I . Symmetry in mathematics :

    Basic concepts and definitions.

    Axial symmetry (definitions, construction plan, examples)

    Central symmetry (definitions, construction plan, whenmeasures)

    Summary table (all properties, features)

II . Applications of symmetry:

1) in mathematics

2) in chemistry

3) in biology, botany and zoology

4) in art, literature and architecture

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1. Basic concepts of symmetry and its types.

The concept of symmetry R goes back through the entire history of mankind. It is found already at the origins of human knowledge. It arose in connection with the study of a living organism, namely man. And it was used by sculptors back in the 5th century BC. e. The word “symmetry” is Greek and means “proportionality, proportionality, sameness in the arrangement of parts.” It is widely used by all areas of modern science without exception. Many great people have thought about this pattern. For example, L.N. Tolstoy said: “Standing in front of a black board and drawing different figures on it with chalk, I was suddenly struck by the thought: why is symmetry clear to the eye? What is symmetry? This is an innate feeling, I answered myself. What is it based on?” The symmetry is truly pleasing to the eye. Who hasn’t admired the symmetry of nature’s creations: leaves, flowers, birds, animals; or human creations: buildings, technology, everything that surrounds us since childhood, everything that strives for beauty and harmony. Hermann Weyl said: “Symmetry is the idea through which man throughout the ages has tried to comprehend and create order, beauty and perfection.” Hermann Weyl is a German mathematician. His activities span the first half of the twentieth century. It was he who formulated the definition of symmetry, established by what criteria one can determine the presence or, conversely, absence of symmetry in a given case. Thus, a mathematically rigorous concept was formed relatively recently - at the beginning of the twentieth century. It's quite complicated. Let us turn and once again remember the definitions that were given to us in the textbook.

2. Axial symmetry.

2.1 Basic definitions

Definition. Two points A and A 1 are called symmetrical with respect to line a if this line passes through the middle of segment AA 1 and is perpendicular to it. Each point of a line a is considered symmetrical to itself.

Definition. The figure is said to be symmetrical about a straight line A, if for each point of the figure there is a point symmetrical to it relative to the straight line A also belongs to this figure. Straight A called the axis of symmetry of the figure. The figure is also said to have axial symmetry.

2.2 Construction plan

And so, to construct a symmetrical figure relative to a straight line, from each point we draw a perpendicular to this straight line and extend it to the same distance, mark the resulting point. We do this with each point and get symmetrical vertices of a new figure. Then we connect them in series and get a symmetrical figure of a given relative axis.

2.3 Examples of figures with axial symmetry.

3. Central symmetry

3.1 Basic definitions

Definition. Two points A and A 1 are called symmetrical with respect to point O if O is the middle of the segment AA 1. Point O is considered symmetrical to itself.

Definition. A figure is said to be 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.

3.2 Construction plan

Construction of a triangle symmetrical to the given one relative to the center O.

To construct a point symmetrical to a point A relative to the point ABOUT, it is enough to draw a straight line OA(Fig. 46 ) and on the other side of the point ABOUT set aside a segment equal to the segment OA. In other words , points A and ; In and ; C and symmetrical about some point O. In Fig. 46 a triangle is constructed that is symmetrical to a triangle ABC relative to the point ABOUT. These triangles are equal.

Construction of symmetrical points relative to the center.

In the figure, points M and M 1, N and N 1 are symmetrical relative to point O, but points P and Q are not symmetrical relative to this point.

In general, figures that are symmetrical about a certain point are equal .

3.3 Examples

Let us give examples of figures that have central symmetry. The simplest figures with central symmetry are the circle and parallelogram.

Point O is called the center of symmetry of the figure. In such cases, the figure has central symmetry. The center of symmetry of a circle is the center of the circle, and the center of symmetry of a parallelogram is the point of intersection of its diagonals.

A straight line also has central symmetry, but unlike a circle and a parallelogram, which have only one center of symmetry (point O in the figure), a straight line has an infinite number of them - any point on the straight line is its center of symmetry.

The pictures show an angle symmetrical relative to the vertex, a segment symmetrical to another segment relative to the center A and a quadrilateral symmetrical about its vertex M.

An example of a figure that does not have a center of symmetry is a triangle.

4. Lesson summary

Let us summarize the knowledge gained. Today in class we learned about two main types of symmetry: central and axial. Let's look at the screen and systematize the knowledge gained.

Summary table

Axial symmetry

Central symmetry

Peculiarity

All points of the figure must be symmetrical relative to some straight line.

All points of the figure must be symmetrical relative to the point chosen as the center of symmetry.

Properties

    1. Symmetrical points lie on perpendiculars to a line.

    3. Straight lines turn into straight lines, angles into equal angles.

    4. The sizes and shapes of the figures are preserved.

    1. Symmetrical points lie on a line passing through the center and a given point of the figure.

    2. The distance from a point to a straight line is equal to the distance from a straight line to a symmetrical point.

3. The sizes and shapes of the figures are preserved.

II. Application of symmetry

Mathematics

In algebra lessons we studied the graphs of the functions y=x and y=x

The pictures show various pictures depicted using the branches of parabolas.

(a) Octahedron,

(b) rhombic dodecahedron, (c) hexagonal octahedron.

Russian language

The printed letters of the Russian alphabet also have different types of symmetries.

There are “symmetrical” words in the Russian language - palindromes, which can be read equally in both directions.

A D L M P T F W– vertical axis

V E Z K S E Y - horizontal axis

F N O X- both vertical and horizontal

B G I Y R U C CH SCHY- no axis

Radar hut Alla Anna

Literature

Sentences can also be palindromic. Bryusov wrote a poem “The Voice of the Moon”, in which each line is a palindrome.

Look at the quadruples of A.S. Pushkin “ Bronze Horseman" If we draw a line after the second line we can notice elements of axial symmetry

And the rose fell on Azor's paw.

I come with the sword of the judge. (Derzhavin)

"Search for a taxi"

"Argentina beckons the Negro"

“The Argentinean appreciates the black man,”

“Lesha found a bug on the shelf.”

The Neva is dressed in granite;

Bridges hung over the waters;

Dark green gardens

Islands covered it...

Biology

The human body is built on the principle of bilateral symmetry. Most of us view the brain as a single structure; in reality, it is divided into two halves. These two parts - two hemispheres - fit tightly to each other. In full accordance with the general symmetry of the human body, each hemisphere is an almost exact mirror image of the other

Control of the basic movements of the human body and its sensory functions is evenly distributed between the two hemispheres of the brain.

The left hemisphere controls the right side of the brain, and the right hemisphere controls the left side.

Botany A flower is considered symmetrical when each perianth consists of an equal number of parts. Flowers having paired parts are considered flowers with double symmetry, etc. Triple symmetry is common in monocotyledons, and quintuple symmetry in dicotyledons. Characteristic feature

The structure of plants and their development is helicity.

Pay attention to the leaf arrangement of the shoots - this is also a peculiar type of spiral - a helical one. Even Goethe, who was not only a great poet, but also a natural scientist, considered spirality to be one of the characteristic features of all organisms, a manifestation of the innermost essence of life. The tendrils of plants twist in a spiral, the growth of tissues in tree trunks occurs in a spiral, the seeds in a sunflower are arranged in a spiral, and spiral movements are observed during the growth of roots and shoots.

A characteristic feature of the structure of plants and their development is spirality. Look at the pine cone. The scales on its surface are arranged strictly regularly - along two spirals that intersect approximately at a right angle. The number of such spirals is

pine cones

equals 8 and 13 or 13 and 21. Zoology Symmetry in animals means correspondence in size, shape and outline, as well as the relative arrangement of body parts located on opposite sides of the dividing line. With radial or radial symmetry, the body has the shape of a short or long cylinder or vessel with a central axis, from which parts of the body extend radially. These are coelenterates, echinoderms,

Axial symmetry

Various types of symmetry of physical phenomena: symmetry of electric and magnetic fields (Fig. 1)

In mutually perpendicular planes, the propagation of electromagnetic waves is symmetrical (Fig. 2)


Fig.1 Fig.2

Art

Mirror symmetry can often be observed in works of art. Mirror" symmetry is widely found in works of art of primitive civilizations and in ancient paintings. Medieval religious paintings are also characterized by this type of symmetry.

One of Raphael’s best early works, “The Betrothal of Mary,” was created in 1504. Under a sunny blue sky lies a valley topped by a white stone temple. In the foreground is the betrothal ceremony.



The High Priest brings Mary and Joseph's hands together. Behind Mary is a group of girls, behind Joseph is a group of young men. Both parts of the symmetrical composition are held together by the counter-movement of the characters.

For modern tastes, the composition of such a painting is boring, since the symmetry is too obvious.

ChemistryA water molecule has a plane of symmetry (straight vertical line). DNA molecules (deoxyribonucleic acid) play an extremely important role in the world of living nature. It is a double-chain high-molecular polymer, the monomer of which is nucleotides.

DNA molecules have a double helix structure built on the principle of complementarity.

Archite culture Man has long used symmetry in architecture. The ancient architects made especially brilliant use of symmetry in architectural structures. Moreover, the ancient Greek architects were convinced that in their works they were guided by the laws that govern nature. By choosing symmetrical forms, the artist thereby expressed his understanding of natural harmony as stability and balance. Communication with other types of animals: bilateral symmetry, secondary cavity, deuterostome, metamerism. Subtype... . Concept about the breed and her structure. Biological properties of animals. Fertility.

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    • , Competition "Presentation for the lesson"

    Presentation for the lesson
















    Back forward

    Attention! Slide previews are for informational purposes only and may not represent all the features of the presentation. If you are interested this work, please download the full version.

    Goals and objectives:

    • improving knowledge about axial symmetry;
    • introduce the concept of central symmetry;
    • teach to recognize figures with axial symmetry and central symmetry;
    • improving knowledge and skills when working with drawing and measuring instruments;
    • develop spatial imagination, design skills and creativity;
    • promote the development of interest in technical creativity;
    • broadening your horizons.

    Materials and tools:

    • Teacher's computer (laptop), multimedia projector, screen; slide presentation for the lesson;

    compass for the board; student's compasses, triangles, colored cardboard and paper, scissors, glue.

    Lesson plan:

    Organizational part (preparation for work).

    Updating basic knowledge.

    Repetition of geometric material.

    Practical work, explanation and demonstration of basic methods of performing work, competitions.

    Summing up the lesson, discussing the work done.

    Cleaning workplaces.

    Progress of the lesson

    Organizing time. Checking readiness for class.

    Task No. 1. "Divide the triangle" Slide 2

    ANSWER (Fig. 2):

    rice. 2

    Divide the equilateral triangle shown in the figure as follows:

    1. Three lines into four equal parts.

    2. Three lines into six equal parts.

    3. Three lines into three equal parts.

    4. One line into four arbitrary parts

    Task No. 2. Slide 3

    In ancient times, the word "SYMMETRY" was used to mean "harmony", "beauty". Indeed, translated from Greek this word means “proportionality, proportionality, uniformity in the arrangement of parts.”

    We encounter symmetry everywhere - in nature, technology, art, science. The concept of symmetry runs through the entire centuries-old history of human creativity. It is found already at the origins of human development. Man has long used symmetry in architecture. It gives harmony and completeness to ancient temples, towers of medieval castles, and modern buildings. What is symmetry? Why does symmetry literally permeate the entire world around us?

    We will consider the symmetry that can be directly seen - the symmetry of positions, shapes, structures. It can be called geometric symmetry.

    AXIAL SYMMETRY Slide 4

    An isosceles (but not equilateral) triangle also has one line symmetry. A equilateral triangle - three lines symmetry.

    U unexpanded of an angle there is one line of symmetry - a straight line on which the bisector of the angle is located.

    A rectangle and a rhombus that are not squares have two lines of symmetry, A square - four lines of symmetry.

    Speech "Mirror (axial) symmetry" Appendix No. 1

    Find figures that have a line of symmetry (Task No. 1) Appendix No. 2

    CENTRAL SYMMETRY Slide 8

    The simplest figures with central symmetry are the circle and parallelogram.

    The center of symmetry of a circle is the center of the circle, and the center of symmetry of a parallelogram is the point of intersection of its diagonals.

    A straight line also has central symmetry, but unlike a circle and a parallelogram, which have only one center of symmetry, a straight line has an infinite number of them - any point on a straight line is its center of symmetry.

    An example of a figure that does not have a center of symmetry is triangle.

    Find figures with central symmetry (Task No. 2) Appendix No. 2

    Find figures that have both axes of symmetry (Task No. 3) Appendix No. 2

    Speech "Symmetry in Letters" Appendix No. 3

    Once - hands waved up
    And at the same time we sighed
    Two or three bent down and reached the floor
    And four - stood up straight and repeat first.
    The air we breathe in is strong
    When bending, exhale in a friendly manner
    But you don't need to bend your knees.
    So that your hands don't get tired,
    We'll put them on our belts.
    We jump like balls
    Girls and boys.

    Practical work "Flying saucer" Appendix No. 5

    What geometric body does a flying saucer resemble? (cylinder)

    What tool will we use? (compass)

    Safety rules when working with compasses.

    We're starting now practical work(Fig. 10):

    1. To make a flying saucer we use cardboard of any color.
    2. On the wrong side of the cardboard we draw a circle R55 (1 piece) and R36 (2 pieces).
    3. Along the length of the cardboard we lay out a rectangle 220 mm long and 12 mm wide (we mark the valves along the length).
    4. Cut out all the details.
    5. We glue parts No. 2 and No. 3, we get a cylinder.
    6. Glue the cylinder to part No. 1
    7. The result was "Flying Saucer".
    8. Design according to your own design.
    9. Competitions.
    10. Summarizing

    Lesson summary

    Today in class we repeated and studied axial and central symmetries.

    • How many axes of symmetry does a line segment have?
    • (2 each).
    • Do a line segment, a straight line, or a square have a center of symmetry? (2 each)
    • Which of these letters have an axis of symmetry? (M, A, N, E) Which of these letters have a center of symmetry? (BUT)

    Appendix No. 6

    Everything is correct.

    Today everyone did a good job and figured out symmetry, but if anyone still doubts, I have prepared this hint for you

    Summing up the lesson, discussing the work done.

    Awarding and congratulations to the winners of the competition.

    1. Literature.
    2. Tarasov L. This amazing symmetrical world.
    3. M., 1982

    Sharygin I.F., Erganzhieva L.N. Visual geometry. M., 1995 Internet resources.(means “proportionality”) - the property of geometric objects to be combined with themselves under certain transformations. By “symmetry” we mean any regularity in

    internal structure bodies or figures.

    Central symmetry— symmetry about a point.

    relative to the point O, if for each point of a figure a point symmetrical to it relative to point O also belongs to this figure. Point O is called the center of symmetry of the figure. IN

    one-dimensional space (on a straight line) central symmetry is mirror symmetry. On a plane (in

    2-dimensional space) symmetry with center A is a rotation of 180 degrees with center A. Central symmetry on a plane, like rotation, preserves orientation. Central symmetry in

    relative to the point three-dimensional space is also called spherical symmetry. It can be represented as a composition of reflection relative to a plane passing through the center of symmetry, with a rotation of 180° relative to a straight line passing through the center of symmetry and perpendicular to the above-mentioned plane of reflection.

    4-dimensional space, central symmetry can be represented as a composition of two 180° rotations around two mutually perpendicular planes passing through the center of symmetry.

    The figure is called symmetrical relatively straight a, if for each point of a figure a point symmetrical to it relative to the line a also belongs to this figure. Straight line a is called the axis of symmetry of the figure.

    4-dimensional has two definitions:

    - Reflective symmetry.

    In mathematics, axial symmetry is a type of motion (mirror reflection) in which the set of fixed points is a straight line, called the axis of symmetry. For example, a flat rectangle is asymmetrical in space and has 3 axes of symmetry, if it is not a square.

    - Rotational symmetry.

    In natural sciences, axial symmetry is understood as rotational symmetry, relative to rotations around a straight line. In this case, bodies are called axisymmetric if they transform into themselves at any rotation around this straight line. In this case, the rectangle will not be an axisymmetric body, but the cone will be.

    Images on a plane of many objects in the world around us have an axis of symmetry or a center of symmetry. Many tree leaves and flower petals are symmetrical about the average stem.

    We often encounter symmetry in art, architecture, technology, and everyday life. The facades of many buildings have axial symmetry. In most cases, patterns on carpets, fabrics, and indoor wallpaper are symmetrical about the axis or center. Many parts of mechanisms, such as gears, are symmetrical.

    « Symmetry" - a word of Greek origin. It means proportionality, the presence of a certain order, patterns in the arrangement of parts.

    Since ancient times, people have used symmetry in drawings, ornaments, and household items.
    Symmetry is widespread in nature. It can be observed in the form of leaves and flowers of plants, in the arrangement of various organs of animals, in the form of crystalline bodies, in a fluttering butterfly, a mysterious snowflake, a mosaic in a temple, a starfish.
    Symmetry is widely used in practice, in construction and technology. This is strict symmetry in the form of ancient buildings, harmonious ancient Greek vases, the Kremlin building, cars, airplanes and much more. (slide 4) Examples of using symmetry are parquet and borders. (see hyperlink on the use of symmetry in borders and parquet floors) Let's look at several examples where you can see symmetry in various objects using a slide show (include icon).

    Definition: – is symmetry about a point.
    Definition: Points A and B are symmetrical about some point O if point O is the midpoint of segment AB.
    Definition: Point O is called the center of symmetry of the figure, and the figure is called centrally symmetrical.
    Property: Figures that are symmetrical about a certain point are equal.
    Examples:

    Algorithm for constructing a centrally symmetrical figure
    1. Let’s construct a triangle A 1B 1 C 1, symmetrical to the triangle ABC, relative to the center (point) O. To do this, connect points A, B, C with center O and continue these segments;
    2. Measure the segments AO, BO, CO and lay off on the other side of point O, segments equal to them (AO=A 1 O 1, BO=B 1 O 1, CO=C 1 O 1);

    3. Connect the resulting points with segments A 1 B 1; A 1 C 1; B1 C 1.
    We got ∆A 1 B 1 C 1 symmetrical ∆ABC.


    – this is symmetry about the drawn axis (straight line).
    Definition: Points A and B are symmetrical about a certain line a if these points lie on a line perpendicular to this one and at the same distance.
    Definition: An axis of symmetry is a straight line when bent along which the “halves” coincide, and a figure is called symmetrical about a certain axis.
    Property: Two symmetrical figures are equal.
    Examples:

    Algorithm for constructing a figure symmetrical with respect to some straight line
    Let's construct a triangle A1B1C1, symmetrical to triangle ABC with respect to straight line a.
    For this:
    1. Let us draw straight lines from the vertices of triangle ABC perpendicular to straight line a and continue them further.
    2. Measure the distances from the vertices of the triangle to the resulting points on the straight line and plot the same distances on the other side of the straight line.
    3. Connect the resulting points with segments A1B1, B1C1, B1C1.

    We obtained ∆A1B1C1 symmetrical ∆ABC.

    The concept of “central symmetry” of a figure presupposes the existence of a certain point - the center of symmetry. On both sides of it there are points belonging to each of them. Each of them has a symmetrical point to itself.

    It should be said that the concept of a center is absent in Euclidean geometry. Moreover, in the eleventh book, in the thirty-eighth sentence, there is a definition of the spatial symmetrical axis. The concept of a center first appeared in the 16th century.

    Central symmetry is present in such well-known figures as a parallelogram and a circle. Both the first and second figures have the same center. The center of symmetry of a parallelogram is located at the point of intersection of lines emerging from opposite points; in a circle is the center of itself. A straight line is characterized by the presence of an infinite number of such sections. Each of its points can be a center of symmetry. A right parallelepiped has nine planes. Of all the symmetrical planes, three are perpendicular to the ribs. The other six pass through the diagonals of the faces. However, there is a figure that does not have it. It is an arbitrary triangle.

    In some sources, the concept of “central symmetry” is defined as follows: a geometric body (figure) is considered symmetrical with respect to the center C if each point A of the body has a point E lying within the same figure, such that the segment AE, passing through center C, cut in half in it. For corresponding pairs of points there are equal segments.

    The corresponding angles of the two halves of a figure in which central symmetry is present are also equal. Two figures lying on either side of the central point can in this case be superimposed on each other. However, it must be said that the imposition is carried out in a special way. Unlike mirror symmetry, central symmetry involves rotating one part of the figure one hundred and eighty degrees around the center. Thus, one part will be in a mirror position relative to the second. Two parts of the figure can thus be superimposed on each other without being removed from the common plane.

    In algebra, odd and even functions are studied using graphs. For the graph is constructed symmetrically with respect to the coordinate axis. For an odd function, it is relative to the point of origin, that is, O. Thus, an odd function is characterized by central symmetry, and an even function is characterized by axial symmetry.

    Central symmetry suggests that a flat figure has a second order. In this case, the axis will lie perpendicular to the plane.

    The central one is quite common. Among the variety of forms, the most perfect examples can be found in abundance. Such examples that attract the eye include different kinds plants, mollusks, insects, many animals. A person admires the beauty of individual flowers and petals, he is surprised by the ideal structure of the honeycomb, the arrangement of seeds on the sunflower cap, and leaves on the plant stem. Central symmetry is found everywhere in life.