as sets of points in Euclidean space that have a fractional metric dimension (in the sense of Minkowski or Hausdorff), or a metric dimension different from the topological one. As an example, I will insert a picture depicting four different fractals.
I'll tell you a little about the history of fractals. The concepts of fractal and fractal geometry, which appeared in the late 70s, have become firmly established among mathematicians and programmers since the mid-80s. The word "fractal" was coined by Benoit Mandelbrot in 1975 to refer to the irregular but self-similar structures with which he was concerned. The birth of fractal geometry is usually associated with the publication of Mandelbrot’s book The Fractal Geometry of Nature in 1977. His works used the scientific results of other scientists who worked in the period 1875-1925 in the same field (Poincaré, Fatou, Julia, Cantor, Hausdorff). But only in our time has it been possible to combine their work into a single system.
There are a lot of examples of fractals, because, as I said, they surround us everywhere. In my opinion, even our entire Universe is one huge fractal. After all, everything in it, from the structure of the atom to the structure of the Universe itself, exactly repeats each other. But there are, of course, more specific examples of fractals from different areas. Fractals, for example, are present in complex dynamics. There they naturally appear in the study of nonlinear dynamic systems. The most studied case is when the dynamic system is specified by iterations polynomial or holomorphic function of a complex of variables on surface. Some of the most famous fractals of this type are the Julia set, Mandelbrot set and Newton pools. Below, in order, the pictures depict each of the above fractals.
Another example of fractals is fractal curves. It is best to explain how to construct a fractal using the example of fractal curves. One of these curves is the so-called Koch Snowflake. There is a simple procedure for obtaining fractal curves on a plane. Let us define an arbitrary broken line with a finite number of links, called a generator. Next, we replace each segment in it with a generator (more precisely, a broken line similar to a generator). In the resulting broken line, we again replace each segment with a generator. Continuing to infinity, in the limit we get a fractal curve. Below is the Koch Snowflake (or Curve).
There are also a huge variety of fractal curves. The most famous of them are the already mentioned Koch Snowflake, as well as the Levy curve, the Minkowski curve, the Dragon's broken line, the Piano curve and the Pythagorean tree. I think you can easily find an image of these fractals and their history on Wikipedia if you wish.
The third example or type of fractals are stochastic fractals. Such fractals include the trajectory of Brownian motion on a plane and in space, the Schramm-Löwner evolution, various types of randomized fractals, that is, fractals obtained using a recursive procedure into which a random parameter is introduced at each step.
There are also purely mathematical fractals. These are, for example, the Cantor set, the Menger sponge, the Sierpinski Triangle and others.
But perhaps the most interesting fractals are natural ones. Natural fractals are objects in nature that have fractal properties. And here the list is already big. I won’t list everything, because it’s probably impossible to list them all, but I’ll tell you about some. For example, in living nature, such fractals include our circulatory system and lungs. And also the crowns and leaves of trees. This also includes starfish, sea urchins, corals, sea shells, and some plants such as cabbage or broccoli. Several such natural fractals from living nature are clearly shown below.
If we consider inanimate nature, then there are much more interesting examples there than in living nature. Lightning, snowflakes, clouds, well-known to everyone, patterns on windows on frosty days, crystals, mountain ranges - all these are examples of natural fractals from inanimate nature.
We looked at examples and types of fractals. As for the use of fractals, they are used in a variety of fields of knowledge. In physics, fractals naturally arise when modeling nonlinear processes, such as turbulent fluid flow, complex diffusion-adsorption processes, flames, clouds, etc. Fractals are used when modeling porous materials, for example, in petrochemistry. In biology, they are used to model populations and to describe internal organ systems (the blood vessel system). After the creation of the Koch curve, it was proposed to use it in calculating the length of the coastline. Fractals are also actively used in radio engineering, information science and computer technology, telecommunications and even economics. And, of course, fractal vision is actively used in modern art and architecture. Here is one example of fractal patterns:
And so, with this I think to complete my story about such an unusual mathematical phenomenon as a fractal. Today we learned about what a fractal is, how it appeared, about the types and examples of fractals. I also talked about their application and demonstrated some of the fractals visually. I hope you enjoyed this little excursion into the world of amazing and fascinating fractal objects.
Fractal example
“Fractal” was introduced into use by mathematicians less than half a century ago, and soon became, along with synergetics and attractor, one of the “three pillars” of the young Theory of Deterministic Chaos, and today is already recognized as one of the fundamental elements of the structure of the universe.
WITH the Latin word fractus is translated as "broken", modern Latin languages gave it the meaning "torn". A fractal is something that is identical to the whole/larger of which it is a part, and, at the same time, copies each of its own constituent parts. Thus, “fractality” is the infinite similarity of “everything” to its components, that is, it is self-similarity at any level. Each level of a fractal branch is called an “iteration”; the more developed the described or graphically depicted system is, the more fractal iterations the observer sees. In this case, the point at which division occurs (for example, a trunk into branches, a river into two streams, etc.) is called the bifurcation point.
The term fractus was chosen by mathematician Benoit Mandelbrot in 1975 to describe a scientific discovery and became popular a few years later after he developed the topic for a wider audience in his book The Fractal Geometry of Nature.
Today, fractal is widely known as the fantastic patterns of so-called “fractal art” created by computer programs. But with the help of a computer you can generate not only beautiful abstract pictures, but also very believable natural landscapes - mountains, rivers, forests. Here, in fact, is the point of transition between science and real life, or vice versa, if we assume that it is generally possible to separate them.
The fact is that fractal principle suitable not only for describing discoveries in the exact sciences. This is, first of all, the principle of the structure and development of nature itself. Everything around us is fractals! The most obvious group of examples are rivers with tributaries, the venous system with capillaries, lightning, frost patterns, trees... More recently, scientists, testing fractal theory, have experimentally verified that, based on the diagram of one tree, one can draw conclusions about the forest area where these trees grow. Other examples of fractal groups: atom - molecule - planetary system - solar system - galaxies - universe... Minute - hour - day - week - month - year - century... Even a community of people organizes itself according to the principles of fractality: I - family - clan - nationality - nationalities - races... Individual - group - party - state. Employee - department - department - enterprise - concern... Even the divine pantheons of different religions are built on the same principle, including Christianity: God the Father - Trinity - saints - church - believers, not to mention the organization of divine pantheons of pagan religions.
Story states that self-similar sets were first noticed in the 19th century in the works of scientists - Poincaré, Fatou, Julia, Cantor, Hausdorff, but the truth is that already the pagan Slavs left us proof that people understood individual existence as a small detail in infinity of the universe. This is a folk culture object called a “spider”, studied by art historians of Belarus and Ukraine. It is a kind of prototype of sculpture in the modern “mobile” style (the parts are in constant motion relative to each other). The “spider” is often made of straw, consists of small, medium, and large elements of the same shape, suspended from each other so that each smaller part exactly repeats the larger one and the entire structure as a whole. This design was hung in the main corner of the home, as if denoting one’s home as an element of the whole world.
The theory of fractality works everywhere today, including in philosophy, which says that during each life, and any and all life as a whole is fractal, there are “bifurcation points” when development can take different paths to higher levels and a moment when a person “finds himself before a choice”, is a real “bufurcation point” in the fractals of his life.
The theory of Deterministic Chaos says that the development of each fractal is not infinite. Scientists believe that at a certain moment there comes a limit beyond which the growth of iterations stops and the fractal begins to “narrow”, gradually reaching its original unit measure, and then the process again goes in a circle - similar to inhalation and exhalation, the changes of morning and night, winter and summer in nature.