Mobius strip and its surprises. science toys

Imagine a surface and an ant sitting on it. Will the ant be able to crawl to the other side of the surface - figuratively speaking, to its underside - without climbing over the edge? Of course not!

August Ferdinand Möbius (1790-1868)

The first example of a one-sided surface, in which an ant can crawl to any place without climbing over the edge, was given by Möbius in 1858.

The Möbius strip, which is also called a loop, surface or leaf, is an object of study in such a mathematical discipline as topology, which studies the general properties of figures that are preserved under such continuous transformations as twisting, stretching, compression, bending, and others not related to integrity violations. . An amazing and unique feature of such a tape is that it has only one side and edge and is in no way connected with its location in space. The Möbius strip is topological, that is, a continuous object with the simplest one-sided surface with a boundary in the usual Euclidean space (3-dimensional), where it is possible to get from one point of such a surface without crossing the edges to any other.

August Ferdinand Möbius (1790-1868) - a student of the "king" of mathematicians Gauss. Möbius was originally an astronomer, like Gauss and many others, to whom mathematics owes its development. In those days, mathematics was not supported, and astronomy gave enough money not to think about them, and left time for one's own reflections. And Möbius became one of the greatest geometers of the 19th century.

At the age of 68, Mobius managed to make a discovery of striking beauty. This is the discovery of one-sided surfaces, one of which is the Möbius strip (or strip). Mobius came up with the ribbon while watching a maid put her handkerchief on the wrong way around her neck.
In Euclidean space, in fact, there are two types of half-turned Möbius strip: one is turned clockwise, the other is turned counterclockwise.

The Möbius strip has the following properties that do not change when it is compressed, cut along or crushed:

1. The presence of one side. A. Möbius in his work "On the Volume of Polyhedra" described a geometric surface, then named after him, having only one side. Checking this is quite simple: we take a strip or a Möbius strip and try to paint over the inside with one color, and the outside with another. It does not matter in what place and direction the coloring was started, the whole figure will be painted over with one color.
2. Continuity is expressed in the fact that any point of this geometric figure can be connected to any other point without crossing the boundaries of the Möbius surface.
3. Connectivity, or two-dimensionality, lies in the fact that when cutting the tape along, several different figures will not come out of it, and it remains whole.

4. It lacks such an important property as orientation. This means that a person walking along this figure will return to the beginning of his path, but only in a mirror image of himself. So an infinite Möbius strip can lead to eternal travel.
5. A special chromatic number showing what is the maximum possible number of regions on the Möbius surface that can be created so that any of them has a common border with all others. The Möbius strip has a chromatic number of 6, but the paper ring has a chromatic number of 5.

Today, the Möbius strip and its properties are widely used in science, serving as the basis for building new hypotheses and theories, conducting research and experiments, and creating new mechanisms and devices. So, there is a hypothesis according to which the Universe is a huge Mobius loop. This is indirectly evidenced by Einstein's theory of relativity, according to which even a ship flying straight can return to the same time and space point from which it started.

Another theory sees DNA as part of the Möbius surface, which explains the difficulty in reading and deciphering the genetic code. Among other things, such a structure provides a logical explanation for biological death - a spiral closed on itself leads to the self-destruction of the object. According to physicists, many optical laws are based on the properties of the Möbius strip. So, for example, a mirror reflection is a special transfer in time and a person sees his mirror double in front of him.

If you are interested in the Möbius strip, how to make its model, you will be prompted by a small instruction:
1. For the manufacture of its model, you will need: - a sheet of plain paper;
- scissors;
- ruler.
2. Cut off the strip from a sheet of paper so that its width is 5-6 times less than the length.
3. Lay out the resulting paper strip on a flat surface. We hold one end with our hand, and turn the other 180 * so that the strip is twisted and the wrong side becomes the front side.
4. We glue the ends of the twisted strip as shown in the figure.

The Möbius strip is ready.
5. Take a pen or marker and start drawing a path in the middle of the tape. If you did everything right, you will return to the same point where you started drawing the line.

In order to get a visual confirmation that the Möbius strip is a one-sided object, try to paint over any of its sides with a pencil or pen. After a while, you will see that you have painted over it completely.

The Möbius strip served as inspiration for sculptures and for graphic art. Escher was one of the artists who was especially fond of it and dedicated several of his lithographs to this mathematical object. One of the famous ones, "Möbius Strip II", shows ants crawling on the surface of the Möbius strip.

The Möbius strip is the emblem of a series of popular science books in the Quantum Library series. It is also recurring in science fiction, such as in Arthur C. Clarke's short story "The Wall of Darkness". Sometimes science fiction stories (following theoretical physicists) suggest that our universe may be some generalized Möbius strip. Also, the Möbius ring is constantly mentioned in the works of the Ural writer Vladislav Krapivin, the cycle “In the depths of the Great Crystal” (for example, “Outpost on the Anchor Field. Tale”). In A.J. Deitch's short story "Möbius Strip", the Boston subway builds a new line whose route becomes so confusing that it becomes a Möbius strip, and trains begin to disappear on the line. Based on the story, the fantastic film "Mobius" directed by Gustavo Mosquera was shot. Also, the idea of ​​the Möbius strip is used in the story of M. Clifton "On the Möbius strip".

The Mobius strip is used as a way to move through space and time by Harry Keefe, the protagonist of Brian Lumley's novel Necroscope.

The Möbius strip plays an important role in R. Zelazny's science fiction novel Doors in the Sand.

In E. Naumov's book "Half-life" (1989), an alcoholic intellectual travels around the country, standing on a Möbius strip.

The course of the novel by the modern Russian writer Alexei Shepelev "Echo" is compared with the Möbius strip. From the annotation to the book: ""Echo" is a literary analogy of the Möbius ring: two storylines - "boys" and "girls" - are intertwined, flow into each other, but do not intersect."

The Möbius strip is also found in Haruki Murakami's essay "Obladi Possess" from the 2010 Radio Murakami collection book, where the Möbius strip is figuratively compared to infinity.

In the visual novel CHARON "Makoto Mobius", the main character Wataro tries to save a classmate from death using a magical artifact - the Mobius strip.

In 1987, the Soviet jazz pianist Leonid Chizhik recorded the album Moebius Tape, which also included the composition of the same name.

The racing track in one of the episodes (season 7, episode 14, 11 minutes) of the Futurama animated series is a Mobius strip.

There are technical applications of the Möbius strip. A conveyor belt strip made in the form of a Möbius strip will last longer because the entire surface of the belt wears out evenly. Continuous tape systems also use Möbius strips (to double the recording time). In many dot-matrix printers, the ink ribbon also has the form of a Möbius strip to increase its resource.

Also above the entrance to the Institute of CEMI RAS is a mosaic high relief "Möbius Ribbon" by architect Leonid Pavlov in collaboration with artists E. A. Zharenova and V. K. Vasiltsov (1976)

Architectural solutions using the Mobius strip idea:

Jewelry in the form of a Mobius strip:

There are technical applications of the Möbius strip. The conveyor belt strip is made in the form of a Möbius belt, which allows it to work longer, because the entire surface of the belt wears out evenly. Continuous tape systems also use Möbius strips (to double the recording time). In many dot-matrix printers, the ink ribbon also has the form of a Möbius strip to increase its resource.

A device called the Möbius resistor is a newly invented electronic element that has no inductance of its own. Möbius strips are also used in continuous film recording systems (to double the recording time), in dot-matrix printers, the ink ribbon was also in the form of a Möbius strip to increase shelf life.

A Möbius strip is a three-dimensional surface with only one side and one boundary, which has the mathematical property of non-orientability. It was discovered independently simultaneously by two German mathematicians August Ferdinand Möbius and Johann Benedict Listing in 1858.

A Möbius strip model can be easily created from a strip of paper by turning one end of the strip half-turn and connecting it to the other end to form a closed shape. If you start drawing a line with a pencil on the surface of the tape, then the line will go deep into the figure and pass under the starting point of the line, as if it had gone to the "other side" of the tape. If you continue the line, it will return to the starting point. In this case, the length of the drawn line will be twice the length of the strip of paper. This example shows that the Möbius strip has only one side and one boundary.

In Euclidean space, in fact, there are two types of half-turned Möbius strip: one is turned clockwise, the other is turned counterclockwise.

Geometry and mathematics

The Möbius strip can be represented by a parametric system of equations:

where and . These equations describe a Möbius strip of width 1 lying in the plane x-y; whose inner circle radius is 1, the center of the inner circle is at the origin (0,0,0). Parameter u moves along the tape, and the parameter v from one border to another.

In another way, the tape can be represented by an expression in polar coordinates:

Topologically, a Möbius strip can be defined as a square x whose top is connected to the bottom in the relation ( x,0) ~ (1-x,1) for 0 ≤ x≤ 1, as shown in the figure on the right.

close objects

Closely related to the Möbius strip is a mysterious object, the Klein bottle. A Klein bottle can be created by gluing two Möbius strips together along their boundaries. This operation cannot be performed in 3D space without creating intersections within the figure.

One of the basic impossible figures impossible triangle can be represented as a Möbius strip if some of its edges are smoothed. In this case, a Möbius strip will be obtained, describing three turns.

Art

The Power Architecture logo

Also, the Möbius strip is often used in images of various logos and trademarks. The most striking example is the international symbol for reuse.

Appendix. Paintings with Möbius strips

The painting below by Paul Bielaczyc is called As the author says, this painting is an amalgamation of various aspects of his life. Celtic knots surround him in his work, paintings by M.K. Escher is always a source of inspiration, and the Möbius strip is relevant to the subject studied by the artist.

The Möbius strip is a simple but amazing thing. You can make it in a couple of seconds, and there are a lot of surprises, patterns and properties of this phenomenon. To make it clearer in practice, take a regular strip of paper, glue, connect its ends. But it is necessary so that one end is turned upside down relative to the other by half a turn. So the famous Möbius strip is ready.

You can talk endlessly about the resulting mysterious surface. Ask yourself how many surfaces a paper ring has. Two? And here and there - one. It is very easy to check this. Take a felt-tip pen or pencil and try to paint over one of the sides of the tape without tearing off and without moving to the other side. Happened? Where is the unpainted side? That's what it is...

The name of the tape was given by its inventor: August Ferdinand Möbius, a professor at the University of Leipzig. He devoted his long and fruitful life to scientific work (and this is 78 years), and he maintained clarity of mind until his death. At the age of 75, the professor described the unique properties of a one-sided surface with an apparent two-layer structure. Since then, the best minds of geometry, physics and even spirituality have explored this object up and down.

You can independently conduct several experiments by picking up a Möbius strip. Try to cut it along, having previously drawn a middle line over the entire surface. What do you think will happen? Two smaller rings? Wrong again - one thing! Twice as long as the previous one, but already twisted twice. Here he will just have two surfaces, and not one, as in the first case. Such a curl is called the Afghan ribbon, it is also widely known to researchers. By the way, in spirituality this effect is called a symbol of duality and is interpreted as an illusory perception of the one.

And if you again draw a longitudinal line, but not in the middle, but closer to the edge by a third of the width of the tape? Cut the resulting ring, and you will already have two of them in your hands: the Mobius strip and the Afghan strip, and in an incomprehensible way they will be linked to each other.

But these are not all surprises. When gluing the tape into a ring, try to take not one, but two paper strips. And then three or even four. I guarantee: the result will surprise you even more!

An interesting experiment can be put hypothetically. Taking a double Möbius strip (that is, glued from two strips) and sticking a finger between them (a pencil, a wooden stick - whatever), we can drive it between the strips indefinitely, thereby proving that the figure consists of two separate parts. Now imagine that a fly is crawling between these ribbons. The bottom strip for her will be the “floor”, the top one will be the “ceiling”, and so on ad infinitum.

But in reality, everything is not as simple as it seems. After all, if you put the mark of the start of the fly's journey "on the floor", then when the insect makes a circle, this very mark will already be "on the ceiling". And in order to go “on the floor” again, you will need to complete one more circle.

Imagine a fly crawling down the street. To the right of it are houses with even numbers, and to the left, respectively, under odd numbers. While walking, at some point our traveler will notice with surprise that the odd numbers are on the right, and the even numbers are on the left! It’s scary to imagine such a situation on our real roads with right-hand traffic, because soon we will have to face other walking “head-on-head”. Here it is - the Möbius strip ...

The application of this and other regularities was found not only in hypothetical, but also in real life. For example, belts in printing devices, automatic transmissions, an abrasive ring in sharpening mechanisms, and much more that you don’t even suspect are created on the basis of tape. Truly, the Möbius strip is a mystery that can be studied indefinitely!

MOU "Budagovskaya secondary school" Subject: Completed by: Ivan Shalygin Pupil of the 5th grade Head: Kalash G.V. Mathematics teacher Budagovo 2012 1 EPIGRAPH: In three-dimensional space We live, Walk, play and go to school So it would not hurt to learn more about him Explore everything About space first. Everything around us is familiar and cute. The maid opened the way to science for us. The ribbon was sewn with a mistake, It acquired meaning for posterity. So Möbius found a sheet for science, He acquired his section in mathematics. The branch that studies the surfaces of bodies Since then, everyone calls topology. How can a fly on a tape not turn off the path? Alas, there is an endless journey ahead of her. 2 Contents I. Mobius strip 1. Contents………………………………………………………………………………………………………. 2.Introduction.……………………………………………………………………………………………………….4 3.Historical background… …………………………………………………………………………………..5 …………………………………………………….5 II. Research Experiments with paper: 1. Painting the surface of the Möbius Strip………………………………………………………………………………………………………..7 2. Cutting the Möbius Strip:……………………………………………………… ………….8 a) along the sheet into two equal parts…………………………………………..……….9 b) during the operation of twisting the tape………………… ………………………10 c) several tapes glued at right angles…………………………………………11 d) several cuts along the sheet into 3; 4; 5; parts.…………………….12 3. Based on the results of the experiments, fill in the tables….………………..12 4. Draw conclusions based on the results of the studies………………………… …………………………………………………………………………………………………………………………………………………………………………………………. ……..13 6. Experiments with a rope and a vest. ………………………………………… 14 III. Practical application of the Möbius strip ……………………………………….15 IV Conclusion……………………………………………………………………… ……………………….16 V. List of used literature……………………………………………………..17 VI. Application…………………………………………………………………………………………….18 Practical lesson of a circle of mathematics on the study of the Mobius Strip in grade 5 ( photographs and video footage taken by Shalygin Ivan)……………………………………………………………………………………………………………… 17 3 Introduction General characteristics of the project: 1. The project "Geometry in space" is long (Designed for the second and third quarters) 2. The project is educational, research. (Research and experiment, systematization and practical application). 3. Group project (Work at the meetings of the circle with students of grade 5) 4. Extended project. (It is carried out within the framework of the school with the subsequent defense of the project section in the form of an abstract and presentation at the district conference “Behind the pages of a mathematics textbook”) 5. According to the results of the section of the project on the topic: “Secrets of the Möbius strip”, an abstract was prepared and the head of the IV group, Ivan Shalygin, spoke. The purpose of the work: 1. To get acquainted with a new section of mathematics - "Topology", with its basic concepts and tasks, to carry out research for practical purposes and make discoveries for oneself. 2. Form the first idea of ​​the Möbius Strip. To get acquainted with the basic techniques of the mathematical approach to the world around. 3. Learn to conduct research, describe the results, fill in the tables and carry out the drawings and drawings of the models obtained during the experiment. 4. Learn to draw reasoned conclusions, generate ideas for resolving situations, apply knowledge to solving new tasks and problems. 5. Conduct practical experiments. 6. Establish the connection of the considered material with life. 4 Historical Background August Ferdinand Möbius (1790-1868) It was raining outside. A pipe was smoked, a cup of favorite coffee with milk was drunk. The view from the window was depressing. A man was sitting in a chair. Thoughts were different, but somehow nothing special came to mind. Only in the air there was a feeling that this particular day would bring glory and perpetuate the name of August Ferdinand Möbius. A beloved wife appeared on the threshold of the room. True, she was not in a good mood. It would be more correct to say that she was angry, which was almost as unbelievable for the peaceful house of Mobius, as seeing a parade of planets three times a year, and categorically demanded the immediate dismissal of the maid, who is so mediocre that she is not even able to properly sew the ribbon. Frowningly examining the unfortunate ribbon, the professor exclaimed: “Ah yes, Martha! The girl is not so stupid. After all, this is a one-sided annular surface. The ribbon has no reverse side!” The open surface was mathematically justified and named after the mathematician and astronomer who described it.Topology - "Geometry of position" Since the German mathematician August Ferdinand Möbius discovered the existence of an amazing one-sided sheet of paper, a whole new branch of mathematics called topology began to develop. mainly studies the surfaces of bodies, and she finds a mathematical relationship between objects that, it would seem, are not related to each other in any way.For example, from the point of view of topology, a pasta nut and a mug have in common that each of these objects has a hole, although in all other respects they are different.5 The Möbius strip marked the beginning of a new science, topology.The word was coined by Johann Benedikt Listing, a professor at the University of Göttingen, who, almost at the same time as his Leipzig colleague, proposed as the first example of a one-sided surface already familiar to us, once twisted, tape. This science is young and therefore mischievous. You can't say otherwise about the rules of the game that are accepted in it. The topologist has the right to bend, twist, compress and stretch any figure - to do anything with it, just not tear it or glue it together. And at the same time, he will assume that nothing happened, all its properties remained unchanged. For him, neither distances, nor angles, nor areas matter. What interests him? The most general properties of figures that do not change under any transformations, unless a catastrophe occurs - an "explosion" of a figure. Therefore, topology is sometimes called "continuity geometry". It is also known as "rubber geometry", because it costs nothing for a topologist to place all his figures on the surface of a children's inflatable ball and endlessly change its shape, making sure that the ball does not burst. And the fact that at the same time straight lines , for example, the sides of a triangle, turn into curves, for a topologist it is deeply indifferent. What are the unusual properties of figures that topology studies? Until now, we have been talking about only one property - one-sidedness. If you move along the surface of the Möbius Strip in one direction, without crossing its boundaries, then, unlike two-sided surfaces (for example, a sphere and a cylinder), you get to a place that is upside down with respect to the original one.If you move a circle along this tape, simultaneously going around it clockwise, then in the initial position the direction of the bypass will become counterclockwise. Other properties that topology studies are continuity, connectedness, orientation For example, continuity is another topological property O. If you compare the scheme of aircraft routes and a geographical map, then 6 make sure that the scale of the Aeroflot is far from sustained - say, Sverdlovsk may be halfway from Moscow to Vladivostok. And all the same, there is something in common between the geographical map. Moscow is indeed connected with Sverdlovsk, and Sverdlovsk with Vladivostok. And, therefore, the topologist can deform the map in any way, as long as the points that were previously neighbors remain one next to the other and further. And, therefore, from a topological point of view, a circle is indistinguishable from a square or a triangle, because it is easy to transform them into one another without breaking continuity. On the Möbius strip, any point can be connected to any other point, and at the same time, the ant in Escher's engraving will never have to crawl over the edge of the "tape". There are no gaps - the continuity is complete. Experiments with paper. To make a Möbius strip, you need to take a sufficiently elongated paper strip and connect the ends of the strip, after turning one of them over. Being on the surface of the Möbius strip, one could walk on it forever. We will now consider some experiments with surfaces and holes obtained from a paper strip. It is most convenient to use strips about 30 - 40 cm long and 3 cm wide. First of all, we glue two rings - one simple and one twisted. 7 The rings are, of course, very similar; but what happens if you draw a continuous line along one side of the ring? When Möbius did this on the twisted ring, he found that the line ran on both sides, even though his pencil did not leave the paper. Does this mean that our ring has only one side? Test your rings now. 1. Completely paint over only one side of each of them. How many surfaces do they have? Try painting one side of the Möbius strip, piece by piece, without going over the edge of the tape. And what? You will color the entire Mobius strip! What is interesting about this sheet? And the fact that the Möbius strip has only one side. We are used to the fact that every surface we deal with (a sheet of paper, a bicycle or volleyball tube) has two sides. 8 2. Place a dot on one side of each ring and draw a continuous line along it until you come back to the marked point. How many edges does a Möbius strip have? Surprise number two: the Möbius strip has one border, and does not consist of two parts, like a regular ring. Let's test the rings by cutting them into two parts along. Now you get two separate rings. But what is it? Instead of two rings, you get one! Moreover, it is larger and thinner than the original ring. Record the results of further twisting and cutting in the table. Several twists. 9 And what happens if you make a full turn? How many edges does the resulting ring have? How many surfaces? What happens if you cut it in half lengthwise? Let's do some research with half-turn twisting. Full turn, half turn. Let's describe the properties and make sketches of the results. The Möbius strip has interesting properties. If you try to cut the ribbon in half along a line equidistant from the edges, instead of two Möbius strips, you get one long double-sided (twice as twisted as the Möbius strip) ribbon, which magicians call the "Afghan ribbon". If now this tape is cut in the middle, you will get two wound on top of each other. Other interesting band combinations can be obtained from Möbius bands with two or more half-turns in them. For example, if you cut a ribbon with three half-turns, you get a ribbon curled into a shamrock knot. A section of the Möbius strip with extra turns gives unexpected figures called paradromic rings. Let's record the results of twisting and cutting in the research table. Table of studies No. 1 With one tape No. p / p Number of half-turns 1 0 The result of one cut in half along Two rings Properties 2 1 One ring The ring is twice as long 3 2 Two rings The rings of the same length are linked to each other 4 3 One ring The ring is twice as long connected knot Rings twice the same length 10 Sketch Conclusions: What happens if you twist the tape twice before gluing it (ie 4 half-turns by 360 degrees)? Such a surface will already be two-sided. And to paint over the entire ring, you will certainly have to turn the tape over to the other side. The properties of this surface are no less amazing. After all, if you cut it along the middle, then you will get two identical rings, but again linked together. Cutting each of them again along the middle, you will find four rings connected to each other. You can now tear the rings in turn - and each time the remaining ones will still be linked together. If you take not a paper tape, but a strip of any fabric, turn one of the ends of the strip three full turns, i.e. 540 degrees, sew both ends. Then take the scissors and carefully cut the strip in the middle, then cut it again, you get three identical rings linked together. A Few Ribbons We'll be amazed at what happens when we cut a double ring. Prepare two rings: one ordinary and one Möbius. Glue them at a right angle, and then cut both lengthwise. Table of studies No. 2 No. p / p Number of rings 1 Two rings located perpendicular to each other. Result of cutting along each tape Three rings Properties Two rings of the same length, the third is twice as long. Two rings of smaller length are intertwined in a pair with a third ring 11 Sketch Additional question Several cuts If you cut the tape at a distance of 1/3 of its width from the edge, you get two rings. But! One large one and a small one attached to it. Table of studies No. 3 No. of p / p Number of cuts 1 Three parts The result of cutting along each tape Two rings Properties One ring of the same length, the second is twice as long linked to each other 12 Sketch 2 Four parts Two rings Both rings are twice as long as the cut, linked to each other friend. One of the rings intertwined the other 3 Five parts Three rings Two rings twice as long are intertwined with each other and linked together in a pair with a third short ring of the original length Conclusions: If you also cut the small ring along the middle, then you will have a very "intricate" weave two rings - identical in size, but different in width. Tricks with the Möbius strip. Physicists say that all optical laws are based on the properties of the Möbius strip, in particular, reflection in a mirror is a kind of transfer in time, short-term, lasting hundredths of a second, because we see in front of us ... right, our mirror double! Due to its unusual properties, the Möbius strip has been widely used over the past 75 years by magicians. If you try to cut the tape along a line equidistant from the edges, instead of two Möbius strips, you get one long double-sided (twice as twisted as a Möbius strip) a ribbon that magicians call the "Afghan ribbon". The research we have done with twisted ribbon rings can show a series of tricks. Here is one of them: We give the viewer three large paper rings, each of which was obtained by gluing the ends of a paper tape. (Study Table 1). The viewer cuts the rings along the middle of the ribbon with scissors until they return to the starting point. As a result, two hotel rings will be obtained from the first one. From the second - one ring, but twice as long, and from the third - two rings linked to each other. 13 If we pass the twisted ribbon three times through the ring, glue the ends, and then cut it along the middle, we will get one large ring with a knot tied around the ring. Similarly, research tables 2 and 3 can be used for tricks. Rope and vest experiments. Mobius strip tricks are part of topological tricks, which require flexible materials that do not change under continuous transformations: stretching and squeezing. To perform experiments, you need a scarf, vest, ropes. First, let's take a problem situation. With the help of experiments, we are looking for a way out of this situation. Experiment 1. The problem of tying knots. How to tie a knot on a scarf without letting go of its ends? It can be done like this. Put the scarf on the table. Cross your arms over your chest. Continuing to hold them in this position, bend down to the table and take one end of the scarf alternately with each hand. After the hands are separated, a knot will turn out by itself in the middle of the scarf. Using topological terminology, we can say that the viewer's arms, his body and scarf form a closed curve in the form of a "three-leaf" knot. When the arms are spread, the knot only moves from the arms to the handkerchief. Experiment 2. Turning the vest inside out without removing it from the person. To the owner the vest must be clasped behind the back.People around must turn the vest inside out without separating the wearer's hands.To demonstrate this experience, it is necessary to unfasten the vest and pull it by the hands behind the wearer's back.The vest will hang in the air, but, of course, will not be removed, because hands are clasped.Now you need to take the left half of the vest and, trying not to wrinkle the vest, insert it as far as possible into the right armhole.Then take the right armhole and insert it into the same armhole and in the same direction.It remains to straighten the vest and pull it over the wearer "The vest will be turned inside out. The same experiment can be carried out without unbuttoning the vest. The only inconvenience will be that the vest with too narrow to remove over the head. Therefore, the vest can be replaced with a sweater. Manipulations with the sweater are exactly the same. This experiment can also be demonstrated on yourself, for which you need to connect your hands with 14 cords, leaving 40 centimeters between them to ensure freedom of movement, and clasp your hands in front. Experiment 3. Untangling rope rings. Two participants are tied with ropes by the hands. Thus, the hands and ropes form two interlocked rings. It is necessary, without untying the ropes, to unravel. The clue to this experience lies in the fact that the participants have two more loops in their hands. It is necessary to stretch one rope through one of the loops on the hands of the other rope and remove the loop through the hand. III. Practical applications of the Möbius strip Its most amazing property is that it is one-sided, it cannot be painted with two colors, and insects crawling on it will go around both sides without crossing the edge. This property has found practical application: many devices have been patented, for example, a sharpening belt, an ink ribbon for printing devices, a belt drive and other technical solutions. The property of one-sidedness of the Möbius strip was used in technology: if the belt is made in the form of a Möbius strip for a belt drive, then its surface wears out twice as slowly as that of a conventional ring. This gives tangible savings. The properties that the Möbius strip possesses can be used in the clothing industry with original cutting of fabric. The spring mechanism of children's clockwork toys most often fails, because children often try to start the spring when it is already twisted to the limit. An annular twisted spring can become a "perpetuum mobile" for children's toys. Another example of the possible use of a new mechanism is the slotted shutter of a photo or movie camera (not digital). In traditional designs, after the shutter is released, it is necessary to close the shutter curtain slot, and then only return it to its original position, simultaneously cocking the spring. Otherwise, the frame will light up when passing through the shutter slit in the opposite direction. The shutter device is very complex. The use of the Möbius strip made it possible to simplify the design, increased its reliability, durability and speed. In many dot-matrix printers, the ink ribbon also has the form of a Möbius strip to increase its resource. Thanks to the Möbius strip, a wide variety of inventions arose. So, for example, special cassettes for a tape recorder were created, which made it possible to listen to tape cassettes from “two sides” without changing their places. How many people were delighted with the rollercoaster rides. This toy is very fond of not only mathematicians. It is not for nothing that, probably, now at the entrance to the Museum of History and Technology in Washington there is a monument to the Möbius strip - on a pedestal, a steel ribbon, twisted for half a turn, slowly rotates. A whole series of sculptures in the form of a Möbius strip was created by sculptor Max Bill. Quite a lot of various drawings were left by Maurits Escher. IV. Conclusion Despite the fact that Möbius made his amazing discovery a long time ago, it is still very popular today. A simple strip of paper, but twisted only once and then glued into a ring, immediately turns into a mysterious Mobius strip and acquires amazing properties. Such properties of surfaces and spaces are studied by a special branch of mathematics - Topology. This science is so complicated that it is not passed at school. Only in institutions. But who knows, maybe in time we will become famous topologists and make wonderful discoveries. And perhaps some intricate surface will be called by our names. Working together with the guys in my group on the project “Secrets of the Möbius Strip” I learned a lot of new and interesting things: I learned how to find literature on the topic proposed by the teacher in the library, read and choose the right material; use articles on the Internet, select the necessary illustrations for the abstract, build tables and fill them out; perform studies of the “Mobius strip” (make the required number of turns, glue and cut); photograph the resulting rings and enter in the table; make a presentation and shoot videos of experiments; speak at a conference and show tricks. All this is quite difficult and time consuming, but very interesting. 16 "Topology, the youngest and most powerful branch of geometry, clearly demonstrates the fruitful influence of contradictions between intuition and logic" R. Courant. 17 Literature 1. Gardner M "Mathematical miracles and secrets", Moscow, "Nauka" 1986 2. Gromov A.S. "Extracurricular assignments in mathematics Grade 8-9" Moscow, Enlightenment 3. N. Langdon, C. Snape "On the road with mathematics" Moscow, Pedagogy, 1987 4. Popular science magazine "Quantum" 1975 No. 7, 1977 No. 7 . 5. Savin A.P. “Encyclopedic Dictionary of a Young Mathematician”, M, Education, 1985 6. Yakusheva G.M. “Great Encyclopedia of a Schoolchild. Mathematics, Moscow, SLOVO, Eksmo, 2006 7. w.w.w.Rambler.ru 18 Appendix Laboratory work "Möbius strip" in the classroom 19 Try to paint one side of the Möbius strip - piece by piece, without going over the edge of the tape. And what? You will color the entire Mobius strip! 20 Put a dot on one side of each ring and draw a continuous line along it until you come to the marked point again. 21 Let's test the rings by cutting them into two parts along. 22 You will now have two separate rings. But what is it? Instead of two rings, you get one! Moreover, it is larger and thinner than the original ring. 23 Let's record the results of twisting and cutting in the study table. 24 Both rings are twice as long as the cut one, linked to each other. One of the rings intertwined the other 25 One ring of the same length, the second is twice as long linked to each other 26 A section of the Möbius strip with additional turns gives unexpected figures called paradromic rings. 27

One of the simplest and at the same time the most complex and strange objects is the Möbius strip. Despite all the originality of this figure, you can easily make it yourself and carry out all the experiments that are described in this article.

The Möbius strip is the simplest non-orientable surface that is one-sided in three-dimensional space. It is often called the Möbius surface and is referred to as continuous (topological) objects.

According to legend, the German astronomer, mathematician and mechanic August Ferdinand Möbius discovered this object after a maid working in his house sewed a fabric ribbon into a ring, inadvertently turning one of its ends upside down. Seeing the result, instead of scolding the unlucky girl, Möbius said: “Oh yes, Martha! The girl is not that stupid. After all, this is a one-sided annular surface. The ribbon has no back!

August Ferdinand Möbius.

Having studied the properties of the tape, Möbius wrote an article about it and sent it to the Paris Academy of Sciences, but it never got published. His materials were published after the death of the mathematician, and an unusual topological surface was named after him.

Making a Möbius strip is very simple: take the strip ABCD, and then fold it so that points A and D connect with B and C.

Making a Möbius strip. It turns out a figure that is ordinary at first glance, which has very interesting properties.

Unusual properties of the Möbius strip

One-sidedness
We are all used to the fact that the surfaces of all objects that we encounter in the real world (for example, a piece of paper) have two sides. But the surface of the Möbius strip is one-sided. This can be easily checked by painting the tape. If you take a pencil and start coloring the tape from anywhere without turning over, then in the end, the tape will be completely painted over.

If someone tries to color only one side of the surface of the Möbius strip, then let him immediately immerse it in a bucket of paint, the surface of the Möbius strip is continuous

This can be easily verified as follows: if you put a point anywhere on the tape, then it can be connected to any other point on the surface of the tape without cutting off the edges. Thus, it turns out that the surface of this object is continuous.

The Möbius strip has no orientation
If you could go through the entire Mobius strip, then at the moment you return to the starting point of the journey, you would turn into a mirror image of yourself.

If the tape is cut along the middle, then in this case only one tape is obtained, although the logic says that there should be two of them, and if you cut it, stepping back from the edge by a third of the width of the tape, you will already get two rings linked together - small and large . Having then made a longitudinal section of a small ring in the middle, as a result, we will get two intertwined rings of the same size, but different in width.

Practical use of the Möbius strip
There are already quite a few inventions based on the properties of this unusual topological object. For example, an ink ribbon in dot-matrix printers, twisted into a Mobius strip, lasts much longer, since wear in this case occurs evenly over its entire surface. And the blades of a kitchen mixer or concrete mixer twisted in the shape of this geometric object reduce energy costs by 20%, and at the same time the quality of the resulting mixture improves.

There is a hypothesis that the DNA polymer, which is a double helix, is a fragment of the Mobius strip and for this reason the DNA code is so difficult to decipher and understand.

Some physicists say that optical effects are based on the same properties that this paradoxical object has, so our reflection in the mirror is a special case of one of the properties of the Möbius strip.

Another hypothesis related to this mathematical object is that our Universe itself may be closed in such a tape and it has its own mirror copy. Because if we move all the time in one direction along the Möbius strip, then, in the end, we will find ourselves at the starting point of our journey, but already in our mirror image.

Mysterious Klein bottle
Based on the Möbius strip, there is another amazing figure - the Klein bottle. It is a bottle with a hole in the bottom. The neck of the bottle is elongated and bent, passing into one of the walls of the bottle itself.

Klein bottle

Such a figure cannot be reproduced in ordinary three-dimensional space, because the neck should not touch the bottle wall and is connected to a hole in its bottom. Thus, a surface is obtained that has only one side. The Klein bottle and the Möbius strip still attract the attention of scientists as well as writers.

A. Deutsch in one of his stories wrote about how one day the paths crossed in the New York subway and the entire subway began to resemble a Möbius strip, and the electric trains running along the tracks began to disappear, reappearing only a few months later.

In Alexander Mitch's book The Giveaway Game, the characters find themselves in a space that resembles a Klein bottle.

The world is still a huge mystery to us, and who knows what other quirks of space scientists will discover in the near future.