Niels ole block biography. Nobel laureates: Niels Bohr

Bohr Niels Hendrik David (1885-1962), Danish theoretical physicist.

Born on October 7, 1885 in Copenhagen, he graduated from the university there in 1908. He worked for some time in Cambridge (England) in the laboratory of the luminary in the field of physics J. Thomson, then was invited to Manchester in the laboratory of another celebrity - E. Rutherford.

A few years later he created and headed the Institute of Theoretical Physics in Copenhagen. This scientific center gave Bohr the opportunity to bring together all the outstanding physicists of that time. The main merit of the scientist is considered to be the formulation of a fundamentally new approach to presenting the physical picture of atomic processes. By this time, massive and contradictory experimental material, the works of M. Planck, A. Einstein, analysis of the emission spectra of atoms showed the unusual patterns of the microworld.

Bohr proposed a new model of the hydrogen-like atom and discovered the conditions for its stability. He developed Planck's idea of ​​quantizing energy and, based on Rutherford's atomic model, created the first quantum model of the atom. At the same time, the physicist established that there are stationary orbits in the atom, moving along which the electron does not emit energy (contrary to the laws of electrodynamics), but can abruptly move to an orbit closer to the atom. At the moment of the jump, it emits a quantum (portion) of energy equal to the difference in the energies of the atom in stationary states.

In addition, Bohr also developed some quantization rules. He also determined the basic laws of spectral lines and electron shells of atoms, explained the features of D.I. Mendeleev’s periodic system of chemical elements and developed his own version of the image of these elements, and came to the idea of ​​​​the existing shell around the atom.

Starting to work on this article, I remembered the time when we, students of a secondary physics and mathematics school, heard about the era of the creation of modern physics, about the heated discussions of the Solvay congresses, about the struggle of ideas in which a new picture of the world was born. The names of the creators of science of the twentieth century: Planck, Einstein, Bohr, Heisenberg, Schrödinger, Pauli - sounded like a call to dare. We worshiped the greats and dreamed of following them in the search for order and law in the chaos of experimental data.

Imperfect photographs of the first half of the twentieth century, even in combination with the printing of popular publications, still brought to us the image of a physicist-thinker with a calm, large, slightly elongated, “horselike” face, with intelligent, understanding eyes. Niels Bohr really was a philosopher who sought answers to the eternal questions of existence by studying the phenomena of the physical world around us.

His interest in philosophy began from childhood. Niels and his brother Harald, a famous mathematician, grew up in the family of a professor at the University of Copenhagen, a member of the Danish Academy of Sciences, and physiologist Christian Bohr. The special spirit of this family was created by the father and his friends, primarily the philosopher Harald Hoeffding. From them, Nils learned to dig into the essence of things, to look for what is hidden behind external forms. While still a student at the University of Copenhagen, Niels and his friends, also students of Hoeffding’s seminar, created a philosophical club called “Ecliptic.” Among its members were a physicist, a mathematician, a lawyer, a psychologist, a historian, an entomologist, a linguist, an art critic... The difference in scientific languages ​​and approaches was not an obstacle for young men who were looking for answers to questions about the relationship between Providence and free will, and about the knowability of the world. According to Leon Rosenfeld, Bohr's friend and biographer, Niels "was about 16 years old when he rejected the spiritual claims of religion and became deeply obsessed with the nature of our thinking and language." These questions did not leave him all his life.

And his life, of course, was devoted to physics. But not the physics that stops at a formal statement of a fact or a mathematical recording of the relationship between physical quantities. He was always interested in the cause, the internal mechanism, “the way the world really works,” rather than how it could be plausibly described. His main successes were in finding connections between facts that no one had connected before: he saw a commonality in the deceleration of particles in the medium and in the weakening of light; in the magnitude of the charge of the nucleus of an atom and the periodicity of the properties of chemical elements of the periodic table. These provisions, obvious to today's physics students, were by no means obvious at the beginning of the twentieth century, and their confirmation required a careful analysis of many facts. Bohr's early work formed the basis of the method that physics lives to this day - when a hypothesis put forward to explain each known fact is examined, checked to see if there are any contradictions in it, and the logical coherence of the emerging theory is the main criterion for its truth, no matter what. She didn’t seem strange at all.

A planetary model of the atom was also created. It would seem how wonderful and beautiful! Like the planets revolving around the sun, the electrons in a Bohr atom revolve around the nucleus - who would object to that? Moreover, after Rutherford’s experiments on the scattering of alpha particles on gold nuclei, which showed that matter is mainly concentrated in compact nuclei located at considerable distances from one another. However, a contradiction arises with the classical theory of radiation: an electron rotating in an orbit should emit an electromagnetic wave and, therefore, lose energy, and as a result, “fall” onto the nucleus. The solution at first glance is simple: it is necessary to “prohibit” the electron from emitting while moving in orbit. But this is the revolution of natural science: the recognition that the laws of the micro-level differ from the laws of the world on a large scale! This needs to be convinced, which means selecting evidence from experiments on electricity, magnetism, spectroscopy, and so on; it is also necessary to explain where the border between the micro and macro worlds extends and how the laws of the micro world flow into classical laws.

Bohr does this, but he doesn’t just build a physical theory, he receives a philosophical principle - the Principle of Correspondence: the “new” theory must be interfaced with the “old” one, and this interface must be thoroughly traced step by step.

Another philosophical principle of Niels Bohr is the Principle of Complementarity. It arose, in particular, from attempts to describe the strange behavior of light: either as waves in diffraction experiments, or as particles in experiments on the photoelectric effect. Light, therefore, can be described using two classical images, but absolutely incompatible ones! And Bohr raises this to a principle: the phenomenon must be described from different sides, albeit in a contradictory (from the point of view of conventional ideas) way. After all, “no matter how far beyond the capabilities of classical analysis quantum events lie... we are forced to register the results obtained in ordinary language.” To describe true reality, a figurative language of special power is needed; Bohr compares the work of a physicist on its creation with the work of a poet - both are looking for images that reflect reality: “The poet is also concerned not so much with the accurate depiction of things, but with the creation of images and the consolidation of mental associations in the heads their listeners." But Bohr's physical reality differs from poetic reality. This is not the inner world of the poet, but a unity of interconnected facts and natural phenomena; to describe it we need concepts that complement each other. Reflecting on the principles of quantum theory as a unified system of ideas, he writes: “For me this is not at all a question of trifling didactic tricks, but a problem of serious attempts to achieve such internal consistency in these ideas as would allow us to hope for the creation of an unshakable basis for subsequent constructive work "

Perhaps this is the most important discovery of science of the twentieth century - the discovery that the world of natural phenomena cannot be described by simple concepts obtained from experience and fixed in the terms of classical science. A world beyond the usual scale is difficult to understand: “We are faced with difficulties that lie so deep that we have no idea of ​​the path leading to overcoming them; in accordance with my view of things, these difficulties are such in nature that they hardly leave us the right to hope that we will be able to construct a description of events in time and space in the same way in the atomic world as we have usually done so far por." To comprehend it, you need to get away from habits and stereotypes and try to see the world with an unclouded gaze, the gaze of a child.

And Niels Bohr successfully copes with this. He is helped by a well-developed sense of humor. Let me remind you, for example, of his judgment about his student who failed in science: “He became a poet - he had too little imagination for physics.” No less famous is Bohr’s statement about one of the physical theories: “There is no doubt that we have before us a crazy theory, but the whole question is whether it is crazy enough to also be true!” At one of the dramatic moments in the formation of a new quantum theory, when each participant in the discussion offered one or another argument, a thought experiment, or simply an image designed to show the correctness of one or another point of view, Einstein found an expression remarkable in its power: “God does not play dice! » It is, indeed, absurd to imagine a Creator guided by chance, but this is precisely the mechanism of quantum phenomena proposed by the Copenhagen interpretation. Niels Bohr retorted: “But, really, it is not our sadness to prescribe to the Lord God how he should rule this world!” An illustration of Niels Bohr's paradoxical thinking can be seen in his classification of “thoughts by depth”: he believed that a statement is trivial and shallow if the exact opposite is nonsense; if the exact opposite is full of meaning, then the judgment is non-trivial.

Philosophical understanding of open laws helped Bohr find answers to important questions of existence. Thus, Heisenberg’s uncertainty relation seemed to him to be the physical basis for the answer to the question that interested him back in the days of the Ecliptic - the question of free will. He saw the entire world of living organisms, as well as mental phenomena, as similar to the world of atomic particles: the same principles operate in both.

When Niels Bohr was granted the dignity of nobility in recognition of his scientific merits, he had to choose a coat of arms and a motto for himself. Seeing deep analogies between Eastern philosophy and the ideas of the science to which he devoted his life, Bohr chose the symbol of Taiji, expressing the relationship between the opposite principles of yin and yang, and as the motto the Latin phrase “Contraria sunt complementa” (“Opposites complement each other”).

for the magazine "Man Without Borders"

Their home was the center of very lively discussions on pressing scientific and philosophical issues, and throughout his life B. reflected on the philosophical implications of his work. He studied at the Gammelholm Grammar School in Copenhagen and graduated in 1903. B. and his brother Harald, who became a famous mathematician, were avid football players during their school years; Nils later became interested in skiing and sailing.

When B. was a physics student at the University of Copenhagen, where he became a bachelor in 1907, he was recognized as an unusually capable researcher. His thesis project, in which he determined the surface tension of water from the vibration of a water jet, earned him a gold medal from the Royal Danish Academy of Sciences. He received his master's degree from the University of Copenhagen in 1909. His doctoral dissertation on the theory of electrons in metals was considered a masterful theoretical study. Among other things, it revealed the inability of classical electrodynamics to explain magnetic phenomena in metals. This research helped Bohr realize early in his scientific career that classical theory could not fully describe the behavior of electrons.

After receiving his doctorate in 1911, B. went to the University of Cambridge, England, to work with J.J. Thomson, who discovered the electron in 1897. However, by that time Thomson had already begun to study other topics, and he showed little interest in B.’s dissertation and the conclusions contained therein. But B., meanwhile, became interested in the work of Ernest Rutherford at the University of Manchester. Rutherford and his colleagues studied issues of radioactivity of elements and the structure of the atom. B. moved to Manchester for several months at the beginning of 1912 and energetically plunged into these studies. He drew many consequences from the nuclear model of the atom proposed by Rutherford, which has not yet received wide recognition. In discussions with Rutherford and other scientists, B. worked out ideas that led him to the creation of his own model of the structure of the atom.

In the summer of 1912, B. returned to Copenhagen and became an assistant professor at the University of Copenhagen. In the same year he married Margret Norlund. They had six sons, one of whom, Oge Bohr, also became a famous physicist.

Over the next two years, B. continued to work on problems arising in connection with the nuclear model of the atom. Rutherford proposed in 1911 that the atom consists of a positively charged nucleus around which negatively charged electrons orbit. This model was based on ideas that were experimentally confirmed in solid state physics, but it led to one intractable paradox. According to classical electrodynamics, an orbiting electron must constantly lose energy, giving it back in the form of light or another form of electromagnetic radiation. As its energy is lost, the electron must spiral toward the nucleus and eventually fall onto it, which would destroy the atom. In fact, atoms are very stable, and therefore there is a gap in the classical theory. Bohr was particularly interested in this apparent paradox of classical physics because it was too reminiscent of the difficulties he had encountered during his dissertation work. A possible solution to this paradox, he believed, could lie in quantum theory.

In 1900, Max Planck proposed that electromagnetic radiation emitted by hot matter does not come in a continuous stream, but in well-defined discrete portions of energy. Having called these units quanta in 1905, Albert Einstein extended this theory to electron emission that occurs when light is absorbed by certain metals (photoelectric effect). Applying the new quantum theory to the problem of atomic structure, B. suggested that electrons have certain allowed stable orbits in which they do not emit energy. Only when an electron moves from one orbit to another does it gain or lose energy, and the amount by which the energy changes is exactly equal to the energy difference between the two orbits. The idea that particles could only have certain orbits was revolutionary because, according to classical theory, their orbits could be located at any distance from the nucleus, just as planets could, in principle, revolve in any orbit around the Sun.

Although Bohr's model seemed strange and a little mystical, it solved problems that had long puzzled physicists. In particular, it provided the key to separating the spectra of elements. When light from a luminous element (such as a heated gas of hydrogen atoms) passes through a prism, it produces not a continuous, all-color spectrum, but a sequence of discrete bright lines separated by broader dark regions. According to B.'s theory, each bright colored line (that is, each individual wavelength) corresponds to the light emitted by electrons as they move from one allowed orbit to another lower-energy orbit. B. derived a formula for the frequencies of lines in the spectrum of hydrogen, which contained Planck’s constant. The frequency multiplied by Planck's constant is equal to the energy difference between the initial and final orbits between which the electrons make the transition. B.'s theory, published in 1913, brought him fame; his model of the atom became known as the Bohr atom.

Immediately appreciating the importance of B.'s work, Rutherford offered him a lecturer position at the University of Manchester - a post that Bohr held from 1914 to 1916. In 1916, he took up the post of professor created for him at the University of Copenhagen, where he continued to work on the structure of the atom . In 1920 he founded the Institute of Theoretical Physics in Copenhagen; With the exception of the period of the Second World War, when B. was not in Denmark, he led this institute until the end of his life. Under his leadership, the institute played a leading role in the development of quantum mechanics (the mathematical description of the wave and particle aspects of matter and energy). During the 20s. Bohr's model of the atom was replaced by a more complex quantum mechanical model, based mainly on the research of his students and colleagues. Nevertheless, Bohr's atom played an essential role as a bridge between the world of atomic structure and the world of quantum theory.

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B. was awarded the Nobel Prize in Physics in 1922 “for his services to the study of the structure of atoms and the radiation emitted by them.” At the presentation of the laureate, Svante Arrhenius, a member of the Royal Swedish Academy of Sciences, noted that B.’s discoveries “led him to theoretical ideas that differ significantly from those that underlay the classical postulates of James Clerk Maxwell.” Arrhenius added that the principles laid down by B. “promise abundant fruits in future research.”

B. wrote many works devoted to problems of epistemology (cognition) arising in modern physics. In the 20s he made a decisive contribution to what was later called the Copenhagen interpretation of quantum mechanics. Based on Werner Heisenberg's uncertainty principle, the Copenhagen interpretation assumes that the rigid laws of cause and effect that we are familiar with in the everyday, macroscopic world do not apply to intra-atomic phenomena, which can only be interpreted in probabilistic terms. For example, it is not even possible in principle to predict in advance the trajectory of an electron; instead, one can specify the probability of each of the possible trajectories.

B. also formulated two of the fundamental principles that determined the development of quantum mechanics: the principle of correspondence and the principle of complementarity. The correspondence principle states that a quantum mechanical description of the macroscopic world must correspond to its description within classical mechanics. The principle of complementarity states that the wave and particle nature of matter and radiation are mutually exclusive properties, although both of these concepts are necessary components of understanding nature. Wave or particle behavior may appear in a certain type of experiment, but mixed behavior is never observed. Having accepted the coexistence of two obviously contradictory interpretations, we are forced to do without visual models - this is the idea expressed by B. in his Nobel lecture. In dealing with the world of the atom, he said, "we must be modest in our demands and content with concepts that are formal in the sense that they lack the visual picture so familiar to us."

In the 30s B. turned to nuclear physics. Enrico Fermi and his colleagues studied the results of bombarding atomic nuclei with neutrons. B., together with a number of other scientists, proposed a droplet model of the nucleus, corresponding to many of the observed reactions. This model, which compared the behavior of an unstable heavy atomic nucleus to a fissile drop of liquid, enabled Otto R. Frisch and Lise Meitner to develop a theoretical framework for understanding nuclear fission in late 1938. The discovery of fission on the eve of the Second World War immediately gave rise to speculation about how it could be used to release colossal energy. During a visit to Princeton in early 1939, B. determined that one of the common isotopes of uranium, uranium-235, is a fissionable material, which had a significant impact on the development of the atomic bomb.

In the first years of the war, B. continued to work in Copenhagen, under the conditions of the German occupation of Denmark, on the theoretical details of nuclear fission. However, in 1943, warned of the impending arrest, B. and his family fled to Sweden. From there, he and his son Auge flew to England in the empty bomb bay of a British military aircraft. Although B. considered the creation of an atomic bomb technically infeasible, work on creating such a bomb had already begun in the United States, and the Allies needed his help. At the end of 1943, Nils and Aage went to Los Alamos to participate in work on the Manhattan Project. The elder B. made a number of technical developments in creating the bomb and was considered an elder among the many scientists who worked there; However, at the end of the war he was extremely worried about the consequences of the use of the atomic bomb in the future. He met with US President Franklin D. Roosevelt and British Prime Minister Winston Churchill, trying to persuade them to be open and frank with the Soviet Union regarding new weapons, and also pushed for the establishment of a system of arms control in the post-war period. However, his efforts were unsuccessful.

After the war, B. returned to the Institute of Theoretical Physics, which expanded under his leadership. He helped found CERN (European Center for Nuclear Research) and played an active role in its scientific program in the 50s. He also took part in the founding of the Nordic Institute for Theoretical Atomic Physics (Nordita) in Copenhagen, the joint scientific center of the Scandinavian states. During these years, B. continued to speak out in the press for the peaceful use of nuclear energy and warned about the dangers of nuclear weapons. In 1950, he sent an open letter to the UN, repeating his wartime call for an “open world” and international arms control. For his efforts in this direction, he received the first Atom for Peace Prize, established by the Ford Foundation in 1957.

Having reached the mandatory retirement age of 70 in 1955, B. resigned as a professor at the University of Copenhagen, but remained head of the Institute of Theoretical Physics. In the last years of his life he continued to contribute to the development of quantum physics and took great interest in the new field of molecular biology.

A tall man with a great sense of humor, B. was known for his friendliness and hospitality. “The benevolent interest in people shown by B. made personal relationships at the institute in many ways reminiscent of similar relationships in the family,” recalled John Cockroft in his biographical memoirs about B. Einstein once said: “What is surprisingly attractive about B. as a scientific thinker is this is a rare fusion of courage and caution; few people had such an ability to intuitively grasp the essence of hidden things, combining this with keen criticism. He is without a doubt one of the greatest scientific minds of our century." B. died on November 18, 1962 at his home in Copenhagen as a result of a heart attack.

B. was a member of more than two dozen leading scientific societies and was president of the Royal Danish Academy of Sciences from 1939 until the end of his life. In addition to the Nobel Prize, he received the highest honors from many of the world's leading scientific societies, including the Max Planck Medal of the German Physical Society (1930) and the Copley Medal of the Royal Society of London (1938). He has held honorary degrees from leading universities including Cambridge, Manchester, Oxford, Edinburgh, Sorbonne, Princeton, McGill, Harvard and Rockefeller Center.

Soviet film actress Tatyana Andreevna Bozhok was born into the family of a railway worker and a housewife in 1957 in Moscow. In the family she was the youngest daughter and the sixth child. Since childhood, Tanya not only studied well, but was also interested in theatrical art: she went to the drama club of the Palace of Pioneers on Shabolovka.

At the age of 15, the assistant director of the film “Every Day of Doctor Kalinnikova” noticed her in the studio and invited Tatyana to film. This film became the debut in the film career of the young actress.

Movies

In Viktor Titov’s drama, dedicated to the work and scientific discoveries of Dr. G. Ilizarov, Tatyana Bozhok played the role of patient Tanechka. Famous artists became her partners in the workshop.

Photos of young Tatyana ended up in the Mosfilm file, and immediately after the first work, the young artist received an offer from Sergei Bondarchuk himself, who was selecting the cast for his film “They Fought for the Motherland.” A short girl with large, sensitive eyes and a thin voice was cast in the role of a nurse in the epic drama of the master.

After a successfully performed role, he invites Tatyana Bozhok to study in his workshop at VGIK, which he runs together with his wife. Since his students have already completed 1 year of study, the girl is taken straight into the 2nd year without exams.

Thanks to her youthful appearance, even the matured and already married actress often got the roles of young ladies. These are young teachers who have barely graduated from college (“The Adventures of Petrov and Vasechkin”, “Citizens of the Universe”, “Beware, Cornflower!”), and pioneer leaders (“Yeralash”, “Everything is the other way around”), and young secretaries or telephone operators (“Wick” ). Each role played by Tatyana Bozhok was quickly remembered by the audience thanks to her gift of transformation.

After graduating from the institute, the actress manages to star in the comedy “Ladies Invite Gentlemen,” where she played the main role. Her partners on the stage were actors who were venerable by that time and.

The 23-year-old inexperienced graduate of VGIK was somewhat complex at first, but her partner supported the girl and often gave wise advice on working on her image. And Leonid Kuravlev treated the young artist very tenderly and fatherly. It happened that while filming in another city, he even fed her tasty supplies. Tatiana Bozhok still has a good relationship with him.

Another significant work of the early period was her role as Masha from the film “The Singles Are Provided with a Hostel.” And again Tatyana Bozhok finds herself in the company of famous stars of Soviet cinema:,. The actress also played the main role in Arnold Agababov’s film “There, Beyond the Seven Mountains,” about the love of a Russian girl from the Siberian wilderness and a native Caucasian Armenian from Yerevan.

Particularly memorable roles of the actress were her work in films for children and in the almanac “Jumble,” in which she starred for 30 years, starting in 1973. Many fans often thought that Tatyana Bozhok was the mother of Fedya Stukov, the actor who played Tom Sawyer. But in real life, the actress is not a relative of Fedor.

Even the episodic appearance of Tatyana Bozhok in the frame was remembered by the audience. And her phrase “me too, James Bond has been found!” from the epic film “Guest from the Future,” where she played Kolya’s mother, became a catchphrase.

One of the popular roles of the actress is the role of teachers. These are often naive, eccentric people who may suffer from their absent-mindedness and indecisiveness. Such teachers, performed by Tatyana Bozhok, can be found in children's films “The Adventures of Petrov and Vasechkin”, “Beware, Cornflower!”. And for her role as a teacher in the film “Citizens of the Universe,” Tatyana Bozhok even received an award for best actress at the Moscow Festival of Young Filmmakers in 1984.

Voice acting

During the period of stagnation in Russian cinema, Tatyana Bozhok switched to working on scoring cartoons and foreign films. At first, she used more of her voice, which was given to her by nature. A high, almost childish timbre allowed her to voice funny cartoon characters, as well as children. But for adult roles, Tatyana Andreevna needed to change her voice, making it lower.

Hello! Let's assume this is an equilateral triangle. And I want to create another shape from this equilateral triangle. I want to do this by dividing each side of the triangle into three equal parts... Three equal parts... This equilateral triangle may not be drawn perfectly, but I think you'll understand. And in each middle part I want to build one more equilateral triangle. So in the middle part, right here, I'm going to make another equilateral triangle... Here too... And here's another equilateral triangle. And from an equilateral triangle it turned out something like a Star of David. And I want to do this again, i.e. I will divide each side into three equal parts, and in each middle part I will draw another equilateral triangle. An equilateral triangle in each middle part... I'll do this for each side. Here and here... I think you get the idea... Here, here, here... I'm almost done with this step... This is what the figure will look like now. And I can do this again - once again divide each segment into three equal parts and in each middle part draw one equilateral triangle: here, here, here, here, and so on. I think you understand where this is going... And I could continue to do this forever. In this lesson I want to think about what will happen to this figure. What I'm drawing now, i.e. if we continue to do this indefinitely, at each step we will divide each side of the figure into three equal parts, and then add one equilateral triangle to each middle part - this figure presented here is called a Koch snowflake. Koch's snowflake... It was first described by this gentleman, a Swedish mathematician whose name was Niels Fabian Helge von Koch. And this snowflake is one of the earliest examples of fractals. Those. this is a fractal. Why is it considered a fractal? Because it looks very much like itself at any scale at which you view it. For example, if you look at it on this scale, then in this part you see a bunch of triangles, but if you enlarge, for example, this part, then you will still see something like this figure. And if you enlarge it again, you will see the same figure. Those. A fractal is a figure made up of several parts that, at any scale, look similar to the entire figure. What’s especially interesting (and why I included such a lesson in the geometry playlist) is that the perimeter of this figure is equal to infinity. Those. If you build a figure like the Koch snowflake, you will have to add another small equilateral triangle to each small triangle an infinite number of times. And to show that the perimeter of such a figure is equal to infinity, let's look at one of its sides here... Here is one of its sides. If we started with the original triangle, this is where this side would be. And suppose its length is equal to S. If we divide this side into three equal parts, then the length of this part will be equal to S/3, the length of this part will also be S/3... Actually, I’d better write below: S/3, S/ 3, S/3. Then we draw an equilateral triangle to the middle part. Like this. Those. the length of each side is now S/3. And the length of this entire new part... It can no longer be called just a line, because there is now a triangle on it... The length of this part, this side, is now equal not to S, but [(S/3)*4 ]. Previously, the length was equal to [(S/3)*3], but now we have one, two, three, four segments of length S/3. Now, after we have added one triangle to the original side, the length of our new side will be equal to 4 times S/3, i.e. (4/3)*S. So, if the original perimeter (i.e. if there was only one triangle) was P₀, then after adding one set of triangles, the perimeter of P1 would be 4/3 times the original perimeter. Because the length of each side of the figure will now be 4/3 times greater than originally. Those. the original perimeter Р₀ consisted of three sides, then each of their sides began to have a length 4/3 times greater, which means that the new perimeter Р₁ will be equal to 4/3 times Р₀. And after adding the second set of triangles, the perimeter of P₂ will be equal to 4/3 times P₁. Those. after each addition of new triangles, the perimeter of the figure becomes 4/3 times larger than the previous perimeter. And if you add new triangles an infinite number of times, then it turns out that when calculating the perimeter, you multiply some number by 4/3 an infinite number of times - therefore, you get an infinite perimeter value. This means that the perimeter with the index “infinity” P∞ (the perimeter of the figure if you add triangles to it an infinite number of times) is equal to infinity. Well, it's interesting, of course, to imagine a figure that has an infinite perimeter, but what's more interesting is that this figure actually has a finite area. When I say finite area, I mean a limited amount of space. I can draw some shape around and this Koch snowflake will never go beyond its boundaries. And to think... Well, I won't give a formal proof. Let's just think about what happens on either side of the figure. So, for the first time, at the first separation step, this triangle appears... At the second step, these two triangles appear, and also these two. And then triangles appear here, here, here, here, etc. But notice that you can keep adding more and more triangles, essentially an infinite number of them, but you will never get beyond this point here. The same restriction will be observed for this side, also for this side, and for this, for this, and also for this. Those. even if you add triangles an infinite number of times, the area of ​​this figure, this Koch snowflake, will never be greater than the area of ​​this bounding hexagon... Well, or greater than the area of ​​this figure... I draw an arbitrary figure that extends beyond the hexagon. You could draw a circle that extends beyond it... So, this figure drawn in blue or this hexagon drawn in purple, of course, have a certain area. And the area of ​​this Koch snowflake will always be limited, even if you add triangles to it an infinite number of times. So there's a lot of interesting stuff here. The first is that it is a fractal. You can increase it in size and at the same time we will see the same figure. The second is an infinite perimeter. And the third is the final area. Now you may say: “But these are too abstract things, they don’t exist in the real world!” But there's this fun fractal experiment that people talk about. This is a calculation of the perimeter of England (well, actually, this can be done for any country). The outline of England looks something like this... So the first way you could approximate the perimeter is to measure this distance, plus this distance, plus this distance, plus this distance, plus this distance and this distance . Then you might think, well, this shape has a finite perimeter. It is clear that its area is finite. But it is still clear that this is not the best way to calculate the perimeter; you can use a better method. Instead of this approximate calculation, you can draw smaller lines around the border, and this will be more accurate. Then you'll think, okay, this is a much better approximation. But, suppose, if you enlarge this figure... If you enlarge it well, then the border will look something like this. .. It will have curves like this... And, in fact, when you calculated the perimeter here, you simply calculated its height, like this. Of course, this will not be a perimeter, and you will need to divide the border into many parts, approximately like this, to get an accurate perimeter. But even in this case, we can say that this is not an entirely accurate calculation of the perimeter, because If you enlarge this part of the line, it will turn out that in the enlarged version it looks different - for example, like this. Accordingly, the division lines will look different - like this. Then you will say: “Eh, no, we need to be more precise!” And you will divide this line into parts even more. And this can be done endlessly, with millimeter precision. The real border of an island or continent (or anything else) is actually a fractal, i.e. a figure with an infinite perimeter, the calculation of which can reach, so to speak, the atomic level. But still the perimeter will not be accurate. But this is almost the same phenomenon as Koch's snowflake, and it can be interesting to think about it. That's all for today. See you in the next lesson!