The axes of symmetry. Figures having an axis of symmetry. What is the vertical axis of symmetry

The life of people is full of symmetry. It's convenient, beautiful, you do not have to invent new standards. But what is it really and is it so beautiful in nature, as it is commonly believed?

Symmetry

Since ancient times people have been trying to order the world around them. Therefore, something is considered beautiful, and something is not very. From an aesthetic point of view, both attractive gold and silver cross-sections are considered, as well as, of course, symmetry. This term is of Greek origin and literally means "proportionality". Of course, this is not just a matter of coincidence on this basis, but also of some others. In general, symmetry is a property of an object when, as a result of certain formations, the result is equal to the original data. This is found both in living and in inanimate nature, as well as in objects made by man.

First of all, the term "symmetry" is used in geometry, but finds application in many scientific fields, and its value remains generally unchanged. This phenomenon is often encountered and is considered interesting, as several of its species, as well as elements, are different. The use of symmetry is also interesting, because it is found not only in nature, but also in ornaments on fabrics, curbs of buildings and many other man-made objects. It is worth considering this phenomenon in more detail, because it is extremely exciting.

The 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 under different conditions. From what science this term refers to, for example, the classification depends. So, the division into types seriously varies, although some basic ones, perhaps, remain unchanged everywhere.

Classification

There are several basic types of symmetry, of which the most common are three:

• Mirror - observed relative to one or more planes. Also, the term is used to denote the type of symmetry, when a transformation such as reflection is used.
• Radial, radial or axial - there are several options in different Sources, in the general sense - symmetry with respect to a straight line. It can be regarded as a special case of the rotational variety.
• Central - there is symmetry about a certain point.

In addition, in geometry, the following types are also distinguished, they are much less common, 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 subdivision into certain groups occurs on the basis of the presence or absence, as well as the number of certain elements, such as centers, planes and symmetry axes. They should be considered separately and in more detail.

Basic Elements

In the phenomenon, some features are highlighted, one of which is necessarily present. The so-called basic elements include planes, centers and symmetry axes. It is in accordance with their presence, lack and quantity determined by the type.

The center of symmetry is the point inside the figure or crystal, in which the lines joining all pairs parallel to each other converge. Of course, it does not always exist. If there are sides to which there is no parallel pair, then such a point can not be found, since it does not exist. In accordance with the definition, it is obvious that the center of symmetry is that through which the figure can be reflected by itself. An example is, for example, a circle and a point in its middle. This element is usually referred to as C.

The plane of symmetry, of course, is imaginary, but it is she who 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 usually 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 separate consideration.

Axes

Often, an element relative to which a figure can be called symmetrical,
A straight line or a line appears. In any case, it is not a point or a plane. Then the axes of symmetry of the figures are considered. There can be a lot of them, and they can be arranged as you like: to divide sides or be parallel to them, and also to cross corners or not to do it. The axes of symmetry are usually denoted as L.

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

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

Examples in geometry

Conditionally it is possible to divide all set of objects of studying of mathematicians on the figures having an axis of symmetry, and such at which it is not present. The first category automatically includes all regular polygons, circles, ovals, and also some special cases, while the others fall into the second group.

As in the case of the symmetry axis of a triangle, this element does not always exist for a quadrilateral. For a square, a rectangle, a rhombus or a parallelogram, it is, and for an irregular figure, respectively, there is none. For a circle, the axis of symmetry is the set of straight lines that pass through its center.

In addition, it is interesting to consider the three-dimensional figures from this point of view. At least one axis of symmetry besides all regular polygons and a ball will have some cones, as well as pyramids, parallelograms and some others. Each case must be considered separately.

Examples in nature

Mirror symmetry in life is called bilateral, it occurs most
often. Any person and so many animals are an example of this. The axial is called radial and occurs much less often, as a rule, in the plant world. And yet they are. For example, it is worthwhile to consider how many axes of symmetry a star has, and whether it has any at all. Of course, we are talking about marine life, and not about the subject of studying astronomers. And the correct answer is 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, cornflower, sunflowers, etc. There are a lot of examples, they are literally everywhere.

Arrhythmia

This term, above all, reminds the majority of medicine and cardiology, but 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 seen as an accident, and sometimes it can be a wonderful device, for example, in clothes or architecture. After all, there are a lot of symmetrical buildings, but the famous Leaning Tower of Pisa is slightly inclined, and although she is not the only one, but this 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 humans and animals are also not completely symmetrical. There were even studies, according to which the "right" people were regarded as inanimate or simply unattractive. All the same, the perception of symmetry and this phenomenon is in itself surprising and has not yet been fully studied, and therefore extremely interesting.