« 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 the points A, B, C with the 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.
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, mark where elements of symmetry are present.
Consider axial and central symmetries as properties of some geometric figures; Consider axial and central symmetries as properties of some geometric figures; Be able to construct symmetrical points and be able to recognize figures that are symmetrical with respect to a point or line; Be able to construct symmetrical points and be able to recognize figures that are symmetrical with respect to a point or line; Improving problem solving skills; Improving problem solving skills; Continue to work on accurately recording and completing geometric drawings; Continue to work on accurately recording and completing geometric drawings;
Oral work “Gentle questioning” Oral work “Gentle questioning” What point is called the middle of the segment? Which triangle is called isosceles? What properties do the diagonals of a rhombus have? State the bisector property of an isosceles triangle. Which lines are called perpendicular? Which triangle is called equilateral? What properties do the diagonals of a square have? What figures are called equal?
What new concepts did you learn about in class? What new concepts did you learn about in class? What new things have you learned about geometric shapes? What new things have you learned about geometric shapes? Give examples of geometric shapes that have axial symmetry. Give examples of geometric shapes that have axial symmetry. Give an example of figures that have central symmetry. Give an example of figures that have central symmetry. Give examples of objects from the surrounding life that have one or two types of symmetry. Give examples of objects from the surrounding life that have one or two types of symmetry.
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, absolute symmetry cannot be found in the organic (living) world! The halves of the sheet copy each other far from perfectly, the same applies to the 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 works of their labor (things) - clothes, cars... In nature, it is characteristic of inorganic formations, for example, crystals.
But let's move on to practice. It’s not worth starting with complex objects like people and animals; let’s try, as the first exercise in a new field, to finish drawing the mirror half of the sheet.
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 to see if 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.
It also has an interesting shape - heart-shaped and with ears at the base, you'll have to puff:
Here's 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.
Axial symmetry and the concept of perfection
Axial symmetry is inherent in all forms in nature and is one of the fundamental principles of beauty. Since ancient times, man has tried
to comprehend the meaning of perfection. This concept was first substantiated by artists, philosophers and mathematicians of Ancient Greece. And the word “symmetry” itself was invented by them. It denotes proportionality, harmony and identity of the parts of the whole. The ancient Greek thinker Plato argued that only an object that is symmetrical and proportionate can be beautiful. Indeed, those phenomena and forms that are proportional and complete “please the eye.” We call them correct.
Axial symmetry as a concept
Symmetry in the world of living beings is manifested in the regular arrangement of identical parts of the body relative to the center or axis. More often in
Axial symmetry occurs in nature. It determines not only the general structure of the organism, but also the possibilities of its subsequent development. The geometric shapes and proportions of living beings are formed by “axial symmetry”. Its definition is formulated as follows: this is the property of objects to be combined under various transformations. The ancients believed that the sphere possesses the principle of symmetry to the fullest extent. They considered this form harmonious and perfect.
Axial symmetry in living nature
If you look at any living creature, the symmetry of the structure of the body immediately catches your eye. Human: two arms, two legs, two eyes, two ears and so on. Each animal species has a characteristic color. If a pattern appears in the coloring, then, as a rule, it is mirrored on both sides. This means that there is a certain line along which animals and people can be visually divided into two identical halves, that is, their geometric structure is based on axial symmetry. Nature creates any living organism not chaotically and senselessly, but according to the general laws of the world order, because nothing in the Universe has a purely aesthetic, decorative purpose. The presence of various forms is also due to natural necessity.
Axial symmetry in inanimate nature
In the world, we are surrounded everywhere by such phenomena and objects as: typhoon, rainbow, drop, leaves, flowers, etc. Their mirror, radial, central, axial symmetry is obvious. It is largely due to the phenomenon of gravity. Often the concept of symmetry refers to the regularity of changes in certain phenomena: day and night, winter, spring, summer and autumn, and so on. In practice, this property exists wherever order is observed. And the laws of nature themselves - biological, chemical, genetic, astronomical - are subject to the principles of symmetry common to us all, since they have an enviable systematicity. Thus, balance and identity as a principle have a universal scope. Axial symmetry in nature is one of the “cornerstone” laws on which the universe as a whole is based.