Axes of symmetry of a rectangle drawing. What is an axis of symmetry. Use of the term in other scientific fields

TRIANGLES.

§ 17. SYMMETRY RELATIVELY TO THE RIGHT STRAIGHT.

1. Figures that are symmetrical to each other.

Let's draw some figure on a sheet of paper with ink, and with a pencil outside it - an arbitrary straight line. Then, without allowing the ink to dry, we bend the sheet of paper along this straight line so that one part of the sheet overlaps the other. This other part of the sheet will thus produce an imprint of this figure.

If you then straighten the sheet of paper again, then there will be two figures on it, which are called symmetrical relative to a given line (Fig. 128).

Two figures are called symmetrical with respect to a certain straight line if, when bending the drawing plane along this straight line, they are aligned.

The straight line with respect to which these figures are symmetrical is called their axis of symmetry.

From the definition of symmetrical figures it follows that all symmetrical figures are equal.

You can obtain symmetrical figures without using bending of the plane, but with the help of geometric construction. Let it be necessary to construct a point C" symmetrical to a given point C relative to straight line AB. Let us drop a perpendicular from point C
CD to straight line AB and as its continuation we will lay down the segment DC" = DC. If we bend the drawing plane along AB, then point C will align with point C": points C and C" are symmetrical (Fig. 129).

Suppose now we need to construct a segment C "D", symmetrical to a given segment CD relative to the straight line AB. Let's construct points C" and D", symmetrical to points C and D. If we bend the drawing plane along AB, then points C and D will coincide, respectively, with points C" and D" (Drawing 130). Therefore, segments CD and C "D" will coincide , they will be symmetrical.

Let us now construct a figure symmetrical to the given polygon ABCDE relative to the given axis of symmetry MN (Fig. 131).

To solve this problem, let’s drop the perpendiculars A A, IN b, WITH With, D d and E e to the axis of symmetry MN. Then, on the extensions of these perpendiculars, we plot the segments
A A" = A A, b B" = B b, With C" = Cs; d D"" =D d And e E" = E e.

The polygon A"B"C"D"E" will be symmetrical to the polygon ABCDE. Indeed, if you bend the drawing along a straight line MN, then the corresponding vertices of both polygons will align, and therefore the polygons themselves will align; this proves that the polygons ABCDE and A" B"C"D"E" are symmetrical about the straight line MN.

2. Figures consisting of symmetrical parts.

Often there are geometric figures that are divided by some straight line into two symmetrical parts. Such figures are called symmetrical.

So, for example, an angle is a symmetrical figure, and the bisector of the angle is its axis of symmetry, since when bent along it, one part of the angle is combined with the other (Fig. 132).

In a circle, the axis of symmetry is its diameter, since when bending along it, one semicircle is combined with another (Fig. 133). The figures in drawings 134, a, b are exactly symmetrical.

Symmetrical figures are often found in nature, construction, and jewelry. The images placed on drawings 135 and 136 are symmetrical.

It should be noted that symmetrical figures can be combined simply by moving along a plane only in some cases. To combine symmetrical figures, as a rule, it is necessary to turn one of them with the opposite side,

People's lives are filled with symmetry. It’s convenient, beautiful, and there’s no need to invent new standards. But what is it really and is it as beautiful in nature as is commonly believed?

Symmetry

Since ancient times, people have sought to organize the world around them. Therefore, some things are considered beautiful, and some are not so much. From an aesthetic point of view, the golden and silver ratios are considered attractive, as well as, of course, symmetry. This term is of Greek origin and literally means “proportionality.” Of course, we are talking not only about coincidence on this basis, but also on some others. In a general sense, symmetry is a property of an object when, as a result of certain formations, the result is equal to the original data. It is found in both living and inanimate nature, as well as in objects made by man.

First of all, the term "symmetry" is used in geometry, but finds application in many scientific fields, and its meaning remains generally unchanged. This phenomenon occurs quite often and is considered interesting, since several of its types, as well as elements, differ. The use of symmetry is also interesting, because it is found not only in nature, but also in patterns on fabric, borders of buildings and many other man-made objects. It is worth considering this phenomenon in more detail, because it is extremely fascinating.

Use of the term in other scientific fields

In the following, symmetry will be considered from the point of view of geometry, but it is worth mentioning that this word is used not only here. Biology, virology, chemistry, physics, crystallography - all this is an incomplete list of areas in which this phenomenon is studied from different angles and under different conditions. For example, the classification depends on what science this term refers to. Thus, the division into types varies greatly, although some basic ones, perhaps, remain unchanged throughout.

Classification

There are several main types of symmetry, of which three are the most common:

In addition, the following types are also distinguished in geometry; they are much less common, but no less interesting:

• sliding;
• rotational;
• point;
• progressive;
• screw;
• fractal;
• etc.

In biology, all species are called slightly differently, although in essence they may be the same. Division into certain groups occurs on the basis of the presence or absence, as well as the quantity of certain elements, such as centers, planes and axes of symmetry. They should be considered separately and in more detail.

Basic elements

The phenomenon has certain features, one of which is necessarily present. The so-called basic elements include planes, centers and axes of symmetry. It is in accordance with their presence, absence and quantity that the type is determined.

The center of symmetry is the point inside a figure or crystal at which the lines connecting in pairs all sides parallel to each other converge. Of course, it does not always exist. If there are sides to which there is no parallel pair, then such a point cannot be found, since it does not exist. According to the definition, it is obvious that the center of symmetry is that through which a figure can be reflected onto itself. An example would be, for example, a circle and a point in its middle. This element is usually designated as C.

The plane of symmetry, of course, is imaginary, but it is precisely it that divides the figure into two parts equal to each other. It can pass through one or more sides, be parallel to it, or divide them. For the same figure, several planes can exist at once. These elements are usually designated as P.

But perhaps the most common is what is called “axis of symmetry”. This is a common phenomenon that can be seen both in geometry and in nature. And it is worthy of separate consideration.

Axles

Often the element in relation to which a figure can be called symmetrical is

a straight line or segment appears. In any case, we are not talking about a point or a plane. Then the figures are considered. There can be a lot of them, and they can be located in any way: dividing the sides or being parallel to them, as well as intersecting corners or not doing so. Axes of symmetry are usually designated as L.

Examples include isosceles and In the first case, there will be a vertical axis of symmetry, on both sides of which there are equal faces, and in the second, the lines will intersect each angle and coincide with all bisectors, medians and altitudes. Ordinary triangles do not have this.

By the way, the totality of all the above elements in crystallography and stereometry is called the degree of symmetry. This indicator depends on the number of axes, planes and centers.

Examples in geometry

Conventionally, we can divide the entire set of objects of study by mathematicians into figures that have an axis of symmetry and those that do not. All circles, ovals, as well as some special cases automatically fall into the first category, while the rest fall into the second group.

As in the case when we talked about the axis of symmetry of a triangle, this element does not always exist for a quadrilateral. For a square, rectangle, rhombus or parallelogram it is, but for an irregular figure, accordingly, it is not. For a circle, the axis of symmetry is the set of straight lines that pass through its center.

In addition, it is interesting to consider three-dimensional figures from this point of view. In addition to all regular polygons and the ball, some cones, as well as pyramids, parallelograms and some others, will have at least one axis of symmetry. Each case must be considered separately.

Examples in nature

In life it is called bilateral, it occurs most
often. Any person and many animals are an example of this. The axial one is called radial and is found much less frequently, as a rule, in the plant world. And yet they exist. For example, it is worth thinking about how many axes of symmetry a star has, and does it have any at all? Of course, we are talking about marine life, and not about the subject of study by astronomers. And the correct answer would be: it depends on the number of rays of the star, for example five, if it is five-pointed.

In addition, radial symmetry is observed in many flowers: daisies, cornflowers, sunflowers, etc. There are a huge number of examples, they are literally everywhere around.

Arrhythmia

This term, first of all, reminds most of medicine and cardiology, but it initially has a slightly different meaning. In this case, the synonym will be “asymmetry”, that is, the absence or violation of regularity in one form or another. It can be found as an accident, and sometimes it can become a wonderful technique, for example in clothing or architecture. After all, there are a lot of symmetrical buildings, but the famous one is slightly tilted, and although it is not the only one, it is the most famous example. It is known that this happened by accident, but this has its own charm.

In addition, it is obvious that the faces and bodies of people and animals are not completely symmetrical either. There have even been studies that show that “correct” faces are judged to be lifeless or simply unattractive. Still, the perception of symmetry and this phenomenon in itself are amazing and have not yet been fully studied, and therefore are extremely interesting.

If all the angles in a quadrilateral are right angles, then it is called a rectangle.

Figure 125 shows rectangle ABCD.

Sides AB and BC have a common vertex B. They are called neighboring sides of rectangle ABCD. Also adjacent are, for example, sides BC and CD.

The adjacent sides of a rectangle are called length And width.

Sides AB and CD do not have common vertices. They are called opposite sides of rectangle ABCD. Also opposite are sides BC and AD.

The opposite sides of a rectangle are equal.

In Figure 125, AB = CD, BC = AD. If the length of a rectangle is a and its width is b, then its perimeter is calculated using the formula already familiar to you:

P = 2 a + 2 b

A rectangle with all sides equal is called square(Fig. 126).

Let us draw a straight line l passing through the midpoints of two opposite sides of the rectangle (Fig. 127). If a sheet of paper is folded along a straight line l, then the two parts of the rectangle lying on opposite sides of the straight line l will coincide.

The figures shown in Figure 128 have a similar property. Such figures are called symmetrical about a straight line . The straight line l is called axis of symmetry of the figure .

So, a rectangle is a figure that has an axis of symmetry. Also, the axis of symmetry has an isosceles triangle (Fig. 129).

A figure can have more than one axis of symmetry. For example, a rectangle other than a square has two axes of symmetry (Fig. 130), and a square has four axes of symmetry (Fig. 131). An equilateral triangle has three axes of symmetry (Fig. 132).

While studying the world around us, we often encounter symmetry. Examples of symmetry in nature are shown in Figure 133.

Objects that have an axis of symmetry are easy to perceive and pleasing to the eye. It is not without reason that in Ancient Greece the word “symmetry” served as a synonym for the words “harmony” and “beauty”.

The idea of ​​symmetry is widely used in fine arts and architecture (Fig. 134).

Goals:

• educational:
  • give an idea of ​​symmetry;
  • introduce the main types of symmetry on the plane and in space;
  • develop strong skills in constructing symmetrical figures;
  • expand your understanding of famous figures by introducing properties associated with symmetry;
  • show the possibilities of using symmetry in solving various problems;
  • consolidate acquired knowledge;
• general education:
  • teach yourself how to prepare yourself for work;
  • teach how to control yourself and your desk neighbor;
  • teach to evaluate yourself and your desk neighbor;
• developing:
  • intensify independent activity;
  • develop cognitive activity;
  • learn to summarize and systematize the information received;
• educational:
  • develop a “shoulder sense” in students;
  • cultivate communication skills;
  • instill a culture of communication.

DURING THE CLASSES

In front of each person are scissors and a sheet of paper.

Exercise 1(3 min).

- Let's take a sheet of paper, fold it into pieces and cut out some figure. Now let's unfold the sheet and look at the fold line.

Question: What function does this line serve?

Suggested answer: This line divides the figure in half.

Question: How are all the points of the figure located on the two resulting halves?

Suggested answer: All points of the halves are at an equal distance from the fold line and at the same level.

– This means that the fold line divides the figure in half so that 1 half is a copy of 2 halves, i.e. this line is not simple, it has a remarkable property (all points relative to it are at the same distance), this line is an axis of symmetry.

Task 2 (2 minutes).

– Cut out a snowflake, find the axis of symmetry, characterize it.

Task 3 (5 minutes).

– Draw a circle in your notebook.

Question: Determine how the axis of symmetry goes?

Suggested answer: Differently.

Question: So how many axes of symmetry does a circle have?

Suggested answer: A lot of.

– That’s right, a circle has many axes of symmetry. An equally remarkable figure is a ball (spatial figure)

Question: What other figures have more than one axis of symmetry?

Suggested answer: Square, rectangle, isosceles and equilateral triangles.

– Consider three-dimensional figures: cube, pyramid, cone, cylinder, etc. These figures also have an axis of symmetry. Determine how many axes of symmetry do the square, rectangle, equilateral triangle and the proposed three-dimensional figures have?

I distribute halves of plasticine figures to students.

Task 4 (3 min).

– Using the information received, complete the missing part of the figure.

Note: the figure can be both planar and three-dimensional. It is important that students determine how the axis of symmetry runs and complete the missing element. The correctness of the work is determined by the neighbor at the desk and evaluates how correctly the work was done.

A line (closed, open, with self-intersection, without self-intersection) is laid out from a lace of the same color on the desktop.

Task 5 (group work 5 min).

– Visually determine the axis of symmetry and, relative to it, complete the second part from a lace of a different color.

The correctness of the work performed is determined by the students themselves.

Elements of drawings are presented to students

Task 6 (2 minutes).

– Find the symmetrical parts of these drawings.

To consolidate the material covered, I suggest the following tasks, scheduled for 15 minutes:

Name all equal elements of the triangle KOR and KOM. What type of triangles are these?

2. Draw several isosceles triangles in your notebook with a common base of 6 cm.

3. Draw a segment AB. Construct a line segment AB perpendicular and passing through its midpoint. Mark points C and D on it so that the quadrilateral ACBD is symmetrical with respect to the straight line AB.

– Our initial ideas about form date back to the very distant era of the ancient Stone Age - the Paleolithic. For hundreds of thousands of years of this period, people lived in caves, in conditions little different from the life of animals. People made tools for hunting and fishing, developed a language to communicate with each other, and during the late Paleolithic era they embellished their existence by creating works of art, figurines and drawings that reveal a remarkable sense of form.
When there was a transition from simple gathering of food to its active production, from hunting and fishing to agriculture, humanity entered a new Stone Age, the Neolithic.
Neolithic man had a keen sense of geometric form. Firing and painting clay vessels, making reed mats, baskets, fabrics, and later metal processing developed ideas about planar and spatial figures. Neolithic ornaments were pleasing to the eye, revealing equality and symmetry.
– Where does symmetry occur in nature?

Suggested answer: wings of butterflies, beetles, tree leaves...

– Symmetry can also be observed in architecture. When constructing buildings, builders strictly adhere to symmetry.

That's why the buildings turn out so beautiful. Also an example of symmetry is humans and animals.

Homework:

1. Come up with your own ornament, draw it on an A4 sheet (you can draw it in the form of a carpet).
2. Draw butterflies, note where elements of symmetry are present.

What is an axis of symmetry? This is a set of points that form a straight line, which is the basis of symmetry, that is, if a certain distance is set aside from a straight line on one side, then it will be reflected in the other direction in the same size. The axis can be anything - a point, a straight line, a plane, and so on. But it’s better to talk about this with clear examples.

Symmetry

In order to understand what an axis of symmetry is, you need to delve into the very definition of symmetry. This is the correspondence of a certain fragment of the body relative to any axis, when its structure is unchanged, and the properties and shape of such an object remain the same relative to its transformations. We can say that symmetry is the property of bodies to display. When a fragment cannot have such a correspondence, this is called asymmetry or arrhythmia.

Some figures do not have symmetry, which is why they are called irregular or asymmetrical. These include various trapezoids (except isosceles), triangles (except isosceles and equilateral) and others.

Types of symmetry

We will also discuss some types of symmetry in order to fully explore this concept. They are divided like this:

1. Axial. The axis of symmetry is a straight line passing through the center of the body. Like this? If you superimpose the parts around the axis of symmetry, they will be equal. This can be seen in the example of a sphere.
2. Mirror. The axis of symmetry here is a straight line, relative to which the body can be reflected and the inverse image obtained. For example, the wings of a butterfly are mirror symmetrical.
3. Central. The axis of symmetry is the point in the center of the body, relative to which, for all transformations, the parts of the body are equal when superimposed.

History of symmetry

The very concept of symmetry is often the starting point in the theories and hypotheses of scientists of ancient times, who were confident in the mathematical harmony of the universe, as well as in the manifestation of the divine principle. The ancient Greeks firmly believed that the Universe was symmetrical, because symmetry is magnificent. Man has long used the idea of ​​symmetry in his knowledge of the picture of the universe.

In the 5th century BC, Pythagoras considered the sphere to be the most perfect form and thought that the Earth was shaped like a sphere and moved in the same way. He also believed that the Earth moved in the form of some kind of “central fire”, around which 6 planets (known at that time), the Moon, the Sun and all other stars were supposed to revolve.

And the philosopher Plato considered polyhedra to be the personification of the four natural elements:

• tetrahedron is fire, since its apex is directed upward;
• cube - earth, since it is the most stable body;
• octahedron - air, there is no explanation;
• icosahedron - water, since the body does not have rough geometric shapes, angles, and so on;
• The image of the entire Universe was the dodecahedron.

Because of all these theories, regular polyhedra are called Platonic solids.

The architects of Ancient Greece used symmetry. All their buildings were symmetrical, as evidenced by images of the ancient temple of Zeus at Olympia.

The Dutch artist M.C. Escher also used symmetry in his paintings. In particular, a mosaic of two birds flying towards them became the basis of the painting “Day and Night”.

Also, our art critics did not neglect the rules of symmetry, as can be seen in the example of Vasnetsov’s painting “Bogatyrs”.

What can we say, symmetry has been a key concept for all artists for many centuries, but in the 20th century its meaning was also appreciated by all workers in the exact sciences. Accurate evidence is provided by physical and cosmological theories, for example, the theory of relativity, string theory, and absolutely all quantum mechanics. From the times of Ancient Babylon and ending with the advanced discoveries of modern science, the ways of studying symmetry and the discovery of its basic laws are traced.

Symmetry of geometric shapes and bodies

Let's take a closer look at geometric bodies. For example, the axis of symmetry of a parabola is a straight line passing through its vertex and cutting the given body in half. This figure has one single axis.

But with geometric figures the situation is different. The axis of symmetry of a rectangle is also a straight line, but there are several of them. You can draw the axis parallel to the width segments, or you can draw it parallel to the length segments. But it's not that simple. Here the straight line has no axes of symmetry, since its end is not defined. Only central symmetry could exist, but, accordingly, there will not be such.

You should also know that some bodies have many axes of symmetry. This is not difficult to guess. There is no need to even talk about how many axes of symmetry a circle has. Any straight line passing through the center of a circle is such, and there are an infinite number of these straight lines.

Some quadrilaterals may have two axes of symmetry. But the second ones must be perpendicular. This happens in the case of a rhombus and a rectangle. In the first, the axes of symmetry are diagonals, and in the second, the middle lines. Only a square has many such axes.

Symmetry in nature

Nature amazes with many examples of symmetry. Even our human body is symmetrical. Two eyes, two ears, a nose and a mouth are located symmetrically relative to the central axis of the face. The arms, legs and the whole body in general are arranged symmetrically to an axis passing through the middle of our body.

And how many examples surround us all the time! These are flowers, leaves, petals, vegetables and fruits, animals and even honeycombs of bees that have a pronounced geometric shape and symmetry. All of nature is arranged in an orderly manner, everything has its place, which once again confirms the perfection of the laws of nature, in which symmetry is the main condition.

Conclusion

We are constantly surrounded by some phenomena and objects, for example, a rainbow, a drop, flowers, petals, and so on. Their symmetry is obvious, to some extent it is due to gravity. Often in nature, the concept of “symmetry” is understood as the regular change of day and night, seasons, and so on.

Similar properties are observed wherever there is order and equality. Also, the laws of nature themselves - astronomical, chemical, biological and even genetic - are subject to certain principles of symmetry, since they are perfectly systematic, which means that balance has an all-encompassing scale. Consequently, axial symmetry is one of the fundamental laws of the universe as a whole.