People's lives are filled with symmetry. It’s convenient, beautiful, and there’s no need to invent new standards. But what is it really and is it as beautiful in nature as is commonly believed?
Symmetry
Since ancient times, people have sought to organize the world around them. Therefore, some things are considered beautiful, and some are not so much. From an aesthetic point of view, the golden and silver ratios are considered attractive, as well as, of course, symmetry. This term has Greek origin and literally means “proportionality”. Of course we're talking about not only about coincidence on this basis, but also on some others. IN in a general sense symmetry is a property of an object when, as a result of certain formations, the result is equal to the original data. This occurs 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 meaning remains generally unchanged. This phenomenon occurs quite often and is considered interesting, since several of its types, as well as elements, differ. The use of symmetry is also interesting, because it is found not only in nature, but also in patterns on fabric, borders of buildings and many other man-made objects. It is worth considering this phenomenon in more detail, because it is extremely fascinating.
Use of the term in other scientific fields
In what follows, symmetry will be considered from a geometric point of view, but it is worth mentioning that given word used not only here. Biology, virology, chemistry, physics, crystallography - all this is an incomplete list of areas in which this phenomenon studied with various sides and in different conditions. For example, the classification depends on what science this term refers to. Thus, the division into types varies greatly, although some basic ones, perhaps, remain unchanged throughout.
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Classification
There are several main types of symmetry, of which three are the most common:
In addition, in geometry there are also following types, they are much less common, but no less interesting:
- sliding;
- rotational;
- point;
- progressive;
- screw;
- fractal;
- etc.
In biology, all species are called slightly differently, although in essence they may be the same. Division into certain groups occurs on the basis of the presence or absence, as well as the quantity of certain elements, such as centers, planes and axes of symmetry. They should be considered separately and in more detail.
Basic elements
The phenomenon has certain features, one of which is necessarily present. 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.
The center of symmetry is the point inside a figure or crystal at which the lines connecting everything in pairs converge parallel friend to the other side. 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 it does not exist. According to the definition, it is obvious that the center of symmetry is that through which a figure can be reflected onto itself. An example would be, for example, a circle and a point in its middle. This element is usually designated as C.
The plane of symmetry, of course, is imaginary, but it is precisely it that divides the figure into two parts equal to each other. It can pass through one or more sides, be parallel to it, or divide them. For the same figure, several planes can exist at once. These elements are usually designated as P.
But perhaps the most common is what is called “axis of symmetry”. This is a common phenomenon that can be seen both in geometry and in nature. And it is worthy of separate consideration.
Axles
Often the element in relation to which a figure can be called symmetrical is 
a straight line or segment appears. In any case, we are not talking about 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 located in any way: dividing the sides or being parallel to them, as well as intersecting corners or not doing so. Axes of symmetry are usually designated as L.
Examples include isosceles and equilateral triangles. In the first case there will be vertical axis symmetry, on both sides of which equal faces, and in the second the lines will intersect each angle and coincide with all bisectors, medians and heights. Ordinary triangles do not have this.
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.
Examples in geometry
Conventionally, we can divide the entire set of objects of study by mathematicians into figures that have an axis of symmetry and those that do not. 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 it was said about the axis of symmetry of a triangle, this element for a quadrilateral does not always exist. For a square, rectangle, rhombus or parallelogram it is, and for irregular figure, accordingly, no. 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 volumetric figures from this point of view. At least one axis of symmetry in addition to all regular polygons and the 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 is most common
often. Any person and many animals are an example of this. Axial is called radial and is much less common, usually in flora. And yet they exist. For example, it is worth thinking about how many axes of symmetry a star has, and does it have any at all? Of course, we are talking about marine life, and not about the subject of study by astronomers. And the correct answer would be: 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: daisies, cornflowers, sunflowers, etc. There are a huge number of examples, they are literally everywhere around.
Arrhythmia
This term, first of all, reminds most of medicine and cardiology, but it initially has a slightly different meaning. IN in this case a synonym would be “asymmetry”, that is, the absence or violation of regularity in one form or another. It can be found as an accident, and sometimes it can become a wonderful technique, 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 accident, but this has its own charm.
In addition, it is obvious that the faces and bodies of people and animals are not completely symmetrical either. There have even been studies that show that “correct” faces are judged to be lifeless or simply unattractive. Still, the perception of symmetry and this phenomenon in itself are amazing and have not yet been fully studied, and therefore are extremely interesting.
Let us now consider the axes of symmetry of the sides of the triangle. Recall that the axis of symmetry of a segment is the perpendicular to the segment in its middle.
Any point of such a perpendicular is equally distant from the ends of the segment. Let now be the perpendiculars drawn through the midpoints of the sides BC and AC triangle ABC(Fig. 220) to these sides, i.e. the axis of symmetry of these two sides. The point of their intersection Q is equally distant from the vertices B and C of the triangle, since it lies on the axis of symmetry of side BC, and it is also equally distant from the vertices A and C. Consequently, it is equally distant from all three vertices of the triangle, including vertices A and B. This means that it lies on the axis of symmetry of the third side AB of the triangle. So, the axes of symmetry of the three sides of the triangle intersect at one point. This point is equally distant from the vertices of the triangle. Therefore, if you draw a circle with a radius equal to the distance of this point from the vertices of the triangle, with the center at the found point, then it will pass through all three vertices of the triangle. Such a circle (Fig. 220) is called a circumscribed circle. Conversely, if you imagine a circle passing through the three vertices of a triangle, then its center must be on equal distances from the vertices of the triangle and therefore belongs to each of the axes of symmetry of the sides of the triangle.
Therefore, a triangle has only one circumscribed circle: around given triangle you can describe a circle, and only one; its center lies at the point of intersection of three perpendiculars raised to the sides of the triangle at their midpoints.
In Fig. 221 shows circles described around an acute, rectangular and obtuse triangles; the center of the circumscribed circle lies in the first case inside the triangle, in the second - in the middle of the hypotenuse of the triangle, in the third - outside the triangle. This follows most simply from the properties of angles supported by an arc of a circle (see paragraph 210).
Since any three points that do not lie on the same line can be considered the vertices of a triangle, it can be argued that a single circle passes through any three points that do not belong to the line. Therefore, two circles have at most two points in common.
An axis of symmetry is a straight line, when rotated around it through a certain angle, the figure aligns with itself.
The smallest angle of rotation that brings the figure into self-alignment is called elementary axis rotation angle. The elementary rotation angle of the axis is an integer times 360 :
where n is an integer.
The number n, showing how many times the elementary angle of rotation of the axis is contained in 360 0, is called axis order.
Geometric figures can contain axes of any order, starting from an axis of the first order and ending with an axis of infinite order.
The elementary angle of rotation of the first order axis (n = 1) is equal to 360 0. Since each figure, being rotated around any direction by 360 0, is combined with itself, then each figure has an infinite number of first-order axes. Such axes are not characteristic, so they are usually not mentioned.
An axis of infinite order corresponds to an infinitely small elementary angle of rotation. This axis is present in all rotation figures as the axis of rotation.
Examples of axes of the third, fourth, fifth, sixth, etc. orders are perpendiculars to the drawing plane, passing through the centers of regular polygons, triangles, squares, pentagons, etc.
Thus, in geometry there is an infinite number of axes of different orders.
In crystalline polyhedra, not any symmetry axes are possible, but only axes of the first, second, third, fourth and sixth order.
Symmetry axes of the fifth and higher than the sixth order are impossible in crystals. This position is one of the basic laws of crystallography and is called the law of symmetry of crystals.
Like others geometric laws crystallography, the law of symmetry of crystals is explained by the lattice structure of the crystalline substance. Indeed, since the symmetry of a crystal is a manifestation of the symmetry of its internal structure, then only such symmetry elements are possible in crystals that do not contradict the properties of the spatial lattice.
Let us prove that the fifth order axis does not satisfy the laws of the spatial lattice and thereby prove its impossibility in crystalline polyhedra.
Let us assume that a fifth order axis in the spatial lattice is possible. Let this axis be perpendicular to the drawing plane, intersecting it at point O (Fig. 2.9). In a particular case, point O may coincide with one of the lattice nodes.
Rice. 2.9. A symmetry axis of the fifth order is impossible in spatial lattices
Let's take the lattice node A 1 closest to the axis, lying in the plane of the drawing. Since everything is repeated five times around the fifth-order axis, there should be only five nodes closest to it in the drawing plane: A 1, A 2, A 3, A 4, A 5. Located at equal distances from point O at the vertices of a regular pentagon, they are aligned with each other when rotated around O by 360/5 = 72°.
These five nodes, lying in the same plane, form a flat mesh of the spatial lattice and therefore all the basic properties of the lattice are applicable to them. If nodes A 1 and A 2 belong to a row of a flat grid with a gap A 1 A 2, then a row can be drawn through any lattice node parallel to the row A 1 A 2. Let's draw such a row through node A 3. This row, which also passes through node A 5, must have a gap equal to A 1 A 2, since in a spatial lattice all parallel rows have the same density.
Therefore, at a distance A 3 A x = A 1 A 2 from node A 3 there must be another node A x. However, the additional node A x turns out to lie closer to point O than node A 1, taken by condition to be closest to the fifth-order axis.
Thus, the assumption we made about the possibility of a fifth-order axis in spatial lattices led us to obvious absurdity and is therefore erroneous.
Since the existence of a fifth-order axis is incompatible with the basic properties of the spatial lattice, such an axis is impossible in crystals.
In a similar way, the impossibility of the existence of symmetry axes higher than the sixth order in crystals is proven and, conversely, the possibility of axes of the second, third, fourth and sixth order in crystals, the presence of which does not contradict the properties of spatial lattices.
To designate axes of symmetry, the letter L is used, and the order of the axis is indicated by a small number located to the right of the letter (for example, L 4 is a fourth-order axis).
In crystalline polyhedra, symmetry axes can pass through the centers of opposite faces perpendicular to them, through the midpoints of opposite edges perpendicular to them (only L 2) and through the vertices of the polyhedron. In the latter case, symmetrical faces and edges are equally inclined to a given axis.
A crystal can have several symmetry axes of the same order, the number of which is indicated by the coefficient in front of the letter. For example, in a rectangular parallelepiped there is 3L 2, i.e., three axes of symmetry of the second order; in the cube there are 3L 4, 4L 3 and 6L 2, i.e. three axes of symmetry of the fourth order, four axes of the third order and six axes of the second order, etc.
How many different axes of symmetry a triangle can have depends on its geometric shape. If this is an equilateral triangle, then it will have as many as three axes of symmetry.
And if it is an isosceles triangle, it will have only one axis of symmetry.
My sister's son is studying this topic in geometry lessons at school. An axis of symmetry is a straight line, when rotated around it by a specific angle symmetrical figure will take the same position in space that it occupied before the rotation, and some of its parts will be replaced by the same others. In an isosceles triangle there are three, in a right triangle there is one, in the others there are none, since their sides are not equal to each other.
It depends on what kind of triangle it is. An equilateral triangle has three axes of symmetry that pass through its three vertices. Isosceles triangle, accordingly, has one axis of symmetry. The remaining triangles do not have axes of symmetry.
The simplest thing you can remember is that an equilateral triangle has three equal sides and has three axes of symmetry
This makes it easier to remember the following
There are no equal sides, that is, all sides are different, which means there are no axes of symmetry
And in an isosceles triangle there is only one axis
You cannot simply answer how many axes of symmetry a triangle has without understanding which particular triangle we are talking about.
An equilateral triangle has three axes of symmetry, respectively.
An isosceles triangle has only one axis of symmetry.
Any other triangles with sides of different lengths do not have any axis of symmetry at all.
A triangle in which all sides are different in size does not have axes of symmetry.
A right triangle can have one axis of symmetry if its legs are equal.
In a triangle in which two sides are equal (isosceles), one axis can be drawn, and in which all three sides are equal (equilateral) - three.
Before answering the question of how many axes of symmetry a triangle has, you first need to remember what an axis of symmetry is.
So, to put it simply, in geometry, the axis of symmetry is a line along which if you bend a figure, you get identical halves.
but it is worth remembering that triangles are also different.
So, isosceles triangle (triangle with two equal sides) has one axis of symmetry.
Equilateral a triangle accordingly has 3 axes of symmetry, since all sides of this triangle are equal.
And here versatile A triangle has no axes of symmetry at all. No matter how you fold it and no matter where you draw straight lines, but since the sides are different, you won’t get two identical halves.
As far as I remember geometry, an equilateral triangle has three axes of symmetry passing through its vertices, these are its bisectors. U right triangle, as well as versatile, obtuse and acute triangles There are no axes of symmetry at all, but an isosceles has one.
And it’s easy to check - just imagine a line along which it can be cut in half so as to get two identical triangles.
Since triangles can be different, their axes of symmetry are respectively different quantities. For example, a triangle with different sides no axes of symmetry at all. And the equilateral has three of them. There is another type of triangle that has one axis of symmetry. It has two equal sides and one right angle.
An arbitrary triangle has no axes of symmetry. An isosceles triangle has one axis of symmetry - the median to the single side. Equilateral triangle has three axes of symmetry - these are its three medians.
People's lives are filled with symmetry. It’s convenient, beautiful, and there’s no need to invent new standards. But what is it really and is it as beautiful in nature as is commonly believed?
Symmetry
Since ancient times, people have sought to organize the world around them. Therefore, some things are considered beautiful, and some are not so much. From an aesthetic point of view, the golden and silver ratios are considered attractive, as well as, of course, symmetry. This term is of Greek origin and literally means “proportionality.” Of course, we are talking not only about coincidence on this basis, but also on some others. In a general sense, symmetry is a property of an object when, as a result of certain formations, the result is equal to the original data. It is found in both living and 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 meaning remains generally unchanged. This phenomenon occurs quite often and is considered interesting, since several of its types, as well as elements, differ. The use of symmetry is also interesting, because it is found not only in nature, but also in patterns on fabric, borders of buildings and many other man-made objects. It is worth considering this phenomenon in more detail, because it is extremely fascinating.
Use of the term in other scientific fields
In the following, 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. For example, the classification depends on what science this term refers to. Thus, the division into types varies greatly, although some basic ones, perhaps, remain unchanged throughout.
Classification
There are several main types of symmetry, of which three are the most common:
In addition, the following types are also distinguished in geometry; they are much less common, but no less interesting:
- sliding;
- rotational;
- point;
- progressive;
- screw;
- fractal;
- etc.
In biology, all species are called slightly differently, although in essence they may be the same. Division into certain groups occurs on the basis of the presence or absence, as well as the quantity of certain elements, such as centers, planes and axes of symmetry. They should be considered separately and in more detail.
Basic elements
The phenomenon has certain features, 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.
The center of symmetry is the point inside a figure or crystal at which the lines connecting in pairs all sides 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 cannot be found, since it does not exist. According to the definition, it is obvious that the center of symmetry is that through which a figure can be reflected onto itself. An example would be, for example, a circle and a point in its middle. This element is usually designated as C.
The plane of symmetry, of course, is imaginary, but it is precisely it that divides the figure into two parts equal to each other. It can pass through one or more sides, be parallel to it, or divide them. For the same figure, several planes can exist at once. These elements are usually designated as P.
But perhaps the most common is what is called “axis of symmetry”. This is a common phenomenon that can be seen both in geometry and in nature. And it is worthy of separate consideration.
Axles
Often the element in relation to which a figure can be called symmetrical is
a straight line or segment appears. In any case, we are not talking about a point or a plane. Then the figures are considered. There can be a lot of them, and they can be located in any way: dividing the sides or being parallel to them, as well as intersecting corners or not doing so. Axes of symmetry are usually designated as L.
Examples include isosceles and 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, the lines will intersect each angle and coincide with all bisectors, medians and altitudes. Ordinary triangles do not have this.
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.
Examples in geometry
Conventionally, we can divide the entire set of objects of study by mathematicians into figures that have an axis of symmetry and those that do not. All 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 talked about the axis of symmetry of a triangle, this element does not always exist for a quadrilateral. For a square, rectangle, rhombus or parallelogram it is, but for an irregular figure, accordingly, it is not. 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 three-dimensional figures from this point of view. In addition to all regular polygons and the ball, some cones, as well as pyramids, parallelograms and some others, will have at least one axis of symmetry. Each case must be considered separately.
Examples in nature
In life it is called bilateral, it occurs most
often. Any person and many animals are an example of this. The axial one is called radial and is found much less frequently, as a rule, in the plant world. And yet they exist. For example, it is worth thinking about how many axes of symmetry a star has, and does it have any at all? Of course, we are talking about marine life, and not about the subject of study by astronomers. And the correct answer would be: 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: daisies, cornflowers, sunflowers, etc. There are a huge number of examples, they are literally everywhere around.
Arrhythmia
This term, first of all, reminds most of medicine and cardiology, but it initially has a slightly different meaning. In this case, the synonym will be “asymmetry”, that is, the absence or violation of regularity in one form or another. It can be found as an accident, and sometimes it can become a wonderful technique, for example in clothing or architecture. After all, there are a lot of symmetrical buildings, but the famous one is slightly tilted, and although it is not the only one, it is the most famous example. It is known that this happened by accident, but this has its own charm.
In addition, it is obvious that the faces and bodies of people and animals are not completely symmetrical either. There have even been studies that show that “correct” faces are judged to be lifeless or simply unattractive. Still, the perception of symmetry and this phenomenon in itself are amazing and have not yet been fully studied, and therefore are extremely interesting.