Trace without lifting your hand. How to draw an open envelope
I. Statement of the problem situation.
Probably everyone remembers from childhood that the following task was very popular: without lifting the pencil from the paper and without drawing along the same line twice, draw “ open envelope”:
Try drawing an “open envelope”.
As you can see, some people succeed and some don't. Why is this happening? How to draw correctly to make it work? And what is it for? To answer these questions, I will tell you one historical fact.
The city of Koenigsberg (after the World War it was called Kaliningrad) stands on the Pregol River. Once there were 7 bridges that connected the shores and two islands. Residents of the city noticed that they could not take a walk across all seven bridges, walking on each of them exactly once. This is how the puzzle arose: “Is it possible to cross all seven Königsberg bridges exactly once and return to the starting place?”
Try it too, maybe someone else will succeed.
In 1735, this problem became known to Leonhard Euler. Euler found out that there is no such way, i.e. he proved that this problem is unsolvable. Of course, Euler solved not only the problem of the Königsberg bridges, but a whole class of similar problems, for which he developed a solution method. You can see that the task is to draw a route on the map - a line, without lifting the pencil from the paper, go around all seven bridges and return to the starting point. Therefore, Euler began to consider a diagram of points and lines instead of a map of bridges, discarding bridges, islands and shores as non-mathematical concepts. Here's what he got:
A, B are islands, M, N are shores, and seven curves are seven bridges.
Now the task is to go around the contour in the figure so that each curve is drawn exactly once.
Nowadays, such diagrams of points and lines are called graphs, the points are called the vertices of the graph, and the lines are called the edges of the graph. Several lines converge at each vertex of the graph. If the number of lines is even, then the vertex is called even; if the number of vertices is odd, then the vertex is called odd.
Let us prove the unsolvability of our problem.
As we can see, in our graph all vertices are odd. First, let's prove that if the traversal of a graph does not start from an odd point, then it must end at this point
Let's take an example of a vertex with three lines. If we came along one line, left along another, and returned again along the third. There is nowhere to go further (there are no more ribs). In our problem, we said that all the points are odd, which means that when we leave one of them, we must end up at the other three odd points at once, which cannot happen.
Before Euler, no one had thought that the bridge puzzle and other path-traversal puzzles had anything to do with mathematics. Euler's analysis of such problems “is the first germ of a new branch of mathematics, today known as topology.”
Topology is a branch of mathematics that studies the properties of figures that do not change during deformations performed without tearing or gluing.
For example, from the point of view of topology, a circle, ellipse, square and triangle have the same properties and are the same figure, since one can be deformed into another, but a ring does not apply to them, since in order to deform it into a circle, gluing is required.
II. Signs of drawing a graph.
1. If there are no odd points in the graph, then it can be drawn with one stroke, without lifting the pencil from the paper, starting from any place.
2. If there are two odd vertices in the graph, then it can be drawn with one stroke, without lifting the pencil from the paper, and you need to start drawing at one odd point and end at the other.
3. If there are more than two odd points in a graph, then it cannot be drawn with one stroke of a pencil.
Let's return to our open envelope problem. Let's count the number of even and odd points: 2 odd and 3 even, which means this figure can be drawn with one stroke, and you need to start at the odd point. Try it, now everyone succeeded?
Let's consolidate the acquired knowledge. Determine which figures can be built and which cannot.
a) All points are even, so this figure can be constructed starting from any place, for example:
b) This figure has two odd points, so it can be constructed without lifting the pencil from the paper, starting from the odd point.
c) This figure has four odd points, so it cannot be constructed.
d) All points here are even, so it can be constructed starting from any place.
Let's check how you have learned new knowledge.
III. Independent work on cards with individual tasks.
Exercise: check whether it is possible to walk across all the bridges by walking on each of them exactly once. And if possible, then draw a path.
IV. Results of the lesson.
Instructions
It is assumed that the given figure consists of points connected by straight or curved segments. Consequently, at each such point a certain segment converges. Such figures are usually called graphs.
If an even number of segments converge at a point, then such a point itself is called an even vertex. If the number of segments is odd, then the vertex is called odd. For example, a square in which both are drawn has four odd vertices and one even vertex at the point of intersection of the diagonals.
By definition, a segment has two , and therefore it always connects two vertices. Therefore, by summing all the incoming segments for all vertices of the graph, only an even number can be obtained. Therefore, no matter what the graph is, there will always be odd vertices in it even number(zero in that one).
A graph in which there are no odd vertices at all can always be drawn without lifting your hand from the paper. It doesn’t matter which peak you start from.
If there are only two odd vertices, then such a graph is also unicursal. The path must begin at one of the odd vertices and end at another of them.
A figure in which there are four or more odd vertices is not unicursal, and it cannot be drawn without repeating lines. For example, the same square with drawn diagonals is not unicursal, since it has four odd vertices. But a square with one diagonal or an “envelope” - a square with diagonals and a “lid” - can be drawn with one line.
To solve the problem, you need to imagine that each drawn line disappears from the figure - it is impossible to go through it a second time. Therefore, when depicting a unicursal figure, you need to ensure that the rest of the work does not fall apart into unrelated parts. If this happens, it will no longer be possible to complete the matter.
Sources:
- How to draw a closed envelope without lifting your hand?
Square is an equilateral and rectangular quadrilateral. It's very easy to draw. Start your workout first on a squared notebook. By using a simple pencil and an invisible square from, learn to draw a square without lifting your hand from the paper.
You will need
- - a simple pencil;
- - checkered leaf;
- - sheet A4;
- - ruler.
Instructions
You can try this: without using a ruler or dots. Draw a square in the middle of the sheet. Don't try to draw it with four perfect lines at first. Draw the sides of the square right through, drawing additional lines until the square turns out to be a square. At the same time, do not take your hand off the paper. Draw lines parallel to the edges of the paper. Do several of these training exercises. This one will teach you straight lines and without tearing off the square hands.
Sources:
- drawing with squares
In painted urban or rural landscapes various bridges. This special building may look elegant and weightless, or, on the contrary, it may create the impression of a strict and heavy structure.
You will need
- pencil, paper, paints
Instructions
Equal and equal figures
Equal-sized and equally-composed figures should not be confused with equal figures, despite the closeness of these concepts.
Equal-sized figures are those that have equal area, if these are figures on a plane, or equal volume, if we're talking about about three-dimensional bodies. The coincidence of all elements that make up these figures is not required. Equal figures will always be equal in size, but not all equal-sized figures can be called equal.
The concept of equiparity is most often applied to polygons. It implies that polygons can be divided into the same number of correspondingly equal figures. Equally sized polygons are always equal in area.
Sources:
- What are equal figures
It is difficult to captivate modern children with anything. They love watching cartoons and playing computer games. But smart parents are always able to interest their child. For example, they might ask him to find a way to draw an envelope without lifting his hand. Read below about some of the tricks of this task.
Warm-up
Before you start tormenting your child with logical tasks, you need to spend some time with him preparatory work. Why is it needed? So that the child does not cheat when he starts to puzzle over the question of how to draw an envelope without lifting his hand. After all, the most interesting thing in this problem is that the line must go from point to point continuously.
What tasks can be offered to a child as a warm-up? Of course, the first thing should be eights. Drawing this number relieves stress, cleanses the brain, and trains the hand. All in all, useful exercise. After this, you can move on to drawing rounded shapes. These can be curls or any other squiggles, the main thing is that during the drawing process the child does not lift the pencil and depicts everything in one smooth line.
How to draw a closed envelope
Many parents themselves spent more than one hour before offering such a task to their child. You can try it too. But we can immediately disappoint you - it is simply impossible to complete such a task without cheating a little. Therefore, we will tell you a method that will help you and your child go a little beyond ordinary logic in order to understand how to draw a closed envelope without lifting your hand.
Take a sheet of paper and bend its edge. We bend it back. Now our task is to draw the top edge closed envelope just on the fold line. To make it easier to understand, let's place dots at the ends of the rectangle. Let's number them starting from the upper left corner. The number one will appear here and further clockwise. From the number 4 to 1 we draw a line, now we connect 1 to 2 and now we draw a diagonal to 4. From 4 to 3 we draw a straight line, and then again a diagonal to 1.
Now let's get to the fun part. We bend the edge of our sheet and draw a zigzag, which forms, as it were, the head of our envelope. It will go from 1 to 2. All that remains is to connect 2 and 3 with a straight line - and the puzzle is solved. Bend part of the sheet back. The riddle of how to draw an envelope without lifting your hand can be offered not only to children, but also to friends or colleagues.
How to draw an open envelope
Those who carefully read the previous paragraph and created their own drawing based on the description already understood how to answer the question posed above. After all, the solution to the riddle of how to draw an open envelope without lifting your hand will be similar to that written in the previous paragraph. Only here you won’t have to bend and bend parts of the sheet. The entire image will be made with one line according to the same pattern.
But if you don’t want to repeat yourself, then we offer another method that will lead to the same result. How to draw an envelope without taking your hands off using the second method? To begin with, we again draw a rectangle with dots and number it again, as in the previous paragraph. From the number 4 to 2 we draw a diagonal, from 2 to 3 we draw a straight line, and from 3 to 1 we draw a diagonal again. Next you need to draw a corner. From 1 to 2 we draw a zigzag, which means top part envelope. From 2 we return to 1 with a straight line and complete our construction by alternately drawing straight lines from 1 to 4 and from 4 to 3.
Why are such tasks needed?
These should be done not only for children, but also for adults. Thanks to them, the human brain tenses up and begins to work. If you train yourself to perform a similar task every day, within a month you will notice that in critical situations solutions are generated faster and less effort is spent on it. It is especially useful for schoolchildren to study logic problems. In this way, they train creativity and learn to approach standard issues in an unconventional way.
