Scientific and practical conference
Municipal educational institution "Secondary" comprehensive school No. 23"
city of Vologda
section: natural science
design and research work
TYPES OF SYMMETRY
The work was completed by an 8th grade student
Kreneva Margarita
Head: higher mathematics teacher
year 2014
Project structure:
1. Introduction.
2. Goals and objectives of the project.
3. Types of symmetry:
3.1. Central symmetry;
3.2. Axial symmetry;
3.3. Mirror symmetry(symmetry relative to the plane);
3.4. Rotational symmetry;
3.5. Portable symmetry.
4. Conclusions.
Symmetry is the idea through which man has tried for centuries to comprehend and create order, beauty and perfection.
G. Weil
Introduction.
The topic of my work was chosen after studying the section “Axial and central symmetry” in the course “8th grade Geometry”. I was very interested in this topic. I wanted to know: what types of symmetry exist, how they differ from each other, what are the principles of construction symmetrical figures in each type.
Goal of the work : Introduction to different types of symmetry.
Tasks:
Study the literature on this issue.
Summarize and systematize the studied material.
Prepare a presentation.
In ancient times, the word “SYMMETRY” was used to mean “harmony”, “beauty”. Translated from Greek, this word means “proportionality, proportionality, sameness in the arrangement of parts of something on opposite sides of a point, straight line or plane.
There are 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 symmetry physical phenomena and the laws of nature. This symmetry lies at the very core natural science picture world: it can be called physical symmetry.
I'll stop studyinggeometric symmetry .
In turn, there are also several types of geometric symmetry: central, axial, mirror (symmetry relative to the plane), radial (or rotary), portable and others. Today I will look at 5 types of symmetry.
Central symmetry
Two points A and A 1 are called symmetrical with respect to point O if they lie on a straight line passing through point O and are located along different sides at the same distance from it. Point O is called the center of symmetry.
The figure is said to be symmetrical about the pointABOUT , if for each point of the figure there is a point symmetrical to it relative to the pointABOUT also belongs to this figure. DotABOUT called the center of symmetry of a figure, the figure is said to have central symmetry.
Examples of figures with central symmetry are a circle and a parallelogram.
The figures shown on the slide are symmetrical relative to a certain point
2. Axial symmetry
Two pointsX And Y are called symmetrical about a straight linet , if this line passes through the middle of the segment XY and is perpendicular to it. It should also be said that each point is a straight linet is considered symmetrical to itself.
Straightt – axis of symmetry.
The figure is said to be symmetrical about a straight linet, if for each point of the figure there is a point symmetrical to it relative to the straight linet also belongs to this figure.
Straighttcalled the axis of symmetry of a figure, the figure is said to have axial symmetry.
An undeveloped angle, an isosceles angle, and an angle have axial symmetry. equilateral triangles, rectangle and rhombus,letters (see presentation).
Mirror symmetry (symmetry about a plane)
Two points P 1 And P are said to be symmetrical with respect to the plane, and if they lie on a straight line, perpendicular to the plane a, and are at the same distance from it
Mirror symmetry well known to every person. It connects any object and its reflection in flat mirror. They say that one figure is mirror symmetrical to another.
On a plane, a figure with countless axes of symmetry was a circle. In space, a ball has countless planes of symmetry.
But if a circle is one of a kind, then in the three-dimensional world there is whole line bodies with an infinite number of planes of symmetry: a straight cylinder with a circle at the base, a cone with a circular base, a ball.
It is easy to establish that every symmetrical plane figure can be aligned with itself using a mirror. It is surprising that such complex figures, like a five-pointed star or an equilateral pentagon, are also symmetrical. As this follows from the number of axes, they are distinguished by high symmetry. And vice versa: it is not so easy to understand why such a seemingly correct figure, like an oblique parallelogram, is asymmetrical.
4. P rotational symmetry (or radial symmetry)
Rotational symmetry - this is symmetry, the preservation of the shape of an objectwhen rotating around a certain axis through an angle equal to 360°/n(or a multiple of this value), wheren = 2, 3, 4, … Specified axis called the rotary axisn-th order.
Atn=2 all points of the figure are rotated through an angle of 180 0 ( 360 0 /2 = 180 0 ) around the axis, while the shape of the figure is preserved, i.e. each point of the figure goes to a point of the same figure (the figure transforms into itself). The axis is called the second-order axis.
Figure 2 shows a third-order axis, Figure 3 - 4th order, Figure 4 - 5th order.
An object can have more than one rotation axis: Fig. 1 - 3 axes of rotation, Fig. 2 - 4 axes, Fig. 3 - 5 axes, Fig. 4 – only 1 axis
The well-known letters “I” and “F” have rotational symmetry. If you rotate the letter “I” 180° around an axis perpendicular to the plane of the letter and passing through its center, the letter will align with itself. In other words, the letter “I” is symmetrical with respect to a rotation of 180°, 180°= 360°: 2,n=2, which means it has second-order symmetry.
Note that the letter “F” also has second-order rotational symmetry.
In addition, the letter has a center of symmetry, and the letter F has an axis of symmetry
Let's return to examples from life: a glass, a cone-shaped pound of ice cream, a piece of wire, a pipe.
If we take a closer look at these bodies, we will notice that all of them, one way or another, consist of a circle, through infinite set whose axes of symmetry pass through countless planes of symmetry. Most of these bodies (they are called bodies of rotation) also have, of course, a center of symmetry (the center of a circle), through which at least one rotational axis of symmetry passes.
For example, the axis of the ice cream cone is clearly visible. It runs from the middle of the circle (sticking out of the ice cream!) to the sharp end of the funnel cone. We perceive the totality of symmetry elements of a body as a kind of symmetry measure. The ball, without a doubt, in terms of symmetry, is an unsurpassed embodiment of perfection, an ideal. The ancient Greeks perceived it as the most perfect body, and the circle, naturally, as the most perfect flat figure.
To describe the symmetry of a particular object, it is necessary to indicate all the rotation axes and their order, as well as all planes of symmetry.
Consider, for example, geometric body, composed of two identical regular quadrangular pyramids.
It has one rotary axis of the 4th order (axis AB), four rotary axes of the 2nd order (axes CE,DF, MP, NQ), five planes of symmetry (planesCDEF, AFBD, ACBE, AMBP, ANBQ).
5 . Portable symmetry
Another type of symmetry isportable With 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 occurs is called the transfer axis, and the distance “a” is called the elementary transfer, period or symmetry step.
A
A periodically repeating pattern on a long strip is called a border. In practice, borders are found in various forms (wall painting, cast iron, plaster bas-reliefs or ceramics). Borders are used by painters and artists when decorating a room. To make these ornaments, a stencil is made. We move the stencil, turning it over or not, tracing the outline, repeating the pattern, and we get an ornament (visual demonstration).
The border is easy to build using a stencil (the starting element), moving or turning it over and repeating the pattern. The figure shows five types of stencils:A ) asymmetrical;b, c ) having one axis of symmetry: horizontal or vertical;G ) centrally symmetrical;d ) having two axes of symmetry: vertical and horizontal.
To construct borders, the following transformations are used:
A
) parallel transfer;b
) symmetry about the vertical axis;V
) central symmetry;G
) symmetry about the horizontal axis.
You can build sockets in the same way. To do this, the circle is divided inton equal sectors, in one of them a sample pattern is made and then the latter is sequentially repeated in the remaining parts of the circle, rotating the pattern each time by an angle of 360°/n .
A clear example The fence shown in the photograph can serve as an application of axial and portable symmetry.
Conclusion: Thus, there are different kinds symmetry, symmetrical points in each of these types of symmetries are built according to certain laws. In life, we encounter one type of symmetry everywhere, and often in the objects that surround us, several types of symmetry can be noted at once. This creates order, beauty and perfection in the world around us.
LITERATURE:
Guide to elementary mathematics. M.Ya. Vygodsky. – Publishing house “Nauka”. – Moscow 1971 – 416 pages.
Modern dictionary foreign words. - M.: Russian language, 1993.
History of mathematics in schoolIX - Xclasses. G.I. Glaser. – Publishing house “Prosveshcheniye”. – Moscow 1983 – 351 pages.
Visual geometry 5th – 6th grades. I.F. Sharygin, L.N. Erganzhieva. – Publishing house “Drofa”, Moscow 2005. – 189 pages
Encyclopedia for children. Biology. S. Ismailova. – Avanta+ Publishing House. – Moscow 1997 – 704 pages.
Urmantsev Yu.A. Symmetry of nature and the nature of symmetry - M.: Mysl arxitekt / arhkomp2. htm, , ru.wikipedia.org/wiki/
In this lesson we will look at another characteristic of some figures - axial and central symmetry. We encounter axial symmetry every day when we look in the mirror. Central symmetry is very common in living nature. At the same time, figures that have symmetry have a number of properties. In addition, we subsequently learn that the axial and central symmetry are types of movements with the help of which a whole class of problems is solved.
This lesson is devoted to axial and central symmetry.
Definition
The two points are called symmetrical relatively straight if:
In Fig. 1 shows examples of points symmetrical with respect to a straight line and , and .
Rice. 1
Let us also note the fact that any point on a line is symmetrical to itself relative to this line.
Figures can also be symmetrical relative to a straight line.
Let us formulate a strict definition.
Definition
The figure is called symmetrical relative to straight, if for each point of the figure the point symmetrical to it relative to this straight line also belongs to the figure. In this case the line is called axis of symmetry. The figure has axial symmetry.
Let's look at a few examples of figures that have axial symmetry and their axes of symmetry.
Example 1
The angle has axial symmetry. The axis of symmetry of the angle is the bisector. Indeed: let’s lower a perpendicular to the bisector from any point of the angle and extend it until it intersects with the other side of the angle (see Fig. 2).
Rice. 2
(because - common side, 
(property of a bisector), and triangles are right-angled). Means, . Therefore, the points are symmetrical with respect to the bisector of the angle.
It follows from this that isosceles triangle has axial symmetry relative to the bisector (height, median) drawn to the base.
Example 2
An equilateral triangle has three axes of symmetry (bisectors/medians/altitudes of each of the three angles (see Fig. 3).
Rice. 3
Example 3
A rectangle has two axes of symmetry, each of which passes through the midpoints of its two opposite sides(see Fig. 4).
Rice. 4
Example 4
A rhombus also has two axes of symmetry: straight lines, which contain its diagonals (see Fig. 5).
Rice. 5
Example 5
A square, which is both a rhombus and a rectangle, has 4 axes of symmetry (see Fig. 6).
Rice. 6
Example 6
For a circle, the axis of symmetry is any straight line passing through its center (that is, containing the diameter of the circle). Therefore, a circle has infinitely many axes of symmetry (see Fig. 7).
Rice. 7
Let us now consider the concept central symmetry.
Definition
The points are called symmetrical relative to the point if: - the middle of the segment.
Let's look at a few examples: in Fig. 8 shows the points and , as well as and , which are symmetrical with respect to the point , and the points and are not symmetrical with respect to this point.
Rice. 8
Some figures are symmetrical about a certain point. Let us formulate a strict definition.
Definition
The figure is called symmetrical about the point, if for any point of the figure the point symmetrical to it also belongs to this figure. The point is called center of symmetry, and the figure has central symmetry.
Let's look at examples of figures with central symmetry.
Example 7
For a circle, the center of symmetry is the center of the circle (this is easy to prove by recalling the properties of the diameter and radius of a circle) (see Fig. 9).
Rice. 9
Example 8
For a parallelogram, the center of symmetry is the point of intersection of the diagonals (see Fig. 10).
Rice. 10
Let's solve several problems on axial and central symmetry.
Task 1.
How many axes of symmetry does the segment have?
A segment has two axes of symmetry. The first of them is a line containing a segment (since any point on a line is symmetrical to itself relative to this line). The second is the perpendicular bisector to the segment, that is, a straight line, perpendicular to the segment and passing through its middle.
Answer: 2 axes of symmetry.
Task 2.
How many axes of symmetry does a straight line have?
A straight line has infinitely many axes of symmetry. One of them is the line itself (since any point on the line is symmetrical to itself relative to this line). And also the axes of symmetry are any lines perpendicular to a given line.
Answer: there are infinitely many axes of symmetry.
Task 3.
How many axes of symmetry does the beam have?
The ray has one axis of symmetry, which coincides with the line containing the ray (since any point on the line is symmetrical to itself relative to this line).
Answer: one axis of symmetry.
Task 4.
Prove that the lines containing the diagonals of a rhombus are its axes of symmetry.
Proof:
Consider a rhombus. Let us prove, for example, that the straight line is its axis of symmetry. It is obvious that the points are symmetrical to themselves, since they lie on this line. In addition, the points and are symmetrical with respect to this line, since 
. Let's choose now arbitrary point and prove that the point symmetrical with respect to it also belongs to the rhombus (see Fig. 11).
Rice. eleven
Draw a perpendicular to the line through the point and extend it until it intersects with . Consider triangles and . These triangles are right-angled (by construction), in addition, they have: - a common leg, and 
(since the diagonals of a rhombus are its bisectors). So these triangles are equal: 
. This means that all their corresponding elements are equal, therefore: . From the equality of these segments it follows that the points and are symmetrical with respect to the straight line. This means that it is the axis of symmetry of the rhombus. This fact can be proven similarly for the second diagonal.
Proven.
Task 5.
Prove that the point of intersection of the diagonals of a parallelogram is its center of symmetry.
Proof:
Consider a parallelogram. Let us prove that the point is its center of symmetry. It is obvious that the points and , and are pairwise symmetrical with respect to the point , since the diagonals of a parallelogram are divided in half by the point of intersection. Let us now choose an arbitrary point and prove that the point symmetric with respect to it also belongs to the parallelogram (see Fig. 12).
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 has Greek origin and literally means “proportionality”. Of course we're talking about not only about coincidence on this basis, but also on some others. IN 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. This occurs both in living and in 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 what follows, symmetry will be considered from a geometric point of view, but it is worth mentioning that given word used not only here. Biology, virology, chemistry, physics, crystallography - all this is an incomplete list of areas in which this phenomenon studied with various sides and in 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, in geometry there are also following types, 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. 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 everything in pairs converge parallel friend to the other side. 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 vertical axis symmetry, on both sides of which equal faces, and in the second the lines will intersect each angle and coincide with all bisectors, medians and heights. 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 it was said about the axis of symmetry of a triangle, this element for a quadrilateral does not always exist. For a square, rectangle, rhombus or parallelogram it is, and for irregular figure, accordingly, no. 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 volumetric figures from this point of view. At least one axis of symmetry in addition to all regular polygons and the ball will have some cones, as well as pyramids, parallelograms and some others. 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. Axial is called radial and is much less common, usually in flora. 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 in this case a synonym would 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.
Today we will talk about a phenomenon that each of us constantly encounters in life: symmetry. What is symmetry?
We all roughly understand the meaning of this term. The dictionary says: symmetry is proportionality and complete correspondence of the arrangement of parts of something relative to a straight line or point. There are two types of symmetry: axial and radial. Let's look at the axial one first. This is, let’s say, “mirror” symmetry, when one half of an object is completely identical to the second, but repeats it as a reflection. Look at the halves of the sheet. They are mirror symmetrical. The halves of the human body are also symmetrical (front view) - identical arms and legs, identical eyes. But let’s not be mistaken; in fact, in the organic (living) world, absolute symmetry cannot be found! The halves of the sheet copy each other far from perfectly, the same applies to human body(take a closer look for yourself); The same is true for other organisms! By the way, it is worth adding that any symmetrical body is symmetrical relative to the viewer only in one position. It’s worth, say, turning a sheet of paper, or raising one hand, and what happens? – you see for yourself.
People achieve true symmetry in the products of their labor (things) - clothes, cars... In nature, it is characteristic inorganic formations, for example, crystals.
But let's move on to practice. You shouldn’t start with complex objects like people and animals; let’s try to finish drawing the mirror half of the sheet as the first exercise in a new field.
Drawing a symmetrical object - lesson 1
We make sure that it turns out as similar as possible. To do this, we will literally build our soul mate. Don’t think that it’s so easy, especially the first time, to draw a mirror-corresponding line with one stroke!
Let's mark several reference points for the future symmetrical line. We proceed like this: with a pencil, without pressing, we draw several perpendiculars to the axis of symmetry - the midrib of the leaf. Four or five is enough for now. And on these perpendiculars we measure to the right the same distance as on the left half to the line of the edge of the leaf. I advise you to use a ruler, don’t rely too much on your eye. As a rule, we tend to reduce the drawing - this has been observed from experience. We do not recommend measuring distances with your fingers: the error is too large.
Let's connect the resulting points with a pencil line:
Now let’s look meticulously at whether the halves are really the same. If everything is correct, we will circle it with a felt-tip pen and clarify our line:
The poplar leaf has been completed, now you can take a swing at the oak leaf.
Let's draw a symmetrical figure - lesson 2
In this case, the difficulty lies in the fact that the veins are marked and they are not perpendicular to the axis of symmetry and not only the dimensions but also the angle of inclination will have to be strictly observed. Well, let’s train our eye:
So a symmetrical oak leaf has been drawn, or rather, we built it according to all the rules:
How to draw a symmetrical object - lesson 3
And let’s consolidate the theme - we’ll finish drawing a symmetrical lilac leaf.
He has too interesting shape- heart-shaped and with ears at the base, you’ll have to puff:
This is what they drew:
Take a look at the resulting work from a distance and evaluate how accurately we were able to convey the required similarity. Here's a tip: look at your image in the mirror and it will tell you if there are any mistakes. Another way: bend the image exactly along the axis (we have already learned how to bend it correctly) and cut out the leaf along the original line. Look at the figure itself and at the cut paper.
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 ideas about famous figures, introducing properties associated with symmetry;
- show the possibilities of using symmetry when solving various tasks;
- 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 on 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 minutes).
– 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 next tasks provided for 15 minutes:
Name them all equal elements triangle KOR and COM. What type of triangles are these?
2. Draw several isosceles triangles in your notebook with common ground equal to 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, in 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.