The mathematical shapes known as fractals originate from the genius of the eminent scientist Benoit Mandelbrot. He spent most of his life in the United States, where he taught mathematics at Yale University. In 1977 and 1982, Mandelbrot published scientific works devoted to the study of "fractal geometry" or "geometry of nature", in which he broke down seemingly random mathematical forms into constituent elements that, on closer inspection, turned out to be repeating - which proves the existence of some kind of pattern for copying . Mandelbrot's discovery had significant positive consequences in the development of physics, astronomy and biology.
How does a fractal work?
A fractal (from the Latin “fractus” - broken, crushed, broken) is a complex geometric figure that is made up of several infinite sequences of parts, each of which is similar to the entire figure, and is repeated as the scale decreases.
The structure of the fractal on all scales is non-trivial. Here we need to clarify what is meant. So, regular figures, such as a circle, an ellipse or the graph of a smooth function, are arranged in such a way that when considering a small fragment of a regular figure on a sufficiently large scale, it will be similar to a fragment of a straight line. For fractals, an increase in scale does not lead to a simplification of the structure of the figure, and on all scales we see a uniformly complex picture.
In nature, many objects have fractal properties, for example: tree crowns, cauliflower, clouds, the circulatory and alveolar systems of humans and animals, crystals, snowflakes, the elements of which are arranged into one complex structure, coastlines (the fractal concept allowed scientists to measure the coastline of the British Isles and other previously unmeasurable objects).
Let's look at the structure of cauliflower. If you cut one of the flowers, it is obvious that the same cauliflower remains in your hands, only smaller in size. We can keep cutting again and again, even under a microscope - but all we get are tiny copies of the cauliflower. In this simplest case, even a small part of the fractal contains information about the entire final structure.
A striking example of a fractal in nature is “Romanescu”, also known as “Romanescu broccoli” or “coral cauliflower”. The first mention of this exotic vegetable dates back to Italy in the 16th century. The buds of this cabbage grow in a logarithmic spiral. 3D artists, designers and chefs never cease to admire her. The latter, moreover, especially value the vegetable for the most refined taste (sweet and nutty, not sulfurous) that cabbage can have, and for the fact that it is less crumbly than ordinary cauliflower. In addition, romaine broccoli is rich in vitamin C, antioxidants and carotenoids.
Fractals in digital technology
Fractal geometry has made an invaluable contribution to the development of new technologies in the field of digital music, and also made it possible to compress digital images. Existing fractal image compression algorithms are based on the principle of storing a compressed image instead of the digital image itself. For a compressed image, the main image remains a fixed point. Microsoft used one of the variants of this algorithm when publishing its encyclopedia, but for one reason or another this idea was not widely used.
The principle of fractal compression of information for compact storage of information about network nodes “Netsukuku” is used by the system for assigning IP addresses. Each node stores 4 kilobytes of information about the state of neighboring nodes. Any new node connects to the general Internet without requiring central regulation of the distribution of IP addresses. We can conclude that the principle of fractal compression of information ensures decentralized operation of the entire network, and therefore work in it proceeds as stably as possible.
Fractals are widely used in computer graphics - when constructing images of trees, bushes, sea surfaces, mountain landscapes, and other natural objects. Thanks to fractal graphics, an effective way was invented to implement complex non-Euclidean objects whose images are similar to natural ones: these are algorithms for synthesizing fractal coefficients, which make it possible to reproduce a copy of any picture as close as possible to the original. Interestingly, in addition to fractal “painting”, there are also fractal music and fractal animation. In the fine arts, there is a direction that deals with obtaining an image of a random fractal - “fractal monotype” or “stochatypy”.
The mathematical basis of fractal graphics is fractal geometry, where the principle of inheritance from the original “parent objects” is the basis for the methods for constructing “heir images”. The very concepts of fractal geometry and fractal graphics appeared only about 30 years ago, but have already become firmly established in the everyday life of computer designers and mathematicians.
The basic concepts of fractal computer graphics are:
- Fractal triangle - fractal figure - fractal object (hierarchy in descending order)
- Fractal line
- Fractal composition
- “Parent object” and “Successor object”
How to build a fractal?
The creator of fractals plays the role of an artist, photographer, sculptor, and scientist-inventor at the same time. What are the upcoming stages of creating a drawing from scratch?
- set the shape of the drawing using a mathematical formula
- investigate the convergence of the process and vary its parameters
- select image type
- choose a color palette
Among fractal graphic editors and other graphic programs we can highlight:
"Art Dabbler"
“Painter” (without a computer, no artist will ever achieve the capabilities laid down by programmers only through a pencil and a brush pen)
“Adobe Photoshop” (but here the image is not created “from scratch”, but, as a rule, only processed)
Send your good work in the knowledge base is simple. Use the form below
Students, graduate students, young scientists who use the knowledge base in their studies and work will be very grateful to you.
Posted on http://www.allbest.ru/
Ministry of Education and Science of the Russian Federation
Subject: Fractals- specialobjectsaliveAndnonlivingpeace
Khabarovsk TOGU 2015
- Table of contents
- fractal geometric fractal graphics
- History of fractals
- Classification of fractals
- Geometric fractals
- Algebraic fractals
- Application of fractals
- Fractals and the world around us
- Fractal graphics
- Application of fractals
- Natural Sciences
- Radio engineering
- Computer science
- Economics and finance
History of fractals
Very often we come across special objects, but few people know that these are fractals. Fractals are unique objects generated by the unpredictable movements of the chaotic world. They are found in both small objects, such as the cell membrane, and large ones, such as the Solar System and Galaxy. In everyday life, we can see fractals in wallpaper, on fabric, on a desktop screensaver on a computer, and in nature - these are plants, sea animals, and natural phenomena.
Scientists have been fascinated by fractals since ancient times, and programmers and computer graphics specialists also love these objects. The discovery of fractals was a revolution in human perception of the world and the discovery of a new aesthetics of art and science.
So what are fractals? Fractal- 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 as a whole.
The term fractal was proposed in 1975. Benoit Mandelbrot to designate the irregular, self-similar structures with which he was concerned. The birth of fractal geometry is the publication of his book “The Fractal Geometry of Nature” in 1977. His work was based on the works of scientists Poincaré, Fatou, Julia, Cantor and Hausdorff, who worked in 1875? 1925 in the same area. But it was only in our time that they were able to combine their work into a single system.
The concept of “fractal” is derived from the Latin “fractus”? consisting of fragments. One of the definitions is: “A fractal is a structure consisting of parts that, in some sense, are similar to the whole.”
Benoit Mandelbrot in his works gave vivid examples of the use of fractals to explain some natural phenomena. He paid great attention to an interesting property that many fractals have. The fact is that often a fractal can be divided into arbitrarily small parts so that each part turns out to be simply a reduced copy of the whole. In other words, if we look at a fractal through a microscope, we will be surprised to see the same picture as without a microscope. This property of self-similarity sharply distinguishes fractals from objects of classical geometry.
For modern scientists, studying fractals? not just a new area of knowledge. This is the discovery of a new type of geometry that describes the world around us and which can be seen not only in textbooks, but also in nature and in the boundless Universe. Now, Mandelbrot and other scientists have expanded the field of fractal geometry so that it can be applied to almost everything in the world, from predicting stock market prices to making new discoveries in theoretical physics.
Classification of fractals
There are different classifications of fractals.
The main classification of fractals is the division into geometric and algebraic.
Geometric fractals have exact self-similarity, and algebraic fractals have approximate self-similarity.
There is also a division into natural and man-made fractals.
Man-made fractals include those that were invented by scientists; they have fractal properties at any scale. Natural fractals are subject to a limitation on the area of existence - that is, the maximum and minimum size at which an object exhibits fractal properties.
The simplest fractals are geometric fractals.
Geometric fractals
Geometric fractals are also called classical, deterministic or linear. They are the most visual, since they have so-called rigid self-similarity, which does not change when the scale changes. This means that no matter how close you zoom in on the fractal, you still see the same pattern.
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 of a given polyline (initiator) is replaced by a generator polyline on the appropriate scale. As a result of endless repetition of this procedure, a fractal curve is obtained. Despite the apparent complexity of this curve, its shape is determined only by the shape of the generator.
The most famous geometric fractals: Koch curve, Minkowski curve, Levy curve, dragon curve, Sierpinski napkin and carpet, Durer pentagon.
Construction of some geometric fractals
1). Koch curve.
It was invented in 1904 by a German mathematician named Helge von Koch. To construct it, a single segment is taken, divided into three equal parts, and the middle link is replaced by an equilateral triangle without this link. In the next step, we repeat the operation for each of the four resulting segments. As a result of endless repetition of this procedure, a fractal curve is obtained.
2). Sierpinski's napkin.
In 1915, Polish mathematician Waclaw Sierpinski came up with an interesting object. To construct it, take a solid equilateral triangle. In the first step, an inverted equilateral triangle is removed from the center. The second step removes three inverted triangles from the remaining three triangles, and so on. According to the theory, there will be no end to this process, and there will be no living space left in the triangle, but it will not fall apart either - the result will be an object consisting of only holes.
3). Harter-Hathway's Dragon.
Harter's dragon, also known as the Harter-Haithaway dragon, was first studied by NASA physicists? John Haithway, William Harter and Bruce Banks. It was described in 1967 by Martin Gardner in the "Mathematical Games" column of Scientific American.
In the next step, each of the line segments is replaced by two segments that form the lateral sides of an isosceles right triangle, for which the original segment would be the hypotenuse. As a result, the segment seems to bend at a right angle. The direction of deflection alternates. The first segment bends to the right (as it moves from left to right), the second - to the left, the third - to the right again, etc.
Examples of geometric fractals
CurveKochNapkinSierpinski
The DragonHarter-Hathway
The second large group of fractals are algebraic. They got their name because they are built on the basis of algebraic formulas.
Algebraic fractals
Complex (algebraic) fractals cannot be created without the help of a computer. To obtain colorful results, this computer must have a powerful mathematical coprocessor and a high-resolution monitor. They got their name because they are built on the basis of algebraic formulas. As a result of mathematical processing of this formula, a point of a certain color is displayed on the screen. The result is a strange figure in which straight lines turn into curves, and self-similarity effects appear at various scale levels, although not without deformations. Almost every point on a computer screen is like a separate fractal.
The most famous algebraic fractals: Mandelbrot and Julia sets, Newton pools.
Algebraic fractals have approximate self-similarity. In fact, if you magnify a small area of any complex fractal, and then do the same on a small portion of that area, the two magnifications will be significantly different from each other. The two images will be very similar in detail, but they will not be completely identical.
ALGEBRAIC FRACTALS
Mandelbrot set approximations
Fractals are finding more and more applications in science. The main reason is that they describe the real world better than traditional physics and mathematics.
Application of fractals
1). Chaos theory: Fractals are always associated with the word chaos. Chaos theory is defined as the study of complex nonlinear dynamic systems. Chaos is the absence of predictability. It occurs in dynamic systems when, for two very close initial values, the system behaves completely differently. An example of a chaotic dynamic system is the weather. Examples of such systems are turbulent flows, biological populations, society and its subsystems: economic, political and other social systems. One of the central concepts in this theory is the impossibility of accurately predicting the state of a system. Chaos theory focuses not on the disorder of a system (the hereditary unpredictability of the system), but on the order it inherits (the common behavior of similar systems). Thus, the science of chaos is a system of ideas about various forms of order, where randomness becomes the organizing principle.
2). Economics: analysis of the securities market.
3). Astrophysics: description of galaxy clustering processes in the Universe.
4). Geology: study of mineral roughness;
5). Cartography: study of coastline shapes; study of an extensive network of river channels.
6). Mechanics of liquids and gases, physics of surfaces:
- dynamics and turbulence of complex flows.
- modeling of flames;
7). Biology and Medicine:
- modeling of animal populations and bird migration;
- modeling of epidemics;
- analysis of the structure of the circulatory system;
- consideration of complex surfaces of cell membranes;
- description of processes inside the body, for example, heartbeat.
8). 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. He cut out a Koch curve shape from aluminum foil and pasted it onto a piece of paper, then attached it to the receiver. It turned out that such an antenna works no worse than a regular one. And although the physical principles of operation of such an antenna have not yet been studied, this did not stop Cohen from founding his own company and launching their serial production.
9). Image compression: the advantages of fractal image compression algorithms are a very small packed file size and short image recovery time. Another advantage of fractal compression is that when the image is enlarged, there is no pixelation effect (increasing the size of the dots to sizes that distort the image). With fractal compression, after enlargement, the picture often looks even better than before.
10). Computer graphics: Computer graphics is going through a period of intense development today. She was able to recreate an endless variety of fractal shapes and landscapes on the monitor screen, immersing the viewer in an amazing virtual space. Nowadays, with the help of relatively simple algorithms, it has become possible to create three-dimensional images of fantastic landscapes and shapes that can be transformed over time into even more exciting pictures. The tendency of fractals to resemble mountains, flowers and trees is exploited by some graphic editors (for example, fractal clouds from 3D studio MAX, fractal mountains in World Builder). Fractal models are widely used today in computer games, creating an environment in them that is difficult to distinguish from reality.
The end of the twentieth century was marked not only by the discovery of amazingly beautiful and infinitely diverse structures called fractals, but also by the awareness of the fractal nature of nature. The world around us is very diverse, and its objects do not fit into the rigid framework of Euclidean lines and surfaces.
Fractals and the world around us
« Beauty is always relative...We should not assume that the shores of the ocean are truly shapeless just because their shape is different from the regular shape of the piers we have built; the shape of mountains cannot be considered irregular on the basis that they are not regular cones or pyramids; just because the distances between the stars are unequal, it does not follow that they were scattered across the sky by an inept hand. These wrongs only exist in our imagination , in fact, they are not such and do not in any way interfere with the true manifestations of life on Earth, neither in the kingdom of plants and animals, nor among people.” These words of the English scientist of the 17th century. Richard Bentley indicate that the idea of combining the forms of coasts, mountains and celestial objects and contrasting them with Euclidean constructions arose in the minds of people for a very long time.
Galileo Galilei said that “the great book of Nature is written in the language of geometry.” Now we can confidently say that it is written in the language of fractal geometry.
What we observe in nature often intrigues us with the endless repetition of the same pattern, increased or decreased as many times as desired. Bizarre shapes of coastlines and intricate bends of rivers, broken surfaces of mountain ranges and outlines of clouds, spreading branches of trees and coral reefs, the timid flicker of a candle and foamy streams of mountain rivers - all these are fractals. Some of them, such as clouds or stormy streams, constantly change their shape, others, like trees or mountain ranges, maintain their structure unchanged. Common to all types of fractal structures is their self-similarity - the main property that ensures the fulfillment of the basic law in fractals - the law of unity in the diversity of the universe.
Human systems and organs are also fractal structures. For example, blood vessels branch multiple times, i.e. have a fractal nature. The electrical activity of the heart is a fractal process. Cardiologists have discovered that the spectral characteristics of heartbeats obey fractal laws, just like earthquakes and economic phenomena. In the tissues of the digestive tract, one wavy surface is embedded in another. The lungs also represent an example of a large area being squeezed into a small space. In fact, the entire structure of the human body is fractal in nature; this has already been recognized by scientists. The principle of a single simple, defining a diverse complex, is embedded in the human genome, when one cell of a living organism contains information about the entire organism as a whole.
Fractal structures in nature
Here are a few sample photos:
As biologist John Haldane said, “The world is not only weirder than we think, but weirder than we can imagine.” Fractals are not Mandelbrot's inventions. They exist objectively. In natural forms and processes, in science and art, which reflect and understand this world. It was “for changing our view of the world thanks to the ideas of fractal geometry” that Benoit Mandelbrot was awarded the honorary Wolf Prize in physics in 1993.
Currently, fractal paintings are very popular. They make an absolutely fantastic impression. Many thin lines forming one whole, or unusual elements intertwined into a single picture. Flashes of bright light and moderate smooth lines. The fractal seems alive. It burns, it glows, it attracts, and you cannot take your eyes off it, studying even the tiniest and most insignificant details.
Fractal graphics
Fractal paintings in the interior
Application of fractals
Natural Sciences
In physics, fractals naturally arise when modeling nonlinear processes, such as turbulent fluid flow, complex diffusion-adsorption processes, flames, clouds, and the like. Fractals are used in modeling porous materials, for example, in petrochemicals. In biology, they are used to model populations and to describe internal organ systems (the blood vessel system). After the creation of the Koch curve, it was proposed to use it when calculating the length of the coastline.
Radio engineering
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
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 by Microsoft when publishing its encyclopedia, but these algorithms were not widely used.
Computer graphics
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 available to generate fractal images.
Decentralized networks
The IP address assignment system in the Netsukuku network (this network is a project for creating a distributed self-organizing peer-to-peer network capable of ensuring the interaction of a huge number of nodes with minimal load on the central processor and memory) 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.
Economics and finance
A. A. Almazov in his book “Fractal Theory. How to change your view of the markets” suggested a way to use fractals when analyzing stock quotes, in particular in the Forex market.
Every time you look at fractals, you think about how beautiful the real world and the world of mathematics are, and that mathematics really is a language that can describe almost everything that exists in the Universe.
Bibliography
1. Mandelbrot B. Fractal geometry of nature. M.: “Institute of Computer Research”, 2002. 656 p.
2. Morozov A.D. Introduction to the theory of fractals. N. Novgorod: Publishing house Nizhny Novgorod. University, 1999, 140 p.
3. Peitgen H.-O., Richter P. H. The beauty of fractals. M.: “Mir”, 1993. - 176 p.
4. Tikhoplav V.Yu., Tikhoplav T.S. Harmony of chaos, or fractal reality. St. Petersburg: Publishing House “Ves”, 2003. 340 p.
5. Feder E. Fractals. M: “Mir”, 1991. 254 p.
6. Schroeder M. Fractals, chaos, power laws. Miniatures from endless paradise. Izhevsk: “RKhD”, 2001. 528 p.
List of sites about fractals
1. http://www.fractals.nsu.ru.
2. http://www.fractalworld.xaoc.ru.
3. http://www.multifractal.narod.ru.
4. http://algolist.manual.ru.
Posted on Allbest.ru
Similar documents
Consideration of the fractal dimension as one of the characteristics of an engineering surface. Description of natural fractals. Measuring the length of a non-smooth (broken) line. Similarity and scaling, self-similarity and self-affinity. Perimeter-area relationship.
test, added 12/23/2015
The history of the emergence of the theory of fractals. A fractal is a self-similar structure whose image does not depend on scale. This is a recursive model, each part of which repeats in its development the development of the entire model as a whole. Practical application of the theory of fractals.
scientific work, added 05/12/2010
Classic fractals. Self-similarity. Snowflake Koch. Sierpinski carpet. L-systems. Chaotic dynamics. Lorentz attractor. Mandelbrot and Julia sets. Application of fractals in computer technology.
course work, added 05/26/2006
Features of some quadrangles. Implementation of models of geometric situations in dynamic geometry environments. Features of the dynamic environment "Living Geometry", features of the construction of parallelogram, rhombus, rectangle and square models in it.
course work, added 05/28/2013
Geometric picture of the world and the prerequisites for the emergence of the theory of fractals. Elements of a deterministic L-system: alphabet, initialization word and a set of generating rules. Fractal properties of social processes: synergetics and chaotic dynamics.
course work, added 03/22/2014
Studying the manifestations of geometric laws in living nature and their use in educational practical activities. Description of geometric laws and the essence of geometric constructions. Graphic education and its place in the modern world.
thesis, added 06/24/2010
Definition of the concept of a model, the need for their application in science and everyday life. Characteristics of material and ideal modeling methods. Classification of mathematical models (deterministic, stochastic), stages of the process of their construction.
abstract, added 08/20/2015
A study of the concepts of symmetry, proportionality, proportionality and uniformity in the arrangement of parts. Characteristics of the symmetrical properties of geometric figures. Descriptions of the role of symmetry in architecture, nature and technology, in solving logical problems.
presentation, added 12/06/2011
History of mathematization of science. Basic methods of mathematization. Limits and problems of mathematization. Problems of applying mathematical methods in various sciences are associated with mathematics itself (the mathematical study of models), with the field of modeling.
abstract, added 05/24/2005
The concept and history of the study of the golden ratio. Features of its reflection in mathematics, nature, architecture and painting. The order and principles of construction, structure and areas of practical application of the golden section, mathematical justification and meaning.
Often, brilliant discoveries made in science can radically change our lives. For example, the invention of a vaccine can save many people, but the creation of new weapons leads to murder. Literally yesterday (on the scale of history) man “tamed” electricity, and today he can no longer imagine his life without it. However, there are also discoveries that, as they say, remain in the shadows, despite the fact that they also have one or another impact on our lives. One of these discoveries was the fractal. Most people have never even heard of this concept and will not be able to explain its meaning. In this article we will try to understand the question of what a fractal is and consider the meaning of this term from the perspective of science and nature.
Order in chaos
In order to understand what a fractal is, we should begin the debriefing from the position of mathematics, but before delving into it, we will philosophize a little. Every person has a natural curiosity, thanks to which he learns about the world around him. Often, in his quest for knowledge, he tries to use logic in his judgments. Thus, by analyzing the processes that occur around him, he tries to calculate relationships and derive certain patterns. The greatest minds on the planet are busy solving these problems. Roughly speaking, our scientists are looking for patterns where there are none, and there should not be any. And yet, even in chaos there is a connection between certain events. This connection is what the fractal is. As an example, consider a broken branch lying on the road. If we look closely at it, we will see that with all its branches and twigs it itself looks like a tree. This similarity of a separate part with a single whole indicates the so-called principle of recursive self-similarity. Fractals can be found all over the place in nature, because many inorganic and organic forms are formed in a similar way. These are clouds, sea shells, snail shells, tree crowns, and even the circulatory system. This list can be continued indefinitely. All these random shapes are easily described by a fractal algorithm. Now we have come to consider what a fractal is from the perspective of exact sciences.
Some dry facts
The word “fractal” itself is translated from Latin as “partial”, “divided”, “fragmented”, and as for the content of this term, there is no formulation as such. It is usually interpreted as a self-similar set, a part of the whole, which repeats its structure at the micro level. This term was coined in the seventies of the twentieth century by Benoit Mandelbrot, who is recognized as the father. Today, the concept of fractal means a graphic image of a certain structure, which, when enlarged, will be similar to itself. However, the mathematical basis for the creation of this theory was laid even before the birth of Mandelbrot himself, but it could not develop until electronic computers appeared.
Historical background, or How it all began
At the turn of the 19th and 20th centuries, the study of the nature of fractals was sporadic. This is explained by the fact that mathematicians preferred to study objects that could be researched on the basis of general theories and methods. In 1872, the German mathematician K. Weierstrass constructed an example of a continuous function that is not differentiable anywhere. However, this construction turned out to be entirely abstract and difficult to perceive. Next came the Swede Helge von Koch, who in 1904 constructed a continuous curve that had no tangent anywhere. It's fairly easy to draw and turns out to have fractal properties. One of the variants of this curve was named after its author - “Koch snowflake”. Further, the idea of self-similarity of figures was developed by the future mentor of B. Mandelbrot, the Frenchman Paul Levy. In 1938, he published the article "Plane and spatial curves and surfaces consisting of parts similar to the whole." In it, he described a new type - the Lewy C-curve. All of the above figures are conventionally classified as geometric fractals.
Dynamic or algebraic fractals
The Mandelbrot set belongs to this class. The first researchers in this direction were the French mathematicians Pierre Fatou and Gaston Julia. In 1918, Julia published a paper based on the study of iterations of rational complex functions. Here he described a family of fractals that are closely related to the Mandelbrot set. Despite the fact that this work glorified the author among mathematicians, it was quickly forgotten. And only half a century later, thanks to computers, Julia’s work received a second life. Computers made it possible to make visible to every person the beauty and richness of the world of fractals that mathematicians could “see” by displaying them through functions. Mandelbrot was the first to use a computer to carry out calculations (such a volume cannot be done manually) that made it possible to construct an image of these figures.
A person with spatial imagination
Mandelbrot began his scientific career at IBM Research Center. While studying the possibilities of transmitting data over long distances, scientists were faced with the fact of large losses that arose due to noise interference. Benoit was looking for ways to solve this problem. Looking through the measurement results, he noticed a strange pattern, namely: the noise graphs looked the same on different time scales.
A similar picture was observed both for a period of one day and for seven days or for an hour. Benoit Mandelbrot himself often repeated that he does not work with formulas, but plays with pictures. This scientist was distinguished by imaginative thinking; he translated any algebraic problem into the geometric area, where the correct answer is obvious. So it is not surprising that he is distinguished by his wealth and became the father of fractal geometry. After all, awareness of this figure can only come when you study the drawings and think about the meaning of these strange swirls that form the pattern. Fractal patterns do not have identical elements, but they are similar at any scale.
Julia - Mandelbrot
One of the first drawings of this figure was a graphic interpretation of the set, which was born out of the work of Gaston Julia and was further developed by Mandelbrot. Gaston tried to imagine what a set would look like based on a simple formula that was iterated through a feedback loop. Let's try to explain what has been said in human language, so to speak, on the fingers. For a specific numerical value, we find a new value using a formula. We substitute it into the formula and find the following. The result is large. To represent such a set it is necessary to perform this operation a huge number of times: hundreds, thousands, millions. This is what Benoit did. He processed the sequence and transferred the results to graphical form. Subsequently, he colored the resulting figure (each color corresponds to a certain number of iterations). This graphic image was named “Mandelbrot fractal”.
L. Carpenter: art created by nature
The theory of fractals quickly found practical application. Since it is very closely related to the visualization of self-similar images, artists were the first to adopt the principles and algorithms for constructing these unusual forms. The first of them was the future founder of Pixar, Lauren Carpenter. While working on a presentation of aircraft prototypes, he came up with the idea of using an image of mountains as a background. Today, almost every computer user can cope with such a task, but in the seventies of the last century, computers were not able to perform such processes, because there were no graphic editors or applications for three-dimensional graphics at that time. And then Loren came across Mandelbrot’s book “Fractals: Form, Randomness and Dimension.” In it, Benoit gave many examples, showing that fractals exist in nature (fyva), he described their varied shapes and proved that they are easily described by mathematical expressions. The mathematician cited this analogy as an argument for the usefulness of the theory he was developing in response to a barrage of criticism from his colleagues. They argued that a fractal is just a pretty picture, has no value, and is a by-product of the work of electronic machines. Carpenter decided to try this method in practice. After carefully studying the book, the future animator began to look for a way to implement fractal geometry in computer graphics. It took him only three days to render a completely realistic image of the mountain landscape on his computer. And today this principle is widely used. As it turns out, creating fractals does not take much time and effort.
Carpenter's solution
The principle Lauren used was simple. It consists of dividing larger ones into small elements, and those into similar smaller ones, and so on. Carpenter, using large triangles, split them into 4 small ones, and so on, until he had a realistic mountain landscape. Thus, he became the first artist to use a fractal algorithm in computer graphics to construct the required image. Today this principle is used to imitate various realistic natural forms.
The first 3D visualization using a fractal algorithm
A few years later, Lauren applied his developments in a large-scale project - the animated video Vol Libre, shown on Siggraph in 1980. This video shocked many, and its creator was invited to work at Lucasfilm. Here the animator was able to realize his full potential; he created three-dimensional landscapes (an entire planet) for the feature film "Star Trek". Any modern program (“Fractals”) or application for creating 3D graphics (Terragen, Vue, Bryce) uses the same algorithm for modeling textures and surfaces.
Tom Beddard
Formerly a laser physicist and now a digital artist and artist, Beddard created a number of very intriguing geometric shapes, which he called Fabergé fractals. Outwardly, they resemble decorative eggs from a Russian jeweler; they have the same brilliant, intricate pattern. Beddard used a template method to create his digital renderings of the models. The resulting products amaze with their beauty. Although many refuse to compare a handmade product with a computer program, it must be admitted that the resulting forms are extremely beautiful. The highlight is that anyone can build such a fractal using the WebGL software library. It allows you to explore various fractal structures in real time.
Fractals in nature
Few people pay attention, but these amazing figures are present everywhere. Nature is created from self-similar figures, we just don’t notice it. It is enough to look through a magnifying glass at our skin or a leaf of a tree, and we will see fractals. Or take, for example, a pineapple or even a peacock's tail - they consist of similar figures. And the Romanescu broccoli variety is generally striking in its appearance, because it can truly be called a miracle of nature.
Musical pause
It turns out that fractals are not only geometric shapes, they can also be sounds. Thus, musician Jonathan Colton writes music using fractal algorithms. It claims to correspond to natural harmony. The composer publishes all of his works under a CreativeCommons Attribution-Noncommercial license, which provides for free distribution, copying, and transfer of works to others.
Fractal indicator
This technique has found a very unexpected application. On its basis, a tool for analyzing the stock exchange market was created, and, as a result, it began to be used in the Forex market. Nowadays, the fractal indicator is found on all trading platforms and is used in a trading technique called price breakout. This technique was developed by Bill Williams. As the author comments on his invention, this algorithm is a combination of several “candles”, in which the central one reflects the maximum or, conversely, the minimum extreme point.
Finally
So we looked at what a fractal is. It turns out that in the chaos that surrounds us, there actually exist ideal forms. Nature is the best architect, ideal builder and engineer. It is arranged very logically, and if we cannot find a pattern, this does not mean that it does not exist. Maybe we need to look on a different scale. We can say with confidence that fractals still hold many secrets that we have yet to discover.
Completed by 7th grade student Polina Karpyuk
Prioda is created from self-similar figures, we just don’t notice it. In this gallery we have collected images in which fractality is clearly visible.
Download:
Preview:
To use presentation previews, create a Google account and log in to it: https://accounts.google.com
Slide captions:
Fractals in nature Completed by: 7th “B” class student Polina Karpyuk Supervisor: Molchanova Irina Pavlovna Rubtsovsk-2015
Mathematics, when looked at correctly, reflects not only truth, but also incomparable beauty. Bertrand Russell
What do a tree, a seashore, a cloud, or the blood vessels in our hand have in common? There is one property of structure that is inherent in all of the listed objects: they are self-similar. From a branch, as from a tree trunk, smaller shoots extend, from them even smaller ones, etc., that is, a branch is similar to the whole tree. The circulatory system is structured in a similar way: arterioles depart from the arteries, and from them the smallest capillaries through which oxygen enters the organs and tissues. The American mathematician Benoit Mandelbrot called this property of objects fractality, and such objects themselves - fractals. The word “fractal” itself is translated from Latin as “partial”, “divided”, “fragmented”, and as for the content of this term, there is no formulation as such. It is usually interpreted as a self-similar set, a part of the whole, which repeats its structure at the micro level. .
Space photographs of Earth's landscapes often provide excellent examples of fractals.
Coastlines usually have a fractal shape, but vary in the degree to which they are rugged. This example shows two characteristic properties of natural fractals: Individual channels are not a copy of each other, but have similar curvilinear outlines, as if they were drawn by the same pattern. Large ducts are similar in outline to small and very small ducts. If we enlarge, for example, the lower left corner of the picture, we will get something similar to the entire picture
The interaction of water and land gives rise to fractal structures in landscapes - be they mountains, rivers or coastlines.
Probably everyone knows the painting by the Japanese artist Hokusai “The Great Wave”, where a tsunami wave is depicted against the backdrop of Fuji. If you look closely at this picture, you will notice that when drawing the crest of a wave, the artist used a fractal, as if consisting of numerous predatory water paws. Therefore, this picture is often used as an illustration for books on chaos theory and fractals.
When a sand dune is eroded by water, it replicates on a tiny scale what gives fractal shape to larger Earth landscapes.
Lightning discharge is one example of natural fractals.
This picture illustrates not only the fractal nature of tree crowns, it suggests another interesting consideration: the forest as a biological community is also a fractal. Individual trees - large and small - then act as branches of the fractal. They are similar, but do not repeat each other.
Leaf veins are a flat natural fractal. For each plant, the characteristic pattern is unique, just as the papillary pattern on a person’s hand is unique. Goethe (poet and scientist) believed that the leaf is the most expressive part of the plant, which reflects its entire morphology.
Ferns are an example of natural fractals that are very similar to computer fractals. Moreover, they are also interesting because ferns are one of the most evolutionarily ancient plants, along with various mosses and other lower plants
This is another famous and very impressive example of a natural fractal that has mathematically clear forms. There are at least three levels of self-similar ingenious pyramids Romanesco cabbage
A magically beautiful fractal that could well inspire some artist. Meanwhile, take a closer look: this is just a tight bunch of cabbage leaves.
These are interesting examples of fractal structure in the mineral world. Carbonate Apatite Gold Nugget is an exquisite treasure crafted by nature itself.
Have you ever thought that we literally think in fractals? There is something to think about here - who would argue that the brain is one of the most amazing and unique creations of nature. And it turns out that outwardly it has the same fractal features as atmospheric clouds or the root system of nettles.
Here everything is even more complicated: two separate fractal trees are intertwined - venous blood is supplied to one, and oxygen-enriched arterial blood is discharged to the other. And in totality, the lung is an amazingly complex system of three fractals - one respiratory and two circulatory.
The retina contains light-sensitive cells that allow us to see. In this photo they are yellowish-green. They do form a network (the retina), but this network is chaotic and fractal.
This is the belly of a pig. Its color patterns also seem to follow fractal rules. This is an interesting topic and, most importantly, has many applications, including military significance. By what rules should a camouflage pattern be drawn up so that its wearer blends with natural forms - landscape and vegetation?
Thank you for your attention!!!
Send your good work in the knowledge base is simple. Use the form below
Students, graduate students, young scientists who use the knowledge base in their studies and work will be very grateful to you.
Posted on http://www.allbest.ru
- Introduction
- 1. The concept of a fractal
- 2. Classification of fractals
- 4. Application of fractals
- Conclusion
- List of used literature
Introduction
The appearance of self-similar mathematical objects a hundred or more years ago was of no interest to almost anyone; they were of interest only to the authors of these objects. Moreover, some scientists dubbed them “monsters” and did not believe that they had anything to do with the real world and science.
Attitudes towards self-similar mathematical objects changed with the advent of computers, when the first images of algebraic and stochastic fractals appeared. Immediately after this, they interested not only mathematicians, but also physicists, biologists, acousticians, and everyone who came across natural objects in their work. Mathematicians were attracted to fractals by the simplicity of the formulas that describe such complex structures, physicists by the opportunity to reconsider physics from a new position, biologists by the correspondence of fractal images with various biological objects.
Fractals have not yet exhausted themselves; fractal objects are still being found in new areas of science. They are used by physicists, biologists, sociologists, economists and many others. Fractals have not been fully studied; new applications are being found for them, changing our attitude both to fractals themselves and to nature.
The object of the work is the phenomenon of fractals.
The subject of the work is the place of fractals in modern science.
The purpose of the work is to consider fractals as both a simple and complex phenomenon.
Objectives of the work: consider the concept of fractals, types of fractals, the history of the emergence and study of fractals, the application of fractals in practice.
1. The concept of a fractal
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 of the 20th century. The word fractal is derived from the Latin fractus and translated means consisting of fragments. Mandelbrot B. Fractal geometry of nature, p. 5 - M.: Institute of Computer Research, 2002. . It was proposed by Benoit Mandelbrot in 1975 to designate irregular but self-similar structures, which he studied. Mandelbrot B. Fractal geometry of nature, p. 5 - M.: Institute for Computer Research, 2002. . The birth of fractal geometry is usually associated with the publication of Mandelbrot’s book “The Fractal Geometry of Nature” in 1977. His works used the scientific results of other scientists who worked in the period 1875-1925 in the same field (Poincaré, Fatou, Julia, Cantor, Hausdorff But only in our time has it been possible to combine their work into a single system.
The role of fractals in computer graphics today is quite large. They come to the rescue, for example, when it is necessary, using several coefficients, to define lines and surfaces of very complex shapes. From the point of view of computer graphics, fractal geometry is indispensable when generating artificial clouds, mountains, and sea surfaces. In fact, a way has been found to easily represent complex non-Euclidean objects, the images of which are very similar to natural ones.
One of the main properties of fractals is self-similarity. In the simplest case, a small part of a fractal contains information about the entire fractal.
The definition of a fractal given by Mandelbrot is as follows: “A fractal is a structure consisting of parts that are in some sense similar to the whole.” Feder E. Fractals: World 1991, p.67.
It should be noted that the word “fractal” is not a mathematical term and does not have a generally accepted strict mathematical definition. It can be used when the figure in question has any of the following properties:
1. Has a non-trivial structure on 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.
2. Is self-similar or approximately self-similar.
3. Has a fractional metric dimension or a metric dimension that exceeds the topological one.
4. Can be constructed using a recursive procedure Feder E. Fractals: World 1991, p. 133.
Many objects in nature have fractal properties, for example coasts, clouds, tree crowns, 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.
Fractals are primarily the language of geometry. However, their main elements are not directly observable. In this respect, they are fundamentally different from the usual objects of Euclidean geometry, such as a straight line or a circle. Fractals are expressed not in primary geometric forms, but in algorithms, sets of mathematical procedures.
These algorithms are transformed into geometric shapes using a computer. The repertoire of algorithmic elements is inexhaustible. Once you master the language of fractals, you can describe the shape of a cloud as clearly and simply as an architect describes a building using drawings that use the language of traditional geometry.
2. Classification of fractals
Geometric fractals. Fractals of this class are the most visual. In the two-dimensional case, they are obtained using some broken line (or surface in the three-dimensional case), called a generator. In one step of the algorithm, each of the segments that make up the polyline is replaced with a generator polyline, on the appropriate scale. As a result of endless repetition of this procedure, a geometric fractal is obtained.
Algebraic fractals. This is the largest group of fractals. They are obtained using nonlinear processes in n-dimensional spaces. Two-dimensional processes are the most studied. When interpreting a nonlinear iterative process as a discrete dynamic system, one can use the terminology of the theory of these systems: phase portrait, steady-state process, attractor, etc.
It is known that nonlinear dynamic systems have several stable states. The state in which the dynamic system finds itself after a certain number of iterations depends on its initial state. Therefore, each stable state (or, as they say, attractor) has a certain region of initial states, from which the system will necessarily fall into the final states under consideration. Thus, the phase space of the system is divided into areas of attraction of attractors. If the phase space is a two-dimensional space, then by coloring the areas of attraction with different colors, one can obtain a color phase portrait of this system (iterative process). By changing the color selection algorithm, you can get complex fractal patterns with bizarre multicolor patterns. A surprise for mathematicians was the ability to generate very complex non-trivial structures using primitive algorithms.
Scholastic fractals. Natural objects that arise as a result of complex random processes often have a fractal shape. Stochastic (random) fractals can be used to model them. Examples of stochastic fractals:
1. trajectory of Brownian motion on the plane and in space;
2. boundary of the trajectory of Brownian motion on a plane. In 2001, Lawler, Schramm and Werner proved Mandelbort's conjecture that its dimension is 4/3.
3. Schramm-Löwner evolutions - conformally invariant fractal curves arising in critical two-dimensional models of statistical mechanics, for example in the Ising model and percolation.
4. 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.
Fractal monotype, or stochatypy, are trends in the visual arts that consist in obtaining an image of a random fractal. Schroeder M. Fractals, chaos, power laws. Miniatures from endless paradise. - Izhevsk: RHD, 2001, p.26.
3. The history of fractals
It is noteworthy that the appearance of fractals (not yet given this name) in the mathematical literature about a hundred years ago was met with regrettable hostility, as has happened in the history of the development of many other mathematical ideas. One famous mathematician, Charles Hermite, even dubbed them monsters. At least the general consensus recognized them as a pathology of interest only to researchers who abuse mathematical fads, and not to real scientists.
As a result of the efforts of Benoit Mandelbrot, this attitude changed, and fractal geometry became a respected applied science. Mandelbrot coined the term fractal based on Hausdorff's theory of fractal (fractional) dimension, proposed in 1919. Many years before the appearance of his first book on fractal geometry, Mandelbrot began researching the appearance of monsters and other pathologies in nature. He found a niche for the disreputable Cantor sets, Peano curves, Weierstrass functions and their many variations, which were considered nonsense. He and his students discovered many new fractals, such as fractal Brownian motion for modeling forest and mountain landscapes, river level fluctuations and heartbeats. With the publication of his books, applications of fractal geometry began to appear like mushrooms after rain. This affected both many applied sciences and pure mathematics. Even the film industry was not left out. Millions of people admired the mountain landscape in the film “Star Migration II: The Wrath of Khan,” constructed using fractals Peitgen H.-O., Richter P. H. The beauty of fractals. - M.: Mir 1993, p.45.
French mathematician Henri Poincaré initiated research into nonlinear dynamics around 1890, leading to modern chaos theory. Interest in the subject increased markedly when Edward Lorenz, a nonlinear weather modeler, discovered in 1963 that long-term weather forecasts were impossible. Lorenz noted that even small errors in measuring the current state of weather conditions can lead to completely incorrect predictions about future weather conditions. This essential dependence on initial conditions underlies the mathematical theory of chaos.
Particle trajectories of Brownian motion, which were studied by Robert Brown as early as 1828 and Albert Einstein in 1905, are an example of fractal curves, although their mathematical description was not given until 1923 by Norbert Wiener. In 1890, Peano constructed his famous curve - a continuous mapping that transforms a segment into a square and, therefore, increases its dimension from one to two. The Koch snowflake boundary (1904), whose dimension d » 1.2618, is another well-known dimension-increasing curve.
The fractal, in no way resembling a curve, which Mandelbrot called dust, is the classical Cantor set (1875 or earlier). This set is so sparse that it does not contain intervals, but nevertheless has the same number of points as the interval. Mandelbrot used such “dust” to model stationary noise in telephony. Fractal dust of one kind or another appears in numerous situations. In fact, it is a universal fractal in the sense that any fractal - an attractor of a system of iterated functions - is either fractal dust or its projection onto a space with a lower dimension Peitgen H.-O., Richter P., p. 22.
Various tree-like fractals were used not only to model tree-plants, but also the bronchial tree (air-bearing branches in the lungs), the functioning of the kidneys, the circulatory system, etc. It is interesting to note Leonardo da Vinci’s assumption that all branches of a tree at a given height, added together , equal in thickness to the trunk (below their level). This implies a fractal model for the tree crown in the form of a fractal surface.
Many remarkable properties of fractals and chaos are revealed by studying iterated mappings. In this case, they start with some function y = f(x) and consider the behavior of the sequence f(x), f(f(x)), f(f(f(x))),... In the complex plane, work of this kind ascends , apparently, to the name of Cayley, who investigated Newton's method of finding the root as applied to complex, and not just real, functions (1879). Remarkable progress in the study of iterated complex mappings was made by Gaston Julia and Pierre Fatou (1919). Naturally, everything was done without the help of computer graphics. These days, many have already seen colorful posters depicting *Julia sets and the Mandelbrot set, which is closely related to them. It is natural to start mastering the mathematical theory of chaos with iterated mappings.
The study of fractals and chaos opens up wonderful possibilities, both in the study of an infinite number of applications and in the field of pure mathematics. But at the same time, as often happens in the so-called new mathematics, the discoveries are based on the pioneering work of the great mathematicians of the past. Sir Isaac Newton understood this when he said, “If I have seen further than others, it is because I have stood on the shoulders of giants.”
4. Application of fractals
Computer graphics
Fractals are widely used in computer graphics to construct images of natural objects, such as trees, bushes, mountain landscapes, sea surfaces, etc.
Physics and other natural sciences
In physics, fractals naturally arise when modeling nonlinear processes, such as turbulent fluid flow, complex random diffusion-adsorption processes, flames, clouds, etc. Fractals are also 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).
Literature
Among literary works there are those that have a textual, structural or semantic fractal nature. In text fractals, elements of text are potentially endlessly repeated:
1. A non-branching infinite tree, identical to themselves from any iteration (“The priest had a dog...”, “The parable of a philosopher who dreams that he is a butterfly who dreams that she is a philosopher who dreams...”, “It is a false statement that the statement is true, the statement is false...").
2. Non-branching endless texts with variations (“Peggy had a funny goose…”) and texts with extensions (“The House That Jack Built”).
3. In structural fractals, the text layout is potentially fractal
4. Wreath of sonnets (15 poems), wreath of sonnets (211 poems), wreath of wreaths of sonnets (2455 poems).
5. “Stories within a story” (“The Book of One Thousand and One Nights”, J. Pototsky “Manuscript Found in Saragossa”).
6. Prefaces that hide the authorship (U. Eco “The Name of the Rose”).
In semantic and narrative fractals, the author talks about the infinite similarity of a part to the whole
H. L. Borges “In the circle of ruins”
J. Cortazar “Yellow Flower”
J. Perek “Kunstkamera”
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 glued it onto a piece of paper, then attached it to the receiver. It turned out that such an antenna works no worse than a regular one. And although the physical principles of operation of such an antenna have not yet been studied, this did not stop Cohen from founding his own company and launching their serial production.
Image compression.
There are algorithms for compressing images using fractals. They are based on the idea that instead of an image, one can store a compression mapping for which the image is a fixed point.
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.
Conclusion
Most people believe that fractals are just beautiful pictures that please the eye. Fortunately, this is not the case, and fractals are used in many areas of human activity. There is already a theoretical basis for creating new areas of their application, such as diagnosing diseases, predicting damage during dynamic impact, and many others. But, despite the theoretical inexhaustibility of the use of fractals, it can be assumed that over time the main directions of their application will emerge.
Only a few decades have passed since Benoit Mandelbrot declared: “The geometry of nature is fractal!” Today we can already assume much more, namely, that fractality is the primary principle of construction of all natural objects without exception.
Conclusions:
1. The nature of fractals is carefully studied by scientists
2. In the future, many problems in medicine, the computer industry, science, etc. will be solved with the help of fractals.
List of used literature
fractal natural graphics
1. Mandelbrot B. Fractal geometry of nature. - M.: Institute of Computer Research, 2002.
2. Peitgen H.-O., Richter P. H. The beauty of fractals. - M.: Mir, 1993.
3. Feder E. Fractals-M.: Mir, 1991.
4. Schroeder M. Fractals, chaos, power laws. Miniatures from endless paradise. - Izhevsk: RHD, 2001.
Posted on Allbest.ru
Similar documents
A fractal is a set whose dimension differs from the usual dimension, called topological. Principles and conditions for the formation of an appropriate system according to Mandelbrot’s research. Types and meaning of fractals, the main stages of their evolution.
test, added 02/19/2015
The essence of modern concepts of the relativity of space and time is in the special and general theories. Hyperchronological historical space, acceleration of historical time. Explanation of the concepts of bifurcations, fractals, attractors, randomness factors.
test, added 12/10/2009
Humanitarian, technical, mathematical types of knowledge and natural science in the modern knowledge system. The role and significance of mathematics and physics in understanding the world. Attitude to nature in the natural sciences and humanities. The problem of confrontation between science and religion.
abstract, added 11/26/2011
The development of natural sciences in the Middle Ages, the place and role of the church in the state. Construction of a theory of atomic structure based on the planetary model. Development of astronomy, characteristics of galaxies. Theories of the origin of life on Earth. Hypotheses of the origin of races.
test, added 09/14/2009
Hippocrates as the founder of modern clinical medicine. The merit of ancient scientists in the development of natural sciences. The content of the basic laws of dialectics, the application of dialectical research methods. The law of the transition of quantity to quality.
test, added 04/03/2011
Synergetics as a theory of self-organizing systems in the modern scientific world. History and logic of the emergence of a synergetic approach in natural science. The influence of this approach on the development of science. Methodological significance of synergetics in modern science.
abstract, added 12/27/2016
General characteristics of bacteria. Their structure, reproduction and nutrition. The concept of natural resources and their characteristics. The structure and significance of the digestive system. Economic classification of natural resources. The structure of the wall of the digestive canal.
test, added 10/09/2012
Current trends in the development of industry, energy, and the national economy. Transformations in the field of science. Consequences of the development of biotechnology and developments in the natural sciences. Chemical processes and energy. Preservation of the ozone layer.
abstract, added 11/18/2009
Application of the principle of absolute objectivity and certainty of empirical data in quantum physics. Using compasses and ruler in Euclidean geometry. Analysis of the periodic system of chemical elements D.I. Mendeleev. Properties of bifurcation points.
test, added 06/12/2015
The concept of bioelectric phenomena. The emergence of modern membrane excitation theory. The main types of bioelectric potentials, the mechanism of their occurrence and application in medical and biological laboratories, in clinical practice in diagnostics.