Principal axes of symmetry. Math lesson

points M and M 1 are called symmetric with respect to a given line L if this line is the perpendicular bisector of the segment MM 1 (Figure 1). Each point of the line L symmetrical to itself. Plane transformation in which each point is mapped to a point symmetrical to it with respect to a given line L, is called axially symmetrical with the L axis and denoted S L :S L (M) = M 1 .

points M and M 1 are mutually symmetrical with respect to L, That's why S L (M 1 )=M. Therefore, the transformation inverse of axial symmetry is the same axial symmetry: S L -1=S L , S L° S L =E. In other words, the axial symmetry of a plane is involutive transformation.

The image of a given point with axial symmetry can be simply constructed using only one compass. Let L- axis of symmetry, A and B- arbitrary points of this axis (Fig. 2). If S L (M) = M 1 , then by the property of the points of the perpendicular bisector to the segment we have: AM=AM 1 and BM=BM one . So the point M 1 belongs to two circles: circles with center A radius AM and circles with center B radius BM (M- given point). Figure F and her image F 1 with axial symmetry are called symmetrical figures with respect to a straight line L(Figure 3).

Theorem. The axial symmetry of a plane is movement.

If A and V- any points of the plane and S L (A)=A 1 , S L (B)=B 1 , then we have to prove that A 1 B 1 = AB. To do this, we introduce a rectangular coordinate system OXY so that the axis OX coincides with the axis of symmetry. points A and V have coordinates A(x 1 ,-y 1 ) and B(x 1 ,-y 2 ) .Points A 1 and V 1 have coordinates A 1 (x 1 ,y 1 ) and B 1 (x 1 ,y 2 ) (Figure 4 - 8). Using the formula for the distance between two points, we find:

From these relations it is clear that AB=A 1 V 1 , which was to be proved.

From a comparison of the orientations of the triangle and its image, we obtain that the axial symmetry of the plane is movement of the second kind.

Axial symmetry maps each line to a line. In particular, each of the lines perpendicular to the axis of symmetry is mapped by this symmetry onto itself.

Theorem. A straight line other than a perpendicular to the axis of symmetry and its image under this symmetry intersect on the axis of symmetry or are parallel to it.

Proof. Let a straight line not perpendicular to the axis be given L symmetry. If m? L=P and S L (m)=m 1 , then m 1 ?m and S L (P)=P, That's why Pm1(Figure 9). If m || L, then m 1 || L, since otherwise the direct m and m 1 would intersect at a point on the line L, which contradicts the condition m||L(Figure 10).

By virtue of the definition of equal figures, straight lines, symmetrical about a straight line L, form with a straight line L equal angles (Figure 9).

Straight L called the axis of symmetry of the figure F, if with symmetry with the axis L figure F displayed on itself: S L (F)=F. They say that the figure F symmetrical about a straight line L.

For example, any straight line containing the center of a circle is the axis of symmetry of this circle. Indeed, let M- arbitrary point of the circle SCH centered O, OL, S L (M)=M one . Then S L (O)=O and OM 1 =OM, i.e. M 1 є u. So, the image of any point of a circle belongs to this circle. Hence, S L (u)=u.

The axes of symmetry of a pair of non-parallel lines are two perpendicular lines containing the bisectors of the angles between these lines. The axis of symmetry of a segment is the line containing it, as well as the perpendicular bisector to this segment.

Axial symmetry properties

1. With axial symmetry, the image of a straight line is a straight line, the image of parallel lines is parallel lines
3. Axial symmetry preserves the simple ratio of three points.
3. With axial symmetry, the segment passes into a segment, a ray into a ray, a half-plane into a half-plane.
4. With axial symmetry, the angle goes into an equal angle.
5. With axial symmetry with the d-axis, any straight line perpendicular to the d-axis remains in place.
6. With axial symmetry, the orthonormal frame goes over into the orthonormal frame. In this case, the point M with coordinates x and y relative to the frame R goes to the point M` with the same coordinates x and y, but relative to the frame R`.
7. The axial symmetry of the plane translates the right orthonormal frame into the left one and, conversely, the left orthonormal frame into the right one.
8. The composition of two axial symmetries of a plane with parallel axes is a parallel translation by a vector perpendicular to the given lines, the length of which is twice the distance between the given lines

You will need

- properties of symmetrical points;
- properties of symmetrical figures;
- ruler;
- square;
- compass;
- pencil;
- paper;
- a computer with a graphics editor.

Instruction

Draw a line a, which will be the axis of symmetry. If its coordinates are not given, draw it arbitrarily. On one side of this line, put an arbitrary point A. you need to find a symmetrical point.

Useful advice

Symmetry properties are constantly used in the AutoCAD program. For this, the Mirror option is used. To build an isosceles triangle or an isosceles trapezoid, it is enough to draw the lower base and the angle between it and the side. Mirror them with the specified command and extend the sides to the required size. In the case of a triangle, this will be the point of their intersection, and for a trapezoid - set value.

You constantly come across symmetry in graphic editors when you use the “flip vertically / horizontally” option. In this case, a straight line corresponding to one of the vertical or horizontal sides of the picture frame is taken as the axis of symmetry.

Sources:

how to draw central symmetry

The construction of a section of a cone is not so difficult task. The main thing is to follow a strict sequence of actions. Then this task will be easy to do and will not require much effort from you.

You will need

- paper;
- pen;
- circle;
- ruler.

Instruction

When answering this question, you first need to decide what parameters the section is set to.
Let this be the line of intersection of the plane l with the plane and the point O, which is the point of intersection with its section.

The construction is illustrated in Fig.1. The first step in constructing a section is through the center of the section of its diameter, extended to l perpendicular to this line. As a result, point L is obtained. Further, through point O, draw a straight line LW, and build two directing cones lying in the main section O2M and O2C. At the intersection of these guides lie the point Q, as well as the already shown point W. These are the first two points of the required section.

Now draw a perpendicular MC at the base of the cone BB1 and build the generators of the perpendicular section O2B and O2B1. In this section, draw a straight line RG through t.O, parallel to BB1. T.R and t.G - two more points of the desired section. If the cross section of the ball were known, then it could be constructed already at this stage. However, this is not an ellipse at all, but something elliptical, having symmetry with respect to the segment QW. Therefore, you should build as many points of the section as possible in order to connect them in the future with a smooth curve to get the most reliable sketch.

Construct an arbitrary section point. To do this, draw an arbitrary diameter AN at the base of the cone and build the corresponding guides O2A and O2N. Through PO draw a straight line passing through PQ and WG, until it intersects with the newly constructed guides at points P and E. These are two more points of the desired section. Continuing in the same way and further, you can arbitrarily desired points.

True, the procedure for obtaining them can be slightly simplified using symmetry with respect to QW. To do this, it is possible to draw straight lines SS' parallel to RG in the plane of the desired section, parallel to RG until they intersect with the surface of the cone. The construction is completed by rounding the constructed polyline from chords. It suffices to construct half of the required section due to the already mentioned symmetry with respect to QW.

Tip 3: How to plot trigonometric function

You need to draw schedule trigonometric functions? Master the algorithm of actions using the example of building a sinusoid. To solve the problem, use the research method.

You will need

- ruler;
- pencil;
- Knowledge of the basics of trigonometry.

Instruction

1. Basic concepts of symmetry and its types.

The concept of symmetry n R runs throughout the 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 as early as the 5th century BC. e. The word "symmetry" is Greek, it means "proportionality, proportionality, the sameness in the arrangement of parts." It is widely used by all areas of modern science without exception. Many great people 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 understandable to the eye? What is symmetry? This is an innate feeling, I answered myself. What is it based on?" The symmetry is really pleasing to the eye. Who has not admired the symmetry of nature's creations: leaves, flowers, birds, animals; or human creations: buildings, technology, - all that surrounds us from childhood, that strives for beauty and harmony. Hermann Weyl said: "Symmetry is the idea through which man has tried for centuries to comprehend and create order, beauty and perfection." Hermann Weyl is a German mathematician. Its activity falls on the first half of the twentieth century. It was he who formulated the definition of symmetry, established by what signs to see the presence or, conversely, the absence of symmetry in a particular case. Thus, a mathematically rigorous representation was formed relatively recently - at the beginning of the 20th century. It is quite complex. We will turn and once again recall the definitions that are 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 the line a if this line passes through the midpoint of the segment AA 1 and is perpendicular to it. Each point of the line a is considered symmetrical to itself.

Definition. The figure is said to be symmetrical with respect to a straight line. a, if for each point of the figure the point symmetrical to it with respect 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 build a symmetrical figure relative to a straight line from each point, we draw a perpendicular to this straight line and extend it by the same distance, mark the resulting point. We do this with each point, we get the symmetrical vertices of the new figure. Then we connect them in series and get a symmetrical figure of this 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 the point O if O is the midpoint of the segment AA 1. Point O is considered symmetrical to itself.

Definition. A figure is called symmetric with respect to the point O if for each point of the figure the point symmetric to it with respect to the point O also belongs to this figure.

3.2 Construction plan

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

To construct a point symmetrical to a point A relative to the point O, it suffices to draw a straight line OA(Fig. 46 ) and on the other side of the point O set aside a segment equal to a segment OA. In other words , points A and ; In and ; C and are symmetrical with respect to some point O. In fig. 46 built a triangle symmetrical to a triangle ABC relative to the point O. These triangles are equal.

Construction of symmetrical points about the center.

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

In general, figures that are symmetrical about some point are equal to .

3.3 Examples

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

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

The figures show an angle symmetrical about the vertex, a segment symmetrical to another segment about 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. Summary of the lesson

Let's summarize the knowledge gained. Today in the lesson we got acquainted with 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 with respect to some straight line.	All points of the figure must be symmetrical about the point chosen as the center of symmetry.
Properties	1. Symmetric points lie on perpendiculars to the line. 3. Straight lines turn into straight lines, angles into equal angles. 4. The sizes and shapes of the figures are saved.	1. Symmetrical points lie on a straight line passing through the center and the 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 saved.

II. Application of symmetry

Mathematics

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

The figures show various pictures depicted with the help of branches of parabolas.

(a) Octahedron,

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

Russian language

Printed letters Russian alphabet also have different types of symmetries.

There are "symmetrical" words in Russian - palindromes, which can be read the same way in both directions.

A D L M P T V- vertical axis

B E W K S E Yu - horizontal axis

W N O X- both vertical and horizontal

B G I Y R U C W Y Z- no axis

Radar hut Alla Anna

Literature

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

Look at the quadruplets of A.S. Pushkin's "The Bronze Horseman". If we draw a line after the second line, we can see the elements of axial symmetry

And the rose fell on Azor's paw.

I go with the judge's sword. (Derzhavin)

"Look for a taxi"

"Argentina Manit Negro",

"Appreciates the Negro Argentine",

"Lesha found a bug on the shelf."

The Neva is dressed in granite;

Bridges hung over the waters;

Dark green gardens

The islands were covered with it ...

Biology

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

The 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, while 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 for monocots, five - for dicots. characteristic feature structure of plants and their development is helicity.

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

A characteristic feature of the structure of plants and their development is helicity.

Look at the pine cone. The scales on its surface are arranged in a strictly regular manner - along two spirals that intersect approximately at a right angle. The number of such spirals in pine cones is 8 and 13 or 13 and 21.

Zoology

Symmetry in animals is understood as correspondence in size, shape and outline, as well as the relative location of body parts located on opposite sides of the dividing line. With radial or radiative symmetry, the body has the form of a short or long cylinder or a vessel with a central axis, from which parts of the body extend in a radial order. These are coelenterates, echinoderms, starfish. With bilateral symmetry, there are three axes of symmetry, but only one pair of symmetrical sides. Because the other two sides - the abdominal and dorsal - are not similar to each other. This kind of symmetry is characteristic of most animals, including insects, fish, amphibians, reptiles, birds, and mammals.

Axial symmetry

Different kinds symmetry physical phenomena: symmetry of electric and magnetic fields (Fig. 1)

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

fig.1 fig.2

Art

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

One of the best early works Raphael - "Betrothal of Mary" - created in 1504. A valley topped with a white-stone temple stretches out under the sunny blue sky. In the foreground is the betrothal ceremony. The High Priest brings the hands of Mary and Joseph closer 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 oncoming movement of the characters. For modern tastes, the composition of such a picture is boring, because the symmetry is too obvious.

Chemistry

The water molecule has a plane of symmetry (straight vertical line). DNA molecules (deoxyribonucleic acid) play an extremely important role in the world of wildlife. It is a double-stranded high molecular weight polymer whose monomer is nucleotides. DNA molecules have a double helix structure built on the principle of complementarity.

architewho

Since ancient times, man has used symmetry in architecture. Ancient architects used symmetry especially brilliantly in architectural structures. Moreover, the ancient Greek architects were convinced that in their works they are guided by the laws that govern nature. Choosing symmetrical forms, the artist thus expressed his understanding of natural harmony as stability and balance.

The city of Oslo, the capital of Norway, has an expressive ensemble of nature and art. This is Frogner - park - a complex of landscape gardening sculpture, which was created over 40 years.

Pashkov House Louvre (Paris)

How many different axes of symmetry a triangle can have depends on its geometric shape. If it is an equilateral triangle, then it will immediately have as many as three axes of symmetry.

And if it is an isosceles triangle, it will have only one axis of symmetry.

The sister's son is just going through this topic at school in geometry lessons. The axis of symmetry is a straight line, when rotated around which at a specific angle, the symmetrical figure will take the same position in space that it occupied before the rotation, and some of its parts will be replaced by the same others. In an isosceles triangle - three, in a rectangular one - one, in the rest - no, since their sides are not equal to each other.

It depends on which triangle. An equilateral triangle has three axes of symmetry that pass through its three vertices. An isosceles triangle, respectively, has one axis of symmetry. The remaining triangles do not have axes of symmetry.

The simplest thing to remember is that an equilateral triangle has three sides equal and it has three axes of symmetry

This makes it easier to remember the following

There are no equal sides, that is, all sides are different, which means there are no axes of symmetry

An isosceles triangle has only one axis.

You can’t just answer how many axes of symmetry a triangle has without understanding which particular triangle we are talking about.

An equilateral triangle has three axes of symmetry, respectively.

An isosceles triangle has only one axis of symmetry.

Any other triangles with sides of different lengths do not have any axis of symmetry at all.

A triangle, in which all sides are different in size, has no axes of symmetry.

A right triangle can have one axis of symmetry if its legs are equal.

In a triangle in which two sides are equal (isosceles) one axis can be drawn, and in which all three sides are equal (equilateral) - three.

Before answering the question of how many axes of symmetry a triangle has, you first need to remember what an axis of symmetry is.

So, simply put, in geometry, the axis of symmetry is a line, if you bend a figure along which, we get the same halves.

but it is worth remembering that triangles are also different.

So here it is isosceles a triangle (a triangle with two equal sides) has one axis of symmetry.

Equilateral the triangle, respectively, has 3 axes of symmetry, since all sides of this triangle are equal.

But versatile A triangle has no axes of symmetry at all. No matter how you fold it, and even where you draw straight lines, but since the sides are different, then two identical halves will not work.

As far as I remember the geometry, an equilateral triangle has three axes of symmetry passing through its vertices, these are its bisectors. At right triangle, as well as scalene, obtuse-angled and acute-angled triangles, there are no axes of symmetry at all, while an isosceles one has one.

And it's easy to check this - just imagine a line along which it can be cut in two so as to get two identical triangles.

Since triangles are different, then their axes of symmetry are respectively in different quantities. For example, a triangle with different sides no axes of symmetry at all. And the equilateral has three of them. There is another type of triangle that has one axis of symmetry. It has two equal sides and one right angle.

An arbitrary triangle has no axes of symmetry. An isosceles triangle has one axis of symmetry - this is the median to a single side. An equilateral triangle has three axes of symmetry - these are its three medians.

Today we will talk about a phenomenon that each of us constantly encounter in life: about symmetry. What is symmetry?

Approximately we all understand the meaning of this term. The dictionary says: symmetry is the proportionality and full correspondence of the arrangement of parts of something relative to a line or point. There are two types of symmetry: axial and radial. Let's look at the axis first. This is, let's say, "mirror" symmetry, when one half of the 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 (full face) are also symmetrical - the same arms and legs, the same eyes. But let's not be mistaken, in fact, in the organic (living) world, absolute symmetry cannot be found! The halves of the sheet do not copy each other perfectly, the same applies to the human body (look at it for yourself); the same is true of other organisms! By the way, it is worth adding that any symmetrical body is symmetrical relative to the viewer in only one position. It is necessary, say, to turn the sheet, or raise one hand, and what? - see for yourself.

People achieve true symmetry in the products 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 to finish the mirror half of the sheet as the first exercise in a new field.

Draw a symmetrical object - lesson 1

Let's try to make it as similar as possible. To do this, we will literally build our soul mate. Do not think that it is 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 act like this: we draw with a pencil without pressure several perpendiculars to the axis of symmetry - the middle vein of the sheet. Four or five is enough. 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 the ruler, do not really rely on the eye. As a rule, we tend to reduce the drawing - it has been noticed in experience. We do not recommend measuring distances with your fingers: the error is too large.

Connect the resulting points with a pencil line:

Now we look meticulously - are the halves really the same. If everything is correct, we will circle it with a felt-tip pen, clarify our line:

The poplar leaf has been completed, now you can swing at the oak one.

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 exactly observed. Well, let's train the eye:

So a symmetrical oak leaf was drawn, or rather, we built it according to all the rules:

How to draw a symmetrical object - lesson 3

And we will fix the topic - we will finish drawing a symmetrical leaf of lilac.

He has too interesting shape- heart-shaped and with ears at the base you have to puff:

Here is what they drew:

Look at the resulting work from a distance and evaluate how accurately we managed to convey the required similarity. Here's a tip for you: 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 correctly) and cut the leaf along the original line. Look at the figure itself and at the cut paper.