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Axis of symmetry

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Sections: Maths

Objectives:

  • educational:
    • give an idea of ​​symmetry,
    • introduce the main types of symmetry on the plane and in space,
    • develop strong skills in building symmetrical figures,
    • expand ideas about famous figures by introducing symmetry-related properties,
    • show the possibilities of using symmetry in solving various problems,
    • to consolidate the acquired knowledge,
  • general education:
    • teach you how to set yourself up for work,
    • teach you how to control yourself and your deskmate,
    • teach you to evaluate yourself and your neighbor at the desk,
  • developing:
    • step up independent activities,
    • to develop cognitive activity,
    • learn to summarize and systematize the information received,
  • educational:
    • to educate students with a “sense of shoulder”,
    • to cultivate communication,
    • to instill a culture of communication.

Before each lie scissors and a sheet of paper.

- Take a piece of paper, fold it in half and cut some shape. Now expand the sheet and look at the fold line.

Question: What function does this line perform?

Estimated answer: This line divides the figure in half.

Question: How are all the points of the figure on the two halves?

Estimated answer: All points of the halves are equally spaced from the fold line and at the same level.

- So, the fold line divides the figure in half so that 1 half is a copy of 2 halves, i.e. this line is not simple, it has a remarkable property (all points relative to it are at the same distance), this line is the axis of symmetry.

- Cut a snowflake, find the axis of symmetry, characterize it.

- Draw a circle in the notebook.

Question: Determine how the axis of symmetry passes?

Estimated answer: Differently.

Question: So how many axes of symmetry does a circle have?

Estimated answer: A lot of.

- That's right, a circle has many axes of symmetry. An equally remarkable figure is a ball (spatial figure)

Question: What other shapes have more than one axis of symmetry?

Estimated answer: Square, rectangle, isosceles and equilateral triangles.

- Consider volumetric figures: cube, pyramid, cone, cylinder, etc. These figures also have an axis of symmetry. Determine how many axes of symmetry have a square, a rectangle, an equilateral triangle, and the proposed volumetric shapes?

I distribute to the students half the figures from plasticine.

- Using the information received, add the missing part of the figure.

Note: the figure can be both planar and volumetric. It is important that the students determine how the axis of symmetry goes and add the missing element. The correctness of the implementation determines the neighbor on the desk, evaluates how well the work is done.

From the lace of the same color on the desktop laid out a line (closed, open, with self-intersection, without self-intersection).

Task 5(group work 5 min).

- Visually determine the axis of symmetry and, with respect to it, complete the second part from the lace of a different color.

The correctness of the work performed is determined by the students themselves.

Students are presented with elements of drawings.

- Find the symmetrical parts of these patterns.

To consolidate the material, I propose the following tasks provided for 15 minutes:

1. The direct OP is the axis of symmetry of the KOM triangle.

Name all equal elements of the triangle KOR and KOM. What is the appearance of these triangles?

2. In the notebook, draw several isosceles triangles with a common base of 6 cm.

3. Draw line AB. Build a straight line perpendicular to the segment AB and passing through its middle. Mark the points C and D on it so that the quadrilateral ACBD is symmetric with respect to the line AB.

- Our initial ideas about the form belong to a very distant era of the ancient Stone Age - the Paleolithic. For hundreds of millennia of this period, people lived in caves, in conditions that differed little from the life of animals. People made tools for hunting and fishing, developed a language for communication with each other, and in the Late Paleolithic era they adorned their existence, creating works of art, figurines and drawings in which a wonderful sense of form is found.
When there was a transition from the simple collection of food to its active production, from hunting and fishing to agriculture, humanity enters the new Stone Age, the Neolithic.
The Neolithic man had a keen sense of geometric shape. Roasting and painting clay vessels, making reed mats, baskets, fabrics, later - metal processing developed ideas about planar and spatial figures. Neolithic ornaments delighted the eye, revealing equality and symmetry.
- And where is symmetry found in nature?

Estimated answer: wings of butterflies, beetles, tree leaves ...

- Symmetry can be observed in architecture. When building buildings, builders clearly adhere to symmetry.

Therefore, the buildings are so beautiful. Also an example of symmetry is man, animals.

1. To invent your own ornament, depict it on a sheet of A4 format (you can draw in the form of a carpet).
2. Draw butterflies, note where there are elements of symmetry.

Use of the term in other scientific fields

In the future, symmetry will be considered from the point of view of geometry, but it is worth mentioning that this word is used not only here. Biology, virology, chemistry, physics, crystallography - all this is an incomplete list of areas in which this phenomenon is studied from different angles and in different conditions. For example, the classification depends on what science this term refers to. So, the division into types varies greatly, although some of the main ones, perhaps, remain unchanged everywhere.

Classification

There are several basic types of symmetry, of which three are most often found:

  • Mirror - observed relative to one or more planes. Also, the term is used to indicate the type of symmetry when a transformation such as reflection is used.
  • Beam, radial or axial - there are several options in various

In addition, the following types are also distinguished in geometry, they are found much less frequently, but no less curious:

  • sliding
  • rotational
  • point
  • progressive
  • screw
  • fractal
  • etc.

In biology, all species are called somewhat differently, although in fact they can be the same. The division into certain groups occurs on the basis of the presence or absence, as well as the number of some elements, such as centers, planes and axes of symmetry. They should be considered separately and in more detail.

Basic elements

Some features are distinguished in the phenomenon, one of which is necessarily present. The so-called basic elements include planes, centers and axes of symmetry. It is in accordance with their presence, absence and quantity that the type is determined.

A center of symmetry is a point inside a figure or crystal at which lines converge, connecting in pairs all sides parallel to each other. Of course, it does not always exist. If there are sides to which there is no parallel pair, then such a point cannot be found, since there is none. According to the definition, it is obvious that the center of symmetry is through which the figure can be reflected on itself. An example is, for example, a circle and a point in its middle. This item is usually referred to as C.

The plane of symmetry, of course, is imaginary, but it is it that divides the figure into two equal parts. It can pass through one or several sides, be parallel to it, and can divide them. For the same figure, several planes can exist at once. These elements are commonly referred to as P.

But perhaps the most common is what is called the "axis of symmetry." This common phenomenon can be seen both in geometry and in nature. And it is worthy of a separate consideration.

Often an element with respect to which the figure can be called symmetrical,

Examples are isosceles and equilateral triangles. In the first case, there will be a vertical axis of symmetry, on both sides of which there are equal faces, and in the second line they will intersect each corner and coincide with all bisectors, medians and heights. Ordinary triangles do not possess it.

By the way, the totality of all the above elements in crystallography and stereometry is called the degree of symmetry. This indicator depends on the number of axes, planes and centers.

Geometry Examples

Conventionally, it is possible to divide the entire set of objects of study of mathematicians into figures having an axis of symmetry, and those that do not have it. All regular polygons, circles, ovals, as well as some special cases automatically fall into the first category, while the rest fall into the second group.

As in the case when we spoke about the axis of symmetry of a triangle, this element for a quadrangle does not always exist. For a square, rectangle, rhombus or parallelogram, it is, but for an irregular figure, respectively, no. For a circle, the axis of symmetry is the set of lines that pass through its center.

In addition, it is interesting to consider volumetric figures from this point of view. In addition to all regular polygons and a ball, some cones, as well as pyramids, parallelograms and some others, will possess at least one axis of symmetry. Each case must be considered separately.

Examples in nature

Mirror symmetry in life is called bilateral, it is found most
often. Any person and so many animals are an example of this. Axial is called radial and is much less common, usually in the plant world. And still they are. For example, it is worth considering how many axes of symmetry a star has, and does it have them at all? Of course, we are talking about marine life, and not about the subject of study of astronomers. And the correct answer would be this: it depends on the number of rays of the star, for example five, if it is five-pointed.

In addition, radial symmetry is observed in many flowers: chamomile, cornflowers, sunflowers, etc. There are a huge number of examples, they are literally everywhere around.

This term, first of all, reminds the majority of medicine and cardiology, however, it initially has a slightly different meaning. In this case, the synonym is "asymmetry", that is, the absence or violation of regularity in one form or another. It can be met as an accident, and sometimes it can be a wonderful welcome, for example, in clothing or architecture. After all, there are a lot of symmetrical buildings, but the famous Leaning Tower of Pisa is slightly tilted, and although it is not the only one, it is the most famous example. It is known that this happened by chance, but this has its own charm.

In addition, it is obvious that the faces and bodies of people and animals are also not completely symmetrical. Even studies were conducted, according to the results of which the “right” persons were regarded as inanimate or simply unattractive. Nevertheless, the perception of symmetry and this phenomenon in itself are amazing and not yet fully understood, and therefore extremely interesting.

Types of symmetry

We also discuss some types of symmetry in order to fully study this concept. They are divided as follows:

  • Axial. The axis of symmetry is the line passing through the center of the body. Like this? If you superimpose the parts around the axis of symmetry, they will be equal. This can be seen in the example of a sphere.
  • Mirror. The axis of symmetry here is the straight line, relative to which the body can be reflected and get the inverse mapping. For example, butterfly wings are mirror symmetrical.
  • Central. The axis of symmetry is a point in the center of the body, with respect to which during all transformations the parts of the body are equal when superimposed.

    History of symmetry

    The very concept of symmetry is often the starting point in the theories and hypotheses of scientists of ancient times who were confident in the mathematical harmony of the universe, as well as in the manifestation of the divine principle. The ancient Greeks sacredly believed that the Universe is symmetrical, because the symmetry is magnificent. Man has long used the idea of ​​symmetry in his knowledge of the picture of the universe.

    In the V century BC, Pythagoras considered the sphere to be the most perfect form and thought that the Earth has the shape of a sphere and moves in the same way. He also believed that the Earth was moving in the form of some kind of "central fire" around which 6 planets (known at that time), the Moon, the Sun and all other stars were to rotate.

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    And the philosopher Plato considered the polyhedra as the personification of four natural elements:

    • tetrahedron - fire, since its top is directed upwards,
    • cube is earth, because it is the most stable body,
    • the octahedron is air, there is no explanation
    • the icosahedron is water, because the body does not have rough geometric shapes, angles, and so on,
    • the image of the entire universe was the dodecahedron.

    Because of all these theories, regular polyhedra are called Plato bodies.

    The architects of ancient Greece used symmetry. All their buildings were symmetrical, this is evidenced by images of the ancient temple of Zeus in Olympia.

    The Dutch artist M.K. Escher also resorted to symmetry in his paintings. In particular, a mosaic of two birds flying towards, became the basis of the painting "Day and Night."

    Also, our art historians did not neglect the rules of symmetry, as can be seen in the example of V. Vasnetsov’s painting “The Heroes”.

    What can I say, symmetry is a key concept for all artists over the course of many centuries, but in the 20th century, all the exact scientists also appreciated its meaning. The exact evidence is physical and cosmological theories, for example, the theory of relativity, string theory, absolutely all quantum mechanics. From the time of Ancient Babylon and ending with the advanced discoveries of modern science, the paths of studying symmetry and the discovery of its basic laws are traced.

    Symmetry of geometric shapes and bodies.

    Let’s take a closer look at geometric bodies. For example, the axis of symmetry of a parabola is a straight line passing through its apex and dissecting this body in half. This figure has one single axis.

    And with geometric figures, the situation is different. The axis of symmetry of the rectangle is also a straight line, but there are several of them. You can draw an axis parallel to the length segments, or you can draw the length. But not so simple. Here the line does not have axes of symmetry, since its end is not defined. Only central symmetry could exist, but, accordingly, there will not be one.

    You should also know that some bodies have many axes of symmetry. This is easy to guess. You don’t even need to talk about how many axes of symmetry a circle has. Any line passing through the center of the circle is such, and these lines are an infinite number.

    Some quadrangles may have two axes of symmetry. But the second should be perpendicular. This happens in the case of a rhombus and a rectangle. In the first axis of symmetry - diagonals, and in the second - the middle lines. Many of these axes are only in the square.

    Symmetry in nature

    Nature strikes with many examples of symmetry. Even our human body is symmetrical. Two eyes, two ears, a nose and a mouth are located symmetrically with respect to the central axis of the face. The arms, legs and the whole body are generally arranged symmetrically to the axis passing through the middle of our body.

    And how many examples constantly surround us! These are flowers, leaves, petals, vegetables and fruits, animals and even honeycombs of bees have a pronounced geometric shape and symmetry. All nature is arranged in an orderly manner, everything has its own place, which once again confirms the perfection of the laws of nature, in which symmetry is the main condition.

    We are constantly surrounded by any phenomena and objects, for example, a rainbow, a drop, flowers, petals and so on. Their symmetry is obvious, to some extent it is due to gravity. Often in nature, the term "symmetry" is understood as the regular change of day and night, seasons, and so on.

    Similar properties are observed wherever there is order and equality. Also, the laws of nature themselves - astronomical, chemical, biological, and even genetic, are subject to certain principles of symmetry, as they have perfect consistency, which means that balance has an all-encompassing scale. Consequently, axial symmetry is one of the fundamental laws of the universe as a whole.

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