Understanding what symmetry is in mathematics is necessary in order to further master the basic and advanced topics of algebra and geometry. This is also important for understanding drawing, architecture, and rules for constructing a drawing. Despite the close relationship with the most accurate science - mathematics, symmetry is important for artists, artists, creators, and for those involved in scientific activities, and in any field.
general information
Not only mathematics, but also the natural sciences are largely based on the concept of symmetry. Moreover, it is found in everyday life, is one of the basic for the nature of our universe. Understanding what symmetry is in mathematics, it is necessary to mention that there are several types of this phenomenon. It is customary to talk about such options:
- Bilateral, that is, when the symmetry is mirror. This phenomenon in the scientific community is usually called "bilateral."
- Uh, okay. For this concept, the key phenomenon is the angle of rotation, calculated by dividing 360 degrees by a given value. In addition, the axis around which these turns are made is determined in advance.
- Padial, when the phenomenon of symmetry is observed, if the turns take place arbitrarily at some random angle. The axis is also independently selected. 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 by choosing arbitrary angles. A specific case of isotropy is distinguished when the phenomenon becomes local, characteristic of the medium or space.
- Rotational, combining the two previously described groups.
- Lorentz-invariant when arbitrary rotations take place. For this type of symmetry, “Minkowski space-time” becomes a key concept.
- Super, defined as the replacement of bosons by fermions.
- Higher, revealed in the course of group analysis.
- Translational, when there are shifts of space for which scientists identify the direction, the distance. Based on the data obtained, a comparative analysis is carried out to reveal symmetry.
- The gauge observed in the case of independence of the gauge theory under the corresponding transformations. Here, special attention is paid to field theory, including focusing on the ideas of Yang-Mills.
- Kaino, belonging to the class of electronic configurations. Mathematics (6th grade) has no idea what such symmetry is, because it is a science of a higher order. The phenomenon is due to the secondary frequency. It was discovered during the scientific work of E. Biron. The terminology is introduced by S. Schukarev.
Mirror
During training at school, students are almost always asked to do the work "Symmetry Around Us" (a project in mathematics). As a rule, it is recommended for completion in the sixth grade of a regular school with a general curriculum for teaching subjects. To cope with the project, you must first familiarize yourself with the concept of symmetry, in particular, to identify what constitutes a mirror type as one of the basic and most understandable for children.
To identify the phenomenon of symmetry, a specific geometric figure is considered, and a plane is also selected. When do they talk about the symmetry of the object in question? First, a certain point is selected on it, and then they are reflected for it. A segment is drawn between the two and it is calculated at what angle to the previously selected plane it passes.
When understanding what symmetry is in mathematics, remember that the plane chosen to identify this phenomenon will be called the plane of symmetry and nothing else. The drawn segment should intersect with it at right angles. The distance from a point to this plane and from it to the second point of the segment must be equal.
Nuances
What else can you learn about by analyzing a phenomenon such as symmetry? Mathematics (Grade 6) tells us that two figures considered symmetrical are not necessarily identical to each other. The concept of equality exists in a narrow and broad sense. So, symmetrical objects in a narrow one are not the same thing.
What life example can be given? Elementary! What can you say about our gloves, mittens? We are all used to wearing them and we know that you can’t lose, because the second one cannot be picked up in a pair, which means that you will have to buy both again. And why? Because paired products, although symmetrical, are designed for the left and right hand. This is a typical example of mirror symmetry. As for equality, such objects are recognized as "mirror equal."
What about the center?
We begin to consider central symmetry by determining the properties of the body, in relation to which it is necessary to evaluate the phenomenon. To call it symmetrical, first choose a certain point located in the center. Next, they select a point (we will arbitrarily call it A) and look for a pair for it (we will arbitrarily denote E).
When determining the symmetry, points A and E are connected together by a straight line, capturing the central point of the body. Next, measure the resulting straight line. If the segment from point A to the center of the object is equal to the segment separating the center from point E, we can say that a center of symmetry is found. Central symmetry in mathematics is one of the key concepts that allow the further development of the theory of geometry.
And if we rotate?
When analyzing what symmetry is in mathematics, one cannot overlook the concept of a rotational subtype of this phenomenon. In order to deal with the terms, they take a body that has a central point, and also determine an integer.
During the experiment, a given body is rotated by an angle equal to the result of dividing 360 degrees by the selected integer index. To do this, you need to know what the axis of symmetry is (grade 2, mathematics, school curriculum). This axis is a straight line connecting two selected points. We can speak about the symmetry of rotation if, at the selected angle of rotation, the body will be in the same position as before the manipulations.
In the case when 2 was chosen by a natural number and a symmetry phenomenon was discovered, they say that axial symmetry is determined in mathematics. This is characteristic of a number of figures. A typical example: a triangle.
More about examples
The practice of many years of teaching mathematics and geometry in high school shows that it is easiest to deal with the phenomenon of symmetry, explaining it with concrete examples.
First, consider the scope. For such a body, symmetry phenomena are simultaneously characteristic:
- central;
- mirrored;
- rotational.
As the main one, choose a point located exactly in the center of the figure. To select a plane, a large circle is defined and as if “cut” into layers. What is math talking about? Rotation and central symmetry in the case of a ball are interrelated concepts, while the diameter of the figure will serve as the axis for the phenomenon under consideration.
Another good example is a round cone. Axial symmetry is characteristic of this figure . In mathematics and architecture, this phenomenon has found wide theoretical and practical application. Please note: the axis of the cone acts as the axis for the phenomenon.
The direct prism demonstrates the phenomenon under study . This figure is characterized by mirror symmetry. The plane selects a “slice” parallel to the base of the figure, removed from them at equal intervals. When creating a geometric, descriptive, architectural project (mathematics, symmetry is no less important than the exact and descriptive sciences), remember the applicability in practice and the usefulness of mirroring when planning the bearing elements.
And if more interesting figures?
What can mathematics tell us (grade 6)? 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, such a parallelogram. For such an object, the central point becomes the one at which its diagonals intersect.
But if we consider the isosceles trapezoid, then it will be a figure with axial symmetry. You can identify it if you select the axis correctly. 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:
As the 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, the symmetry of which would be a key topic, usually we primarily recall the wise words of the great scientist Weil: “Symmetry is an idea that ordinary people have been trying to understand for centuries, because it creates perfect beauty through a unique order.”
As you know, some objects seem beautiful to most, while others repel, even if they have no obvious flaws. Why it happens? The answer to this question shows the relationship of architecture and mathematics in symmetry, because it is this phenomenon that becomes the basis for assessing the subject as aesthetically attractive.
One of the most beautiful women on our planet is the supermodel Brush Tar conflicton. She is sure that she came to success primarily due to a unique phenomenon: her lips are symmetrical.
As you know, nature gravitates to symmetry, and cannot reach it. This is not a general rule, but take a look at the people around you: almost no absolute symmetry can be found in human faces, although the desire for it is obvious. The more symmetrical the face of the interlocutor, the more beautiful he seems.
How symmetry became the idea of ​​beauty
It is surprising that symmetry is based on a person’s perception of the beauty of the surrounding space and objects in it. For centuries, people have been striving to understand what seems 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 are liked by people much less. All this is associated with the concept of "harmony." Over why this is so important for man, from ancient times, wise men, artists, and artists were racking their brains.
It is worth taking a closer look at the geometric shapes, and the phenomenon of symmetry will become obvious and understandable. The most typical symmetrical phenomena in the space surrounding us:
- rocks;
- flowers and leaves of plants;
- paired external organs inherent in living organisms.
The described phenomena have a source of nature itself. But what can be seen symmetrical, looking closely at the products of human hands? It is noticeable that people tend to create just that if they strive to do something beautiful or functional (or both, and such at the same time):
- patterns and ornaments popular since ancient times;
- building elements;
- structural elements of technology;
- needlework.
About terminology
"Symmetry" is a word that came to our language from the ancient Greeks, who first paid close attention to this phenomenon and tried to study it. The term means the presence of a certain system, as well as a harmonious combination of parts of the object. Translating the word "symmetry", you can choose as synonyms:
- proportionality;
- sameness;
- proportionality.
Since ancient times, symmetry has been an important concept for the development of mankind in various fields and industries. The peoples from ancient times had common ideas about this phenomenon, mainly considering it in a broad sense. Symmetry signified harmony and poise. Nowadays, terminology is taught in a regular school. For example, what is the axis of symmetry (Grade 2, mathematics) for children tells the teacher in an ordinary lesson.
As an idea, this phenomenon often becomes the initial message of scientific hypotheses and theories. This was especially popular in previous centuries, when the idea of ​​mathematical harmony inherent in the very system of the universe reigned throughout the world. Connoisseurs of those eras were convinced that symmetry is a manifestation of divine harmony. But in ancient Greece, philosophers assured that the whole universe is symmetrical, and all this was based on the postulate: "Symmetry is beautiful."
Great Greeks and Symmetry
Symmetry excited the minds of the famous scientists of ancient Greece. Evidence has come down to this day that Plato urged to separately admire the regular polyhedrons. In his opinion, such figures are personifications of the elements of our world. The following classification existed:
Element | Figure |
Fire | Tetrahedron, as its peak tends to rise. |
Water | Icosahedron The choice is due to the "rolling" of the figure. |
Air | Octahedron. |
Land | The most stable object, that is, a cube. |
Universe | Dodecahedron. |
In many respects, it is because of this theory that regular polyhedra are called platonic solids.
But the terminology was introduced even earlier, and here the sculptor Poliklet played an important role.
Pythagoras and symmetry
During the life of Pythagoras and later, when his teachings experienced their heyday, the phenomenon of symmetry was clearly framed. It was then that symmetry was subjected to scientific analysis, which yielded important results for practical application.
According to the findings:
- Symmetry is based on the concepts of proportions, uniformity and equality. If one or another concept is violated, the figure becomes less symmetrical, gradually turning into completely asymmetric.
- There are 10 opposite pairs. According to the doctrine, symmetry is a phenomenon that brings together opposites and thereby forms the universe as a whole. For many centuries this postulate had a strong influence on a number of sciences, both exact, philosophical, and also natural.
Pythagoras and his followers singled out "completely symmetrical bodies", which were considered as satisfying conditions:
- each face is a polygon;
- faces are found in corners;
- the figure should have equal sides and angles.
It was Pythagoras who was the first to say that there are only five such bodies. This great discovery laid the foundation for 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 this tiny piece of ice falling from the sky has not only an extremely complex crystalline structure, but also is perfectly symmetrical. Consider it carefully: the snowflake is really beautiful, and its complex lines are fascinating.