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 about a 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 for constructing 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, uniformity in the arrangement of parts of something according to opposite sides from a point, 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 triangle and, 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, … The indicated axis is 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, a 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 of the use of axial and portable symmetry is the fence shown in the photograph.
Conclusion: Thus, there are different kinds symmetries, symmetrical points in each of these types of symmetry are constructed 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/
Let g be a fixed line (Fig. 191). Let's take an arbitrary point X and drop the perpendicular AX to the straight line g. On the continuation of the perpendicular beyond point A, we put aside the segment AX", equal to the segment OH. Point X" is said to be symmetrical to point X relative to straight line g.
If a point X lies on a line g, then the point symmetrical to it is the point X itself. Obviously, the point symmetrical to the point X" is a point X.
The transformation of a figure F into a figure F", in which each of its points X goes to a point X", symmetrical with respect to a given straight line g, is called a symmetry transformation with respect to a straight line g. In this case, the figures F and F" are called symmetrical with respect to straight line g (Fig. 192).
If a symmetry transformation with respect to a line g takes a figure F into itself, then this figure is called symmetric with respect to a line g, and the line g is called the axis of symmetry of the figure.
For example, straight lines passing through the intersection point of the diagonals of a rectangle parallel to its sides are the axes of symmetry of the rectangle (Fig. 193). The straight lines on which the diagonals of a rhombus lie are its axes of symmetry (Fig. 194).
Theorem 9.3. The transformation of symmetry about a straight line is a movement.
Proof. Let's take this straight line as the y-axis Cartesian system coordinates (Fig. 195). Let an arbitrary point A (x; y) of the figure F go to the point A" (x"; y") of the figure F". From the definition of symmetry with respect to a straight line it follows that points A and A" have equal ordinates, and the abscissas differ only in sign:
x"= -x.
Let's take two arbitrary points A(x 1; y 1) and B (x 2; y 2) - They will move to points A" (- x 1, y 1) and B" (-x 2; y 2).
AB 2 = (x 2 - x 1) 2 + (y 2 - y 1) 2
A"B" 2 =(-x 2 + x 1) 2 +(y 2 -y 1) 2.
From this it is clear that AB = A "B". And this means that the transformation of symmetry about a straight line is motion. The theorem has been proven.
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 various types of 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. So, 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 an equilateral triangle has 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 indicates, mirror symmetry connects any object and its reflection in a plane 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.”
Since ancient times, man has developed ideas about beauty. All creations of nature are beautiful. People are beautiful in their own way, animals and plants are amazing. The sight of a precious stone or a salt crystal pleases the eye; it is difficult not to admire a snowflake or a butterfly. But why does this happen? It seems to us that the appearance of objects is correct and complete, the right and left halves of which look the same, as if in a mirror image.
Apparently, people of art were the first to think about the essence of beauty. Ancient sculptors who studied the structure human body, back in the 5th century BC. The concept of “symmetry” began to be used. This word has Greek origin and means harmony, proportionality and similarity in the arrangement of the component parts. Plato argued that only that which is symmetrical and proportionate can be beautiful.
In geometry and mathematics, three types of symmetry are considered: axial symmetry (relative to a straight line), central (relative to a point) and mirror symmetry (relative to a plane).
If each of the points of an object has its own exact mapping within it relative to its center, there is central symmetry. Examples of this are: geometric bodies like a cylinder, ball, correct prism etc.
The axial symmetry of points relative to a straight line provides that this straight line intersects the middle of the segment connecting the points and is perpendicular to it. Examples of the bisector of an undeveloped angle isosceles triangle, any straight line drawn through the center of the circle, etc. If axial symmetry is characteristic, the definition of mirror points can be visualized by simply bending it along the axis and putting equal halves “face to face.” The desired points will touch each other.
At mirror symmetry the points of an object are located equally relative to the plane that passes through its center.
Nature is wise and rational, therefore almost all of its creations have a harmonious structure. This applies to both living beings and inanimate objects. The structure of most life forms is characterized by one of three types of symmetry: bilateral, radial or spherical.
Most often, axial can be observed in plants developing perpendicular to the soil surface. In this case, symmetry results from rotating identical elements around common axis, located in the center. The angle and frequency of their location may be different. Examples are trees: spruce, maple and others. In some animals, axial symmetry also occurs, but this is less common. Of course, nature is rarely characterized by mathematical precision, but the similarity of the elements of an organism is still striking.
Biologists often consider not axial symmetry, but bilateral (bilateral) symmetry. An example of this is the wings of a butterfly or dragonfly, plant leaves, flower petals, etc. In each case, the right and left parts of the living object are equal and are mirror images of each other.
Spherical symmetry is characteristic of the fruits of many plants, some fish, mollusks and viruses. Examples of radial symmetry are some types of worms and echinoderms.
In human eyes, asymmetry is most often associated with irregularity or inferiority. Therefore, in most creations of human hands, symmetry and harmony can be traced.
Definition. Symmetry (means “proportionality”) is the property of geometric objects to combine with themselves under certain transformations. Under symmetry understand every correctness in internal structure bodies or figures.
Symmetry about a point- this is central symmetry (Fig. 23 below), and symmetry about a straight line- this is axial symmetry (Fig. 24 below).
Symmetry about a point assumes that there is something on both sides of a point at equal distances, such as other points or locus points (straight lines, curved lines, geometric shapes).
If you connect symmetrical points (points of a geometric figure) with a straight line through a symmetry point, then the symmetrical points will lie at the ends of the straight line, and the symmetry point will be its middle. If you fix the symmetry point and rotate the straight line, then the symmetrical points will describe curves, each point of which will also be symmetrical to the point of the other curved line.
Symmetry about a straight line(axis of symmetry) assumes that along a perpendicular drawn through each point of the axis of symmetry, two symmetrical points are located at the same distance from it. The same geometric figures can be located relative to the axis of symmetry (straight line) as relative to the point of symmetry.
An example would be a sheet of notebook that is folded in half if a straight line is drawn along the fold line (axis of symmetry). Each point on one half of the sheet will have a symmetrical point on the second half of the sheet if they are located at the same distance from the fold line and perpendicular to the axis.
The line of axial symmetry, as in Figure 24, is vertical, and the horizontal edges of the sheet are perpendicular to it. That is, the axis of symmetry serves as a perpendicular to the midpoints of the horizontal straight lines bounding the sheet. Symmetrical points (R and F, C and D) are located at the same distance from the axial line - perpendicular to the lines connecting these points. Consequently, all points of the perpendicular (axis of symmetry) drawn through the middle of the segment are equidistant from its ends; or any point perpendicular (axis of symmetry) to the middle of a segment is equidistant from the ends of this segment.
6.7.3. Axial symmetry
Points A And A 1 are symmetrical with respect to line m, since line m is perpendicular to the segment AA 1 and passes through its middle.
m– axis of symmetry.
Rectangle ABCD has two axes of symmetry: straight m And l.
If the drawing is bent in a straight line m or in a straight line l, then both parts of the drawing will coincide.
Square ABCD has four axes of symmetry: straight m, l, k And s.
If the square is bent along any of the straight lines: m, l, k or s, then both sides of the square will coincide.
A circle with a center at point O and radius OA has an infinite number of axes of symmetry. These are straight lines: m, m 1, m 2, m 3 .
Exercise. Construct point A 1, symmetrical point A(-4; 2) relative to the Ox axis.
Construct point A 2 symmetrical to point A(-4; 2) relative to the Oy axis.
Point A 1 (-4; -2) is symmetrical to point A (-4; 2) relative to the Ox axis, since the Ox axis is perpendicular to the segment AA 1 and passes through its middle.
For points symmetrical about the Ox axis, the abscissas coincide, and the ordinates are opposite numbers.
Point A 2 (4; -2) is symmetrical to point A (-4; 2) relative to the Oy axis, since the Oy axis is perpendicular to the segment AA 2 and passes through its middle.
For points symmetrical about the Oy axis, the ordinates coincide, and the abscissas are opposite numbers.
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Central and axial symmetries
Central symmetry
Two points A and A 1 are called symmetrical relative to point O if O is the middle of the segment AA 1 (Fig. 1). Point O is considered symmetrical to itself.
Example central symmetry
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 figure is also said to have central symmetry.
Examples of figures with central symmetry are a circle and a parallelogram (Fig. 2).
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. A straight line also has central symmetry, but unlike a circle and a parallelogram, which have only one center of symmetry (point O in Fig. 2), a straight line has an infinite number of them - any point on the straight line is its center of symmetry.
Axial symmetry
Two points A and A 1 are called symmetrical with respect to line a if this line passes through the middle of segment AA 1 and is perpendicular to it (Fig. 3). Each point of a line a is considered symmetrical to itself.
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.
Examples of such figures and their axes of symmetry are shown in Figure 4.
Note that for a circle, any straight line passing through its center is an axis of symmetry.
Comparison of symmetries
Central and axial symmetries
How many axes of symmetry does the figure shown in the figure have?
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Lesson “Axial and central symmetry”
Brief description of the document:
Symmetry is enough interesting topic in geometry, since this very concept is very often encountered not only in the process of human life but also in nature.
The first part of the video presentation “Axial and central symmetry” gives the definition of the symmetry of two points relative to a straight line on a plane. The condition for their symmetry is the possibility of drawing a segment through them, through the middle of which a given straight line will pass. Required condition Such symmetry is the perpendicularity of a segment and a straight line.
The next part of the video tutorial gives clear example definition, which is shown in the form of a drawing, where several pairs of points are symmetrical about a straight line, and any point on this straight line is symmetrical to itself.
After receiving initial concepts about symmetry, students are encouraged to complex definition a figure that is symmetrical about a straight line. The definition is offered in the form of a text rule, and is also accompanied by a voiceover from the speaker. This part concludes with examples of symmetrical and asymmetrical figures, relative to a straight line. Interestingly, there are geometric figures that have several axes of symmetry - all of them are clearly presented in the form of drawings, where the axes are highlighted in a separate color. You can make the proposed material easier to understand in this way: an object or figure is symmetrical if it coincides exactly when folding the two halves around its axis.
In addition to axial symmetry, there is symmetry about one point. It is this concept that is dedicated next part video presentations. First, a definition is given of the symmetry of two points relative to a third, then an example is provided in the form of a figure, which shows a symmetrical and asymmetrical pair of points. This part of the lesson ends with examples. geometric shapes, which have or do not have a center of symmetry.
At the end of the lesson, students are invited to familiarize themselves with the most striking examples symmetries that can be found in the surrounding world. Understanding and ability to build symmetrical figures are simply necessary in the lives of people who are engaged in the most different professions. At its core, symmetry is the basis of everything human civilization, since 9 out of 10 objects surrounding a person have one or another type of symmetry. Without symmetry, the construction of many large architectural structures would not have been possible, it would not have been possible to achieve impressive industrial capacities, and so on. In nature, symmetry is also a very common phenomenon, and if in inanimate objects It is almost impossible to find it, but the living world is literally teeming with it - almost all flora and fauna, with rare exceptions, have either axial or central symmetry.
Regular school program is developed in such a way that it can be understood by any student admitted to the lesson. A video presentation makes this process several times easier, since it simultaneously affects several centers for the development of information, provides material in several colors, thereby forcing students to concentrate their attention on the most important thing during the lesson. Unlike the usual way of teaching in schools, when not every teacher has the opportunity or desire to answer students’ clarifying questions, a video lesson can easily be rewinded to the desired place in order to listen to the speaker again and read necessary information again until it is fully understood. Given the simplicity of presenting the material, a video presentation can be used not only during school activities, but also at home, as independent method training.
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Presentation “Movements. Axial symmetry"
Documents in the archive:
Document's name 8.
Description of the presentation by individual slides:
Central symmetry is one example of movement
Definition: Axial symmetry with the a-axis is a mapping of space onto itself, in which any point K goes into a point K1 symmetrical to it relative to the a-axis
1) Oxyz - rectangular system coordinates Oz - axis of symmetry 2) M(x; y; z) and M1(x1; y1; z1), symmetrical about the Oz axis The formulas will also be true if the point M ⊂ Oz Axial symmetry is the movement Z X Y M(x; y; z) M1(x1; y1; z1) O
Prove: Problem 1, with axial symmetry, a straight line forming an angle φ with the axis of symmetry is mapped onto a straight line, also forming an angle φ with the axis of symmetry. Solution: with axial symmetry, a straight line forming an angle φ with the axis of symmetry is mapped onto a straight line, also forming an angle with the axis of symmetry. axis of symmetry angle φ A F E N m l a φ φ
Given: 2) △ABD - rectangular, according to the Pythagorean theorem: 1) DD1 ⏊ (A1C1D1), 3) △BDD2 - rectangular, according to the Pythagorean theorem: Problem 2 Find: BD2 Solution:
Brief description of the document:
Presentation “Movements. Axial symmetry" represents visual material to explain the main provisions of this topic in a school mathematics lesson. In this presentation, axial symmetry is considered as another type of movement. During the presentation, students are reminded of the studied concept of central symmetry, a definition of axial symmetry is given, the proposition that axial symmetry is movement is proven, and the solution to two problems in which it is necessary to operate with the concept of axial symmetry is described.
Rotational symmetry is a movement, so representing it on a chalkboard is challenging. Clearer, understandable constructions can be made using electronic means. Thanks to this, the structures are clearly visible from any desk in the classroom. In the drawings, it is possible to highlight construction details in color and focus attention on the features of the operation. Animation effects are used for the same purpose. With the help of presentation tools, it is easier for the teacher to achieve learning goals, so presentation is used to increase the effectiveness of the lesson.
The demonstration begins by reminding students of the type of movement they have learned—central symmetry. An example of the application of the operation is the symmetrical display of a drawn pear. A point is marked on the plane relative to which each point of the image becomes symmetrical. The displayed image is thus inverted. In this case, all distances between points of the object are preserved with central symmetry.
The second slide introduces the concept of axial symmetry. The figure shows a triangle, each of its vertices transforms into a symmetrical vertex of the triangle relative to a certain axis. The definition of axial symmetry is highlighted in the box. It is noted that with it, each point of the object becomes symmetrical.
Next, in a rectangular coordinate system, axial symmetry is considered, the properties of the coordinates of an object displayed using axial symmetry, and it is also proven that with this mapping, distances are preserved, which is a sign of movement. On the right side of the slide is a rectangular coordinate system Oxyz. The Oz axis is taken as the axis of symmetry. A point M is marked in space, which, with appropriate mapping, turns into M 1. The figure shows that with axial symmetry, the point retains its applicate.
It is noted that the arithmetic mean of the abscissa and ordinate of this mapping with axial symmetry is equal to zero, that is, (x+ x 1)/2=0; (y+ y 1)/2=0. Otherwise, this indicates that x=-x 1 ; y=-y 1 ; z=z 1 . The rule also applies if point M is marked on the Oz axis itself.
To consider whether the distances between points are preserved with axial symmetry, an operation is described on points A and B. Displayed relative to the Oz axis, the described points go into A1 and B1. To determine the distance between points, we use a formula in which the distance is calculated by coordinates. It is noted that AB=√(x 2 -x 1) 2 +(y 2 -y 1) 2 +(z 2 -z 1) 2), and for the displayed points A 1 B 1 =√(-x 2 +x 1) 2 +(-y 2 +y 1) 2 +(z 2 -z 1) 2). Taking into account the properties of squaring, it can be noted that AB = A 1 B 1. This suggests that distances are maintained between points - main feature movements. This means that axial symmetry is movement.
Slide 5 discusses the solution to problem 1. In it, it is necessary to prove the statement that a straight line passing at an angle φ to the axis of symmetry forms the same angle φ with it. For the problem, an image is given on which the axis of symmetry is drawn, as well as a straight line m, forming an angle φ with the axis of symmetry, and relative to its axis its display is a straight line l. The proof of the statement begins with the construction of additional points. It is noted that straight line m intersects the axis of symmetry at A. If we mark point F≠A on this straight line and drop a perpendicular from it to the axis of symmetry, we obtain the intersection of the perpendicular with the axis of symmetry at point E. With axial symmetry, the segment FE goes into the segment NE. As a result of this construction, right triangles ΔAEF and ΔAEN were obtained. These triangles are equal, since AE is their common side, and FE = NE are equal in construction. Accordingly, the angle ∠EAN=∠EAF. It follows from this that the displayed straight line also forms an angle φ with the axis of symmetry. The problem is solved.
The last slide discusses the solution to Problem 2, in which you are given a cube ABCDA 1 B 1 C 1 D 1 with side a. It is known that after symmetry about the axis containing the edge B 1 D 1, point D goes into D 1. The problem requires finding BD 2. A construction is made for the problem. The figure shows a cube, from which it can be seen that the axis of symmetry is the diagonal of the cube face B 1 D 1. The segment formed by the movement of point D is perpendicular to the plane of the face to which the axis of symmetry belongs. Since the distances between points are maintained during movement, then DD 1 = D 1 D 2 =a, that is, the distance DD 2 =2a. From right triangleΔABD by the Pythagorean theorem it follows that BD=√(AB 2 +AD 2)=a√2. From the right triangle ΔВDD 2 it follows by the Pythagorean theorem BD 2 =√(DD 2 2 +ВD 2) = а√6. The problem is solved.
Presentation “Movements. Axial symmetry" is used to increase efficiency school lesson mathematics. Also, this visualization method will help the teacher implementing distance learning. The material can be offered for independent consideration by students who have not mastered the topic of the lesson well enough.
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