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.

If you think for a minute and imagine any object in your mind, then in 99% of cases the figure that comes to mind will be of the correct shape. Only 1% of people, or rather their imagination, will draw an intricate object that looks completely wrong or disproportionate. This is rather an exception to the rule and refers to unconventionally thinking individuals with a special view of things. But returning to the absolute majority, it is worth saying that a significant proportion of correct items still prevails. The article will talk exclusively about them, namely about symmetrical drawing of them.

Drawing the right objects: just a few steps to the finished drawing

Before you start drawing a symmetrical object, you need to select it. In our version it will be a vase, but even if it doesn’t in any way resemble what you decided to depict, don’t despair: all the steps are absolutely identical. Follow the sequence and everything will work out:

1. All objects of regular shape have a so-called central axis, which should definitely be highlighted when drawing symmetrically. To do this, you can even use a ruler and draw a straight line down the center of the landscape sheet.
2. Next, look carefully at the item you have chosen and try to transfer its proportions onto a sheet of paper. This is not difficult to do if you mark light strokes on both sides of the line drawn in advance, which will later become the outlines of the object being drawn. In the case of a vase, it is necessary to highlight the neck, bottom and the widest part of the body.
3. Do not forget that symmetrical drawing does not tolerate inaccuracies, so if there are some doubts about the intended strokes, or you are not sure of the correctness of your own eye, double-check the laid down distances with a ruler.
4. The last step is connecting all the lines together.

Symmetrical drawing is available to computer users

Due to the fact that most of the objects around us have the correct proportions, in other words, they are symmetrical, computer application developers have created programs in which you can easily draw absolutely everything. You just need to download them and enjoy the creative process. However, remember, a machine will never be a substitute for a sharpened pencil and a sketchbook.

You will need

• - properties of symmetrical points;
• - properties of symmetrical figures;
• - ruler;
• - square;
• - compass;
• - pencil;
• - paper;
• - a computer with a graphics editor.

Instructions

Draw a straight line a, which will be the axis of symmetry. If its coordinates are not specified, draw it arbitrarily. Place an arbitrary point A on one side of this line. You need to find a symmetrical point.

Helpful advice

Symmetry properties are used constantly in AutoCAD. To do this, use the Mirror option. To construct an isosceles triangle or isosceles trapezoid, it is enough to draw the lower base and the angle between it and the side. Reflect them using the specified command and extend the sides to the required size. In the case of a triangle, this will be the point of their intersection, and for a trapezoid, this will be a given value.

You constantly come across symmetry in graphic editors when you use the “flip vertically/horizontally” option. In this case, the axis of symmetry is taken to be a straight line corresponding to one of the vertical or horizontal sides of the picture frame.

Sources:

• how to draw central symmetry

Constructing a cross section of a cone is not such a difficult task. The main thing is to follow a strict sequence of actions. Then this task will be easily accomplished and will not require much labor from you.

You will need

• - paper;
• - pen;
• - circle;
• - ruler.

Instructions

When answering this question, you must first decide what parameters define the section.
Let this be the straight line of intersection of the plane l with the plane and the point O, which is the intersection with its section.

The construction is illustrated in Fig. 1. The first step in constructing a section is through the center of the section of its diameter, extended to l perpendicular to this line. The result is point L. Next, draw a straight line LW through point O, and construct two guide cones lying in the main section O2M and O2C. At the intersection of these guides lie point Q, as well as the already shown point W. These are the first two points of the desired section.

Now draw a perpendicular MS at the base of the cone BB1 ​​and construct generatrices of the perpendicular section O2B and O2B1. In this section, through point O, draw a straight line RG parallel to BB1. Т.R and Т.G are two more points of the desired section. If the cross section of the ball were known, then it could be built already at this stage. However, this is not an ellipse at all, but something elliptical that has symmetry with respect to the segment QW. Therefore, you should build as many section points as possible in order to connect them later with a smooth curve to obtain the most reliable sketch.

Construct an arbitrary section point. To do this, draw an arbitrary diameter AN at the base of the cone and construct the corresponding guides O2A and O2N. Through t.O, draw a straight line passing through PQ and WG until it intersects with the newly constructed guides at points P and E. These are two more points of the desired section. Continuing in the same way, you can find as many points as you want.

True, the procedure for obtaining them can be slightly simplified using symmetry with respect to QW. To do this, you can draw straight lines SS’ in the plane of the desired section, parallel to RG until they intersect with the surface of the cone. The construction is completed by rounding the constructed polyline from chords. It is enough to construct half of the desired section due to the already mentioned symmetry with respect to QW.

Video on the topic

Tip 3: How to graph a trigonometric function

You need to draw schedule trigonometric functions? Master the algorithm of actions using the example of constructing a sinusoid. To solve the problem, use the research method.

You will need

• - ruler;
• - pencil;
• - knowledge of the basics of trigonometry.

Instructions

Video on the topic

note

If the two semi-axes of a single-strip hyperboloid are equal, then the figure can be obtained by rotating a hyperbola with semi-axes, one of which is the above, and the other, different from the two equal ones, around the imaginary axis.

Helpful advice

When examining this figure relative to the Oxz and Oyz axes, it is clear that its main sections are hyperbolas. And when this spatial figure of rotation is cut by the Oxy plane, its section is an ellipse. The neck ellipse of a single-strip hyperboloid passes through the origin of coordinates, because z=0.

The throat ellipse is described by the equation x²/a² +y²/b²=1, and the other ellipses are composed by the equation x²/a² +y²/b²=1+h²/c².

Sources:

• Ellipsoids, paraboloids, hyperboloids. Rectilinear generators

The shape of a five-pointed star has been widely used by man since ancient times. We consider its shape beautiful because we unconsciously recognize in it the relationships of the golden section, i.e. the beauty of the five-pointed star is justified mathematically. Euclid was the first to describe the construction of a five-pointed star in his Elements. Let's join in with his experience.

You will need

• ruler;
• pencil;
• compass;
• protractor.

Instructions

The construction of a star comes down to the construction and subsequent connection of its vertices to each other sequentially through one. In order to build the correct one, you need to divide the circle into five.
Construct an arbitrary circle using a compass. Mark its center with point O.

Mark point A and use a ruler to draw line segment OA. Now you need to divide the segment OA in half; to do this, from point A, draw an arc of radius OA until it intersects the circle at two points M and N. Construct the segment MN. The point E where MN intersects OA will bisect segment OA.

Restore the perpendicular OD to the radius OA and connect points D and E. Make a notch B on OA from point E with radius ED.

Now, using line segment DB, mark the circle into five equal parts. Label the vertices of the regular pentagon sequentially with numbers from 1 to 5. Connect the dots in the following sequence: 1 with 3, 2 with 4, 3 with 5, 4 with 1, 5 with 2. Here is the regular five-pointed star, into a regular pentagon. This is exactly the way I built it

This pair of means determines the location of the elements of the composition relative to the main axis. If it is the same, then the composition appears as symmetrical; if there is a slight deviation to the side, then the composition is disymmetrical. With such a significant deviation, it becomes asymmetrical.

Very often, symmetry, like asymmetry, is expressed in the juxtaposition of several compositional axes. The simplest case is the relationship between the main axis and its subordinate axes, which determine the position of the secondary parts of the composition. If the secondary axes diverge significantly from the main axis, the composition may collapse. To achieve its integrity, various techniques are used: bringing the axes closer together, merging them, adopting a common direction. Figure 17 shows formal compositions (schemes) built on their basis.

Figure 17 - Compositions with different axes of symmetry

Practical task

1 Create a symmetrical composition (different types of symmetry) (Appendix A, Figures 15-16).

2 Create an asymmetrical composition (Appendix A, Figure 17).

Requirements:

7-10 search variants of the composition are performed;

pay close attention to the arrangement of elements; When implementing the main idea, take care of the accuracy of execution.

Pencil, ink, watercolor, colored pencils. Sheet format – A3.

Equilibrium

A correctly constructed composition is balanced.

Equilibrium- this is the placement of composition elements in which each item is in a stable position. There is no doubt about its location and no desire to move it along the pictorial plane. This does not require an exact mirror match between the right and left sides. The quantitative ratio of tonal and color contrasts of the left and right parts of the composition should be equal. If in one part there are more contrasting spots, it is necessary to strengthen the contrast ratios in the other part or weaken the contrasts in the first. You can change the outlines of objects by increasing the perimeter of contrasting relationships.

To establish balance in the composition, the shape, direction, and location of the visual elements are important (Figure 18).

Figure 18 - Balance of contrasting spots in the composition

An unbalanced composition looks random and unreasonable, causing a desire to further work on it (rearrange elements and their details) (Figure 19).

Figure 19 - Balanced and unbalanced composition

A properly constructed composition cannot cause doubts or feelings of uncertainty. It should have a clarity of relationships and proportions that soothes the eye.

Let's consider the simplest schemes for constructing compositions:

Figure 20 – Schemes of composition balance

Image A is balanced. In the combination of his squares and rectangles of various sizes and proportions, life is felt, you don’t want to change or add anything, there is a compositional clarity of proportions.

You can compare the stable vertical line in Figure 20, A with the oscillating one in Figure 20, B. The proportions in Figure B are based on small differences that make it difficult to determine their equivalence, to understand what is depicted - a rectangle or a square.

In Figure 20, B, each disc individually appears unbalanced. Together they form a pair that is at rest. In Figure 20, D, the same pair looks completely unbalanced, because shifted relative to the axes of the square.

There are two types of equilibrium.

Static balance occurs when figures are symmetrically arranged on a plane relative to the vertical and horizontal axes of the format of a composition of symmetrical shape (Figure 21).

Figure 21 - Static equilibrium

Dynamic equilibrium occurs when figures are asymmetrically arranged on a plane, i.e. when they are shifted to the right, left, up, down (Figure 22).

Figure 22 - Dynamic equilibrium

In order for the figure to appear depicted in the center of the plane, it needs to be moved slightly upward relative to the format axes. The circle located in the center seems to be shifted downwards, this effect is enhanced if the bottom of the circle is painted in a dark color (Figure 23).

Figure 23 – Balance of the circle

A large figure on the left side of the plane is able to balance a small contrasting element on the right, which is active due to its tonal relationship with the background (Figure 24).

Figure 24 – Balance of large and small elements

Practical task

1 Create a balanced composition using any motifs (Appendix A, Figure 18).

2 Perform an unbalanced composition (Appendix A, Figure 19).

Requirements:

perform search options (5-7 pcs.) in achromatic design with finding tonal relationships;

the work must be neat.

Material and dimensions of the composition

Mascara. Sheet format – A3.

Today we will talk about a phenomenon that each of us constantly encounters in life: symmetry. What is symmetry?

We all roughly understand the meaning of this term. The dictionary says: symmetry is proportionality and complete correspondence of the arrangement of parts of something relative to a straight line or point. There are two types of symmetry: axial and radial. Let's look at the axial one first. This is, let’s say, “mirror” symmetry, when one half of an object is completely identical to the second, but repeats it as a reflection. Look at the halves of the sheet. They are mirror symmetrical. The halves of the human body are also symmetrical (front view) - identical arms and legs, identical eyes. But let’s not be mistaken; in fact, in the organic (living) world, absolute symmetry cannot be found! The halves of the sheet copy each other far from perfectly, the same applies to the human body (take a closer look for yourself); The same is true for other organisms! By the way, it is worth adding that any symmetrical body is symmetrical relative to the viewer only in one position. It’s worth, say, turning a sheet of paper, or raising one hand, and what happens? – you see for yourself.

People achieve true symmetry in the works of their labor (things) - clothes, cars... In nature, it is characteristic of inorganic formations, for example, crystals.

But let's move on to practice. You shouldn’t start with complex objects like people and animals; let’s try to finish drawing the mirror half of the sheet as the first exercise in a new field.

Drawing a symmetrical object - lesson 1

We make sure that it turns out as similar as possible. To do this, we will literally build our soul mate. Don’t think that it’s so easy, especially the first time, to draw a mirror-corresponding line with one stroke!

Let's mark several reference points for the future symmetrical line. We proceed like this: with a pencil, without pressing, we draw several perpendiculars to the axis of symmetry - the midrib of the leaf. Four or five is enough for now. And on these perpendiculars we measure to the right the same distance as on the left half to the line of the edge of the leaf. I advise you to use a ruler, don’t rely too much on your eye. As a rule, we tend to reduce the drawing - this has been observed from experience. We do not recommend measuring distances with your fingers: the error is too large.

Let's connect the resulting points with a pencil line:

Now let’s look meticulously at whether the halves are really the same. If everything is correct, we will circle it with a felt-tip pen and clarify our line:

The poplar leaf has been completed, now you can take a swing at the oak leaf.

Let's draw a symmetrical figure - lesson 2

In this case, the difficulty lies in the fact that the veins are marked and they are not perpendicular to the axis of symmetry and not only the dimensions but also the angle of inclination will have to be strictly observed. Well, let’s train our eye:

So a symmetrical oak leaf has been drawn, or rather, we built it according to all the rules:

How to draw a symmetrical object - lesson 3

And let’s consolidate the theme - we’ll finish drawing a symmetrical lilac leaf.

It also has an interesting shape - heart-shaped and with ears at the base, you'll have to puff:

This is what they drew:

Take a look at the resulting work from a distance and evaluate how accurately we were able to convey the required similarity. Here's a tip: look at your image in the mirror and it will tell you if there are any mistakes. Another way: bend the image exactly along the axis (we have already learned how to bend it correctly) and cut out the leaf along the original line. Look at the figure itself and at the cut paper.