symmetry architectural facade building

Symmetry is a concept that reflects the order existing in nature, proportionality and proportionality between the elements of any system or object of nature, orderliness, balance of the system, stability, i.e. some element of harmony.

Millennia passed before humanity, in the course of its social and production activities, realized the need to express in certain concepts the two tendencies it had established primarily in nature: the presence of strict orderliness, proportionality, balance and their violation. People have long paid attention to the correct shape of crystals, the geometric rigor of the structure of honeycombs, the sequence and repeatability of the arrangement of branches and leaves on trees, petals, flowers, plant seeds, and reflected this orderliness in their practical activities, thinking and art.

Objects and phenomena of living nature have symmetry. It not only pleases the eye and inspires poets of all times and peoples, but allows living organisms to better adapt to their environment and simply survive.

In living nature, the vast majority of living organisms exhibit different kinds symmetries (shape, similarity, relative location). Moreover, organisms of different anatomical structures can have the same type of external symmetry.

The principle of symmetry states that if space is homogeneous, the transfer of a system as a whole in space does not change the properties of the system. If all directions in space are equivalent, then the principle of symmetry allows the rotation of the system as a whole in space. The principle of symmetry is respected if the origin of time is changed. In accordance with the principle, it is possible to make a transition to another reference system moving relative to this system with constant speed. The inanimate world is very symmetrical. Often symmetry violations in quantum physics elementary particles- this is a manifestation of an even deeper symmetry. Asymmetry is a structure-forming and creative principle of life. In living cells, functionally significant biomolecules are asymmetrical: proteins consist of levorotatory amino acids (L-form), and nucleic acids They contain, in addition to heterocyclic bases, dextrorotatory carbohydrates - sugars (D-form), in addition, DNA itself - the basis of heredity is a right-handed double helix.

The principles of symmetry underlie the theory of relativity, quantum mechanics, physicists solid, nuclear and nuclear physics, particle physics. These principles are most clearly expressed in the invariance properties of the laws of nature. This is not only about physical laws, but also others, for example, biological. An example of a biological law of conservation is the law of inheritance. It is based on invariance biological properties in relation to the transition from one generation to another. It is quite obvious that without conservation laws (physical, biological and others), our world simply could not exist.

Thus, symmetry expresses the preservation of something despite some changes or the preservation of something despite a change. Symmetry presupposes the invariability not only of the object itself, but also of any of its properties in relation to transformations performed on the object. The immutability of certain objects can be observed in relation to various operations - rotations, translations, mutual replacement of parts, reflections, etc.

Let's consider the types of symmetry in mathematics:

• * central (relative to the point)
• * axial (relatively straight)
• * mirror (relative to the plane)
• 1. Central symmetry (Appendix 1)

A figure is said to be symmetrical with respect to point O if, for each point of the figure, a point symmetrical with respect to point O also belongs to this figure. Point O is called the center of symmetry of the figure.

The concept of a center of symmetry was first encountered in the 16th century. In one of Clavius’s theorems, which states: “if a parallelepiped is cut by a plane passing through the center, then it is split in half and, conversely, if a parallelepiped is cut in half, then the plane passes through the center.” Legendre, who first introduced elementary geometry elements of the doctrine of symmetry, shows that right parallelepiped there are 3 planes of symmetry perpendicular to the edges, and the cube has 9 planes of symmetry, of which 3 are perpendicular to the edges, and the other 6 pass through the diagonals of the faces.

Examples of figures that have central symmetry are the circle and parallelogram.

In algebra, when studying even and odd functions, their graphs are considered. When constructed, the graph of an even function is symmetrical with respect to the ordinate axis, and the graph of an odd function is symmetrical with respect to the origin, i.e. point O. This means Not even function has central symmetry, and the even function is axial.

2. Axial symmetry (Appendix 2)

A figure is called symmetrical with respect to line a if, for each point of the figure, a point symmetrical with respect to line a also belongs to this figure. Straight line a is called the axis of symmetry of the figure. The figure is also said to have axial symmetry.

In more in the narrow sense the axis of symmetry is called the axis of symmetry of the second order and speaks of “axial symmetry”, which can be defined as follows: a figure (or body) has axial symmetry about a certain axis if each of its points E corresponds to a point F belonging to the same figure such that the segment EF is perpendicular to the axis, intersects it and at the intersection point is divided in half.

I will give examples of figures that have axial symmetry. An undeveloped angle has one axis of symmetry - the straight line on which the angle's bisector is located. An isosceles (but not equilateral) triangle also has one axis of symmetry, and equilateral triangle-- three axes of symmetry. A rectangle and a rhombus, which are not squares, each have two axes of symmetry, and a square has four axes of symmetry. A circle has an infinite number of them - any straight line passing through its center is an axis of symmetry.

There are figures that do not have a single axis of symmetry. Such figures include a parallelogram, different from a rectangle, and a scalene triangle.

3. Mirror symmetry (Appendix 3)

Mirror symmetry (symmetry relative to a plane) is a mapping of space onto itself in which any point M goes into a point M1 that is symmetrical to it relative to this plane.

Mirror symmetry is well known to every person from everyday observation. As the name itself shows, mirror symmetry connects any object and its reflection in flat mirror. One figure (or body) is said to be mirror symmetrical to another if together they form a mirror symmetrical figure (or body).

Billiards players have long been familiar with the action of reflection. Their “mirrors” are the sides of the playing field, and the role of a ray of light is played by the trajectories of the balls. Having hit the side near the corner, the ball rolls towards the side located at a right angle, and, having been reflected from it, moves back parallel to the direction of the first impact.

It should be noted that two symmetrical figures or two symmetrical parts of one figure with all their similarities, equality of volumes and surface areas, in general case, are unequal, i.e. they cannot be combined with each other. These are different figures, they cannot be replaced with each other, for example, the right glove, boot, etc. not suitable for the left arm or leg. Items can have one, two, three, etc. planes of symmetry. For example, a straight pyramid, the base of which is an isosceles triangle, is symmetrical about one plane P. A prism with the same base has two planes of symmetry. The right one hexagonal prism there are seven of them. Bodies of rotation: ball, torus, cylinder, cone, etc. have infinite number planes of symmetry.

The ancient Greeks believed that the universe was symmetrical simply because symmetry is beautiful. Based on considerations of symmetry, they made a number of guesses. Thus, Pythagoras (5th century BC), considering the sphere the most symmetrical and perfect form, made a conclusion about the sphericity of the Earth and its movement along the sphere. At the same time, he believed that the Earth moves along the sphere of a certain “central fire”. According to Pythagoras, the six planets known at that time, as well as the Moon, Sun, and stars, were supposed to revolve around the same “fire.”

The purpose of the lesson:

• formation of the concept of “symmetrical points”;
• teach children to construct points symmetrical to data;
• learn to construct segments symmetrical to data;
• consolidation of what has been learned (formation of computational skills, division of a multi-digit number by a single-digit number).

On the stand “for the lesson” there are cards:

1. Organizational moment

Greetings.

The teacher draws attention to the stand:

Children, let's start the lesson by planning our work.

Today in mathematics lesson we will take a journey into 3 kingdoms: the kingdom of arithmetic, algebra and geometry. Let's start the lesson with the most important thing for us today, with geometry. I'll tell you a fairy tale, but "A fairy tale is a lie, but there is a hint in it - a lesson for good fellows."

": One philosopher named Buridan had a donkey. Once, leaving for a long time, the philosopher put two identical armfuls of hay in front of the donkey. He placed a bench, and to the left of the bench and to the right of it, at the same distance, he placed completely identical armfuls of hay.

Figure 1 on the board:

The donkey walked from one armful of hay to another, but still did not decide which armful to start with. And, in the end, he died of hunger."

Why didn't the donkey decide which armful of hay to start with?

What can you say about these armfuls of hay?

(The armfuls of hay are exactly the same, they were at the same distance from the bench, which means they are symmetrical).

2. Let's do a little research.

Take a sheet of paper (each child has a sheet of colored paper on their desk), fold it in half. Pierce it with the leg of a compass. Expand.

What did you get? (2 symmetrical points).

How can you be sure they are truly symmetrical? (let's fold the sheet, the dots match)

3. On the desk:

Do you think these points are symmetrical? (No). Why? How can we be sure of this?

Figure 3:

Are these points A and B symmetrical?

How can we prove this?

(Measure the distance from the straight line to the points)

Let's return to our pieces of colored paper.

Measure the distance from the fold line (axis of symmetry) first to one and then to the other point (but first connect them with a segment).

What can you say about these distances?

(The same)

Find the middle of your segment.

Where is it?

(Is the point of intersection of segment AB with the axis of symmetry)

4. Pay attention to the corners, formed as a result of the intersection of segment AB with the axis of symmetry. (We find out with the help of a square, each child works at his own workplace, one studies at the blackboard).

Children's conclusion: segment AB is at right angles to the axis of symmetry.

Without knowing it, we have now discovered a mathematical rule:

If points A and B are symmetrical about a straight line or axis of symmetry, then the segment connecting these points is at a right angle or perpendicular to this straight line. (The word “perpendicular” is written separately on the stand). We say the word “perpendicular” out loud in chorus.

5. Let us pay attention to how this rule is written in our textbook.

Work according to the textbook.

Find symmetrical points relative to the straight line. Will points A and B be symmetrical about this line?

6. Working on new material.

Let's learn how to construct points symmetrical to data relative to a straight line.

The teacher teaches reasoning.

To construct a point symmetrical to point A, you need to move this point from the straight line to the same distance to the right.

7. We will learn to construct segments symmetrical to data relative to a straight line. Work according to the textbook.

Students reason at the board.

8. Oral counting.

This is where we will end our stay in the “Geometry” Kingdom and will do a little mathematical warm-up by visiting the “Arithmetic” Kingdom.

While everyone is working orally, two students are working on individual boards.

A) Perform division with verification:

B) After inserting the required numbers, solve the example and check:

Verbal counting.

1. The lifespan of a birch is 250 years, and an oak is 4 times longer. How long does an oak tree live?
2. A parrot lives on average 150 years, and an elephant is 3 times less. How many years does an elephant live?
3. The bear invited guests to him: a hedgehog, a fox and a squirrel. And as a gift they presented him with a mustard pot, a fork and a spoon. What did the hedgehog give the bear?

We can answer this question if we execute these programs.

• Mustard - 7
• Fork - 8
• Spoon - 6

(The hedgehog gave a spoon)

4) Calculate. Find another example.

• 810: 90
• 360: 60
• 420: 7
• 560: 80

5) Find a pattern and help write down the required number:

3	9	81
2		16

5	10	20
6		24

9. Now let's rest a little.

Let's listen to Beethoven's Moonlight Sonata. A minute of classical music. Students put their heads on the desk, close their eyes, and listen to music.

10. Journey into the kingdom of algebra.

Guess the roots of the equation and check:

Students solve problems on the board and in notebooks. They explain how they guessed it.

11. "Blitz tournament" .

a) Asya bought 5 bagels for a rubles and 2 loaves for b rubles. How much does the entire purchase cost?

Let's check. Let's share our opinions.

12. Summarizing.

So, we have completed our journey into the kingdom of mathematics.

What was the most important thing for you in the lesson?

Who liked our lesson?

It was a pleasure working with you

Thank you for the lesson.

Movement concept

Let us first examine the concept of movement.

Definition 1

A mapping of a plane is called a motion of the plane if the mapping preserves distances.

There are several theorems related to this concept.

Theorem 2

The triangle, when moving, turns into an equal triangle.

Theorem 3

Any figure, when moving, transforms into a figure equal to it.

Axial and central symmetry are examples of motion. Let's look at them in more detail.

Axial symmetry

Definition 2

Points $A$ and $A_1$ are called symmetrical with respect to the line $a$ if this line is perpendicular to the segment $(AA)_1$ and passes through its center (Fig. 1).

Picture 1.

Let's consider axial symmetry using an example problem.

Example 1

Build symmetrical triangle For given triangle regarding any aspect of it.

Solution.

Let us be given a triangle $ABC$. We will construct its symmetry with respect to the side $BC$. The side $BC$ with axial symmetry will transform into itself (follows from the definition). Point $A$ will go to point $A_1$ in the following way: $(AA)_1\bot BC$, $(AH=HA)_1$. Triangle $ABC$ will transform into triangle $A_1BC$ (Fig. 2).

Figure 2.

Definition 3

A figure is called symmetrical with respect to straight line $a$ if every symmetrical point of this figure is contained in the same figure (Fig. 3).

Figure 3.

Figure $3$ shows a rectangle. It has axial symmetry with respect to each of its diameters, as well as with respect to two straight lines that pass through the centers opposite sides given rectangle.

Central symmetry

Definition 4

Points $X$ and $X_1$ are called symmetrical with respect to point $O$ if point $O$ is the center of the segment $(XX)_1$ (Fig. 4).

Figure 4.

Let's consider central symmetry using an example problem.

Example 2

Construct a symmetrical triangle for a given triangle at any of its vertices.

Solution.

Let us be given a triangle $ABC$. We will construct its symmetry relative to the vertex $A$. The vertex $A$ with central symmetry will transform into itself (follows from the definition). Point $B$ will go to point $B_1$ as follows: $(BA=AB)_1$, and point $C$ will go to point $C_1$ as follows: $(CA=AC)_1$. Triangle $ABC$ will transform into triangle $(AB)_1C_1$ (Fig. 5).

Figure 5.

Definition 5

A figure is symmetrical with respect to point $O$ if every symmetrical point of this figure is contained in the same figure (Fig. 6).

Figure 6.

Figure $6$ shows a parallelogram. It has central symmetry about the point of intersection of its diagonals.

Example task.

Example 3

Let us be given a segment $AB$. Construct its symmetry with respect to the line $l$, which does not intersect the given segment, and with respect to the point $C$ lying on the line $l$.

Solution.

Let us schematically depict the condition of the problem.

Figure 7.

Let us first depict axial symmetry with respect to straight line $l$. Since axial symmetry is a movement, then by Theorem $1$, the segment $AB$ will be mapped onto the segment $A"B"$ equal to it. To construct it, we will do the following: draw straight lines $m\ and\n$ through points $A\ and\B$, perpendicular to straight line $l$. Let $m\cap l=X,\ n\cap l=Y$. Next we draw the segments $A"X=AX$ and $B"Y=BY$.

Figure 8.

Let us now depict the central symmetry with respect to the point $C$. Because central symmetry is a motion, then by Theorem $1$, the segment $AB$ will be mapped onto the segment $A""B""$ equal to it. To construct it, we will do the following: draw the lines $AC\ and\ BC$. Next we draw the segments $A^("")C=AC$ and $B^("")C=BC$.

Figure 9.

You will need

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

Instructions

Draw a straight line a, which will be the axis of symmetry. If its coordinates are not specified, draw it arbitrarily. On one side of this straight line place arbitrary point A. it is necessary to find a symmetrical point.

Helpful advice

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

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

Sources:

• how to draw central symmetry

Constructing a cross 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 labor from you.

You will need

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

Instructions

When answering this question, you must first decide what parameters define the section.
Let this be the straight line of intersection of the plane l with the plane and the point O, which is the 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. The result is point L. Next, draw a straight line LW through point O, and construct two guide cones lying in the main section O2M and O2C. At the intersection of these guides lie point Q, as well as the already shown point W. These are the first two points of the desired section.

Now draw a perpendicular MS at the base of the cone BB1 ​​and construct the generators perpendicular section O2B and O2B1. In this section, through point O, draw a straight line RG parallel to BB1. Т.R and Т.G are two more points of the desired section. If the cross section of the ball were known, then it could be built already at this stage. However, this is not an ellipse at all, but something elliptical that has symmetry with respect to the segment QW. Therefore, you should build as many section points as possible in order to connect them later with a smooth curve to obtain the most reliable sketch.

Construct an arbitrary section point. To do this, draw an arbitrary diameter AN at the base of the cone and construct the corresponding guides O2A and O2N. Through t.O, 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, you can find as many points as you want.

True, the procedure for obtaining them can be slightly simplified using symmetry with respect to QW. To do this, you can draw straight lines SS’ 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 is enough to construct half of the desired section due to the already mentioned symmetry with respect to QW.

Video on the topic

Tip 3: How to make a graph trigonometric function

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

You will need

• - ruler;
• - pencil;
• - knowledge of the basics of trigonometry.

Instructions

Video on the topic

note

If the two semi-axes of a single-strip hyperboloid are equal, then the figure can be obtained by rotating a hyperbola with semi-axes, one of which is the above, and the other, different from the two equal ones, around the imaginary axis.

Helpful advice

When examining this figure relative to the Oxz and Oyz axes, it is clear that its main sections are hyperbolas. And when cutting this spatial figure rotation by the Oxy plane, its cross section is an ellipse. The neck ellipse of a single-strip hyperboloid passes through the origin of coordinates, because z=0.

The throat ellipse is described by the equation x²/a² +y²/b²=1, and the other ellipses are composed by the equation x²/a² +y²/b²=1+h²/c².

Sources:

• Ellipsoids, paraboloids, hyperboloids. Rectilinear generators

The shape of a five-pointed star has been widely used by man since ancient times. We consider its shape beautiful because we unconsciously recognize in it the relationships of the golden section, i.e. the beauty of the five-pointed star is justified mathematically. Euclid was the first to describe the construction of a five-pointed star in his Elements. Let's join in with his experience.

You will need

• ruler;
• pencil;
• compass;
• protractor.

Instructions

The construction of a star comes down to the construction and subsequent connection of its vertices to each other sequentially through one. In order to build the correct one, you need to divide the circle into five.
Build arbitrary circle using a compass. Mark its center with point O.

Mark point A and use a ruler to draw line segment OA. Now you need to divide the segment OA in half; to do this, from point A, draw an arc of radius OA until it intersects the circle at two points M and N. Construct the segment MN. The point E where MN intersects OA will bisect segment OA.

Restore the perpendicular OD to the radius OA and connect points D and E. Make a notch B on OA from point E with radius ED.

Now, using line segment DB, mark the circle by five equal parts. Label the vertices of the regular pentagon sequentially with numbers from 1 to 5. Connect the points in next sequence: 1 with 3, 2 with 4, 3 with 5, 4 with 1, 5 with 2. Here is the correct five-pointed star, in regular pentagon. This is exactly the way I built it

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 get symmetrical figures without using bending of the plane, but with the help 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 this segment CD relative to straight AB. Let's build points C" and D", symmetrical to the 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, the segments CD and C "D" will align, they will be symmetrical.

Let us now construct a figure symmetrical given polygon ABCDE relative to this 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 found geometric figures, which 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,