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.
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 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 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 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
In life it is called bilateral, it occurs most
often. Any person and many animals are examples 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 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.
CHAPTER THREE
POLYhedra
V. THE CONCEPT OF SYMMETRY OF SPATIAL FIGURES
99. Central symmetry. Two figures are called symmetrical with respect to any point O in space if each point A of one figure corresponds in the other figure to point A", located on straight line OA on the other side of point O, at a distance equal to the distance point A from point O (Fig. 114). Point O is called center of symmetry figures.
We have already seen an example of such symmetrical figures in space (§ 53), when, by continuing the edges and faces of a polyhedral angle beyond the vertex, we obtained a polyhedral angle symmetrical to the given one. The corresponding segments and angles that make up two symmetrical figures are equal to each other. Nevertheless, the figures as a whole cannot be called equal: they cannot be combined with one another due to the fact that the order of the parts in one figure is different than in the other, as we saw in the example of symmetrical polyhedral angles.
In some cases, symmetrical figures can be combined, but their incongruous parts will coincide. For example, let’s take a right trihedral angle (Fig. 115) with a vertex at point O and edges OX, OY, OZ.
Let's build him symmetrical angle OX"Y"Z". The corner OXYZ can be combined with OX"Y"Z" so that the edge OX coincides with OY", and the edge OY coincides with OX". If we combine the corresponding edges OX with OX" and OY with OY", then the edges OZ and OZ" will be directed in opposite directions.
If symmetrical figures together constitute one geometric body, then this geometric body is said to have a center of symmetry. Thus, if a given body has a center of symmetry, then every point belonging to this body corresponds to a symmetrical point, also belonging to given body. Of the ones we reviewed geometric bodies have a center of symmetry, for example: 1) a parallelepiped, 2) a prism with a regular polygon at its base even number sides
Regular tetrahedron has no center of symmetry.
100. Symmetry relative to the plane. Two spatial figures are called symmetrical with respect to the plane P if each point A in one figure corresponds to a point A in the other, and the segment AA" is perpendicular to the plane P and is divided in half at the point of intersection with this plane.
Theorem. Any two corresponding segments in two symmetrical figures are equal to each other.
Let two figures be given, symmetrical with respect to the plane P. Let us select some two points A and B of the first figure, let A" and B" be the corresponding points of the second figure (Figure 116, the figures are not shown in the drawing).
Let further C be the point of intersection of segment AA" with plane P, D be the point of intersection of segment BB" with the same plane. By connecting points C and D with a straight line, we get two quadrilaterals ABDC and A"B"DC. Since AC = A"C, BD = B"D and
/
ACD = /
A.C.D. /
BDC = /
In "DC, as right angles, then these quadrilaterals are equal (which is easily verified by superposition). Consequently, AB = A"B". It immediately follows from this theorem that the corresponding plane and dihedral angles two figures that are symmetrical relative to a plane are equal to each other. However, it is impossible to combine these two figures with one another so that their corresponding parts match, since the order of the parts in one figure is the opposite of that, which takes place in another (this will be proven below, § 102). The simplest example of two figures that are symmetrical relative to a plane are: any object and its reflection in flat mirror; every figure is symmetrical with its mirror image relative to the plane of the mirror.
If any geometric body can be divided into two parts that are symmetrical with respect to a certain plane, then this plane is called the plane of symmetry of this body.
Geometric bodies with a plane of symmetry are extremely common in nature and in everyday life. The body of humans and animals has a plane of symmetry, dividing it into right and left parts.
This example makes it especially clear that symmetrical figures cannot be combined. Thus, the hands of the right and left hands are symmetrical, but they cannot be combined, which can be seen at least from the fact that the same glove cannot fit both the right and left hands. Big number household items has a plane of symmetry: chair, dinner table, bookcase, sofa, etc. Some, such as the dining table, even have not one, but two planes of symmetry (Fig. 117).
Usually, when considering an object that has a plane of symmetry, we strive to take such a position in relation to it that the plane of symmetry of our body, or at least our head, coincides with the plane of symmetry of the object itself. In this case. symmetrical shape the subject becomes especially noticeable.
101. Symmetry about the axis. Axis of symmetry of the second order. Two figures are called symmetrical with respect to the l axis (the axis is a straight line) if each point A of the first figure corresponds to point A" of the second figure, so that the segment AA" is perpendicular to the l axis, intersects with it and is divided in half at the intersection point. The l axis itself is called the second order axis of symmetry.
From this definition it immediately follows that if two geometric bodies, symmetrical about any axis, are intersected by a plane perpendicular to this axis, then the cross-section will result in two flat figures, symmetrical with respect to the point of intersection of the plane with the axis of symmetry of the bodies.
From here it is further easy to deduce that two bodies that are symmetrical about an axis can be combined with one another by rotating one of them 180° around the axis of symmetry. In fact, let us imagine all possible planes perpendicular to the axis of symmetry.
Each such plane intersecting both bodies contains figures that are symmetrical with respect to the point where the plane meets the axis of symmetry of the bodies. If you force the cutting plane to slide on its own, rotating it around the axis of symmetry of the body by 180°, then the first figure coincides with the second.
This is true for any cutting plane. Rotation of all sections of the body by 180° is equivalent to rotation of the entire body by 180° around the axis of symmetry. This is where the validity of our statement follows.
If after rotation spatial figure around a certain straight line it coincides 180° with itself, then they say that the figure has this straight line as its axis of symmetry of the second order.
The name “second-order symmetry axis” is explained by the fact that during a full revolution around this axis, the body will, in the process of rotation, twice take a position coinciding with the original one (including the original one). Examples of geometric bodies that have an axis of symmetry of the second order are:
1) regular pyramid with an even number of side faces; its axis of symmetry is its height;
2) cuboid; it has three axes of symmetry: straight lines connecting the centers of its opposite faces;
3) correct prism with an even number of side faces. The axis of its symmetry is each straight line connecting the centers of any pair of its opposite faces (the side faces and the two bases of the prism). If the number of lateral faces of the prism is 2 k, then the number of such axes of symmetry will be k+ 1. In addition, the axis of symmetry for such a prism is each straight line connecting the midpoints of its opposite side edges. The prism has such axes of symmetry A.
So the correct one is 2 k-faceted prism has 2 k+1 axes, symmetry.
102. Dependence between various types symmetry in space. There is a relationship between different types of symmetry in space - axial, planar and central - expressed by the following theorem.
Theorem. If the figure F is symmetrical with the figure F" relative to the plane P and at the same time symmetrical with the figure F" relative to the point O lying in the plane P, then the figures F" and F" are symmetrical relative to the axis passing through the point O and perpendicular to the plane R.
Let's take some point A of figure F (Fig. 118). It corresponds to point A" of figure F" and point A" of figure F" (figures F, F" and F" themselves are not shown in the drawing).
Let B be the point of intersection of the segment AA" with the plane P. Let us draw the plane through points A, A" and O. This plane will be perpendicular to the plane P, since it passes through the straight line AA", perpendicular to this plane. In the plane AA"O we will draw straight line OH perpendicular to OB. This straight line OH will also be perpendicular to the plane P. Next, let C be the intersection point of straight lines AA and OH.
In the triangle AA"A"", the segment BO connects the midpoints of the sides AA" and AA", therefore, BO || A"A", but BO_|_OH, which means AA"_|_OH. Further, since O is the midpoint sides AA", and CO || AA", then A"C = A"C. From here we conclude that points A" and A" are symmetrical with respect to the OH axis. The same is true for all other points of the figure. This means that our theorem is proven. It immediately follows from this theorem, that two figures that are symmetrical relative to the plane cannot be combined so that their corresponding parts are combined. In fact, the figure F" is combined with F" by rotating around the axis OH by 180°. But the figures F" and F cannot be combined. as symmetrical about the point, therefore, figures F and F" also cannot be combined.
103. Axes of symmetry of higher orders. A figure that has an axis of symmetry aligns with itself after rotating around the axis of symmetry through an angle of 180°. But there may be cases when the figure comes into alignment with starting position after rotation around a certain axis through an angle less than 180°. Thus, if the body does full turn around this axis, then during the rotation process it will align several times with its original position. This axis of rotation is called the axis of symmetry higher order, and the number of body positions coinciding with the original one is called the order of the symmetry axis. This axis may not coincide with the axis of symmetry of the second order. Thus, a regular triangular pyramid does not have a second-order symmetry axis, but its height serves as a third-order symmetry axis for it. In fact, after rotating this pyramid around the height at an angle of 120°, it aligns with itself (Fig. 119).
When the pyramid rotates around a height, it can occupy three positions that coincide with the original one, including the original one. It is easy to notice that every symmetry axis of even order is at the same time an symmetry axis of second order.
Examples of higher order symmetry axes:
1) Correct n-a carbon pyramid has an axis of symmetry n-th order. This axis is the height of the pyramid.
2) Correct n- a carbon prism has an axis of symmetry n-th order. This axis is a straight line connecting the centers of the bases of the prism.
104. Symmetry of the cube. As for any parallelepiped, the point of intersection of the diagonals of the cube is the center of its symmetry.
The cube has nine planes of symmetry: six diagonal planes and three planes passing through the midpoints of each four of its parallel edges.
The cube has nine axes of symmetry of the second order: six straight lines connecting the midpoints of its opposite edges, and three straight lines connecting the centers of opposite faces (Fig. 120).
These last lines are axes of symmetry of the fourth order. In addition, the cube has four third-order axes of symmetry, which are its diagonals. In fact, the diagonal of the cube AG (Fig. 120) is obviously equally inclined to the edges AB, AD and AE, and these edges are equally inclined to one another. If we connect points B, D and E, we get the correct triangular pyramid ADBE, for which the diagonal of the cube AG serves as its height. When this pyramid aligns with itself when rotated around the height, the entire cube will align with its original position. As is easy to see, the cube has no other axes of symmetry. Let's see how many different ways the cube can be combined with itself. Rotation around the ordinary axis of symmetry gives one position of the cube, different from the original one, in which the cube as a whole is aligned with itself.
Rotation around a third-order axis produces two such positions, and rotation around a fourth-order axis produces three such positions. Since the cube has six axes of the second order (these are ordinary axes of symmetry), four axes of the third order and three axes of the fourth order, there are 6 1 + 4 2 + 3 3 = 23 positions of the cube, different from the original one, at which it is combined with itself yourself.
It is easy to verify directly that all these positions are different from one another, and also from the initial position of the cube. Together with the starting position, they make up 24 ways of combining the cube with itself.
Definition of symmetry;
Definition of symmetry;
Central symmetry;
Axial symmetry;
Symmetry relative to the plane;
Rotation symmetry;
Mirror symmetry;
Symmetry of similarity;
Plant symmetry;
Animal symmetry;
Symmetry in architecture;
Is man a symmetrical creature?
Symmetry of words and numbers;
SYMMETRY
SYMMETRY- proportionality, sameness in the arrangement of parts of something on opposite sides of a point, straight line or plane.
(Ozhegov's Explanatory Dictionary)
So, a geometric object is considered symmetrical if something can be done to it, after which it will remain unchanged.
ABOUT ABOUT ABOUT called center of symmetry of the figure.
The figure is said to be symmetrical about the point ABOUT, if for each point of the figure there is a point symmetrical to it relative to the point ABOUT also belongs to this figure. Dot ABOUT called center of symmetry of the figure.
circle and parallelogram center of the circle ). Schedule Not even function
An example of a figure that does not have a center of symmetry is arbitrary triangle.
Examples of figures that have central symmetry are circle and parallelogram. The center of symmetry of a circle is center of the circle, and the center of symmetry of the parallelogram is the point of intersection of its diagonals. Any straight line also has central symmetry ( any point on a line is its center of symmetry). Schedule odd function symmetrical about the origin.
A A a called axis of symmetry of the figure.
The figure is said to be symmetrical about a straight line A, if for each point of the figure there is a point symmetrical to it relative to the straight line A also belongs to this figure. Straight a called axis of symmetry of the figure.
At an unturned corner one axis of symmetry angle bisector one axis of symmetry three axes of symmetry two axes of symmetry, and the square is four axes of symmetry relative to the y-axis.
There are figures that do not have a single axis of symmetry. Such figures include parallelogram, other than a rectangle, scalene triangle.
At an unturned corner one axis of symmetry- straight line on which it is located angle bisector. An isosceles triangle also has one axis of symmetry, and an equilateral triangle is three axes of symmetry. A rectangle and a rhombus that are not squares have two axes of symmetry, and the square is four axes of symmetry. The circle has an infinite number of them. The graph of an even function is symmetrical when constructed relative to the y-axis.
Points A And A1 A A AA1 And perpendicular A counts symmetrical to itself
Points A And A1 are called symmetrical relative to the plane A(plane of symmetry), if the plane A passes through the middle of the segment AA1 And perpendicular to this segment. Each point of the plane A counts symmetrical to itself. Two figures are called symmetrical relative to the plane (or mirror-symmetrical relative) if they consist of pairwise symmetrical points. This means that for each point of one figure, a point symmetrical (relatively) to it lies in another figure.
The body (or figure) has rotational symmetry, if when turning an angle 360º/n, where n is an integer fully compatible
The body (or figure) has rotational symmetry, if when turning an angle 360º/n, where n is an integer, near some straight line AB (axis of symmetry) it fully compatible with its original position.
Radial symmetry- a form of symmetry that is preserved when an object rotates around a specific point or line. Often this point coincides with the center of gravity of the object, that is, the point at which intersects an infinite number of axes of symmetry. Similar objects can be circle, ball, cylinder or cone.
Mirror symmetry binds anyone
Mirror symmetry binds anyone an object and its reflection in a plane mirror. One figure (or body) is said to be mirror symmetrical to another if together they form a mirror image symmetrical figure(or body). Symmetrically mirrored figures, for all their similarities, differ significantly from each other. Two mirror-symmetrical flat figures can always be superimposed on each other. However, to do this it is necessary to remove one of them (or both) from their common plane.
Symmetry of similarity nesting dolls.
Symmetry of similarity are unique analogues of previous symmetries with the only difference being that they are associated with simultaneous reduction or increase in similar parts of the figure and the distances between them. The simplest example of such symmetry is nesting dolls.
Sometimes figures can have different types of symmetry. For example, some letters have rotational and mirror symmetry: AND, N, M, ABOUT, A.
There are many other types of symmetries that are abstract in nature. For example:
Commutation symmetry, which consists in the fact that if identical particles are swapped, then no changes occur;
Gauge symmetries connected with zoom change. In inanimate nature, symmetry primarily arises in such a natural phenomenon as crystals, from which almost all solids are composed. It is this that determines their properties. The most obvious example of the beauty and perfection of crystals is the well-known snowflake.
We encounter symmetry everywhere: in nature, technology, art, science. The concept of symmetry runs through centuries of history human creativity. The principles of symmetry play important role in physics and mathematics, chemistry and biology, technology and architecture, painting and sculpture, poetry and music. The laws of nature are also subject to the principles of symmetry.
axis of symmetry.
Many flowers have an interesting property: they can be rotated so that each petal takes the position of its neighbor, and the flower aligns with itself. This flower has axis of symmetry.
Helical symmetry observed in the arrangement of leaves on the stems of most plants. Arranging in a spiral along the stem, the leaves seem to spread out in all directions and do not block each other from the light, which is extremely necessary for plant life.
Bilateral symmetry Plant organs are also present, for example, the stems of many cacti. Often found in botany radially symmetrically arranged flowers.
dividing line.
Symmetry in animals means correspondence in size, shape and outline, as well as the relative arrangement of body parts located on opposite sides dividing line.
The main types of symmetry are radial(radial) – it is possessed by echinoderms, coelenterates, jellyfish, etc.; or bilateral(two-sided) - we can say that every animal (be it an insect, fish or bird) consists of two halves- right and left.
Spherical symmetry occurs in radiolarians and sunfishes. Any plane drawn through the center divides the animal into equal halves.
The symmetry of a structure is associated with the organization of its functions. The projection of the plane of symmetry - the axis of the building - usually determines the location of the main entrance and the beginning of the main traffic flows.
Every detail in a symmetrical system exists like a double to your obligatory couple, located on the other side of the axis, and due to this it can only be considered as part of the whole.
Most common in architecture mirror symmetry. The buildings of Ancient Egypt and the temples of ancient Greece, amphitheatres, baths, basilicas and triumphal arches of the Romans, palaces and churches of the Renaissance, as well as numerous buildings of modern architecture are subordinate to it.
accents
To better reflect symmetry, buildings are placed accents- particularly significant elements (domes, spiers, tents, main entrances and staircases, balconies and bay windows).
To design the decoration of architecture, an ornament is used - a rhythmically repeating pattern based on the symmetrical composition of its elements and expressed by line, color or relief. Historically, several types of ornaments have developed based on two sources - natural forms and geometric figures.
But an architect is first and foremost an artist. And therefore even the most “classical” styles were more often used dissymmetry– nuanced deviation from pure symmetry or asymmetry- deliberately asymmetrical construction.
No one will doubt that outwardly a person is built symmetrically: the left hand always corresponds to the right and both hands are exactly the same. But the similarities between our hands, ears, eyes and other parts of the body are the same as between an object and its reflection in a mirror.
right his half rough features, inherent male. Left half
Numerous measurements of facial parameters in men and women have shown that right his half compared to the left, it has more pronounced transverse dimensions, which gives the face a more rough features characteristic of the male sex. Left half the face has more pronounced longitudinal dimensions, which gives it smooth lines and femininity. This fact explains the predominant desire of females to pose in front of artists with the left side of their faces, and males with the right.
Palindrome
Palindrome(from the gr. Palindromos - running back) is an object in which the symmetry of its components is specified from beginning to end and from end to beginning. For example, a phrase or text.
The straight text of a palindrome, read according to the normal reading direction of a given script (usually from left to right), is called upright, reverse – by rover or reverse(from right to left). Some numbers also have symmetry.
Consider axial and central symmetries as properties of some geometric shapes; Consider axial and central symmetries as properties of some geometric figures; Know how to build symmetrical points and be able to recognize figures that are symmetrical with respect to a point or line; Be able to construct symmetrical points and be able to recognize figures that are symmetrical with respect to a point or line; Improving problem solving skills; Improving problem solving skills; Continue to work on accurately recording and completing geometric drawings; Continue to work on accurately recording and completing geometric drawings;
Oral work“Gentle questioning” Oral work “Gentle questioning” What point is called the middle of the segment? Which triangle is called isosceles? What properties do the diagonals of a rhombus have? State the bisector property of an isosceles triangle. Which lines are called perpendicular? Which triangle is called equilateral? What properties do the diagonals of a square have? What figures are called equal?
What new concepts did you learn about in class? What new concepts did you learn about in class? What new things have you learned about geometric shapes? What new things have you learned about geometric shapes? Give examples of geometric shapes that have axial symmetry. Give examples of geometric shapes that have axial symmetry. Give an example of figures that have central symmetry. Give an example of figures that have central symmetry. Give examples of items from surrounding life having one or two types of symmetry. Give examples of objects from the surrounding life that have one or two types of symmetry.
Symmetry is associated with harmony and order. And for good reason. Because the question of what symmetry is, there is an answer in the form of a literal translation from ancient Greek. And it turns out that it means proportionality and immutability. And what could be more orderly than a strict definition of location? And what can be called more harmonious than something that strictly corresponds to size?
What does symmetry mean in different sciences?
Biology. An important component of symmetry in it is that animals and plants have regularly arranged parts. Moreover, there is no strict symmetry in this science. There is always some asymmetry. It admits that the parts of the whole do not coincide with absolute precision.
Chemistry. The molecules of a substance have a certain pattern in their arrangement. It is their symmetry that explains many properties of materials in crystallography and other branches of chemistry.Physics. A system of bodies and changes in it are described using equations. They contain symmetrical components, which simplifies the entire solution. This is accomplished by searching for conserved quantities.
Mathematics. It is there that basically explains what symmetry is. Moreover higher value it is given in geometry. Here, symmetry is the ability to display in figures and bodies. IN in the narrow sense it comes down simply to a mirror image.
How do different dictionaries define symmetry?
No matter which of them we look at, the word “proportionality” will appear everywhere. In Dahl one can also see such an interpretation as uniformity and equality. In other words, symmetrical means the same. It also says that it is boring; something that doesn’t have it looks more interesting.
When asked what symmetry is, Ozhegov’s dictionary already talks about the sameness in the position of parts relative to a point, line or plane.
Ushakov’s dictionary also mentions proportionality, as well as the complete correspondence of two parts of the whole to each other.
When do we talk about asymmetry?
The prefix “a” negates the meaning of the main noun. Therefore, asymmetry means that the arrangement of elements does not lend itself to a certain pattern. There is no immutability in it.
This term is used in situations where the two halves of an item are not completely identical. Most often they are not at all similar.
In living nature, asymmetry plays an important role. Moreover, it can be both useful and harmful. For example, the heart is placed in the left half of the chest. Due to this, the left lung is significantly smaller in size. But it is necessary.
About central and axial symmetry
In mathematics, the following types are distinguished:
- central, that is, made relative to one point;
- axial, which is observed near a straight line;
- specular, it is based on reflections;
- transfer symmetry.
What is an axis and center of symmetry? This is a point or line relative to which any point on the body can find another. Moreover, such that the distance from the original to the resulting one is divided in half by the axis or center of symmetry. As these points move, they describe identical trajectories.
The easiest way to understand what symmetry about an axis is is with an example. The notebook sheet needs to be folded in half. The fold line will be the axis of symmetry. If you draw a perpendicular line to it, then all the points on it will have points lying at the same distance on the other side of the axis.
In situations where it is necessary to find the center of symmetry, you need to do in the following way. If there are two figures, then find their identical points and connect them with a segment. Then divide in half. When there is only one figure, knowledge of its properties can help. Often this center coincides with the intersection point of the diagonals or heights.
What shapes are symmetrical?
Geometric figures can have axial or central symmetry. But it is not required condition, there are many objects that do not possess it at all. For example, a parallelogram has a central one, but it does not have an axial one. But non-isosceles trapezoids and triangles have no symmetry at all.
If central symmetry is considered, there are quite a lot of figures that have it. These are a segment and a circle, a parallelogram and all regular polygons with a number of sides that is divisible by two.
The center of symmetry of a segment (also a circle) is its center, and for a parallelogram it coincides with the intersection of the diagonals. While for regular polygons this point also coincides with the center of the figure.
If a straight line can be drawn in a figure, along which it can be folded, and the two halves coincide, then it (the straight line) will be an axis of symmetry. What's interesting is how many axes of symmetry different shapes have.
For example, spicy or obtuse angle has only one axis, which is its bisector.
If you need to find the axis in isosceles triangle, then you need to draw the height to its base. The line will be the axis of symmetry. And just one. And in an equilateral one there will be three of them at once. In addition, the triangle also has central symmetry relative to the point of intersection of the heights.
The circle may have infinite number axes of symmetry. Any straight line that passes through its center can fulfill this role.
A rectangle and a rhombus have two axes of symmetry. In the first, they pass through the middles of the sides, and in the second, they coincide with the diagonals.
The square combines the previous two figures and has 4 axes of symmetry at once. They are the same as those of a rhombus and a rectangle.