What is symmetry in mathematics? Definition and examples

To understand what symmetry is in mathematics, it is necessary to further master the basic and advanced topics of algebra and geometry. It is also important for understanding drafting, architecture, rules for constructing a drawing. Despite the close connection with the most exact science - mathematics, symmetry is important for artists, artists, creators, and for those engaged in scientific activity, and in any field.

general information

Not only mathematics, but also natural sciences are largely based on the concept of symmetry. Moreover, it occurs in everyday life, is one of the basic for the nature of our universe. Understanding what is symmetry in mathematics, it is necessary to mention that there are several types of this phenomenon. It is customary to talk about such options:

• Two-sided, that is, when the symmetry is mirrored. This phenomenon in the academic environment is usually called "bilateral".
• Nin order. For this concept, the key phenomenon is the rotation angle, calculated by dividing 360 degrees by some specified value. In addition, the axis is determined in advance around which these turns are made.
• Padial, when the phenomenon of symmetry is observed, if rotations are made arbitrarily at some random angle. The axis is also selected in an independent manner. To describe this phenomenon, the SO (2) group is used.
• Spherical. In this case we are talking about three dimensions in which the object is rotated, choosing arbitrary angles. A specific case of isotropy is singled out when the phenomenon becomes local, characteristic of the medium or space.
• Rotational, combining the two groups described earlier.
• Lorentz is invariant when arbitrary rotations take place. For this type of symmetry, the key concept is "Minkowski space-time".
• Super, defined as the replacement of bosons by fermions.
• The highest, revealed in the course of group analysis.
• Translational, when there are shifts in space, for which scientists identify direction, distance. On the basis of the data obtained, a comparative analysis is carried out, which makes it possible to reveal the symmetry.
• Calibration, observed in the case of independence of the gauge theory under appropriate transformations. Here, special attention is paid to field theory, including focusing on the ideas of Yang-Mills.
• Kayno, belonging to the class of electronic configurations. About what this symmetry is, mathematics (6th grade) has no representation, because it is a higher-order science. The phenomenon is due to the secondary periodicity. It was discovered during the scientific work of E. Biron. The terminology was introduced by S. Shchukarev.

Mirror

During schooling, students are almost always asked to do the work "Symmetry around us" (a project in mathematics). As a rule, it is recommended for implementation in the sixth grade of a regular school with a general program of teaching subjects. To cope with the project, it is necessary first to familiarize yourself with the notion of symmetry, in particular, to reveal what is a mirror type as one of the basic and most understandable for children.

To identify the phenomenon of symmetry, consider a specific geometric figure, and also choose a plane. When do they speak of the symmetry of the object under consideration? First, a point is selected on it, and then it is reflected for it. Between them two spend a segment and calculate, at what angle to the previously chosen plane it passes.

Finding out what symmetry is in mathematics, remember that the plane chosen for revealing this phenomenon will be called the plane of symmetry and in no other way. The drawn line must intersect with it at a right angle. The distance from the point to this plane and from it to the second point of the segment must be equal.

Nuances

What else interesting can you learn by analyzing such a phenomenon as symmetry? Mathematics (Grade 6) tells us that two figures, considered to be symmetrical, are not necessarily identical to each other. The concept of equality exists in a narrow and broad sense. So, the symmetric objects in the narrow are not the same thing.

What is the best example of life? Elementary! What about our gloves, mittens? We are all used to wearing them and we know that we can not lose, because the second one can not be picked up in a pair, which means that we will have to buy both again. And why? Because the paired products, although symmetrical, but designed for left and right hand. This is a typical example of mirror symmetry. As to equality, such objects are recognized as "mirror-equal".

And what about the center?

To consider the central symmetry begin with the definition of the properties of the body, in relation to which it is necessary to evaluate the phenomenon. To call it symmetrical, first select some point located in the center. Next, choose a point (conditionally call it A) and look for a pair (conditionally denoted by E).

When determining the symmetry, the points A and E are connected with each other by a straight line that captures the central point of the body. Next, measure the resulting straight line. If the segment from the point A to the center of the object is equal to the segment separating the center from the point E, we can say that a center of symmetry is found. Central symmetry in mathematics is one of the key concepts that allow further development of the theory of geometry.

And if we rotate?

Understanding what symmetry is in mathematics, one can not overlook the concept of the rotational subtype of this phenomenon. In order to understand the terms, take a body that has a central point, and also define an integer.

In the course of the experiment, the given body is rotated by an angle equal to the result of dividing 360 degrees by the selected whole index. To do this, you need to know what the symmetry axis is (class 2, mathematics, school program). This axis is a straight line connecting two selected points. The symmetry of rotation can be said if, at the chosen angle of rotation, the body will be in the same position as before the manipulations.

In the case when the natural number was chosen 2, and the phenomenon of symmetry is discovered, it is said that axial symmetry in mathematics is determined. This is typical for a number of figures. A typical example: a triangle.

About the examples more

The practice of many years of teaching mathematics and geometry in high school shows that the easiest way to deal with the phenomenon of symmetry, explaining it on specific examples.

First, let's look at the sphere. For such a body symmetry phenomena are simultaneously characteristic:

• Central;
• Mirror;
• Rotational.

The main point is the point located exactly in the center of the figure. To pick up a plane, define a large circle and, as it were, "cut" it into strata. What does mathematics say? Rotation and central symmetry in the case of a sphere are interrelated concepts, while the diameter of the figure will serve as an axis for the phenomenon under consideration.

Another example is a round cone. This figure is characterized by axial symmetry. In mathematics and architecture, this phenomenon has found wide theoretical and practical application. Note: the axis of the cone serves as an axis for the phenomenon.

The direct prism demonstrates this phenomenon clearly. This figure is characterized by mirror symmetry. The plane is chosen as the "slice" parallel to the bases of the figure, remote from them at equal intervals. Creating a geometric, descriptive, architectural project (mathematical symmetry is as important as exact and descriptive sciences), remember the applicability in practice and the usefulness of planning the bearing elements of the phenomenon of specularity.

And if more interesting figures?

What can mathematics tell us about (6th grade)? Central symmetry is not only in such a simple and understandable object as a ball. It is also characteristic of more interesting and complex figures. For example, this is a parallelogram. For such an object, the central point becomes the one in which its diagonals intersect.

But if we consider an isosceles trapezoid, then it will be a figure with axial symmetry. You can identify it if you select the right axis. The body is symmetrical about a line perpendicular to the base and intersecting it exactly in the middle.

Symmetry in mathematics and architecture necessarily takes into account the rhombus. This figure is noteworthy in that it simultaneously combines two types of symmetry:

• axial;
• central.

As an axis, you must select the diagonal of the object. In the place where the diagonals of the rhombus intersect, its center of symmetry is located.

About beauty and symmetry

Forming a project for mathematics, whose symmetry would be a key topic, usually recall first and foremost the wise words of the great scientist Weil: "Symmetry is an idea that an ordinary person tries to understand for centuries, because it creates perfect beauty through a unique order."

As you know, other objects seem to most beautiful, while others repel, even if they do not have obvious flaws. Why is this happening? The answer to this question shows the interconnection of architecture and mathematics in symmetry, because it is this phenomenon that becomes the basis for evaluating the subject as aesthetically attractive.

One of the most beautiful women on our planet is the supermodel Brush of Tarlikton. She is sure that the success came first thanks to a unique phenomenon: her lips are symmetrical.

As you know, nature tends towards symmetry, and can not achieve it. This is not a general rule, but take a look at the surrounding people: in human faces, there is practically no absolute symmetry, although obviously the desire for it. The more symmetrical the face of the interlocutor, the more beautiful it seems.

How the symmetry became the idea of the beautiful

Surprisingly, symmetry is based on a person's perception of the beauty of the surrounding space and objects in it. For many centuries people have been trying to understand what is beautiful and what repels impartiality.

Symmetry, proportions - this is what helps to visually perceive an object and evaluate it positively. All elements, parts must be balanced and in reasonable proportions with each other. It has long been found out that asymmetric objects like people much less. All this is associated with the concept of "harmony". Above why it is so important for a person, since ancient times wise men, artists, artists have racked their brains.

It is worth looking closely at geometric figures, and the phenomenon of symmetry will become obvious and accessible for understanding. The most typical symmetrical phenomena in the space around us:

• rocks;
• Flowers and leaves of plants;
• Paired external organs inherent in living organisms.

The phenomena described are the source of the very nature. But what can you see symmetrical, taking a closer look at the products of human hands? It is noticeable that people are attracted to the creation of such a person, if they want to do something beautiful or functional (or both, and this at the same time):

• Patterns and ornaments, popular since ancient times;
• Building elements;
• Elements of technical constructions;
• needlework.

About terminology

"Symmetry" is a word that came into our language from the ancient Greeks, who for the first time paid close attention to this phenomenon and tried to study it. The term denotes the presence of some system, as well as a harmonious combination of parts of the object. Translating the word "symmetry", you can choose as synonyms:

• proportionality;
• Identity;
• Proportionality.

Since ancient times, symmetry is an important concept for the development of mankind in various fields and branches. Peoples from antiquity had common ideas about this phenomenon, mainly considering it in a broad sense. Symmetry meant harmony and balance. Nowadays, vocabulary is taught in an ordinary school. For example, what is the axis of symmetry (class 2, mathematics) is told to the teacher by the teacher in the usual occupation.

As an idea, this phenomenon often becomes the initial premise of scientific hypotheses and theories. This was especially popular in previous centuries, when the idea of mathematical harmony inherent in the system of the universe dominated the whole world. Experts of those epochs were convinced that symmetry is a manifestation of divine harmony. But in ancient Greece, philosophers assured that the entire universe is symmetric, and all this was based on the postulate: "Symmetry is beautiful."

Great Greeks and Symmetry

Symmetry stirred the minds of the most famous scientists of ancient Greece. Up to now, there have been evidences that Plato called separately to admire the correct polyhedra. In his opinion, such figures are the personifications of the elements of our world. There was the following classification:

Element	Figure
Fire	Tetrahedron, since its top tends to skyward.
Water	Icosahedron. The choice is due to the "coil" of the figure.
Air	Octahedron.
Earth	The most stable object, that is, the cube.
Universe	Dodecahedron.

In many ways precisely because of this theory it is customary to call regular polyhedra platonic bodies.

But the terminology was introduced even earlier, and here not the least role was played by the sculptor Poliklet.

Pythagoras and symmetry

During the period of Pythagoras' life and later, when his teaching went through its heyday, the phenomenon of symmetry was clearly formed. It was then that symmetry was subjected to scientific analysis, which gave important results for practical use.

According to the findings:

• Symmetry is based on the concepts of proportions, monotony and equality. When a particular concept is violated, the figure becomes less symmetrical, gradually shifting to completely asymmetric.
• There are 10 opposite pairs. According to the doctrine, symmetry is a phenomenon that reduces to a single opposite and thereby forms the universe as a whole. This postulate for many centuries exerted a strong influence on a number of sciences, both exact and philosophical, as well as natural.

Pythagoras and his followers distinguished "perfectly symmetrical bodies," to which the following were categorized:

• Each face is a polygon;
• Faces meet in the corners;
• The figure must have equal sides and angles.

It was Pythagoras who first said that there are only five such bodies. This great discovery marked the beginning of geometry and is extremely important for modern architecture.

Do you want to see with your own eyes the most beautiful phenomenon of symmetry? Catch a snowflake in winter. Surprisingly, the fact is that a tiny piece of ice falling from the sky has not only an extremely complex crystal structure, but also a perfectly symmetrical one. Consider it carefully: the snowflake is really beautiful, and its complex lines are fascinating.