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:
- 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.
- 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.
- 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.
- 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.
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.
TRIANGLES.
§ 17. SYMMETRY RELATIVELY TO THE RIGHT STRAIGHT.
1. Figures that are symmetrical to each other.
Let's draw some figure on a sheet of paper with ink, and with a pencil outside it - an arbitrary straight line. Then, without allowing the ink to dry, we bend the sheet of paper along this straight line so that one part of the sheet overlaps the other. This other part of the sheet will thus produce an imprint of this figure.
If you then straighten the sheet of paper again, then there will be two figures on it, which are called symmetrical relative to a given line (Fig. 128).
Two figures are called symmetrical with respect to a certain straight line if, when bending the drawing plane along this straight line, they are aligned.
The straight line with respect to which these figures are symmetrical is called their axis of symmetry.
From the definition of symmetrical figures it follows that all symmetrical figures are equal.
You can obtain symmetrical figures without using bending of the plane, but with the help of geometric construction. Let it be necessary to construct a point C" symmetrical to a given point C relative to straight line AB. Let us drop a perpendicular from point C
CD to straight line AB and as its continuation we will lay down the segment DC" = DC. If we bend the drawing plane along AB, then point C will align with point C": points C and C" are symmetrical (Fig. 129).
Suppose now we need to construct a segment C "D", symmetrical to a given segment CD relative to the straight line AB. Let's construct points C" and D", symmetrical to points C and D. If we bend the drawing plane along AB, then points C and D will coincide, respectively, with points C" and D" (Drawing 130). Therefore, segments CD and C "D" will coincide , they will be symmetrical.
Let us now construct a figure symmetrical to the given polygon ABCDE relative to the given axis of symmetry MN (Fig. 131).
To solve this problem, let’s drop the perpendiculars A A, IN b, WITH With, D d and E e to the axis of symmetry MN. Then, on the extensions of these perpendiculars, we plot the segments
A A" = A A, b B" = B b, With C" = Cs; d D"" =D d And e E" = E e.
The polygon A"B"C"D"E" will be symmetrical to the polygon ABCDE. Indeed, if you bend the drawing along a straight line MN, then the corresponding vertices of both polygons will align, and therefore the polygons themselves will align; this proves that the polygons ABCDE and A" B"C"D"E" are symmetrical about the straight line MN.
2. Figures consisting of symmetrical parts.
Often there are geometric figures that are divided by some straight line into two symmetrical parts. Such figures are called symmetrical.
So, for example, an angle is a symmetrical figure, and the bisector of the angle is its axis of symmetry, since when bent along it, one part of the angle is combined with the other (Fig. 132).
In a circle, the axis of symmetry is its diameter, since when bending along it, one semicircle is combined with another (Fig. 133). The figures in drawings 134, a, b are exactly symmetrical.
Symmetrical figures are often found in nature, construction, and jewelry. The images placed on drawings 135 and 136 are symmetrical.
It should be noted that symmetrical figures can be combined simply by moving along a plane only in some cases. To combine symmetrical figures, as a rule, it is necessary to turn one of them with the opposite side,
Goals:
- educational:
- give an idea of symmetry;
- introduce the main types of symmetry on the plane and in space;
- develop strong skills in constructing symmetrical figures;
- expand your understanding of famous figures by introducing properties associated with symmetry;
- show the possibilities of using symmetry in solving various problems;
- consolidate acquired knowledge;
- general education:
- teach yourself how to prepare yourself for work;
- teach how to control yourself and your desk neighbor;
- teach to evaluate yourself and your desk neighbor;
- developing:
- intensify independent activity;
- develop cognitive activity;
- learn to summarize and systematize the information received;
- educational:
- develop a “shoulder sense” in students;
- cultivate communication skills;
- instill a culture of communication.
DURING THE CLASSES
In front of each person are scissors and a sheet of paper.
Exercise 1(3 min).
- Let's take a sheet of paper, fold it into pieces and cut out some figure. Now let's unfold the sheet and look at the fold line.
Question: What function does this line serve?
Suggested answer: This line divides the figure in half.
Question: How are all the points of the figure located on the two resulting halves?
Suggested answer: All points of the halves are at an equal distance from the fold line and at the same level.
– This means that the fold line divides the figure in half so that 1 half is a copy of 2 halves, i.e. this line is not simple, it has a remarkable property (all points relative to it are at the same distance), this line is an axis of symmetry.
Task 2 (2 minutes).
– Cut out a snowflake, find the axis of symmetry, characterize it.
Task 3 (5 minutes).
– Draw a circle in your notebook.
Question: Determine how the axis of symmetry goes?
Suggested answer: Differently.
Question: So how many axes of symmetry does a circle have?
Suggested answer: A lot of.
– That’s right, a circle has many axes of symmetry. An equally remarkable figure is a ball (spatial figure)
Question: What other figures have more than one axis of symmetry?
Suggested answer: Square, rectangle, isosceles and equilateral triangles.
– Consider three-dimensional figures: cube, pyramid, cone, cylinder, etc. These figures also have an axis of symmetry. Determine how many axes of symmetry do the square, rectangle, equilateral triangle and the proposed three-dimensional figures have?
I distribute halves of plasticine figures to students.
Task 4 (3 min).
– Using the information received, complete the missing part of the figure.
Note: the figure can be both planar and three-dimensional. It is important that students determine how the axis of symmetry runs and complete the missing element. The correctness of the work is determined by the neighbor at the desk and evaluates how correctly the work was done.
A line (closed, open, with self-intersection, without self-intersection) is laid out from a lace of the same color on the desktop.
Task 5 (group work 5 min).
– Visually determine the axis of symmetry and, relative to it, complete the second part from a lace of a different color.
The correctness of the work performed is determined by the students themselves.
Elements of drawings are presented to students
Task 6 (2 minutes).
– Find the symmetrical parts of these drawings.
To consolidate the material covered, I suggest the following tasks, scheduled for 15 minutes:
Name all equal elements of the triangle KOR and KOM. What type of triangles are these?
2. Draw several isosceles triangles in your notebook with a common base of 6 cm.
3. Draw a segment AB. Construct a line segment AB perpendicular and passing through its midpoint. Mark points C and D on it so that the quadrilateral ACBD is symmetrical with respect to the straight line AB.
– Our initial ideas about form date back to the very distant era of the ancient Stone Age - the Paleolithic. For hundreds of thousands of years of this period, people lived in caves, in conditions little different from the life of animals. People made tools for hunting and fishing, developed a language to communicate with each other, and during the late Paleolithic era they embellished their existence by creating works of art, figurines and drawings that reveal a remarkable sense of form.
When there was a transition from simple gathering of food to its active production, from hunting and fishing to agriculture, humanity entered a new Stone Age, the Neolithic.
Neolithic man had a keen sense of geometric form. Firing and painting clay vessels, making reed mats, baskets, fabrics, and later metal processing developed ideas about planar and spatial figures. Neolithic ornaments were pleasing to the eye, revealing equality and symmetry.
– Where does symmetry occur in nature?
Suggested answer: wings of butterflies, beetles, tree leaves...
– Symmetry can also be observed in architecture. When constructing buildings, builders strictly adhere to symmetry.
That's why the buildings turn out so beautiful. Also an example of symmetry is humans and animals.
Homework:
1. Come up with your own ornament, draw it on an A4 sheet (you can draw it in the form of a carpet).
2. Draw butterflies, note where elements of symmetry are present.