Infinite Mobius strip. Master class "surprises of the Mobius strip"

Municipal educational institution "Budagovskaya secondary school" Topic: Completed by: Shalygin Ivan 5th grade student Leader: Kalash G.V. Math teacher Budagovo 2012 1 EPIGRAPH: You and I live in three-dimensional space, We walk, play and go to school. So it wouldn’t hurt to learn more about it. Explore everything about space first. Everything around us is familiar and nice to us. The maid opened the way to science for us. The ribbon was sewn with an error, but it acquired meaning for posterity. So Mobius found a worksheet for science, acquired his own section in mathematics. The branch that studies the surfaces of bodies. Since then, everyone has called topology. How can a fly on a tape not deviate from its path? Alas, she faces an endless path. 2 Contents I. Möbius strip 1. Contents……………………………………………………………………………………………………………..3 2.Introduction.………………………………………………………………………………………………………………….4 3.Historical background… …………………………………………………………………………………..5 4.Topology – “Position geometry”….....…… …………………………………………………………………….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…………………………… ………………………………………………………12 5. Tricks with Möbius strip……………………………………………………… ……..13 6. Experiments with rope and vest. …………………………………………14 III. Practical application of the Möbius strip……………………………………………………….15 IV Conclusion………………………………………………………………………………………… ……………………….16 V. List of references……………………………………………………..17 VI. Appendix………………………………………………………………………………………………………….18 Practical lesson of a mathematics club on the study of the Möbius strip in the 5th grade ( photographs and video footage taken by Ivan Shalygin)…………………………………………………………………………………………………………………………………… 17 3 Introduction General characteristics of the project: 1. The project “Geometry in Space” is long-term (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 club meetings with 5th grade students) 4. Extended project. (Conducted within the school with subsequent defense of a section of the project in the form of an abstract and presentation at the regional conference “Behind the Pages of a Mathematics Textbook”) 5. Based on 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. Purpose of the work: 1. To get acquainted with a new branch of mathematics - “Topology”, with its basic concepts and tasks, to carry out research for practical purposes and to make discoveries for oneself. 2. Form a first idea about the Möbius Strip. Get acquainted with the basic techniques of a mathematical approach to the world around you. 3. Learn to conduct research, describe the results, fill out tables and execute the resulting drawings and drawings of models obtained during the experiment. 4. Learn to draw reasoned conclusions, generate ideas for resolving situations, and apply knowledge to solve 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. I smoked a pipe and drank a cup of my favorite coffee with milk. The view from the window was depressing. A man was sitting in a chair. There were different thoughts, but somehow nothing special came to mind. There was only a feeling in the air that this particular day would bring glory and perpetuate the name of August Ferdinand Mobius. His beloved wife appeared on the threshold of the room. True, she was not in a good mood. More correctly, she was angry that for the peaceful house of Mobius it was almost as incredible 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 sew a ribbon correctly. Gloomily looking at the ill-fated ribbon, the professor exclaimed: “Oh yes, Martha! The girl is not so stupid. After all, this is a one-sided annular surface. The ribbon has no back!” The open surface received a mathematical justification and a name in honor of the mathematician and astronomer who described it.Topology - "Position Geometry" From the moment the German mathematician August Ferdinand Möbius discovered the existence of an amazing one-sided sheet of paper, a whole new branch of mathematics began to develop, called topology. mainly studies the surfaces of bodies, and she finds a mathematical relationship between objects that seem to be in no way connected with each other. For example, from a topological point of view, a macaroni nut and a mug are related in that each of these objects has a hole, although in all other respects they are different.5 The Möbius strip laid the foundation for a new science - topology. This word was coined by Johann Benedict 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 the already familiar, once twisted, tape. This science is young and therefore mischievous. There is no other way to say 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 whatever he wants with it, just not tear it or glue it together. And at the same time, he will believe that nothing happened, all its properties remained unchanged. For him, neither distances, nor angles, nor areas matter. What is he interested in? The most general properties of figures that do not change under any transformations, unless a catastrophe occurs - the “explosion” of the figure. Therefore, sometimes topology is called “geometry of continuity”. It is also known under the name “rubber geometry”, because it costs a topologist nothing to place all his figures on the surface of a children’s inflatable ball and endlessly change its shape, making sure only that the ball does not burst. And the fact that at the same time straight lines , for example, the sides of a triangle will turn into curves, it is deeply indifferent to the topologist. What unusual properties of figures does topology study? Until now, we have been talking about only one property - one-sidedness. If you move along the surface of the Mobius strip in one direction, without crossing its boundaries, then, unlike two-sided surfaces (for example, a sphere and a cylinder), you end up in a place that is inverted in relation to the original one.If you move a circle along this tape, while simultaneously going around it clockwise, then in the initial position the direction of the traversal will become counterclockwise. Other properties that topology studies are continuity, connectivity, orientation.For example, continuity is another topological property. If you compare the diagram of plane routes and a geographical map, then 6 you will be convinced that the scale of Aeroflot is far from being consistent - for example, Sverdlovsk may be halfway from Moscow to Vladivostok. And yet, there is something in common between a geographical map. Moscow is indeed connected with Sverdlovsk, and Sverdlovsk with Vladivostok. And, therefore, the topologist can deform the map in any way he wants, as long as the points that were previously neighbors remain one next to the other and further. This means that from a topological point of view, a circle is indistinguishable from a square or a triangle, because they are easy to transform one into the other without breaking continuity. On a 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 “ribbon”. There are no breaks - complete continuity. Experiments with paper. To make a Möbius strip, you need to take a fairly elongated strip of paper and connect the ends of the strip, first turning one of them over. If you were on the surface of a Möbius strip, you could walk on it forever. We will now consider several experiments with surfaces and holes made from paper strips. It is most convenient to use strips approximately 30–40 cm long and 3 cm wide. First of all, let's 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 down both sides, although his pencil did not leave the paper. Does this mean that our ring only has one side? Now try your rings. 1. Completely paint only one side of each of them. How many surfaces do they have? Try painting one side of the Mobius strip, piece by piece, without going over the edge of the strip. And what? You will paint the entire Mobius strip! What is so interesting about this sheet? And the fact that the Möbius strip has only one side. We are accustomed to the fact that every surface we deal with (a sheet of paper, a bicycle tube or a volleyball tube) has two sides. 8 2. Place a point on one side of each ring and draw a continuous line along it until you come to the marked point again. How many edges does a Möbius strip have? Surprise number two: the Möbius strip has only one boundary, and does not consist of two parts, like a regular ring. Let's test the rings by cutting them into two parts lengthwise. Now you will 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. Record the results of further twisting and cutting in a table. Several twists. 9 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 twisting it half a turn. A full turn, one and a half turns. Let's describe the properties and make sketches of the results. The Möbius strip has interesting properties. If you try to cut the tape in half 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) strip, which magicians call an “Afghan strip”. If you now cut this tape in the middle, you will get two wound on top of each other. Other interesting strip combinations can be derived from Möbius strips with two or more half-turns in them. For example, if you cut a ribbon with three half turns, you will get a ribbon curled into a trefoil knot. Cutting a Möbius strip with additional turns produces unexpected figures called paradromic rings. Let's record the results of twisting and cutting in the research table. Research table No. 1 With one tape No. Number of half-turns 1 0 Result of one cut in half lengthwise Two rings Properties 2 1 One ring A ring twice as long 3 2 Two rings Rings of the same length interlocked with each other 4 3 One ring A ring twice as long connected knot Rings twice as narrow as the same length 10 Sketch Conclusions: What happens if you twist it twice before gluing the tape (i.e. 4 half turns of 360 degrees)? Such a surface will already be double-sided. And in order to paint the entire ring, you will definitely 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 lengthwise in the middle, you will get two identical rings, but again interlocked. Cutting each of them again along the middle, you will find four rings connected to each other. You can now tear the rings one by one - 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 scissors and carefully cut the strip in the middle, then cut again, you get three identical rings interlocked with each other. Several ribbons We will be amazed at what happens when we cut a double ring. Prepare two rings: one regular and one Möbius. Glue them at right angles, and then cut both lengthwise. Research table No. 2 No. Number of rings 1 Two rings located perpendicular to each other. Result of cutting along each strip Three rings Properties Two rings of the same length, the third is twice as long. Two rings of shorter length are intertwined in pairs with a third ring 11 Sketch Additional question Several cuts If you cut the ribbon at a distance of 1/3 of its width from the edge, you will get two rings. But! One big one and a small one linked to it. Research table No. 3 No. Number of cuts 1 Three parts Result of cutting along each tape Two rings Properties One ring of the same length, the second twice as long are interlocked with each other 12 Sketch 2 Four parts Two rings Both rings are twice as long as the cut one, interlocked with 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 into a pair by a third short ring of the original length Conclusions: If you also cut a small ring along, in the middle, then you will have a very “intricate” interweaving two rings - identical in size, but different in width. Tricks with Mobius strip. Physicists claim that all optical laws are based on the properties of the Mobius strip, in particular, reflection in a mirror is a kind of transfer in time, short-term, lasting hundredths of a second, after all, we see in front of us... that's right, a mirrored double of ourselves! Due to its unusual properties, the Mobius strip has been widely used over the past 75 years by magicians. If you try to cut the strip along a line equidistant from the edges, instead of two Mobius strips you will get one long double-sided (twice as twisted as a Möbius strip) a strip that magicians call an “Afghan strip.” Based on our research with twisted strip rings, we can perform a series of tricks. Here is one of them: We present the viewer with three large paper rings, each of which was made by gluing the ends of a paper tape. (Studies Table 1). The spectator uses scissors to cut the rings along the ribbon in the middle until he returns to the starting point. As a result, the first one will turn into two separate rings. From the second there is one ring, but twice as long, and from the third there are two rings interlocked with each other. 13 If you pass a thrice-twisted ribbon through the ring, glue the ends together, and then cut it lengthwise in the middle, you will get one large ring with a knot tied around the ring. Similarly, for magic tricks, you can use research tables 2 and 3. Experiments with rope and vest. Tricks with a Möbius strip are part of topological tricks, which require flexible materials that do not change under continuous transformations: stretching and compression. To perform the experiments, you need a scarf, vest, and ropes. First, we pose 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 in a scarf without letting go of its ends? It can be done like this. Place the scarf on the table. Cross your arms over your chest. Continuing to hold them in this position, bend over the table and take one end of the scarf with each hand in turn. After the arms are spread apart, a knot will form automatically in the middle of the scarf. Using topological terminology, we can say that the viewer’s hands, his body and scarf form a closed curve in the form of a “three-leaf” knot. When spreading the hands, the knot only moves from the hands to the scarf. Experiment 2. Turning the vest inside out without removing it from the person. To the owner vest, you need to clasp your fingers behind your back. Those around you must turn the vest inside out, without separating the owner's hands. To demonstrate this experiment, you need to unfasten the vest and pull it by hand behind the owner'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 hem of the vest and, trying not to wrinkle the vest, push it as far as possible into the right armhole. Then take the right armhole and push it into the same armhole and in the same direction. All that remains is to straighten the vest and pull it on the owner The vest will be turned inside out. The same experiment can be carried out without unfastening the vest. The only inconvenience will be that the vest is too narrow to be removed over the head. Therefore, the vest can be replaced with a sweater. The manipulations with the sweater are repeated exactly. This experiment can be demonstrated on yourself, for which you need to connect your hands with a 14 cord, 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 by the hands with ropes. Thus, the hands and ropes form two interlocking rings. It is necessary to unravel without untying the ropes. The answer to this experiment lies in the fact that the participants each have two more loops on their hands. It is necessary to pull one rope through one of the loops on the hands of the other rope and remove the loop through the hand. III. Practical application 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 one-sided property of the Möbius strip was used in technology: if the belt of a belt drive is made in the form of a Möbius strip, then its surface wears out twice as slow as that of a regular ring. This provides significant savings. The properties that the Möbius strip has can be used in the clothing industry for original cutting of fabric. The spring mechanism of children's wind-up toys most often fails because children often try to wind the spring when it is already twisted to the limit. A ring twisted spring can become a “perpetual motion machine” for children's toys. Another example of the possible use of a new mechanism is the slot shutter of a photo or movie camera (not digital). In traditional designs, after releasing the shutter, it is necessary to close the shutter curtain slot, and then only return it to its original position, while simultaneously charging the spring. Otherwise, the frame will light up when passing the shutter slit in the opposite direction. The shutter device is quite complex. The use of a Möbius strip simplified the design, increased its reliability, durability and performance. In many matrix printers, the ink ribbon also has the form of a Mobius strip to increase its resource. Thanks to the Mobius strip, many different inventions arose. For example, special cassette tapes were created for tape recorders, which made it possible to listen to tape cassettes from “both sides” without changing their places. How many people were delighted by the “Roller Coaster” rides. This toy was very popular not only with mathematicians. It’s probably not for nothing that now at the entrance to the Museum of History and Technology in Washington there is a monument to the Mobius strip - a steel ribbon twisted half a turn slowly rotates on a pedestal. A whole series of sculptures in the form of a Mobius strip was created by sculptor Max Bill. Quite a lot of different drawings were left by Maurits Escher. IV. Conclusion Despite the fact that Mobius made his amazing discovery a long time ago, it is still very popular today. A simple strip of paper, twisted just 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 complex that it is not taught in school. Only in institutes. But who knows, maybe over time we will become famous topologists and make wonderful discoveries. And perhaps some intricate surface will be named after us. Working with the guys in my group on the project “Secrets of the Mobius strip”, I learned a lot of new and interesting things: I learned to find literature on a topic proposed by the teacher in the library, read and select the necessary material; use articles on the Internet, select the necessary illustrations for the abstract, build tables and fill them out; carry out research on the “Möbius strip” (make the required number of turns, glue and cut); photograph the resulting rings and enter them into the table; make a presentation and film experiments; speak at a conference and perform magic tricks. All this is quite complicated and time-consuming, but very interesting. 16 “Topology, the youngest and most powerful branch of geometry, clearly demonstrates the fruitful influence of the contradictions between intuition and logic” R. Courant. 17 Literature 1. Gardner M “Mathematical miracles and mysteries”, Moscow, “Science” 1986 2. Gromov A.S. “Extracurricular tasks in mathematics grades 8-9” Moscow, Education 3. N. Langdon, Ch. Snape “On the road with mathematics” Moscow, Pedagogy, 1987 4. Popular scientific magazine “Kvant” 1975 No. 7, 1977 No. 7 . 5. Savin A.P. “Encyclopedic Dictionary of a Young Mathematician”, M, Prosveshchenie, 1985 6. Yakusheva G.M. “Big Encyclopedia for Schoolchildren. Mathematics”, Moscow, “WORD”, Eksmo, 2006 7. w.w.w.Rambler.ru 18 Appendix Laboratory work “Möbius Strip” in a math circle class 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 paint the entire Mobius strip! 20 Place a point on one side of each ring and draw a continuous line along it until you come back to the marked point 21 Let's test the rings by cutting them in two lengthwise. 22 Now you will 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 write down the results of twisting and cutting in the research table. 24 Both rings are twice as long as the cut one, interlocking with each other. One of the rings intertwined the other 25 One ring of the same length, the second twice as long are interlocked with each other 26 Cutting the Möbius strip with additional turns gives unexpected figures called paradromic rings. 27

There is scientific knowledge and phenomena that bring mystery and mystery into the everyday life of our lives. The Mobius strip applies to them fully.

Modern mathematics wonderfully describes all its properties and features using formulas. But ordinary people, who have little understanding of toponymy and other geometric wisdom, almost every day encounter objects made in its image and likeness, without even knowing it.

What it is? Who opened it and when?

A Möbius strip, also called a loop, surface or sheet, is an object of study in the mathematical discipline of 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 a violation of integrity . An amazing and unique feature of such a tape is that it has only one side and edge and is in no way related to its location in space. A Mobius strip is topological, that is, a continuous object with the simplest one-sided surface with a boundary in ordinary Euclidean space (3-dimensional), where it is possible from one point of such a surface to get to any other without crossing the edges.

Such a complex object as a Möbius strip was discovered in a rather unusual way. First of all, we note that two mathematicians, completely unrelated to each other in their research, discovered it at the same time - in 1858. Another interesting fact is that both of these scientists at different times were students of the same great mathematician - Johann Carl Friedrich Gauss. So, until 1858 it was believed that any surface must have two sides. However, Johann Benedict Listing and August Ferdinand Möbius discovered a geometric object that had only one side and describe its properties. The strip was named after Möbius, but topologists consider Listing and his work “Preliminary Studies in Topology” to be the founding father of “rubber geometry.”

Properties

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

1. The presence of one side. A. Mobius in his work “On the Volume of Polyhedra” described a geometric surface, later named in his honor, with only one side. It’s quite simple to check this: take a Mobius strip or strip and try to paint the inside with one color and the outside with another. It doesn’t matter in what place and direction the coloring was started, the entire figure will be painted with the same 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 Mobius surface.

3. Connectedness, or two-dimensionality, lies in the fact that when cutting the tape lengthwise, several different shapes will not turn out from it, and it remains solid.

4. It lacks such an important property as orientation. This means that a person following this figure will return to the beginning of his path, but only in a mirror image of himself. Thus, an infinite Mobius strip can lead to an eternal journey.

5. A special chromatic number showing the maximum possible number of areas on the Mobius surface that can be created so that any of them has a common boundary with all the others. The Möbius strip has a chromatic number of 6, but the paper ring has a chromatic number of 5.

Scientific use

Today, the Mobius strip and its properties are widely used in science, serving as the basis for constructing new hypotheses and theories, conducting research and experiments, and creating new mechanisms and devices.

Thus, 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 views DNA as part of the Mobius 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 Mobius strip. So, for example, a mirror reflection is a special transfer in time and a person sees his mirror double in front of him.

Implementation in practice

The Mobius strip has been used in various industries for a long time. The great inventor Nikola Tesla at the beginning of the century invented the Mobius resistor, consisting of two conductive surfaces twisted into 1800, which can resist the flow of electric current without creating electromagnetic interference.

Based on studies of the surface of the Mobius strip and its properties, many devices and instruments have been created. Its shape is repeated in the creation of conveyor belt strips and ink ribbons in printing devices, abrasive belts for sharpening tools and automatic transfers. This allows you to significantly increase their service life, since wear occurs more evenly.

Not long ago, the amazing features of the Mobius strip made it possible to create a spring that, unlike conventional springs that fire in the opposite direction, does not change the direction of operation. It is used in the stabilizer of the steering wheel drive, ensuring the return of the steering wheel to its original position.

In addition, the Möbius strip sign is used in a variety of brands and logos. The most famous of these is the international symbol of recycling. It is placed on the packaging of goods that are either recyclable or made from recycled resources.

Source of creative inspiration

The Möbius strip and its properties formed the basis for the work of many artists, writers, sculptors and filmmakers. The most famous artist who used the tape and its features in such works as “Mobius Strip II (Red Ants)”, “Riders” and “Knots” is Maurits Cornelis Escher.

Möbius strips, or minimum energy surfaces as they are also called, have become a source of inspiration for mathematical artists and sculptors such as Brent Collins and Max Bill. The most famous monument to the Mobius strip is installed at the entrance to the Washington Museum of History and Technology.

Russian artists also did not stay away from this topic and created their own works. The Mobius Strip sculptures were installed in Moscow and Yekaterinburg.

Literature and topology

The unusual properties of Möbius surfaces have inspired many writers to create fantastic and surreal works. The Mobius loop plays an important role in R. Zelazny’s novel “Doors in the Sand” and serves as a means of movement through space and time for the main character of the novel “Necroscope” by B. Lumley

She also appears in the stories “The Wall of Darkness” by Arthur C. Clarke, “On the Mobius Strip” by M. Clifton and “The Mobius Strip” by A. J. Deitch. Based on the latter, director Gustavo Mosquera made the fantastic film “Mobius”.

We do it ourselves, with our own hands!

If you are interested in the Mobius strip, how to make a model of it, a small instruction will tell you:

1. To make its model you will need:

A sheet of plain paper;

Scissors;

Ruler.

2. Cut a strip from a sheet of paper so that its width is 5-6 times less than its length.

3. Lay out the resulting paper strip on a flat surface. We hold one end with our hand, and turn the other by 1800 so that the strip twists and the wrong side becomes the front side.

4. Glue the ends of the twisted strip together as shown in the figure.

The Mobius strip is ready.

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

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

Budarina Svetlana

Arndt Anastasia

The paper discusses the history of the discovery of the Möbius strip and the experiments that can be carried out with the Möbius strip.

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Municipal budgetary educational institution

"Vesennenskaya secondary school"

Christmas readings

Nomination: “Exact Sciences”

Secrets of the Mobius strip

Arndt Anastasia

5th grade student

Supervisor:

Arndt Irina

Vasilevna,

Mathematic teacher

With. Spring

year 2014

Introduction. ………………………………………………………..…..…..… With. 3

Chapter I. Historical background. .....…………………………………....... With. 3-4

Chapter II. Möbius strip. ………………………………………….....…….With. 4-9

§1. Making a Mobius strip. ………………………………........…..With. 4

§2. Experiments with Möbius strip. ……..………………………........With. 4-6

§3. Application of the Mobius strip in life. …………………………..… p.7-9

Conclusion. ………………………………………..…………………........With. 9

Literature. ……………………………………………………………..….With. 10

Introduction.

Each of us has an intuitive idea of ​​what "surface" is. The surface of a sheet of paper, the surface of the walls of a classroom, the surface of the globe are known to everyone. Could there be anything unexpected and even mysterious in such an ordinary concept? The Moebius sample sheet shows that it can. Many people know what a Möbius strip (strip) is. For those who are not yet familiar with the amazing worksheet that belongs to the “mathematical surprises,” we invite you to explore with us and plunge into the bright feeling of knowledge.

I was very interested in this topic. I decided to deepen my knowledge in this area.

The purpose of my work: to explore the Mobius strip as one of the objects of topology.

Objectives: - collect all possible information about the Mobius strip;

Experimentally investigate the properties of the Mobius strip;

Show the use of Mobius strip in life.

Chapter I. Historical background.

Mysterious and famousThe Möbius strip was discovered independently by German mathematicians August Ferdinand Möbius and Johann Benedict Listing in 1858.

August Ferdinand Mobius(1790-1868), born in the city of Schulpforte, German geometer, student of the “king of mathematicians” the famous K.F. Gauss. Mobius was originally an astronomer. Professor at the University of Leipzig since 1816. He began to conduct independent astronomical observations at the Pleisenburg Observatory in 1818. became its director. Working in quiet solitude, Möbius made many interesting discoveries; he became one of the largest geometers of the 19th century. At the age of 68, he managed to make a discovery of amazing beauty. This is the discovery of one-sided surfaces, one of which is the Möbius strip. This was the most significant event in his life!

They say that Mobius was helped to open his “leaf” by a maid who sewed the ends of the ribbon incorrectly.

There are often cases in history when one idea occurs to several inventors at the same time. This happened with the Mobius strip.

In the same year, 1858, the idea of ​​the tape came to another scientist, a student of K.F. Gauss -Johann Benedict Listing(1808-1882), German mathematician and physicist, professor at the University of Göttingen. He gave the name to the science that studies continuity - topology

Topology studies the properties of geometric shapes that do not change if they are bent, stretched, or compressed. The championship in the discovery of a topological object - a strip - went to August Mobius.

What struck these two German professors? And the fact that the Mobius strip has only one side.

Chapter II. Möbius strip.

§1. Making a Mobius strip.

A Möbius strip is very easy to make, hold in your hands, cut, experiment in some other way. Studying the Möbius strip is a good introduction to the elements of topology.

The Möbius strip is one of those mathematical surprises. To make a Möbius strip, take a rectangular strip ABB 1 A 1 , twist it 180 degrees and glue the opposite sides AB and A 1 in 1 , i.e. so that points A and B coincide 1 and points A 1 and B.

We get a twisted ring.And we ask ourselves: how many sides does this piece of paper have? Two, like anyone else? No. It has ONE side. Don't believe me?

§2. Experiments with the Möbius strip.

To study its properties, I conducted several experiments, which I divided into two groups:

Group I.

Experience No. 1 . I started painting the Mobius strip without turning it over.

Result. The Möbius strip was completely painted over.

“If anyone decides to paint only one side of the surface of a Möbius strip, let him immediately immerse the whole thing in a bucket of paint,” write Richard Courant and Herbert Robins in the excellent book “What is Mathematics?”

Experience No. 2.

Imagine that a shapeshifter travels along a Mobius strip, and after going all the way, he will return to the starting point. At the same time, it will go around both surfaces - external and internal, without intersecting the edges.This proves thata Möbius strip is a one-sided surface. He returned to the starting point. But in what form! Inverted!

And for him to return to the start in a normal position, he needs to make another “round-leaf” trip. The Möbius strip has only one side!

Group II experiments related to cutting the Mobius strip.

I conducted a series of experiments, the results of which were entered into a table.

These are the unexpected things that happen to a simple strip of paper if you glue it together into a Möbius strip.

§3. Application of the Mobius strip in life.

While doing this work, I came to the conclusion that although the Möbius strip was discovered back in the 19th century, it is relevant both in the 20th and 20th centuries.

The amazing properties of the Möbius strip have been and are used in technology, physics, and optics. He inspired the creativity of many writers and artists.

It is curious that the Mobius strip continues to excite the minds of inventors even now. In many countries around the world, amazing mechanisms based on it have been patented.

Möbius strip in technology and physics

On Mobius-spinned magnetic tapes, the volume of recorded information doubles andplays twice as long.Special cassettes were created that made it possible to listen to them from “both sides” without changing places.

This tape works great for tying and carrying cargo in ports. Conveyor belts for moving hot materials, if turned out according to Möbius, will take turns “resting” from the hot materials. As a result, the cooling of the belt improves, and the belt wears out evenly, which means it will last longer.This provides significant savings.

Möbius strip in nature and in life.

There is a hypothesis that the DNA helix itself is also a fragment of a Mobius strip and that is the only reason why the genetic code is so difficult to decipher and perceive. Moreover, such a structure quite logically explains the reason for the onset of biological death - the spiral closes on itself and self-destruction occurs.

Möbius strip in art.

The mysterious Mobius strip has always excited the minds of writers, artists and sculptors. The Möbius strip served as inspiration for sculptures and graphic art. Escher was one of the artists who especially loved it and dedicated several of his lithographs to this mathematical object. One famous one shows ants crawling across the surface of a Möbius strip.

His drawings depicting a Möbius strip are also widely known.

The monuments dedicated to the Möbius strip are very interesting.

The streets of many cities are decorated with sculptures based on the Mobius strip theme.

Jewelers dedicated their works to the Möbius strip.

The Möbius strip is depicted on various emblems, and is depicted on the badge of the Faculty of Mechanics and Mathematics of Moscow University.

The international symbol for recycling is also the Möbius Strip.

In addition, a crater on the far side of the Moon is named after Möbius.

Architects are using the Möbius strip in innovative ways. This is how, for example, the incredible project of a new library in Astana (Kazakhstan) looks like.

Conclusion.

The Möbius strip has many interesting properties.

1. The Möbius strip has one edge.
2. The Möbius strip has one side.
3. A Möbius strip is a topological object. Like any topological figure, a Möbius strip does not change its properties until it is cut, torn, or its individual pieces are glued together.
4. One edge and one side of the Mobius strip are not related to its position in space, and are not related to the concepts of distance.

The Möbius strip is the first one-sided surface to be discovered. Later, mathematicians discovered a whole series of one-sided surfaces. In this work, I tried to describe the properties of a beautiful surface - the Mobius strip, show its significance in practice, and prove that the Mobius strip is a topological figure.

Despite the fact that Möbius made his amazing discovery a long time ago, it is still very popular today:

• Mathematicians are undergoing further research;
• for schoolchildren it is very interesting to experiment with the Möbius strip;
• in technology – new ways of using the Möbius strip are being discovered.

I have not exhausted experiments with the Möbius strip. They are endless, interesting and depend on your own patience. In the future, I plan to continue researching this unpredictable leaf.

Literature.

1. Voloshinov A.V., “Mathematics and Art” M.: “Enlightenment”, 1996.
2. Newspaper "Mathematics" supplement to the publishing house "First of September", No. 14 1999, No. 24 2006.
3. Gardner M. “Mathematical wonders and mysteries”, “Science” 1978.
4. Gusev V.A., Kombarov A.P. “Mathematical warm-up” M.: “Enlightenment”, 1986.
5. Internet site resources:http://ru.wikipedia.
6. Kordemsky B. A. Do-it-yourself topological experiments. Kvant, 1974, No. 3.

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.

A 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 classified as a continuous (topological) object.

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, carelessly turning one of its ends over. Seeing the result, instead of scolding the unlucky girl, Mobius said: “Oh yes, Martha! The girl isn't that stupid. After all, this is a one-sided annular surface. The ribbon has no back!”

August Ferdinand Moebius.

Having studied the properties of the tape, Mobius wrote an article about it and sent it to the Paris Academy of Sciences, but never saw its publication. His materials were published after the mathematician’s death, and an unusual topological surface was named in his honor.

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

Making a Mobius strip. The result is a seemingly ordinary figure that has very interesting properties.

Unusual properties of the Möbius strip

Unilateralism
We are all accustomed 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 over the tape. If you take a pencil and start coloring the tape from any place without turning it over, then in the end the tape will be completely painted over.

If someone tries to paint only one side of the surface of the Möbius strip, then it is better to 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 crossing the edge. Thus, it turns out that the surface of this object is continuous.

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

If the tape is cut lengthwise in the middle, then in this case you get only one tape, although 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 end up with two rings linked together - a small one and a large one . Having then made a longitudinal section of the small ring in the middle, in the end we will get two intertwined rings of the same size, but different in width.

Practical use of Möbius strip
There are already quite a few inventions based on the properties of this unusual topological object. For example, the 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 a 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 Mobius 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 always move in one direction along the Mobius strip, then, in the end, we will find ourselves at the starting point of our journey, but in our own mirror image.

The mysterious Klein bottle
Based on the Möbius strip, there is another amazing figure - the Klein bottle. It is a bottle with a hole at the bottom. The neck of the bottle is elongated and curved, 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 wall of the bottle and should be connected to a hole in its bottom. This results in a surface that has only one side. The Klein bottle and the Möbius strip still attract the attention of scientists and writers.

A. Deitch, in one of his stories, wrote about how one day the tracks crossed in the New York subway and the entire subway began to resemble a Mobius strip, and 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 still remains a huge mystery to us, and who knows what other quirks of space scientists will discover in the near future.

Let's 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!

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

M. Escher "Mobius strip II" "Transition" through the Mobius strip into another dimension

August Ferdinand Möbius (1790-1868) - 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 provided enough money not to think about them, and left time for one’s own thoughts. And Möbius became one of the largest geometers of the 19th century.

At the age of 68, Möbius made a discovery of amazing beauty. This is the discovery of one-sided surfaces, one of which is the Möbius strip (or strip). Möbius came up with the idea of ​​the ribbon when he observed a maid who was wearing her scarf incorrectly around her neck.

M. Escher "Möbius Strip"

Let's make a Mobius strip: take a paper strip - a long narrow rectangle ABCD (convenient dimensions: length 30 cm, width 3 cm). Having twisted one end of the strip 180º, glue a ring from it (points A and C, B and D). The model is ready.

Möbius strip model can be easily created from a strip of paper by turning one end of the strip half a turn and connecting it to the other end into a closed shape. If you start drawing a line with a pencil on the surface of the tape, the line will go deep into the figure and pass under the starting point of the line, as if going 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 a Möbius strip has only one side and one border.

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

The Mobius strip will give you a surprise if you try to cut it. Cut the sheet along the center line. What did you get? Instead of falling apart into two pieces, the tape unfolds into a long, connected, closed strip. Cut the tape obtained after the first cut again along the center line. Before the last squeeze of the scissors, try to guess what will happen?

To get a Möbius strip, we turned the strip of paper 180º, half a turn. Now twist the strip 360º, a full turn. Glue it together, then cut it along the center line. It is difficult to predict what the result will be.

Now let’s try to make such a model: cut a slit in the ABCD strip and thread one end through it. Turn it half a turn and glue it together as shown in the picture.

Now continue the cut along the entire ribbon. What did you get?

The mysterious and famous Moebius strip, which appeared in 1858, worried artists and sculptors. Many drawings depicting the Möbius strip were left by the famous Dutch artist Maurice Escher (see article).

A whole series of variants of the Mobius strip can be found in sculpture.

Romance with a stone. Mobius Sling. S. Karpikov Monument to the Mobius strip in Moscow. A. Nalich

Paradox and perfection. A. Etkalo Geometric sculptures by Merit Rasmussen

Minsk. Square near the Central Scientific Library named after Yakub Kolas.

Architectural solutions using the Moebius strip idea:

Incredible new library project in Astana, Kazakhstan

Table compositions:

There is even furniture in the form of a Mobius strip

Jewelry in the form of a Mobius strip:

There is a hypothesis that the human DNA spiral itself is also a fragment of a Mobius strip.

The international symbol for recycling is a Möbius strip.

The Möbius strip is also a recurring theme in science fiction., for example in Arthur C. Clarke's story "The Wall of Darkness". Sometimes science fiction stories (following theoretical physicists) suggest that our Universe may be some kind of generalized Möbius strip. Also, the Mobius 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 Anchor Field. A Tale”). In the story "The Mobius Strip" by A. J. Deitch, the Boston subway builds a new line whose route becomes so confusing that it becomes a Mobius strip, causing trains to disappear on the line. Based on the story, the science fiction film “Mobius”, directed by Gustavo Mosquera, was shot. Also, the idea of ​​a Mobius strip is used in M. Clifton’s story “On the Mobius Strip.” The course of the novel by the modern Russian writer Alexei A. Shepelev “Echo” (St. Petersburg: Amphora, 2003) is compared with the Möbius strip. From the annotation to the book: ““Echo” is a literary analogy of the Mobius ring: two storylines—“boys” and “girls”—are intertwined, flow into each other, but do not intersect.”