Mathematical patterns in life. Mathematical laws of wildlife

If you carefully look around, the role of mathematics in human life becomes obvious. Computers, modern telephones and other technology accompany us every day, and their creation is impossible without the use of the laws and calculations of great science. However, the role of mathematics in society is not limited to such applications. Otherwise, for example, many artists could say with a clear conscience that the time devoted to solving problems and proving theorems in school was wasted. However, this is not true. Let's try to figure out what mathematics is for.

Base

To begin with, it is worth understanding what mathematics is in general. Translated from ancient Greek, its very name means "science", "study". Mathematics is based on the operations of counting, measuring and describing the shapes of objects. upon which knowledge of structure, order, and relationships is based. They are the essence of science. The properties of real objects in it are idealized and written in a formal language. This is how they are converted into mathematical objects. Some of the idealized properties become axioms (statements that do not require proof). Other true properties are then deduced from them. This is how a real-life object is formed.

Two sections

Mathematics can be divided into two complementary parts. Theoretical science is engaged in a deep analysis of intra-mathematical structures. Applied science provides its models to other disciplines. Physics, chemistry and astronomy, engineering systems, forecasting and logic use the mathematical apparatus all the time. With its help, discoveries are made, patterns are discovered, events are predicted. In this sense, the importance of mathematics in human life cannot be overestimated.

The basis of professional activity

Without knowledge of the basic mathematical laws and the ability to use them in the modern world, it becomes very difficult to learn almost any profession. Not only financiers and accountants deal with numbers and operations with them. Without such knowledge, an astronomer will not be able to determine the distance to a star and the best time to observe it, and a molecular biologist will not be able to understand how to deal with a gene mutation. An engineer will not design a working alarm or video surveillance system, and a programmer will not find an approach to the operating system. Many of these and other professions simply do not exist without mathematics.

Humanitarian knowledge

However, the role of mathematics in the life of a person, for example, who has devoted himself to painting or literature, is not so obvious. And yet traces of the queen of sciences are also present in the humanities.

It would seem that poetry is sheer romance and inspiration, there is no place for analysis and calculation in it. However, it is enough to recall the poetic sizes of amphibrachs), as the understanding comes that mathematics had a hand in this as well. Rhythm, verbal or musical, is also described and calculated using the knowledge of this science.

For a writer or psychologist, such concepts as the reliability of information, an isolated case, generalization, and so on are often important. All of them are either directly mathematical, or are built on the basis of patterns developed by the queen of sciences, exist thanks to her and according to her rules.

Psychology was born at the intersection of the humanities and natural sciences. All its directions, even those that work exclusively with images, are based on observation, data analysis, their generalization and verification. Modeling, forecasting, and statistical methods are used here.

From school

Mathematics in our life is present not only in the process of mastering the profession and implementing the acquired knowledge. One way or another, we use the queen of sciences at almost every moment of time. That is why mathematics is taught early enough. Solving simple and complex problems, the child does not just learn to add, subtract and multiply. He slowly, from the basics comprehends the structure of the modern world. And this is not about technical progress or the ability to check the change in the store. Mathematics forms some features of thinking and influences the attitude to the world.

The simplest, the most difficult, the most important

Probably everyone will remember at least one evening at homework, when you wanted to desperately howl: “I don’t understand what math is for!”, put aside the hated difficult and tedious tasks and run away to the yard with friends. At school and even later, at the institute, the assurances of parents and teachers “it will come in handy later” seem like annoying nonsense. However, they turn out to be right.

It is mathematics, and then physics, that teaches you to find cause-and-effect relationships, lays the habit of looking for the notorious "where the legs grow from." Attention, concentration, willpower - they also train in the process of solving those very hated tasks. If we go further, then the ability to deduce consequences from facts, predict future events, and also do the same is laid during the study of mathematical theories. Modeling, abstraction, deduction and induction are all sciences and at the same time ways the brain works with information.

And psychology again

Often it is mathematics that gives the child the revelation that adults are not omnipotent and know far from everything. This happens when mom or dad, when asked to help solve a problem, only shrug their hands and announce their inability to do it. And the child is forced to look for the answer himself, make mistakes and look again. Sometimes parents just refuse to help. “You have to do it yourself,” they say. And they do it right. After many hours of trying, the child will receive not just a homework done, but the ability to independently find solutions, detect and correct errors. And this is also the role of mathematics in human life.

Of course, independence, the ability to make decisions, be responsible for them, the absence of fear of mistakes are developed not only in the lessons of algebra and geometry. But these disciplines play a significant role in the process. Mathematics brings up such qualities as purposefulness and activity. Of course, a lot depends on the teacher. Incorrect presentation of the material, excessive rigor and pressure can, on the contrary, instill fear of difficulties and mistakes (first in the classroom, and then in life), unwillingness to express one's opinion, passivity.

Mathematics in everyday life

Adults after graduating from university or college do not stop solving mathematical problems every day. How to catch the train? Will it be possible to cook dinner for ten guests from a kilogram of meat? How many calories are in a dish? How long will one bulb last? These and many other questions are directly related to the queen of sciences and cannot be solved without her. It turns out that mathematics is invisibly present in our lives almost constantly. And most of the time we don't even notice it.

Mathematics in the life of society and the individual affects a huge number of areas. Some professions are unthinkable without it, many appeared only thanks to the development of its individual areas. Modern technical progress is closely connected with the complication and development of the mathematical apparatus. Computers and telephones, airplanes and spacecraft would never have appeared if the queen of sciences had not been known to people. However, the role of mathematics in human life is not limited to this. Science helps a child to master the world, teaches him to interact more effectively with it, forms thinking and individual qualities of character. However, mathematics alone would not have coped with such tasks. As mentioned above, the presentation of the material and the personality of the person who introduces the child to the world play a huge role.

In conclusion, we will try to briefly characterize the general laws governing the development of mathematics.

1. Mathematics is not the creation of any one historical epoch, of any one people; it is the product of a number of epochs, the product of the work of many generations. Its first concepts and provisions arose,

as we have seen, in ancient times and already more than two thousand years ago they were brought into a harmonious system. Despite all the transformations of mathematics, its concepts and conclusions are preserved, passing from one era to another, like, for example, the rules of arithmetic or the Pythagorean theorem.

New theories include previous achievements, clarifying, supplementing and generalizing them.

At the same time, as is clear from the brief outline of the history of mathematics given above, its development not only does not come down to a simple accumulation of new theorems, but also includes significant, qualitative changes. Accordingly, the development of mathematics is divided into a number of periods, the transitions between which are precisely indicated by such fundamental changes in the very subject or structure of this science.

Mathematics includes in its sphere all new areas of quantitative relations of reality. At the same time, spatial forms and quantitative relations in the simple, most direct sense of these words have been and remain the most important subject of mathematics, and the mathematical understanding of new connections and relations inevitably takes place on the basis of and in connection with the already established system of quantitative and spatial scientific concepts.

Finally, the accumulation of results within mathematics itself necessarily entails both an ascent to new levels of abstraction, to new generalizing concepts, and a deepening into the analysis of foundations and initial concepts.

Just as an oak in its mighty growth thickens old branches with new layers, throws out new branches, stretches upwards and deepens with its roots downwards, so mathematics in its development accumulates new material in its already established areas, forms new directions, ascends to new heights of abstraction and deepens into their foundations.

2. Mathematics has real forms and relations of reality as its subject, but, as Engels said, in order to study these forms and relations in their pure form, it is necessary to completely separate them from their content, to leave this latter aside as something indifferent. However, there are no forms and relations outside the content; mathematical forms and relations cannot be absolutely indifferent to the content. Mathematics, therefore, by its very nature striving to bring about such a separation, strives to bring about the impossible. This is the fundamental contradiction in the very essence of mathematics. It is a manifestation of the general contradiction of knowledge, specific to mathematics. The reflection by thought of any phenomenon, any side, any moment of reality coarsens, simplifies it, snatching it out of the general connection of nature. When people, studying the properties of space, found that it has a Euclidean geometry, was made exclusively

an important act of cognition, but it also contained a delusion: the real properties of space were [taken in a simplified, schematic way, in abstraction from matter. But without this, there would simply be no geometry, and it was precisely on the basis of this abstraction (both from its internal study and from a comparison of mathematical results with new data from other sciences) that new geometric theories were born and strengthened.

The constant resolution and restoration of this contradiction at the stages of cognition that are getting closer and closer to reality is the essence of the development of cognition. In this case, the determining factor is, of course, the positive content of knowledge, the element of absolute truth in it. Knowledge is on the ascending line, and does not mark time in a simple confusion with delusion. The movement of cognition is the constant overcoming of its inaccuracies and limitations.

This basic contradiction entails others. We have seen this in the contrast between the discrete and the continuous. (In nature, there is no absolute gap between them, and their separation in mathematics inevitably entailed the need to create ever new concepts that reflect reality more deeply and at the same time overcome the internal imperfections of the existing mathematical theory). In exactly the same way, the contradictions of the finite and the infinite, the abstract and the concrete, form and content, etc. appear in mathematics as manifestations of its fundamental contradiction. But its decisive manifestation is that, abstracting from the concrete, revolving in the circle of its abstract concepts, mathematics is thereby separated from experiment and practice, and at the same time it is only insofar as a science (i.e., has cognitive value), since relies on practice, since it turns out to be not pure, but applied mathematics. To use somewhat Hegelian language, pure mathematics constantly "denies" itself as pure mathematics, without which it cannot have scientific significance, cannot develop, cannot overcome the difficulties that inevitably arise within it.

In their formal form, mathematical theories are opposed to the real content as some schemes for specific conclusions. Mathematics acts here as a method for formulating the quantitative laws of natural science, as an apparatus for developing its theories, as a means of solving problems of natural science and technology. The significance of pure mathematics at the present stage lies primarily in the mathematical method. And just as any method does not exist and develop on its own, but only on the basis of its applications, in connection with the content to which it is applied, so mathematics cannot exist and develop without applications. Here again, the unity of opposites is revealed: the general method is opposed to a specific problem, as a means of solving it, but it itself arises from the generalization of specific material and exists

develops and finds its justification only in solving specific problems.

3. Public practice plays a decisive role in the development of mathematics in three respects. It poses new problems for mathematics, stimulates its development in one direction or another, and provides a criterion for the truth of its conclusions.

This is very clearly seen in the example of the emergence of analysis. First, it was the development of mechanics and technology that raised the problem of studying the dependences of variables in their general form. Archimedes, coming close to the differential and integral calculus, remained, however, within the framework of the problems of statics, while in modern times it was the study of motion that gave rise to the concepts of variable and function and forced the formulation of analysis. Newton could not develop mechanics without developing a corresponding mathematical method.

Secondly, it was the needs of social production that prompted the formulation and solution of all these problems. Neither ancient nor medieval society had these incentives yet. Finally, it is quite characteristic that, in its inception, mathematical analysis found justification for its conclusions precisely in applications. That is the only reason why it could develop without those strict definitions of its basic concepts (variable, function, limit) that were given later. The truth of the analysis was established by applications in mechanics, physics and technology.

This applies to all periods of the development of mathematics. Starting from the 17th century. along with mechanics, theoretical physics and the problems of new technology have the most direct influence on its development. Continuum mechanics and then field theory (thermal conduction, electricity, magnetism, gravitational field) guide the development of the theory of partial differential equations. The development of molecular theory and statistical physics in general, since the end of the last century, has served as an important stimulus for the development of probability theory, especially the theory of random processes. The theory of relativity played a decisive role in the development of Riemannian geometry with its analytical methods and generalizations.

At present, the development of new mathematical theories, such as functional analysis, etc., is stimulated by the problems of quantum mechanics and electrodynamics, problems of computer technology, statistical problems of physics and technology, etc., etc. Physics and technology not only pose new tasks, prompt it to new subjects of study, but also awaken the development of the branches of mathematics necessary for them, which initially developed to a greater extent within itself, as was the case with Riemannian geometry. In short, for the intensive development of science, it is necessary that it not only approach the solution of new problems, but that the need for their solution be imposed

development needs of society. In mathematics, many theories have recently arisen, but only those of them are being developed and firmly established in science that have found their applications in natural science and technology or have played the role of important generalizations of those theories that have such applications. At the same time, other theories remain without movement, such as, for example, some refined geometric theories (non-Desarguesian, non-Archimedean geometries), which have not found significant applications.

The truth of mathematical deductions finds its ultimate foundation not in general definitions and axioms, not in the formal rigor of proofs, but in real applications, i.e., ultimately, in practice.

In general, the development of mathematics must be understood primarily as the result of the interaction of the logic of its subject, reflected in the internal logic of mathematics itself, the influence of production and connections with natural science. This difference follows complex paths of struggle of opposites, including significant changes in the basic content and forms of mathematics. In terms of content, the development of mathematics is determined by its subject, but it is motivated mainly and ultimately by the needs of production. This is the basic regularity of the development of mathematics.

Of course, we must not forget that we are talking only about the basic regularity and that the connection between mathematics and production, generally speaking, is complex. From what has been said above, it is clear that it would be naive to try to justify the emergence of any given mathematical theory by a direct "production order". Moreover, mathematics, like any science, has a relative independence, its own internal logic, reflecting, as we emphasized, objective logic, i.e., the regularity of its subject.

4. Mathematics has always experienced the most significant influence not only of social production, but of all social conditions in general. Its brilliant progress during the rise of ancient Greece, the success of algebra in Italy during the Renaissance, the development of analysis in the era following the English Revolution, the success of mathematics in France in the period adjoining the French Revolution - all this convincingly demonstrates the inseparable connection between the progress of mathematics and the general technical, cultural, political progress of society.

This is also clearly seen in the development of mathematics in Russia. The formation of an independent Russian mathematical school, coming from Lobachevsky, Ostrogradsky and Chebyshev, cannot be separated from the progress of Russian society as a whole. The time of Lobachevsky is the time of Pushkin,

Glinka, the time of the Decembrists, and the flourishing of mathematics was one of the elements of the general upsurge.

All the more convincing is the influence of social development in the period after the Great October Socialist Revolution, when studies of fundamental importance appeared one after another with amazing speed in many directions: in set theory, topology, number theory, probability theory, the theory of differential equations, functional analysis, algebra, geometry.

Finally, mathematics has always experienced and is experiencing a noticeable influence of ideology. As in any science, the objective content of mathematics is perceived and interpreted by mathematicians and philosophers within the framework of one ideology or another.

In short, the objective content of science always fits into certain ideological forms; The unity and struggle of these dialectical opposites - objective content and ideological forms - in mathematics, as in any science, play by no means the last role in its development.

The struggle between materialism, which corresponds to the objective content of science, and idealism, which contradicts this content and distorts its understanding, goes through the entire history of mathematics. This struggle was clearly indicated already in ancient Greece, where the materialism of Thales, Democritus and other philosophers who created Greek mathematics was opposed by the idealism of Pythagoras, Socrates and Plato. With the development of the slave system, the top of society broke away from participation in production, considering it the lot of the lower class, and this gave rise to a separation of "pure" science from practice. Only purely theoretical geometry was recognized as worthy of the attention of a true philosopher. It is characteristic that Plato considered the emerging studies of some mechanical curves and even conic sections to remain outside the bounds of geometry, since they “do not bring us into communion with eternal and incorporeal ideas” and “need to use the tools of a vulgar craft.”

A vivid example of the struggle of materialism against idealism in mathematics is the activity of Lobachevsky, who put forward and defended the materialistic understanding of mathematics against the idealistic views of Kantianism.

The Russian mathematical school is generally characterized by a materialistic tradition. Thus, Chebyshev clearly emphasized the decisive importance of practice, and Lyapunov expressed the style of the Russian mathematical school in the following remarkable words: “The detailed development of questions that are especially important from the point of view of application and at the same time present special theoretical difficulties that require the invention of new methods and the ascent to the principles of science , then generalizing the conclusions obtained and creating in this way a more or less general theory. Generalizations and abstractions are not in themselves, but in connection with specific material.

theorems and theories are not in themselves, but in the general connection of science, which ultimately leads to practice - this is what actually turns out to be important and promising.

Such were the aspirations of such great scientists as Gauss and Riemann.

However, with the development of capitalism in Europe, materialistic views, which reflected the advanced ideology of the rising bourgeoisie of the epoch of the 16th and early 19th centuries, began to be replaced by idealistic views. So, for example, Cantor (1846-1918), creating the theory of infinite sets, directly referred to God, speaking in the spirit that infinite sets have an absolute existence in the divine mind. The largest French mathematician of the late XIX - early XX century. Poincare put forward the idealistic concept of "conventionalism", according to which mathematics is a scheme of conditional agreements adopted for the convenience of describing the diversity of experience. So, according to Poincare, the axioms of Euclidean geometry are nothing more than conditional agreements and their meaning is determined by convenience and simplicity, but not by correspondence with reality. Therefore, Poincaré said that, for example, in physics they would sooner abandon the law of rectilinear propagation of light than Euclidean geometry. This point of view was refuted by the development of the theory of relativity, which, despite all the "simplicity" and "convenience" of Euclidean geometry, in full agreement with the materialistic ideas of Lobachevsky and Riemann, led to the conclusion that the real geometry of space is different from Euclidean.

On the basis of the difficulties that arose in set theory, and in connection with the need to analyze the basic concepts of mathematics, among mathematicians at the beginning of the 20th century. different currents emerged. Unity in understanding the content of mathematics was lost; different mathematicians began to differently consider not only the general foundations of science, which was before, but even began to evaluate the meaning and significance of individual concrete results and proofs in different ways. Conclusions that seemed meaningful and meaningful to some, others declared devoid of meaning and significance. Idealistic currents of “logicism”, “intuitionism”, “formalism”, etc., arose.

Logisticians claim that all mathematics is deducible from the concepts of logic. Intuitionists see the source of mathematics in intuition and give meaning only to what is intuitively perceived. Therefore, in particular, they completely deny the significance of Cantor's theory of infinite sets. Moreover, intuitionists deny the simple meaning even of such statements.

as a theorem that every algebraic equation of degree has roots. For them, this statement is empty until a method for calculating the roots is specified. Thus, the complete denial of the objective meaning of mathematics led the intuitionists to discredit, as "devoid of meaning", a significant part of the achievements of mathematics. The most extreme of them have gone so far as to assert that there are as many mathematicians as there are mathematicians.

An attempt in his own way to save mathematics from such attacks was made by the greatest mathematician of the beginning of our century - D. Hilbert. The essence of his idea was to reduce mathematical theories to purely formal operations on symbols according to prescribed rules. The calculation was that with such a completely formal approach, all difficulties would be removed, because the subject of mathematics would be symbols and rules for operating with them without any relation to their meaning. This is the setting of formalism in mathematics. According to the intuitionist Brouwer, for the formalist the truth of mathematics is on paper, while for the intuitionist it is in the head of the mathematician.

It is not difficult, however, to see that both of them are wrong, because mathematics, and at the same time what is written on paper, and what a mathematician thinks, reflects reality, and the truth of mathematics lies in its correspondence with objective reality. Separating mathematics from material reality, all these currents turn out to be idealistic.

Hilbert's idea was defeated as a result of its own development. The Austrian mathematician Gödel proved that even arithmetic cannot be completely formalized, as Hilbert hoped. Godel's conclusion clearly revealed the inner dialectics of mathematics, which does not allow one to exhaust any of its areas by formal calculus. Even the simplest infinity of the natural series of numbers turned out to be an inexhaustible finite scheme of symbols and rules for operating with them. Thus, it was mathematically proved what Engels expressed in general terms when he wrote:

"Infinity is a contradiction... The destruction of this contradiction would be the end of infinity." Hilbert hoped to enclose mathematical infinity within the framework of finite schemes and thereby eliminate all contradictions and difficulties. This turned out to be impossible.

But under the conditions of capitalism, conventionalism, intuitionism, formalism, and other similar trends are not only preserved, but supplemented by new variants of idealistic views on mathematics. Theories connected with the logical analysis of the foundations of mathematics are essentially used in some new variants of subjective idealism. Subjective

idealism now uses mathematics, in particular mathematical logic, no less than physics, and therefore the questions of understanding the foundations of mathematics acquire particular acuteness.

Thus, the difficulties in the development of mathematics under capitalism gave rise to an ideological crisis in this science, similar in its foundations to the crisis in physics, the essence of which was elucidated by Lenin in his brilliant work Materialism and Empirio-Criticism. This crisis does not at all mean that mathematics in the capitalist countries is completely retarded in its development. A number of scientists, standing on clearly idealistic positions, are making important, sometimes outstanding successes in solving specific mathematical problems and developing new theories. It suffices to refer to the brilliant development of mathematical logic.

The fundamental flaw in the view of mathematics that is widespread in capitalist countries lies in its idealism and metaphysics: in the separation of mathematics from reality and neglect of its real development. Logistics, intuitionism, formalism and other similar trends single out some one of its aspects in mathematics - connection with logic, intuitive clarity, formal rigor, etc. - unreasonably exaggerate, absolutize its significance, tear it away from reality and behind a deep analysis of this one feature of mathematics in itself loses sight of mathematics as a whole. It is precisely because of this one-sidedness that none of these currents, with all the subtlety and depth of individual conclusions, can lead to a correct understanding of mathematics. In contrast to various currents and shades of idealism and metaphysics, dialectical materialism considers mathematics, as well as all science as a whole, as it is, in all the richness and complexity of its connections and development. And precisely because dialectical materialism seeks to understand all the richness and complexity of the connections of science with reality, all the complexity of its development, going from a simple generalization of experience to higher abstractions and from them to practice, precisely because it constantly brings its own approach to science in accordance with its objective content, with its new discoveries, precisely for this reason, and ultimately only for this reason, it turns out to be the only truly scientific philosophy leading to a correct understanding of science in general and, in particular, mathematics.

Introduction

We are often told in school that mathematics is the queen of sciences. Once I heard another phrase that one of the school teachers once said and my dad likes to repeat: "Nature is not so stupid as not to use the laws of mathematics." (F. M. Kotelnikov, former professor of mathematics at the Department of Moscow State University). That is what gave me the idea to study this issue.

This idea is confirmed by the following saying: “Beauty is always relative ... One should not ... believe that the shores of the ocean are indeed shapeless just because their shape is different from the correct shape of the piers we have built; the shape of the mountains cannot be considered incorrect on the grounds that they are not regular cones or pyramids; from the fact that the distances between the stars are not the same, it does not yet follow that they were scattered across the sky by an inept hand. These irregularities exist only in our imagination, but in fact they are not and do not interfere with the true manifestations of life on Earth, in the kingdom of plants and animals, or among people. (Richard Bentley, 17th century English scholar)

But studying mathematics, we rely only on the knowledge of formulas, theorems, calculations. And mathematics appears before us as a kind of abstract science operating with numbers. However, as it turns out, mathematics is a beautiful science.

It was as a poet that I set myself the following goal: to show the beauty of mathematics with the help of patterns that exist in nature.

To achieve its goal, it was divided into a number of tasks:

To study the variety of mathematical patterns used by nature.

Give a description of these patterns.

On your own experience, try to find mathematical relationships in the structure of the cat's body (As it was said in one famous movie: train on cats).

Methods used in the work: analysis of literature on the topic, scientific experiment.

1. 1. Search for mathematical patterns in nature.

Mathematical patterns can be sought both in living and inanimate nature.

In addition, it is necessary to determine what patterns to look for.

Since not many patterns were studied in the sixth grade, I had to study high school textbooks. In addition, I had to take into account that very often nature uses geometric patterns. Therefore, in addition to algebra textbooks, I had to turn my attention to geometry textbooks.

Mathematical patterns found in nature:

1. Golden section. Fibonacci numbers (Archimedes spiral). As well as other types of spirals.
2. Various types of symmetry: central, axial, rotary. As well as symmetry in animate and inanimate nature.
3. Angles and geometric shapes.
4. Fractals. The term fractal is derived from the Latin fractus (break, break), i.e. create fragments of irregular shape.
5. Arithmetic and progression geometry.

Let us consider in more detail the identified regularities but in a slightly different sequence.

The first thing that catches the eye is the presence symmetry in nature. Translated from Greek, this word means "proportionality, proportionality, uniformity in the arrangement of parts." A mathematically rigorous idea of ​​symmetry was formed relatively recently - in the 19th century. In the simplest interpretation (according to G. Weyl), the modern definition of symmetry looks like this: an object is called symmetric if it can be somehow changed, resulting in the same thing as it started with. .

In nature, two types of symmetry are most common - “mirror” and “radial” (“radial”) symmetries. However, in addition to one name, these types of symmetry have others. So mirror symmetry is also called: axial, bilateral, leaf symmetry. Radial symmetry is also called radial.

Axial symmetry most common in our world. Houses, various devices, cars (externally), people (!) Everything is symmetrical, well, or almost. People are symmetrical in that all healthy people have two hands, five fingers on each hand, if the palms are folded, it will be like a mirror image.

Checking symmetry is very easy. It is enough to take a mirror and place it approximately in the middle of the object. If that part of the object that is on the matte, non-reflecting side of the mirror corresponds to reflection, then the object is symmetrical.

Radial symmetry .Anything that grows or moves vertically, ie. up or down relative to the earth's surface, subject to radial-beam symmetry.

The leaves and flowers of many plants have radial symmetry. (Fig. 1, applications)

On the cross sections of the tissues that form the root or stem of a plant, radial symmetry is clearly visible (kiwi fruit, tree cut). Radial symmetry is characteristic of sedentary and attached forms (corals, hydra, jellyfish, sea anemones). (Fig. 2, applications)

Rotational symmetry . Rotation by a certain number of degrees, accompanied by translation to a distance along the axis of rotation, generates helical symmetry - the symmetry of a spiral staircase. An example of helical symmetry is the arrangement of leaves on the stem of many plants. The head of a sunflower has processes arranged in geometric spirals that unwind from the center outwards. (Fig. 3, applications)

Symmetry is found not only in wildlife. In inanimate nature there are also examples of symmetry. Symmetry manifests itself in the diverse structures and phenomena of the inorganic world. The symmetry of the external form of a crystal is a consequence of its internal symmetry - the ordered mutual arrangement of atoms (molecules) in space.

The symmetry of the snowflakes is very beautiful.

But it must be said that nature does not tolerate exact symmetry. There are always at least minor deviations. So, our hands, feet, eyes and ears are not completely identical to each other, even if they are very similar.

Golden section.

The golden ratio in the 6th grade is not passed now. But it is known that the golden section, or the golden ratio, is the ratio of the smaller part to the larger one, which gives the same result when dividing the entire segment into a larger part and dividing the larger part into a smaller one. Formula: A/B=B/C

Basically the ratio is 1/1.618. The golden ratio is very common in the animal kingdom.

A person, one might say, completely “consists” of the golden ratio. For example, the distance between the eyes (1.618) and between the eyebrows (1) is the golden ratio. And the distance from the navel to the foot and height will also be the golden ratio. Our entire body is “strewn” with golden proportions. (Fig. 5, applications)

Angles and Geometric Shapes are also common in nature. There are noticeable corners, for example, they are clearly visible in sunflower seeds, in honeycombs, on insect wings, in maple leaves, etc. The water molecule has an angle of 104.7 0 C. But there are also subtle angles. For example, In the inflorescence of a sunflower, the seeds are located at an angle of 137.5 degrees from the center.

Geometric figures they also saw everything in animate and inanimate nature, only paid little attention to them. As you know, a rainbow is a part of an ellipse, the center of which is below ground level. The leaves of plants, the fruits of plums have the shape of an ellipse. Although they can certainly be calculated using some more complex formula. For example, like this (Fig. 6, applications):

Spruce, some types of shells, various cones are cone-shaped. Some inflorescences look like either a pyramid, or an octahedron, or the same cone.

The most famous natural hexagon are honeycombs (bee, wasp, bumblebee, etc.). Unlike many other forms, they have an almost perfect shape and differ only in the size of the cells. But if you pay attention, it is noticeable that the faceted eyes of insects are also close to this form.

Spruce cones are very similar to small cylinders.

In inanimate nature, it is almost impossible to find ideal geometric shapes, but many mountains look like pyramids with different bases, and a sand spit resembles an ellipse.

And there are many such examples.

I have already considered the golden ratio. Now I want to turn my attention to Fibonacci numbers and other spirals, which are closely related to the golden ratio.

Spirals are very common in nature. The shape of the spirally curled shell attracted the attention of Archimedes (Fig. 2). He studied it and deduced the equation of the spiral. The spiral drawn according to this equation is called by his name. The increase in her step is always uniform. At present, the Archimedes spiral is widely used in engineering. (Fig.7 application)

"Golden" spirals are widespread in the biological world. As noted above, animal horns grow from one end only. This growth is carried out in a logarithmic spiral. In the book "Crooked Lines in Life" T. Cook explores various types of spirals that appear in the horns of rams, goats, antelopes and other horned animals.

The spiral and spiral arrangement of leaves on tree branches was noticed long ago. The spiral was seen in the arrangement of sunflower seeds, in pine cones, pineapples, cacti, etc. The joint work of botanists and mathematicians has shed light on these amazing natural phenomena. It turned out that in the arrangement of leaves on a branch - phyllotaxis, sunflower seeds, pine cones, the Fibonacci series manifests itself, and therefore, the law of the golden section manifests itself. The spider spins its web in a spiral pattern. A hurricane is spiraling. A frightened herd of reindeer scatter in a spiral.

And, finally, information carriers - DNA molecules - are also twisted into a spiral. Goethe called the spiral "the curve of life."

The scales of a pine cone on its surface are arranged in a strictly regular manner - along two spirals that intersect approximately at a right angle.

However, let's return to one chosen spiral - the Fibonacci numbers. These are very interesting numbers. The number is obtained by adding the previous two. Here are the initial Fibonacci numbers for 144: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... And let's turn to illustrative examples (slide 14).

fractalswere opened not long ago. The concept of fractal geometry appeared in the 70s of the 20th century. Now fractals have actively entered our lives, and even such a direction as fractal graphics is developing. (Fig. 8, applications)

Fractals are quite common in nature. However, this phenomenon is more typical for plants and inanimate nature. For example, fern leaves, umbrella inflorescences. In inanimate nature, these are lightning strikes, patterns on windows, snow sticking to tree branches, coastline elements, and much more.

Geometric progression.

A geometric progression in its most elementary definition is the multiplication of the previous number by a coefficient.

This progression is present in unicellular organisms. For example, any cell is divided into two, these two are divided into four, and so on. That is, it is a geometric progression with a coefficient of 2. And in simple terms, the number of cells with each division increases by 2 times.

Bacteria are exactly the same. Division, doubling the population.

Thus, I studied the mathematical patterns that exist in nature, and gave relevant examples.

It should be noted that at the moment, mathematical laws in nature are being actively studied, and there is even a science called biosymmetry. It describes much more complex patterns than were considered in the work.

Conducting a scientific experiment.

Rationale for the choice:

The cat was chosen as an experimental animal for several reasons:

I have a cat at home;

I have four of them at home, so the data obtained should be more accurate than when studying one animal.

Experiment sequence:

Measuring the cat's body.

Recording the results obtained;

Search for mathematical patterns.

Conclusions on the results obtained.

List of things to study on a cat:

• Symmetry;
• golden ratio;
• Spirals;
• corners;
• fractals;
• Geometric progression.

The study of symmetry on the example of a cat showed that a cat is symmetrical. The type of symmetry is axial, i.e. it is symmetrical about the axis. As was studied in the theoretical material, for a cat, as for a mobile animal, radial, central, and also rotational symmetry are uncharacteristic.

To study the golden ratio, I took measurements of the cat's body, photographed it. The ratio of the size of the body with a tail and without a tail, the body without a tail to the head really come close in the value of the golden ratio.

65/39=1,67

39/24=1,625

In this case, it is necessary to take into account the measurement error, the relativity of the length of the wool. But in any case, the results obtained are close to the value of 1.618. (Fig. 9, appendix).

The cat stubbornly did not want to let her be measured, so I tried to photograph her, compiled a golden ratio scale and superimposed it on photographs of cats. Some of the results are very interesting.

For example:

• the height of the sitting cat from the floor to the head, and from the head to the "armpit";
• "carpal" and "elbow joints";
• the height of the sitting cat to the height of the head;
• the width of the muzzle to the width of the bridge of the nose;
• muzzle height to eye height;
• nose width to nostril width;

I found only one spiral in a cat - these are claws. A similar spiral is called an involute.

In the body of a cat, you can find various geometric shapes, but I was looking for angles. Only the ears and claws were angular. But claws, as I defined earlier, are spirals. The shape of the ears is more like a pyramid.

The search for fractals on the body of a cat did not give any results, since it does not have anything similar and dividing into the same small details. Still, fractals are more typical for plants than for animals, especially mammals.

But, after reflecting on this issue, I came to the conclusion that there are fractals in the body of a cat, but in the internal structure. Since I have not yet studied the biology of mammals, I turned to the Internet and found the following drawings (Fig. 10, appendices):

Thanks to them, I was convinced that the circulatory and respiratory systems of a cat branch according to the law of fractals.

Geometric progression is characteristic of the process of reproduction, but not of the body. An arithmetic progression is not typical for cats, since a cat gives birth to a certain number of kittens. A geometric progression in cat reproduction can probably be found, but most likely there will be some complex coefficients. I will explain my thoughts.

The cat begins to give birth to kittens at the age of 9 months to 2 years (it all depends on the cat itself). The gestation period is 64 days. A cat nurses kittens for about 3 months, so on average she will have 4 litters per year. The number of kittens is from 3 to 7. As you can see, certain patterns can be caught, but this is not a geometric progression. Too blurry settings.

I got results like this:

In the body of a cat there are: axial symmetry, the golden ratio, spirals (claws), geometric shapes (pyramidal ears).

In appearance, there are no fractals and geometric progression.

The internal structure of a cat belongs more to the field of biology, but it should be noted that the structure of the lungs and circulatory system (as well as other animals) obeys the logic of fractals.

Conclusion

In my work, I researched the literature on the topic and studied the main theoretical issues. On a specific example, he proved that in nature a lot, if not everything, obeys mathematical laws.

Having studied the material, I realized that in order to understand nature, you need to know not only mathematics, you need to study algebra, geometry and their sections: stereometry, trigonometry, etc.

On the example of a domestic cat, I investigated the execution of mathematical laws. As a result, I got that in the body of a cat there is axial symmetry, the golden ratio, spirals, geometric shapes, fractals (in the internal structure). But at the same time, he could not find a geometric progression, although certain patterns were clearly traced in the reproduction of cats.

And now I agree with the phrase: "Nature is not so stupid as not to subordinate everything to the laws of mathematics."

Sometimes it seems that our world is simple and clear. In fact, this is the great mystery of the Universe that created such a perfect planet. Or maybe it was created by someone who probably knows what he is doing? The greatest minds of our time are working on this question.

Each time they come to the conclusion that it is impossible to create everything that we have without the Higher Mind. What an extraordinary, complex and at the same time simple and direct our planet Earth! The world around us is amazing with its rules, shapes, colors.

Nature laws

The first thing you can pay attention to on our vast and amazing planet is that She is found in all forms of the surrounding world, and is also the main principle of beauty, ideality and proportionality. This is nothing but mathematics in nature.

The concept of "symmetry" means harmony, correctness. This is a property of the surrounding reality, systematizing fragments and turning them into a single whole. Even in ancient Greece, signs of this law began to be noticed for the first time. For example, Plato believed that beauty appears solely as a result of symmetry and proportionality. In fact, if we look at things that are proportionate, regular, and complete, then our inner state will be beautiful.

The laws of mathematics in animate and inanimate nature

Let's take a look at any creature, for example, the most perfect - a man. We will see the structure of the body, which looks the same on both sides. You can also list many samples, such as insects, animals, marine life, birds. Each species has its own color.

If any pattern or pattern is present, it is known to be mirrored about the center line. All organisms are created due to the rules of the universe. Such mathematical regularities can also be traced in inanimate nature.

If you pay attention to all phenomena, such as a tornado, a rainbow, plants, snowflakes, you can find a lot in common in them. Relatively, the leaf of the tree is divided in half, and each part will be a reflection of the previous one.

Even if we take as an example a tornado that rises vertically and looks like a funnel, then it can also be conditionally divided into two absolutely identical halves. You can meet the phenomenon of symmetry in the change of day and night, the seasons. The laws of the surrounding world are mathematics in nature, which has its own perfect system. The entire concept of the creation of the Universe is based on it.

Rainbow

We rarely think about natural phenomena. It started to snow or rain, the sun came out or thunder struck - the usual state of changing weather. Consider a multi-colored arc that can usually be found after precipitation. A rainbow in the sky is an amazing natural phenomenon, accompanied by a spectrum of all colors visible only to the human eye. This happens due to the passage of the rays of the sun through the outgoing cloud. Each raindrop serves as a prism that has optical properties. We can say that any drop is a small rainbow.

Passing through a water barrier, the rays change their original color. Every stream of light has a certain length and shade. Therefore, our eye perceives the rainbow as such a multi-colored one. Note the interesting fact that this phenomenon can only be seen by a person. Because it's just an illusion.

types of rainbow

1. A rainbow formed from the sun is the most common. It is the brightest of all varieties. Consists of seven primary colors: red orange, yellow, green, blue, indigo, violet. But if you look at the details, there are much more shades than our eyes can see.
2. A rainbow created by the moon occurs at night. It is believed that it can always be seen. But, as practice shows, basically this phenomenon is observed only in rainy areas or near large waterfalls. The colors of the lunar rainbow are very dull. They are destined to be considered only with the help of special equipment. But even with it, our eye is able to make out only a strip of white.
3. The rainbow, which appeared as a result of fog, is like a wide shining light arch. Sometimes this type is confused with the previous one. From above, the color can be orange, from below it can have a shade of purple. The sun's rays, passing through the fog, form a beautiful natural phenomenon.
4. rarely occurs in the sky. It is not similar to the previous species in its horizontal shape. the phenomenon is possible only over cirrus clouds. They usually extend at an altitude of 8-10 kilometers. The angle at which the rainbow will show itself in all its glory must be more than 58 degrees. The colors usually stay the same as in the solar rainbow.

Golden Ratio (1.618)

Ideal proportion is most often found in the animal world. They are awarded such a proportion, which is equal to the root of the corresponding number of PHI to one. This ratio is the connecting fact of all animals on the planet. The great minds of antiquity called this number the divine proportion. It can also be called the golden ratio.

This rule is fully consistent with the harmony of the human structure. For example, if you determine the distance between the eyes and eyebrows, then it will be equal to the divine constant.

The golden ratio is an example of how important mathematics is in nature, the law of which designers, artists, architects, creators of beautiful and perfect things began to follow. They create with the help of the divine constant their creations, which are balanced, harmonious and pleasant to look at. Our mind is able to consider beautiful those things, objects, phenomena, where there is an unequal ratio of parts. Proportionality is what our brain calls the golden ratio.

DNA helix

As the German scientist Hugo Weil rightly noted, the roots of symmetry came through mathematics. Many noted the perfection of geometric figures and paid attention to them. For example, a honeycomb is nothing more than a hexagon created by nature itself. You can also pay attention to the cones of spruce, which have a cylindrical shape. Also in the surrounding world, a spiral is often found: horns of large and small livestock, mollusk shells, DNA molecules.

Created on the principle of the golden ratio. It is a link between the scheme of the material body and its real image. And if we consider the brain, then it is nothing more than a conductor between the body and the mind. The intellect connects life and the form of its manifestation and allows the life contained in the form to know itself. With the help of this, humanity can understand the surrounding planet, look for patterns in it, which are then applied to the study of the inner world.

division in nature

Cell mitosis consists of four phases:

• Prophase. It increases the core. Chromosomes appear, which begin to twist into a spiral and turn into their ordinary form. A place is formed for cell division. At the end of the phase, the nucleus and its membrane dissolve, and the chromosomes flow into the cytoplasm. This is the longest stage of division.
• metaphase. Here the twisting into a spiral of chromosomes ends, they form a metaphase plate. The chromatids line up opposite each other in preparation for division. Between them there is a place for disconnection - a spindle. This is where the second stage ends.

• Anaphase. The chromatids move in opposite directions. Now the cell has two sets of chromosomes due to their division. This stage is very short.
• Telophase. In each half of the cell, a nucleus is formed, inside which the nucleolus is formed. The cytoplasm is actively dissociated. The spindle gradually disappears.

Meaning of Mitosis

Due to the unique method of division, each subsequent cell after reproduction has the same composition of genes as its mother. The composition of the chromosomes of both cells receive the same. It did not do without such a science as geometry. Progression in mitosis is important because all cells reproduce according to this principle.

Where do mutations come from

This process guarantees a constant set of chromosomes and genetic materials in each cell. Due to mitosis, the development of the organism, reproduction, regeneration occurs. In the event of a violation due to the action of some poisons, the chromosomes may not disperse into their halves, or they may be observed structural disturbances. This will be a clear indicator of incipient mutations.

Summing up

What do mathematics and nature have in common? You will find the answer to this question in our article. And if you dig deeper, then you need to say that with the help of studying the world around you, a person comes to know himself. Without the Creator of all living things, there could be nothing. Nature is exclusively in harmony, in a strict sequence of its laws. Is all this possible without reason?

Let us cite the statement of the scientist, philosopher, mathematician and physicist Henri Poincaré, who, like no one else, will be able to answer the question of whether mathematics is fundamental in nature. Some materialists may not like such reasoning, but they are unlikely to be able to refute it. Poincaré says that the harmony that the human mind wants to discover in nature cannot exist outside of it. which is present in the minds of at least a few individuals, may be available to all mankind. The connection that brings together mental activity is called the harmony of the world. Recently, there has been tremendous progress on the way to such a process, but they are very small. These links connecting the Universe and the individual should be valuable to any human mind that is sensitive to these processes.

Introduction. 2

Chapter 1. Mathematical patterns of living nature. 3

Chapter 2. Principles of shaping in nature 5

Chapter 3

Chapter 4. Escher's Geometric Rhapsody. fifteen

Chapter 5

List of used literature. twenty

Introduction.

From a superficial acquaintance with mathematics, it may seem like an incomprehensible labyrinth of formulas, numerical dependencies and logical paths. Casual visitors who do not know the true value of mathematical treasures are frightened by the dry scheme of mathematical abstractions, through which the mathematician sees the living multicolored reality.

The one who comprehended the wonderful world of mathematics does not remain only an enthusiastic contemplator of its treasures. He himself seeks to create new mathematical objects, looking for ways to solve new problems, or new, more advanced solutions to already solved problems. More than 300 proofs of the Pythagorean theorem, dozens of non-classical circle quadratures, angle trisections and cube doublings have already been found and published.

But restless inquisitive thought leads to new searches. At the same time, even more than the result itself, the search for it attracts. This is natural. After all, the path to solving each sufficiently meaningful problem is always an amazing chain of inferences, cemented by the law of logic.

Mathematical creativity is the true creativity of the mind. Here is what the Soviet mathematician G.D. Suvorov wrote: “The theorem, written down logically flawlessly, really seems to be devoid of any poetic beginning and seems not the fruit of a fiery fantasy, but a gloomy child of mother logic. But no one knows, except the scientist, what a whirlwind of fantasies and poetic upsurges actually gave rise to this theorem. After all, she was a winged, exotic butterfly before she was captured, put to sleep by logic and pinned to paper with proof pins!". It is natural that in their memoirs K.F. Gauss, A. Poincaré, J. Hadamard, A.N. for them were roads into the unknown. Because they were going to these solutions for the first time, and mathematics gave them the full measure of the joy of the discoverers.

In some problems, among the many roads to the answer, there is one, the most unexpected, often carefully "camouflaged" and, as a rule, the most beautiful and desirable. Great happiness to find it and go through it. The search for such solutions, the ability to go beyond the capabilities of already known algorithms is a true aesthetic mathematical creativity.
^

Chapter 1. Mathematical patterns of living nature.

Wildlife demonstrates numerous symmetrical forms of organisms. In many cases, the symmetrical shape of the organism is complemented by colorful symmetrical colors.

A small birch weevil barely reaching 4 mm, of course, does not know higher mathematics. But, making a cradle for his offspring, he "draws", or rather cuts out on a leaf of a tree an evolute - a curve, which is a set of centers of leaf curvature. The edge of the leaf itself will be involute with respect to the curve cut by the weevil.

The architecture of the honeycomb cell is subject to complex geometric patterns.

Theoretical curves and phase curve of fluctuations in the number of populations in the aggregate of two interacting species (biocenosis) "predator-prey".

Vito Voltaire (1860-1940) was an outstanding Italian mathematician. He built a theory of the dynamics of the number of biological populations,

in which he applied the method of differential equations.

Like most mathematical models of biological phenomena, it comes from many simplifying assumptions.

AT jumping, the center of mass of animals describes a well-known figure - a square parabola, the branches of which are turned downwards: y=ax 2 , a>1, a

The contours of the leaves of many plants are beautiful. With great accuracy, their forms are described by elegant equations in a polar or Cartesian coordinate system.

^

Chapter 2

Everything that took on some form formed, grew, strove to take a place in space and preserve itself. This aspiration finds fulfillment mainly in two variants - upward growth or spreading over the surface of the earth and twisting in a spiral.

The shell is twisted in a spiral. If you unfold it, you get a length slightly inferior to the length of the snake. A small ten-centimeter shell has a spiral 35 cm long. Spirals are very common in nature.

The shape of the spirally curled shell attracted the attention of Archimedes. He studied it and deduced the equation of the spiral. The spiral drawn according to this equation is called by his name. The increase in her step is always uniform. At present, the Archimedes spiral is widely used in engineering.

Even Goethe emphasized the tendency of nature to spirality. The spiral and spiral arrangement of leaves on tree branches was noticed long ago. The spiral was seen in the arrangement of sunflower seeds, in pine cones, pineapples, cacti, etc. The spider spins its web in a spiral pattern. A hurricane is spiraling. A frightened herd of reindeer scatter in a spiral. The DNA molecule is twisted into a double helix. Goethe called the spiral "the curve of life."

The shell of Nautilus, Haliotis and others molluscs form in the form of a logarithmic spiral: p=ae b φ .

Leaves on young shoots of plants are arranged in a spatial spiral. And looking at them from above, we find a second spiral, since they are still located so as not to interfere with each other to perceive sunlight. The distances between individual leaves are characterized by the numbers of the Fibonacci series: 1,1,2,3,5,8,…,u n , u n +1 ,…, where u n =u n -1 +u n -2.


In a sunflower, the seeds are arranged along characteristic arcs close to two families of logarithmic spirals.

Nature preferred the logarithmic spiral because of the many wonderful properties of this curve. For example, it does not change during a similarity transformation.

Therefore, the body does not need to rebuild the architecture of its body in the process of growth.

A striking example of the asymmetry of living things at the submolecular level is the secondary form of the material carriers of hereditary information - the double helix of the DNA giant molecule. But DNA is already a helix wound around the nucleosome, it is a doubly helix. Life arises in a subtle, amazingly precise process of realizing the plans of nature-architect, according to which protein molecules are built.

The spider weaves its trap in the form of a complex transcendental curve - a logarithmic spiral p=ae b φ

^

Chapter 3

A person distinguishes objects around him by shape. Interest in the form of an object may be dictated by vital necessity, or it may be caused by the beauty of the form. The form, which is based on a combination of symmetry and the golden ratio, contributes to the best visual perception and the appearance of a sense of beauty and harmony. The whole always consists of parts, parts of different sizes are in a certain relationship to each other and to the whole. The principle of the golden section is the highest manifestation of the structural and functional perfection of the whole and its parts in art, science, technology and nature.

In mathematics, proportion (Latin proportio) is the equality of two ratios: a: b = c: d.

Line segment AB can be divided into two parts in the following ways:

• 
  into two equal parts - AB: AC = AB: BC;
• 
  into two unequal parts in any ratio (such parts do not form proportions);
• 
  thus, when AB: AC = AC: BC.

^ Golden Ratio- this is such a proportional division of the segment into unequal parts, in which the entire segment relates to the larger part in the same way as the larger part itself relates to the smaller one; or in other words, the smaller segment is related to the larger one as the larger one is to everything

a: b = b: c or c: b = b: a.

Geometric representation of the golden ratio

P A practical acquaintance with the golden ratio begins with dividing a straight line segment in the golden ratio using a compass and ruler. Division of a line segment according to the golden ratio. BC = 1/2 AB; CD=BC

From point B, a perpendicular equal to half AB is restored. The resulting point C is connected by a line to point A. On the resulting line, a segment BC is plotted, ending with point D. The segment AD is transferred to the straight line AB. The resulting point E divides the segment AB in the ratio of the golden ratio.

Segments of the golden ratio are expressed as an infinite irrational fraction AE \u003d 0.618 ..., if AB is taken as a unit, BE \u003d 0.382 ... For practical purposes, approximate values ​​\u200b\u200bof 0.62 and 0.38 are often used. If the segment AB is taken as 100 parts, then the larger part of the segment is 62, and the smaller one is 38 parts.

The properties of the golden section are described by the equation:

x 2 - x - 1 \u003d 0.

Solution to this equation:

The properties of the golden section created a romantic aura of mystery and almost mystical worship around this number.
^ History of the golden ratio
It is generally accepted that the concept of the golden division was introduced into scientific use by Pythagoras, an ancient Greek philosopher and mathematician (VI century BC). There is an assumption that Pythagoras borrowed his knowledge of the golden division from the Egyptians and Babylonians. Indeed, the proportions of the Cheops pyramid, temples, bas-reliefs, household items and decorations from the tomb of Tutankhamun indicate that the Egyptian craftsmen used the ratios of the golden division when creating them. The French architect Le Corbusier found that in the relief from the temple of Pharaoh Seti I in Abydos and in the relief depicting Pharaoh Ramses, the proportions of the figures correspond to the values ​​​​of the golden division. The architect Khesira, depicted on a relief of a wooden board from the tomb of his name, holds measuring instruments in his hands, in which the proportions of the golden division are fixed.

The Greeks were skilled geometers. Even arithmetic was taught to their children with the help of geometric figures. The square of Pythagoras and the diagonal of this square were the basis for constructing dynamic rectangles.

^ Dynamic Rectangles

Plato (427...347 BC) also knew about the golden division. His dialogue "Timaeus" is devoted to the mathematical and aesthetic views of the school of Pythagoras and, in particular, to the questions of the golden division.

In the facade of the ancient Greek temple of the Parthenon there are golden proportions. During its excavations, compasses were found, which were used by architects and sculptors of the ancient world. The Pompeian compass (Museum in Naples) also contains the proportions of the golden division.

In the ancient literature that has come down to us, the golden division was first mentioned in Euclid's Elements. In the 2nd book of the "Beginnings" a geometric construction of the golden division is given. After Euclid, Hypsicles (II century BC), Pappus (III century AD) and others were engaged in the study of the golden division. In medieval Europe with the golden division We met through Arabic translations of Euclid's Elements. The translator J. Campano from Navarre (3rd century) commented on the translation. The secrets of the golden division were jealously guarded, kept in strict secrecy. They were known only to the initiates.

During the Renaissance, interest in the golden division among scientists and artists increased in connection with its use both in geometry and in art, especially in architecture Leonardo da Vinci, an artist and scientist, saw that Italian artists had great empirical experience, but little knowledge . He conceived and began to write a book on geometry, but at that time a book by the monk Luca Pacioli appeared, and Leonardo abandoned his idea. According to contemporaries and historians of science, Luca Pacioli was a real luminary, the greatest mathematician in Italy between Fibonacci and Galileo. Luca Pacioli was a student of the artist Piero della Francesca, who wrote two books, one of which was called On Perspective in Painting. He is considered the creator of descriptive geometry.

Luca Pacioli was well aware of the importance of science for art. In 1496, at the invitation of the Duke of Moreau, he came to Milan, where he lectured on mathematics. Leonardo da Vinci also worked at the Moro court in Milan at that time. In 1509, Luca Pacioli's Divine Proportion was published in Venice, with brilliantly executed illustrations, which is why they are believed to have been made by Leonardo da Vinci. The book was an enthusiastic hymn to the golden ratio. Among the many advantages of the golden ratio, the monk Luca Pacioli did not fail to name its “divine essence” as an expression of the divine trinity of God the Son, God the Father and God the Holy Spirit (it was understood that the small segment is the personification of God the Son, the larger segment is the personification of God the Father, and the entire segment - the god of the holy spirit).

Leonardo da Vinci also paid much attention to the study of the golden division. He made sections of a stereometric body formed by regular pentagons, and each time he obtained rectangles with aspect ratios in golden division. Therefore, he gave this division the name of the golden section. So it is still the most popular.

At the same time, in northern Europe, in Germany, Albrecht Dürer was working on the same problems. He sketches an introduction to the first draft of a treatise on proportions. Durer writes. “It is necessary that the one who knows something should teach it to others who need it. This is what I set out to do."

Judging by one of Dürer's letters, he met with Luca Pacioli during his stay in Italy. Albrecht Dürer develops in detail the theory of the proportions of the human body. Dürer assigned an important place in his system of ratios to the golden section. The height of a person is divided in golden proportions by the belt line, as well as by the line drawn through the tips of the middle fingers of the lowered hands, the lower part of the face - by the mouth, etc. Known proportional compass Dürer.

Great astronomer of the 16th century Johannes Kepler called the golden ratio one of the treasures of geometry. He is the first to draw attention to the significance of the golden ratio for botany (plant growth and structure).

In subsequent centuries, the rule of the golden ratio turned into an academic canon, and when, over time, a struggle began in art with the academic routine, in the heat of the struggle, “they threw out the child along with the water.” The golden section was “discovered” again in the middle of the 19th century. In 1855, the German researcher of the golden section, Professor Zeising, published his work Aesthetic Research. With Zeising, exactly what happened was bound to happen to the researcher who considers the phenomenon as such, without connection with other phenomena. He absolutized the proportion of the golden section, declaring it universal for all phenomena of nature and art. Zeising had numerous followers, but there were also opponents who declared his doctrine of proportions to be "mathematical aesthetics".

^ Golden proportions in the human figure
Zeising did a great job. He measured about two thousand human bodies and came to the conclusion that the golden ratio expresses the average statistical law. The division of the body by the navel point is the most important indicator of the golden ratio. The proportions of the male body fluctuate within the average ratio of 13: 8 = 1.625 and approach the golden ratio somewhat closer than the proportions of the female body, in relation to which the average value of the proportion is expressed in the ratio 8: 5 = 1.6. In a newborn, the proportion is 1: 1, by the age of 13 it is 1.6, and by the age of 21 it is equal to the male. The proportions of the golden section are also manifested in relation to other parts of the body - the length of the shoulder, forearm and hand, hand and fingers, etc.

^ Golden proportions in parts of the human body
At the end of XIX - beginning of XX centuries. a lot of purely formalistic theories appeared about the use of the golden section in works of art and architecture. With the development of design and technical aesthetics, the law of the golden ratio extended to the design of cars, furniture, etc.

Among the roadside herbs grows an unremarkable plant - chicory. Let's take a closer look at it. A branch was formed from the main stem. Here is the first leaf.

Chicory

The process makes a strong ejection into space, stops, releases a leaf, but is already shorter than the first one, again makes an ejection into space, but of lesser force, releases a leaf of an even smaller size and ejection again. If the first outlier is taken as 100 units, then the second is equal to 62 units, the third is 38, the fourth is 24, and so on. The length of the petals is also subject to the golden ratio. In growth, the conquest of space, the plant retained certain proportions. Its growth impulses gradually decreased in proportion to the golden section.

^ viviparous lizard

In the lizard, at first glance, proportions that are pleasing to our eyes are caught - the length of its tail relates to the length of the rest of the body as 62 to 38.

Nature has carried out the division into symmetrical parts and golden proportions. In parts, a repetition of the structure of the whole is manifested.
^ Bird egg

The great Goethe, a poet, naturalist and artist (he drew and painted in watercolor), dreamed of creating a unified doctrine of the form, formation and transformation of organic bodies.

Pierre Curie at the beginning of our century formulated a number of profound ideas of symmetry. He argued that one cannot consider the symmetry of any body without taking into account the symmetry of the environment.

The patterns of "golden" symmetry are manifested in the energy transitions of elementary particles, in the structure of some chemical compounds, in planetary and space systems, in the gene structures of living organisms. These patterns, as indicated above, are in the structure of individual human organs and the body as a whole, and are also manifested in biorhythms and the functioning of the brain and visual perception.

The golden ratio cannot be considered in itself, separately, without connection with symmetry. The great Russian crystallographer G.V. Wulff (1863...1925) considered the golden ratio to be one of the manifestations of symmetry.

^

Chapter 4. Escher's Geometric Rhapsody.

The Dutch artist Maur Cornelius Escher (1898-1971) created a whole world of visual images that reveal the fundamental ideas and patterns of mathematics, physics, the psychological characteristics of human perception of objects of reality in the three-dimensional space around us.

Unlimited space, mirror images, contradictions between the plane and space - all these concepts are embodied in memorable, full of special charm images. Lizards visually represent geometric representations studied in high school.

Horsemen give an excellent visual representation of parallel transfer, symmetry, filling the entire plane with figures of complex configuration.

"Cube and Magic Ribbons". Belvedere tapes - not just -

really magical: a geometric joke, and a whole

"Prominences" on them can be a complex of surprises,

consider the sign of both convexity generated by singularities and concavity. human perception of objects

It is enough to change the point of view, in three-dimensional space.

How do the ribbons get twisted?
Maurits Cornelius Escher has created a unique gallery of paintings that belong both to art and science. They illustrate Einstein's theory of relativity, the structure of matter, geometric transformations, topology, crystallography, and physics. This is evidenced by the names of some of the artist's albums: "Unlimited Space", "Mirror Images", "Inversions", "Polyhedrons", "Relativity", "Contradictions between Plane and Space", "Impossible Constructions".

"I often feel closer to mathematicians than to my fellow artists," Escher wrote. And indeed, his paintings are unusual, they are filled with deep philosophical meaning, they convey complex mathematical relationships. Escher's reproductions of paintings are widely used as illustrations in scientific and non-fiction books.

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Chapter 5

The nature of the number  is one of the biggest mysteries of mathematics. Intuition suggested that the circumference of a circle and its diameter are equally comprehensible quantities.

The calculation of hundreds of decimal places  over the past two centuries, many scientists have been engaged in

In the book Nightmares of Eminent Persons, the famous English mathematician and philosopher Bertrand Russell wrote: “Pi's face was hidden by a mask. Everyone understood that no one could break it, while remaining alive. Eyes looked piercingly, ruthlessly, coldly and mysteriously through the slits of the mask. It may be too pathetic to describe a mathematical concept, but, in general, it is true. Indeed, the history of the number  is the exciting pages of the centuries-old victorious march of mathematical thought, the tireless work of the discoverers of truth. There were triumphs of victories along the way, there were bitter defeats, dramatic collisions and comic misunderstandings. Scientists have done a gigantic job of searching, revealing the arithmetic nature of one of the most intractable, mysterious and popular numbers - the number denoted by the Greek letter .

Sumero-Babylonian mathematicians calculated the circumference and area of ​​a circle with approximations that correspond to the value =3, they also knew a more accurate approximation =3 1/8. The papyrus of Rayna (Ahmes) states that the area of ​​a circle is (8/9*2R) 2 =256/81R 2

This means that ≈3.1605….
Archimedes was the first to set the task of calculating the circumference and area of ​​a circle on a scientific basis. So, r =  > 48a 96 ≈3.1410>3 10/71

The scientist calculated the upper limit (3 1/7): 3 10/71≈3.14084... The Uzbek mathematician and astronomer al-Kashi, who worked at the scientific center of the famous mathematician and astronomer Ulugbek, calculated the number 2 with an accuracy of 16 correct decimal places: 2=6.283 185 307 179 5866.

By doubling the number of sides of regular polygons inscribed in a circle, he obtained a polygon with 800,355,168 sides.

The Dutch mathematician Ludolf Van Zeilen (1540-1610) calculated 35 decimal places  and bequeathed to carve this value on his grave monument.

One of the most beautiful quadratures of a circle, made by the Polish mathematician A.A. Kochansky (1631-1700).

All constructions are carried out with the same compass opening and quickly lead to a fairly good approximation of the number.

Johann Heinrich Lambert (1728-1777) was a German mathematician, physicist, astronomer and philosopher. He took a decisive step towards unraveling the number . In 1766

he proved the irrationality of the number . The result of revealing the secret of the number  was summed up by the German mathematician Ferdinand Lindemann (1852-1939).

In 1882 he proved that the number  is transcendental. Thus, the impossibility of squaring the circle in the classical formulation of this problem was proved.

Random events: they were realized by throwing a needle and also helped scientists calculate the number  with a fairly high accuracy.
This task was first set and carried out by the French naturalist Georges Louis Leclerc Buffon (1707-1788).

In the same way, the Swiss astronomer and mathematician Rudolf Wolf (1816-1896), as a result of 5 thousand throws of a needle, found that =3.1596.

Other scientists got the following results: with 3204 throws =3.1533; with 3408 throws =3.141593.

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List of used literature.

1. Encyclopedic dictionary of a young mathematician

2. Vasiliev N.B., Gutenmakher V.L. Straight lines and curves. - M .: Nauka, 1976

3. Markushevich A.I. Remarkable curves. - M., Nauka, 1978

4. Stroyk D.Ya. Brief outline of the history of mathematics. - M., Nauka, 1984

5. Glazer G.I. History of mathematics at school., M., Education, 1982

6. Gardner M. Mathematical miracles and secrets. M., Mir. 1978

1. 
   Kovalev F.V. Golden section in painting. K .: Vyscha school, 1989.
2. 
   Kepler I. About hexagonal snowflakes. - M., 1982.
3. 
   Durer A. Diaries, letters, treatises - L., M., 1957.
4. 
   Tsekov-Karandash Ts. About the second golden section. – Sofia, 1983.
5. 
   Stakhov A. Codes of the golden ratio.