The meaning of symmetry in the knowledge of nature. Amazing symmetry of nature Objects with axial symmetry in nature
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Oh, symmetry! I sing your anthem! Oh, symmetry! I sing your anthem! I recognize you everywhere in the world. You are in the Eiffel Tower, in a small midge, You are in a Christmas tree near a forest path. With you in friendship are both a tulip and a rose, And a snow swarm - the creation of frost! The concept of symmetry is familiar and plays an important role in everyday life. Many human creations are deliberately given a symmetrical shape for both aesthetic and practical reasons. In ancient times, the word “symmetry” was used as “harmony”, “beauty”. Indeed, in Greek it means “proportionality, proportionality, uniformity in the arrangement of parts”
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Central and axial symmetries Central symmetry - A figure is called symmetrical with respect to point O if, for each point of the figure, a point symmetrical with respect to point O also belongs to this figure. Point O is called the center of symmetry of the figure. The figure is also said to have central symmetry. Axial symmetry - A figure is called symmetrical with respect to line a if for each point of the figure a point symmetrical with respect to line a also belongs to this figure. Straight line a is called the axis of symmetry of the figure. The figure is also said to have axial symmetry.
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The manifestation of symmetry in living nature Beauty in nature is not created, but only recorded and expressed. Let us consider the manifestation of symmetry from the “global”, namely from our planet Earth. The fact that the Earth is a ball became known to educated people in ancient times. The earth, in the minds of most well-read people before the era of Copernicus, was the center of the universe. Therefore, they considered the straight lines passing through the center of the Earth to be the center of symmetry of the Universe. Therefore, even the model of the Earth - the globe has an axis of symmetry.
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Almost all living beings are built according to the laws of symmetry; it is not for nothing that the word “symmetry” means “proportionality” when translated from Greek. Almost all living beings are built according to the laws of symmetry; it is not for nothing that the word “symmetry” means “proportionality” when translated from Greek. Among flowers, for example, there is rotational symmetry. Many flowers can be rotated so that each petal takes the position of its neighbor, the flower aligns with itself. The minimum angle of such rotation is not the same for different colors. For the iris it is 120°, for the bellflower – 72°, for the narcissus – 60°.
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There is helical symmetry in the arrangement of leaves on plant stems. Arranging in a screw along the stem, the leaves seem to spread out in different directions and do not block each other from the light), although the leaves themselves also have an axis of symmetry. In the arrangement of leaves on plant stems, screw symmetry is observed. Positioned like a screw along the stem, the leaves seem to spread out in different directions and do not obscure each other from the light), although the leaves themselves also have an axis of symmetry
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Considering the general plan of the structure of any animal, we usually notice a certain regularity in the arrangement of body parts or organs, which are repeated around a certain axis or occupy the same position in relation to a certain plane. This regularity is called body symmetry. The phenomena of symmetry are so widespread in the animal world that it is very difficult to indicate a group in which no symmetry of the body can be noticed. Both small insects and large animals have symmetry. Considering the general plan of the structure of any animal, we usually notice a certain regularity in the arrangement of body parts or organs, which are repeated around a certain axis or occupy the same position in relation to a certain plane. This regularity is called body symmetry. The phenomena of symmetry are so widespread in the animal world that it is very difficult to indicate a group in which no symmetry of the body can be noticed. Both small insects and large animals have symmetry.
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Manifestation of symmetry in inanimate nature Crystals bring the charm of symmetry to the world of inanimate nature. Each snowflake is a small crystal of frozen water. The shape of snowflakes can be very diverse, but they all have rotational symmetry and, in addition, mirror symmetry. What is a crystal? A solid body that has the natural shape of a polyhedron. Salt, ice, sand, etc. consist of crystals. First of all, Romeu-Delisle emphasized the correct geometric shape of crystals based on the law of constancy of angles between their faces. Why are crystals so beautiful and attractive? Their physical and chemical properties are determined by their geometric structure. In crystallography (the science of crystals) there is even a section called “Geometric Crystallography”. In 1867, artillery general, professor at the Mikhailovsky Academy in St. Petersburg A.V. Gadolin strictly mathematically derived all combinations of symmetry elements that characterize crystalline polyhedra. In total, there are 32 types of symmetries of ideal crystal shapes.
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INTRODUCTION: A truly vast literature is devoted to the problem of symmetry. Textbooks and scientific monographs to works that appeal not so much to a drawing and formula, but to an artistic image, and combine scientific authenticity with literary precision. In the Concise Oxford Dictionary, symmetry is defined as “beauty due to the proportionality of the parts of the body or any whole, balance, likeness, harmony, consistency” (the term “symmetry” itself in Greek means “proportionality,” which ancient philosophers understood as a special case of harmony - coordination of parts within the whole). Symmetry is one of the most fundamental and one of the most general patterns of the universe: inanimate, living nature and society. We encounter symmetry everywhere. The concept of symmetry runs through the entire centuries-old history of human creativity. It is found already at the origins of human knowledge; it is widely used by all areas of modern science without exception. What is symmetry? Why does symmetry literally permeate the entire world around us? There are, in principle, two groups of symmetries. The first group includes symmetry of positions, shapes, structures. This is the symmetry that can be directly seen. It can be called geometric symmetry. The second group characterizes the symmetry of physical phenomena and laws of nature. This symmetry lies at the very basis of the natural scientific picture of the world: it can be called physical symmetry. Over thousands of years, in the course of social practice and knowledge of the laws of objective reality, humanity has accumulated numerous data indicating the presence of two tendencies in the world around us: on the one hand, towards strict orderliness and harmony, and on the other, towards their violation. People have long paid attention to the correct shape of crystals, flowers, honeycombs and other natural objects and reproduced this proportionality in works of art, in the objects they created, through the concept of symmetry. “Symmetry,” writes the famous scientist J. Newman, “establishes a funny and surprising relationship between objects, phenomena and theories that outwardly seem to be unrelated to anything: terrestrial magnetism, the female veil, polarized light, natural selection, group theory, invariants and transformations, the working habits of bees in a hive, the structure of space, vase designs, quantum physics, flower petals, X-ray interference patterns, sea urchin cell division, equilibrium configurations of crystals, Romanesque cathedrals, snowflakes, music, the theory of relativity. .." The word “symmetry” has a dual interpretation. In one sense, symmetrical means something very proportional, balanced; symmetry shows the way of coordinating many parts, with the help of which they are combined into a whole. The second meaning of this word is balance. Aristotle also spoke about symmetry as a state that is characterized by the relationship of extremes. From this statement it follows that Aristotle, perhaps, was closest to the discovery of one of the most fundamental laws of Nature - the law of its duality. It is characteristic that science came to the most interesting results precisely then, when the facts of symmetry violation were established. The consequences arising from the principle of symmetry were intensively developed by physicists in the last century and led to a number of important results. Such consequences of the laws of symmetry are, first of all, the laws of conservation of classical physics. Currently, definitions of the categories of symmetry and asymmetries based on the listing of certain characteristics. For example, symmetry is defined as a set of properties: order, uniformity, proportionality, harmony. All signs of symmetry in many of its definitions are considered equal, equally significant, and in certain specific cases, when establishing the symmetry of a phenomenon, any of them can be used. So, in some cases symmetry is homogeneity, in others it is proportionality, etc. The same can be said about the definitions of asymmetry existing in private sciences. THE IMPORTANCE OF SYMMETRY IN THE KNOWLEDGE OF NATURE The idea of symmetry was often the starting point in the hypotheses and theories of past scientists. The orderliness introduced by symmetry manifests itself, first of all, in limiting the variety of possible structures and in reducing the number of possible options. An important physical example is the existence of symmetry-determined restrictions on the diversity of molecular and crystal structures. Let us illustrate this idea with the following example. Let us assume that in some distant galaxy there live highly developed beings who, among other activities, are also fond of games. We may know nothing about the tastes of these creatures, the structure of their bodies and the characteristics of their psyche. However, it is certain that their dice have one of five shapes - tetrahedron, cube, octahedron, dodecahedron, icosahedron. Any other form of dice is, in principle, excluded, since the requirement of equiprobability of any face when playing predetermines the use of the form of a regular polyhedron, and there are only five such forms. The idea of symmetry has often served as a guiding thread for scientists when considering the problems of the universe. Observing the chaotic scattering of stars in the night sky, we understand that behind the external chaos are hidden completely symmetrical spiral structures of galaxies, and within them are symmetrical structures of planetary systems. The symmetry of the external shape of a crystal is a consequence of its internal symmetry - the ordered relative arrangement in space of atoms (molecules). In other words, the symmetry of a crystal is associated with the existence of a spatial lattice of atoms, the so-called crystal lattice. According to the modern point of view, the most fundamental laws of nature are in the nature of prohibitions. They determine what can and cannot happen in nature. Thus, the laws of conservation in elementary particle physics are laws of prohibition. They prohibit any phenomenon in which the “conserved quantity” would change, which is its own “absolute” constant (eigenvalue) of the corresponding object and characterizes its “weight” in the system of other objects. And these values are absolute as long as such an object exists. In modern science, all laws of conservation are considered precisely as laws of prohibition. Thus, in the world of elementary particles, many conservation laws are obtained as rules prohibiting those phenomena that are never observed in experiments. The prominent Soviet scientist Academician V.I. Vernadsky wrote in 1927: “What was new in science was not the identification of the principle of symmetry, but the identification of its universality.” Indeed, the universality of symmetry is amazing. Symmetry establishes internal connections between objects and phenomena that are not externally connected in any way. The universality of symmetry is not only that it is found in a variety of objects and phenomena. The principle of symmetry itself is universal, without which it is essentially impossible to consider a single fundamental problem, be it the problem of life or the problem of contacts with extraterrestrial civilizations. The principles of symmetry underlie the theory of relativity, quantum mechanics, solid state physics, atomic and nuclear physics, and particle physics. These principles are most clearly expressed in the invariance properties of the laws of nature. We are talking not only about physical laws, but also others, for example, biological ones. An example of a biological law of conservation is the law of inheritance. It is based on the invariance of biological properties with respect to the transition from one generation to another. It is quite obvious that without conservation laws (physical, biological and others), our world simply could not exist.
It is necessary to highlight aspects without which symmetry is impossible:
1) the object is the bearer of symmetry; things, processes, geometric figures, mathematical expressions, living organisms, etc. can act as symmetrical objects.
2) some features - quantities, properties, relationships, processes, phenomena - of an object, which remain unchanged during symmetry transformations; they are called invariants or invariants.
3) changes (of an object) that leave the object identical to itself according to invariant characteristics; such changes are called symmetry transformations;
4) the property of an object to transform, according to selected characteristics, into itself after its corresponding changes.
It is important to emphasize that invariance is secondary to change; rest is relative, movement is absolute.
Thus, symmetry expresses the preservation of something despite some changes or the preservation of something despite a change. Symmetry presupposes the invariability not only of the object itself, but also of any of its properties in relation to transformations performed on the object. The immutability of certain objects can be observed in relation to various operations - rotations, translations, mutual replacement of parts, reflections, etc. In this regard, different types of symmetry are distinguished.
ROTARY SYMMETRY. An object is said to have rotational symmetry if it aligns with itself when rotated through an angle of 2?/n, where n can be 2, 3, 4, etc. to infinity. The axis of symmetry is called the axis of the nth order.
TRANSPORTABLE (TRANSLATIONAL) SYMMETRY. Such symmetry is spoken of when, when moving a figure along a straight line to some distance a or a distance that is a multiple of this value, it coincides with itself.
The straight line along which the transfer is carried out is called the transfer axis, and the distance a is called the elementary transfer or period. Associated with this type of symmetry is the concept of periodic structures or lattices, which can be both flat and spatial.
Look at the faces of the people around you: one eye is squinted a little more, the other less, one eyebrow is more arched, the other less; one ear is higher, the other is lower. Let us add to what has been said that a person uses his right eye more than his left. Watch, for example, people who shoot with a gun or a bow.
From the above examples it is clear that in the structure of the human body and his habits there is a clearly expressed desire to sharply highlight any direction - right or left. This is not an accident. Similar phenomena can also be noted in plants, animals and microorganisms.
Scientists have long noticed this. Back in the 18th century. scientist and writer Bernardin de Saint-Pierre pointed out that all seas are filled with single-vave gastropods of countless species, in which all the curls are directed from left to right, similar to the movement of the Earth, if you place them with holes to the north and sharp ends to the Earth.
But before we begin to consider the phenomena of such asymmetry, we will first find out what symmetry is.
In order to understand at least the main results achieved in the study of the symmetry of organisms, we need to start with the basic concepts of the theory of symmetry itself. Remember which bodies are usually considered equal in everyday life. Only those that are completely identical or, more precisely, which, when superimposed, are combined with each other in all their details, such as, for example, the two upper petals in Figure 1. However, in the theory of symmetry, in addition to compatible equality, two more types of equality are distinguished - mirror and compatible-mirror. With mirror equality, the left petal from the middle row of Figure 1 can be accurately aligned with the right petal only after preliminary reflection in the mirror. And if two bodies are compatible-mirror equal, they can be combined with each other both before and after reflection in the mirror. The petals of the bottom row in Figure 1 are equal to each other and compatible and mirror.
From Figure 2 it is clear that the presence of equal parts in a figure alone is not enough to recognize the figure as symmetrical: on the left they are located irregularly and we have an asymmetrical figure, on the right they are uniform and we have a symmetrical rim. This regular, uniform arrangement of equal parts of a figure relative to each other is called symmetry.
Equality and sameness of arrangement of parts of a figure are revealed through symmetry operations. Symmetry operations are rotations, translations, and reflections.
The most important things for us here are rotations and reflections. Rotations are understood as ordinary rotations around an axis by 360°, as a result of which equal parts of a symmetrical figure exchange places, and the figure as a whole is combined with itself. In this case, the axis around which the rotation occurs is called a simple axis of symmetry. (This name is not accidental, since in the theory of symmetry, various types of complex axes are also distinguished.) The number of combinations of a figure with itself during one complete revolution around an axis is called the order of the axis. Thus, the image of a starfish in Figure 3 has one simple fifth-order axis passing through its center.
This means that by rotating the image of a star around its axis by 360°, we will be able to superimpose equal parts of its figure on top of each other five times.
Reflections mean any specular reflections - at a point, line, plane. The imaginary plane that divides the figures into two mirror-like halves is called the plane of symmetry. Consider in Figure 3 a flower with five petals. It has five planes of symmetry intersecting on a fifth-order axis. The symmetry of this flower can be designated as follows: 5*m. The number 5 here means one axis of symmetry of the fifth order, and m is a plane, the point is the sign of the intersection of five planes on this axis. The general formula for the symmetry of similar figures is written in the form n*m, where n is the axis symbol. Moreover, it can have values from 1 to infinity (?).
When studying the symmetry of organisms, it was found that in living nature the most common type of symmetry is n*m. Biologists call the symmetry of this type radial (radial). In addition to the flowers and starfish shown in Figure 3, radial symmetry is inherent in jellyfish and polyps, cross-sections of apples, lemons, oranges, persimmons (Figure 3), etc.
With the emergence of living nature on our planet, new types of symmetry arose and developed, which before either did not exist at all or were few. This is especially clearly seen in the example of a special case of symmetry of the form n*m, which is characterized by only one plane of symmetry, dividing the figure into two mirror-like halves. In biology, this case is called bilateral (two-sided) symmetry. In inanimate nature, this type of symmetry does not have a predominant significance, but it is extremely richly represented in living nature (Fig. 4).
It is characteristic of the external structure of the body of humans, mammals, birds, reptiles, amphibians, fish, many mollusks, crustaceans, insects, worms, as well as many plants, such as snapdragon flowers.
It is believed that such symmetry is associated with differences in the movement of organisms up and down, forward and backward, while their movements to the right and left are exactly the same. Violation of bilateral symmetry inevitably leads to inhibition of the movement of one of the sides and a change in translational movement into a circular one. Therefore, it is no coincidence that actively mobile animals are bilaterally symmetrical.
Bilaterality of immobile organisms and their organs arises due to the dissimilarity of the conditions of the attached and free sides. This appears to be the case with some leaves, flowers, and rays of coral polyps.
It is appropriate to note here that symmetry has not yet been encountered among organisms, which is limited to the presence of only a center of symmetry. In nature, this case of symmetry is perhaps widespread only among crystals; This includes, among other things, blue crystals of copper sulfate growing magnificently from the solution.
Another main type of symmetry is characterized by only one axis of symmetry of the nth order and is called axial or axial (from the Greek word “axon” - axis). Until very recently, organisms whose form is characterized by axial symmetry (with the exception of the simplest, special case, when n = 1) were not known to biologists. However, it has recently been discovered that this symmetry is widespread in the plant kingdom. It is inherent in the corollas of all those plants (jasmine, mallow, phlox, fuchsia, cotton, yellow gentian, centaury, oleander, etc.), the edges of the petals of which lie on top of each other in a fan-like manner clockwise or counterclockwise (Fig. 5).
This symmetry is also inherent in some animals, for example the jellyfish Aurelia insulinda (Fig. 6). All these facts led to the establishment of the existence of a new class of symmetry in living nature.
Objects of axial symmetry are special cases of bodies of dissymmetric, i.e., disordered, symmetry. They differ from all other objects, in particular, in their peculiar relationship to mirror reflection. If the bird’s egg and the body of the crayfish do not change their shape at all after mirror reflection, then (Fig. 7)
an axial pansy flower (a), an asymmetric helical mollusk shell (b) and, for comparison, a clock (c), a quartz crystal (d), and an asymmetric molecule (e) after mirror reflection change their shape, acquiring a number of opposite characteristics. The hands of a real clock and a mirror clock move in opposite directions; the lines on the magazine page are written from left to right, and the mirror ones are written from right to left, all the letters seem to be turned inside out; the stem of a climbing plant and the spiral shell of a gastropod in front of a mirror go from left to top to right, and mirror ones go from right to top to left, etc.
As for the simplest, special case of axial symmetry (n=1), which is mentioned above, it has been known to biologists for a long time and is called asymmetric. As an example, it is enough to refer to the picture of the internal structure of the vast majority of animal species, including humans.
Already from the examples given, it is easy to notice that dissymmetric objects can exist in two varieties: in the form of the original and a mirror reflection (human hands, mollusk shells, pansy corollas, quartz crystals). In this case, one of the forms (no matter which one) is called the right P, and the other left - L. Here it is very important to understand that right and left can and are called not only the arms or legs of a person known in this regard, but also any dissymmetrical bodies - products of human production (screws with right-hand and left-hand threads), organisms, inanimate bodies.
The discovery of P-L forms in living nature immediately raised a number of new and very deep questions for biology, many of which are now being solved by complex mathematical and physicochemical methods.
The first question is the question of the laws of the form and structure of P- and L-biological objects.
More recently, scientists have established the deep structural unity of dissymmetrical objects of living and inanimate nature. The fact is that rightism-leftism is a property equally inherent in living and inanimate bodies. Various phenomena associated with rightism and leftism also turned out to be common to them. Let us point out only one such phenomenon - dissymmetric isomerism. It shows that in the world there are many objects of different structures, but with the same set of parts that make up these objects.
Figure 8 shows the predicted and then discovered 32 buttercup corolla shapes. Here, in each case, the number of parts (petals) is the same - five; only their relative positions are different. Therefore, here we have an example of dissymmetric isomerism of the corollas.
Another example would be objects of a completely different nature, the glucose molecule. We can consider them along with the corollas of the buttercup precisely because of the similarity of the laws of their structure. The composition of glucose is as follows: 6 carbon atoms, 12 hydrogen atoms, 6 oxygen atoms. This set of atoms can be distributed in space in very different ways. Scientists believe that glucose molecules can exist in at least 320 different species.
The second question: how often do P- and L-forms of living organisms occur in nature?
The most important discovery in this regard was made in the study of the molecular structure of organisms. It turned out that the protoplasm of all plants, animals and microorganisms absorbs mainly only P-sugars. Thus, every day we eat the right sugar. But amino acids are found mainly in the L-form, and proteins built from them are found mainly in the P-form.
Let's take two protein products as an example: egg white and sheep's wool. Both of them are right-handed. The wool and egg whites of the “left-hander” have not yet been found in nature. If it were possible to somehow create L-wool, that is, such wool, the amino acids in which would be located along the walls of the screw curling to the left, then the problem of fighting moths would be solved: moths can feed only on P-wool, just like this The same way people digest only the P-protein of meat, milk, and eggs. And this is not difficult to understand. Moths digest wool, and humans digest meat through special proteins - enzymes, which are also right-handed in their configuration. And just as an L-screw cannot be screwed into nuts with a P-thread, it is impossible to digest L-wool and L-meat using P-enzymes, if any were found.
Perhaps this is also the mystery of the disease known as cancer: there is information that in some cases cancer cells build themselves not from right-handed, but from left-handed proteins that are not digestible by our enzymes.
The widely known antibiotic penicillin is produced by mold only in the P-form; its artificially prepared L form is not antibiotically active. Pharmacies sell the antibiotic chloramphenicol, and not its antipode, pravomycetin, since the latter is significantly inferior to the former in its medicinal properties.
Tobacco contains L-nicotine. It is several times more poisonous than P-nicotine.
If we consider the external structure of organisms, then here we will see the same thing. In the vast majority of cases, whole organisms and their organs are found in the P- or L-form. The back part of the body of wolves and dogs moves somewhat to the side when running, so they are divided into right- and left-running. Left-handed birds fold their wings so that the left wing overlaps the right, while right-handed birds do the opposite. Some pigeons prefer to circle to the right when flying, while others prefer to circle to the left. For this reason, pigeons have long been popularly divided into “right-handed” and “left-handed”. The shell of the mollusk Fruticicola lantzi is found mainly in the U-twisted form. It is remarkable that when feeding on carrots, the predominant P-forms of this mollusk grow well, and their antipodes - L-mollusks - sharply lose weight. The ciliate slipper, due to the spiral arrangement of cilia on its body, moves in a drop of water, like many other protozoa, along a left-curling corkscrew. Ciliates that penetrate into the medium along the right corkscrew are rare. Narcissus, barley, cattail, etc. are right-handed: their leaves are found only in a U-helical form (Fig. 9). But beans are left-handed: the leaves of the first tier are often L-shaped. It is remarkable that, compared to P-leaves, L-leaves weigh more, have a larger area, volume, osmotic pressure of cell sap, and growth rate.
The science of symmetry can reveal many interesting facts about humans. As you know, on average on the globe there are approximately 3% of left-handers (99 million) and 97% of right-handers (3 billion 201 million). According to some information, in the USA and on the African continent there are significantly more left-handers than, for example, in the USSR.
It is interesting to note that the speech centers in the brain of right-handers are located on the left, while in left-handers they are located on the right (according to other sources, in both hemispheres). The right half of the body is controlled by the left, and the left by the right hemisphere, and in most cases the right half of the body and the left hemisphere are better developed. In humans, as you know, the heart is on the left side, the liver is on the right. But for every 7-12 thousand people there are people whose all or part of their internal organs are located in a mirror image, that is, vice versa.
The third question is the question about the properties of the P- and L-forms. The examples already given make it clear that in living nature a number of properties of the P- and L-forms are not the same. Thus, using examples with shellfish, beans and antibiotics, the difference in nutrition, growth rate and antibiotic activity in their P- and L-forms was shown.
This feature of the P- and L-forms of living nature is of very great importance: it allows, from a completely new perspective, to sharply distinguish living organisms from all those P- and L-bodies of inanimate nature, which in one way or another are equal in their properties, for example, from elementary particles.
What is the reason for all these features of the dissymmetrical bodies of living nature?
It was found that by growing the microorganisms Bacillus mycoides on agar-agar with P- and L-compounds (sucrose, tartaric acid, amino acids), L-colonies can be converted into P-, and P- into L-forms. In some cases, these changes were long-term, possibly hereditary. These experiments indicate that the external P- or L-form of organisms depends on metabolism and the P- and L-molecules participating in this exchange.
Sometimes transformations from P- to L-forms and vice versa occur without human intervention.
Academician V.I. Vernadsky notes that all the shells of the fossil mollusks Fusus antiquus found in England are left-handed, while modern shells are right-handed. Obviously, the reasons that caused such changes changed over geological eras.
Of course, the change in types of symmetry as life evolved occurred not only in dissymmetric organisms. Thus, some echinoderms were once bilaterally symmetrical mobile forms. Then they switched to a sedentary lifestyle and developed radial symmetry (although their larvae still retained bilateral symmetry). In some echinoderms that switched to an active lifestyle for the second time, radial symmetry was again replaced by bilateral symmetry (irregular urchins, holothurians).
So far we have talked about the reasons that determine the shape of P- and L-organisms and their organs. Why are these forms not found in equal quantities? As a rule, there are more either P- or L-forms. The reasons for this are not known. According to one very plausible hypothesis, the causes may be dissymmetric elementary particles, for example, right-handed neutrinos that predominate in our world, as well as right-handed light, which always exists in slight excess in diffuse sunlight. All this initially could create an unequal occurrence of right and left forms of dissymmetric organic molecules, and then lead to an unequal occurrence of P- and L-organisms and their parts.
These are just some of the questions of biosymmetry - the science of the processes of symmetrization and dissymmetrization in living nature.
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Introduction
While walking in the grove in the fall, I collected beautiful fallen leaves and brought them home. My dad (A. A. Radionov, researcher at the Southern Mathematical Institute of the All-Russian Scientific Center of the Russian Academy of Sciences), looking at them, uttered the phrase: here is another example of symmetry in nature. I became interested and the first thing I did was look in S.I. Ozhegov’s dictionary to see what the word “symmetry” meant, and then I began to pester my father with questions: how did he determine that this is “symmetry” and what types of symmetry are there? This was the reason to study this issue.
The purpose of the work: to show what types of symmetry are observed in nature, and how they are described using mathematics.
My task was:
Describe the different types of symmetry;
Try to independently find mathematical relationships in the structure of tree leaves.
Object of study: maple and grape leaves.
Subject of research: symmetry in natural objects.
Methods used in the work: analysis of literature on the topic, scientific experiment.
This work is classified as abstract-experimental.
The significance of the results obtained lies in the fact that plant leaves can be studied mathematically, measured instrumentally, and the symmetry of these natural objects can be checked.
Symmetry in the nature around us
Symmetry (ancient Greek - “proportionality”) is the regular arrangement of similar (identical) parts of the body or forms of a living organism relative to the center or axis of symmetry. This implies that proportionality is part of harmony, the correct combination of parts of the whole.
Harmony is a Greek word meaning “coherence, proportionality, unity of parts and whole.” Externally, harmony can manifest itself in symmetry and proportionality.
Symmetry is a very common phenomenon; its universality serves as an effective method of understanding nature. In living nature, symmetry is not absolute and always contains some degree of asymmetry. Asymmetry - (Greek "without" and "symmetry") - lack of symmetry.
By carefully examining natural phenomena, you can see the commonality even in the most insignificant things and details, and find manifestations of symmetry. The shape of a tree leaf is not random: it is strictly natural. The sheet seems to be glued together from two more or less identical halves, one of which is located mirror-image relative to the other. The symmetry of a leaf is repeated for all leaves of a given tree. That's an example mirror symmetry- when an object can be divided into right and left or upper and lower halves by an imaginary axis called an axis of mirror symmetry. The halves located on opposite sides of the axis are almost identical to each other. The mirror exactly reproduces what it “sees,” but the order considered is reversed: the right hand of the double in the mirror turns out to be the left. Mirror symmetry can be found everywhere: in the leaves and flowers of plants. Moreover, mirror symmetry is inherent in the bodies of almost all living beings (Appendix No. 1, Fig. a).
Many flowers have radial symmetry: the appearance of the pattern will not change if it is rotated by some angle around its center. This symmetry is called rotational symmetry or axial symmetry. With this symmetry, a leaf or flower, turning around the axis of symmetry, turns into itself. If you cut a plant stem or tree trunk, then radial symmetry in the form of stripes is often clearly visible on the cut (Appendix No. 1, Fig. b).
A rotation by a certain number of degrees, accompanied by an increase in size along the axis of rotation (or a decrease in size or no change in size), generates helical symmetry- symmetry of the spiral staircase (Appendix No. 1, Fig. c).
Symmetry of similarity. Another type of symmetry is the symmetry of similarity, associated with the simultaneous increase or decrease of similar parts of the figure and the distances between them. All growing organisms exhibit this symmetry: the small sprout of any plant contains all the features of a mature plant. The symmetry of similarity is manifested everywhere in nature on everything that grows: in growing objects of plants, animals and crystals (Appendix No. 1, Fig. d).
In mathematics, self-similar geometric objects are called fractals. It is characteristic of fractals that a small part of a geometric curve is similar to the entire curve. The figure shows the process of constructing self-similar Koch curves and Koch snowflakes (the first 4 steps). (Appendix No. 2)
Any segment of a curve constructed in this way has an infinite length. Fractals are characterized by fractal dimension. The term fractal and fractal dimension were introduced by mathematician Benoit Mandelbrot in 1975. Fractal dimension was introduced as a coefficient describing geometrically complex shapes for which details are more important than the complete design.
Dimension 2 means that we can uniquely define any curve by two numbers. The surface of a sphere is two-dimensional (it can be defined using two angles of latitude and longitude). Dimension is defined as follows: for one-dimensional objects, doubling their linear size leads to an increase in size by a factor of two. For two-dimensional objects, doubling the linear dimensions results in a fourfold increase in the size (area of the rectangle). For 3-dimensional objects, doubling the linear dimensions leads to an eight-fold increase in volume.
The dimension D can be determined mathematically using the rule:
where N -N is the number of parts, is the scale factor, D is the dimension.
From here we get the formula for the dimension:
Take a segment, divide it into three equal parts (N = 3), each resulting part will be 3 times less long () than the length of the initial segment:
Therefore, for a segment the dimension is equal to one.
Similarly for area: if you measure the area of a square, and then measure the area of a square with a side longer than the length of the side of the initial square, then it will turn out to be 9 times smaller (N = 9) than the area of the initial square:
for a flat figure the dimension is two. For a spatial figure such as a cube, the calculated dimension is three.
Similar calculations for the Koch curve give the result:
Therefore, fractals correspond not to an integer, but a fractional dimension.
Conducting a scientific experiment
Justification for choice:
Fallen leaves of trees were chosen as experimental material: maple and grape, symmetrical in appearance (axial, mirror symmetry).
Experiment sequence:
Measuring the area of the left and right parts of the sheet;
Measuring the angles between veins on a sheet;
Measuring the lengths of the veins present on the sheet;
Recording the results obtained;
Search for mathematical patterns;
Conclusions based on the results obtained.
List of things to study on a leaf of a tree:
Symmetry;
Fractals;
Geometric progression;
Logarithms.
Examination of fallen leaves showed that the leaves are symmetrical about their axis. A more detailed examination shows that the symmetry is slightly broken at the edges of the sheet, and in some cases within the surface of the sheet.
To make sure how similar the left and right parts of the sheet are, the following measurements were taken:
1) measuring the area of the left and right parts of the sheet;
2) measuring the angles at which the veins intersect in the left and right parts of the sheet;
3) measuring the length of the main veins in the left and right parts of the sheet;
4) measuring the length of secondary veins in the left and right parts of the sheet;
5) measuring the length of the smallest veins of the leaf.
For ease of measurement, all sheets were first scanned and then printed on paper on a black and white printer, accurately preserving the dimensions and details of the image. Measurements were taken on a paper image of the sheet. To measure the area of the left and right parts of the sheet, a grid with a step of 5 mm was additionally superimposed on the image. The areas of the left or right parts of the sheet were calculated by the number of small squares with an area of 5x5 mm 2 filled by the sheet. Some squares turned out to be partially filled: those filled more than half were taken into account in the calculation, and those filled less than half were not taken into account in the calculations.
The photographs show the process of taking measurements (Appendix No. 3).
Maple Leaf
1) measuring the area of the left side showed 317 squares of 25 mm 2 or 79.25 square centimeters. The measurement of the right side showed 312 squares of 25 mm 2 or 78 square centimeters. Taking into account the error in measurement accuracy, the obtained result suggests that approximately the areas of the left and right parts of the sheet are the same (Appendix No. 4, Fig. 1).
2) Determining the angles at which the leaf veins diverge from its base shows that these angles are approximately the same and amount to about 25 degrees. On the right side of the sheet, when moving clockwise from the middle of the sheet, the first vein is spaced at 26 degrees, the second at 52 degrees, and the third at 74 degrees. And on the left side of the sheet, when moving counterclockwise from the sheet axis, the first vein deviates by 24 degrees, the second by 63 degrees, and the third by 80 degrees. Figure 2 of Appendix No. 4 shows these measurements: it can be seen that despite all the symmetry of the sheet, some minor violations of symmetry are observed.
3) Measurements of vein lengths. The figure shows the measured lengths of the main veins along with the corners. In cases where a leaf vein turned out to be strongly curved, its length was measured along the length of a broken curve: the curved vein was divided into three approximately equal parts and each part was measured as a straight line - with a ruler. The length of the main veins on the right side of the sheet was 30.2 cm. On the left side of the sheet - 30.6 cm. The total length together with the central vein was 75 cm.
Additionally, the lengths of all secondary, small leaf veins that do not emerge from the base of the leaf were measured. On the left side of the sheet, their total length is 52.6 cm, and on the right side of the sheet - 51.1 cm. The total length is 103.7 cm (Appendix No. 4, Fig. 3).
Surprisingly, the total length of the minor leaf veins is greater than the length of the main leaf veins. On the left side, the ratio of these lengths is 1.72. On the right side - 1.69. The resulting ratios are close to each other, but not exactly equal.
grape leaf
1) Measuring the angles at which the veins of a grape leaf diverge from its base shows that these angles are approximately the same and amount to about 40 degrees. On the right side of the leaf there are two such veins and when moving clockwise from the middle of the sheet, the first vein is spaced at 41 degrees, the second at 86 degrees. On the left side of the sheet, when moving counterclockwise from the sheet axis, the first vein deviates by 41 degrees, the second by 80 degrees. Figure 1 of Appendix No. 5 shows these measurements. The lengths of the main veins of the leaf are also marked here.
Equally interesting is measuring the angles at which the secondary veins (those that do not extend from the center of the base of the leaf) intersect. These measurements are presented in Figure 2 of Appendix No. 5: for secondary leaf veins, there is a greater variation in the angles at which they intersect with other veins, but on average this angle is approximately 60 degrees. This average angle is the same both on the left side of the sheet and on the right side. The lengths of these secondary veins are also marked here.
2) Measuring the lengths of the veins. The length of the main ones (emanating from the base of the leaf) on the left side of the sheet is 16 cm. On the right side of the sheet - 16.4 cm. The length with the central vein is 44.4 cm.
The length of the secondary veins on the left side of the leaf is 41.2 cm, and on the right side - 43 cm. In total, the total length of the secondary veins is 84.2 cm. For a grape leaf, the length of the secondary veins is approximately twice as long as the length of the main veins of the leaf.
For a grape leaf, it is also possible to measure the length of the network of the smallest veins. They are clearly visible on the back surface of the leaf. Measurements of the lengths of the smallest veins were made by counting their number at half the distance between two secondary veins, after which the number found was multiplied by the length of one of them (approximately half the distance between the two main veins). In this case, small veins that are not connected to the main veins and are located between larger veins could drop out of the count.
The length of the smallest veins measured in this way on the left side of the leaf was 110.7 cm, and on the right side of the leaf - 133.9 cm. The total length of the smallest veins was 244.6 cm (Fig. 3, Appendix No. 5).
The surprising finding is that the smaller the veins, the longer their total length. On the left side of the sheet the ratio of the measured lengths is:
smallest veinlets / secondary veinlets = 110.7 / 41.2 = 2.69;
secondary veins / main veins = 41.2 / 16.0 = 2.57.
On the right side there are similar relationships
133,9 / 43,0 = 3,11,
43,0 / 16,4 = 2,62.
The resulting length ratios are more accurate for the ratio of secondary to primary veinlets because these lengths are measured more accurately. For the left side, the ratio of the length of the smallest veins to the length of the secondary veins also gives approximately the same value of about 2.7. Only on the right side of the sheet is this ratio noticeably greater and equal to 3.11.
From measuring the lengths and intersection angles of the veins, the following conclusions can be drawn.
In the left and right parts of the sheet, approximately equal angles are observed between the main and secondary veins.
Also, in the left and right parts, the lengths of the main and secondary veins are approximately the same.
The ratio of the lengths of secondary veins to the length of the main veins is approximately 2.6. This means that when moving from primary veins to secondary ones, their length increases by 2.6 times. The ratio of the lengths of the smallest veins to the length of secondary veins is 2.7 for the left part of the leaf and 3.1 for the right part of the leaf. This means that when moving from secondary veins to the smallest ones, their length increases by 2.7 times (3.1 for the right side of the leaf).
The found pattern can be explained by the fractal structure of the leaf: when moving from a large scale to a smaller scale, approximately one coefficient of increase in the length of the corresponding veins is observed.
For the intersection angles of veins of different scales, it is impossible to talk about a fractal structure. The primary veins intersect at an angle of 40 degrees, the secondary veins at an angle of 60 degrees, and the smallest at approximately 90 degrees.
Let's apply the fractal dimension formula for a grape leaf.
for the left side of the sheet:
number of main ones: 2;
main length: 16.0 cm;
number of secondary: 12;
secondary length 41.2 cm;
number of smallest veins: 407;
the length of the smallest veins is 110.7 cm;
Calculation of the fractal dimension for a geometric fractal at stages 2) and 3) should give close values. The resulting figures differ by more than two times. This suggests that the veins of a grape leaf do not form a geometric fractal. A similar conclusion follows from a comparison of the angles at which veins of different levels intersect (40, 60, 90 degrees).
Conclusion
In my work, I showed with a concrete example that natural symmetrical tree leaves obey mathematical laws. However, even taking into account the measurement error, the leaves I examined are not completely symmetrical - differences were found in the left and right parts of the leaf, that is, in living nature, symmetry is not absolute and always contains a certain degree of asymmetry. For example, the length of the main veins of a maple leaf on the left side is 30.6 cm, and on the right - 30.2 cm. In percentage terms, this difference is 1.3%. For a grape leaf, the same difference is 2.5%.
When moving from a larger scale of leaf veins to a smaller scale of these veins, approximately the same coefficient of increase in the lengths of the corresponding veins is observed. This coefficient is equal to 2.6 (for a grape leaf) and is maintained when moving from the largest veins to smaller ones, and from them - when moving to the smallest veins.
This behavior of the veins is not the fractal structure of the grape leaf: measuring the fractal dimension gives different values for veins of different levels. The observed complex structure of leaf veins is formed to supply water and nutrients to the entire leaf area of the plant. Apparently, the fractal structure of leaf veins is not always the best (optimal) form for a plant to perform this task.
List of used literature:
1.Paitgen H.O., Richter P.H., The beauty of fractals. Images of complex dynamic systems//Mir.- M., 1993, 206 p. ISBN 5-03-001296-6
2. Tarasov L.V. This amazingly symmetrical world // Enlightenment.-M., 1982-p.176
3. Ozhegov S.I. Dictionary of the Russian language // Russian language.-20th ed. M., 1988-p.585
4.Wikipedia, Fractal dimension. https://ru.wikipedia.org/wiki/Fractal_dimension
5. Fractals are around us. http://sakva.net/fractals_rus/
6. Ivanovsky A. Fractal geometry of the world. http://w-o-s.ru/article/4003
7. Symmetry in nature. http://wonwilworl.blogspot.ru/2014/01/blog-post.html
Appendix No. 1
Appendix No. 2
Koch curve
Koch's snowflakes
Appendix No. 3
Appendix No. 4
Symmetry in nature is an objective property, one of the main ones in modern natural science. This is a universal and general characteristic of our material world.
Symmetry in nature is a concept that reflects the existing order in the world, proportionality and proportionality between the elements of various systems or objects of nature, the balance of the system, orderliness, stability, that is, a certain
Symmetry and asymmetry are opposite concepts. The latter reflects the disorder of the system, the lack of equilibrium.
Shapes of symmetries
Modern natural science defines a number of symmetries that reflect the properties of the hierarchy of individual levels of organization of the material world. Various types or forms of symmetries are known:
- spatiotemporal;
- calibration;
- isotopic;
- mirrored;
- permutable.
All listed types of symmetries can be divided into external and internal.
External symmetry in nature (spatial or geometric) is represented by a huge variety. This applies to crystals, living organisms, molecules.
Internal symmetry is hidden from our eyes. It manifests itself in laws and mathematical equations. For example, Maxwell's equation, which determines the relationship between magnetic and electrical phenomena, or Einstein's property of gravity, which connects space, time and gravity.
Why is symmetry needed in life?
Symmetry in living organisms was formed during the process of evolution. The very first organisms that arose in the ocean had a perfect spherical shape. In order to penetrate into a different environment, they had to adapt to new conditions.
One of the ways of such adaptation is symmetry in nature at the level of physical forms. The symmetrical arrangement of body parts ensures balance during movement, vitality and adaptation. The external forms of humans and large animals have a fairly symmetrical appearance. In the plant world there is also symmetry. For example, the cone-shaped crown of a spruce tree has a symmetrical axis. This is a vertical trunk, thickened at the bottom for stability. The individual branches are also located symmetrically in relation to it, and the shape of the cone allows the crown to rationally use solar energy. The external symmetry of animals helps them maintain balance when moving, enrich themselves with energy from the environment, using it rationally.
In chemical and physical systems, symmetry is also present. Thus, the most stable molecules are those that have high symmetry. Crystals are highly symmetrical bodies; three dimensions of an elementary atom are periodically repeated in their structure.
Asymmetry
Sometimes the internal arrangement of organs in a living organism is asymmetrical. For example, a person’s heart is located on the left, the liver on the right.
In the process of life, plants absorb chemical mineral compounds from molecules of symmetrical shape from the soil and convert them in their bodies into asymmetric substances: proteins, starch, glucose.
Asymmetry and symmetry in nature are two opposing characteristics. These are categories that are always in struggle and unity. Different levels of development of matter can have properties of either symmetry or asymmetry.
If we assume that equilibrium is a state of rest and symmetry, and non-equilibrium movement is caused by asymmetry, then we can say that the concept of equilibrium in biology is no less important than in physics. Biological is characterized by the principle of stability of thermodynamic equilibrium. It is asymmetry, which is a stable dynamic equilibrium, that can be considered a key principle in solving the problem of the origin of life.
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