What shapes can be made from a Mobius sheet. Mobius strip - an amazing discovery

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.

Who opened it and when?

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:

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.

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

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.

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 by econet.ru

sources

Technology - youth 1984-09, page 65

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.

experience	Description of the experience	Result
	A simple ring was cut lengthwise in the middle.	We got two simple rings, the same length, twice as wide.
	The Möbius strip was cut along the middle.	We got 1 ring, the length of which is twice as long, the width is twice as narrow, twisted 1 full turn.
	Cut the Möbius strip, retreating from the edge by about a third of its width.	You get two strips, one is a shorter Möbius strip, the other is a longer one. tape with two half turns.
	Divide a 4cm wide ribbon into four equal parts, start cutting at a distance of 1cm from the edge.	You get two ribbons, one equal to the length of the original, the other long.
	Cut a 5 cm wide Möbius strip lengthwise at a distance of 1 cm from the edge.	You will get two rings interlocked with each other: a Möbius strip 3 cm wide, equal to the length of the original one and 1 cm wide, twice the length of the original one, twisted two full turns.
	Glue the Möbius strip together by twisting it twice.	We get two Mobius strips linked to each other.

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.

The Möbius strip has one edge.
The Möbius strip has one side.
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.
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.

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

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!

August Ferdinand Mobius (1790-1868)

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.

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.

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

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.

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.

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 180* 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.

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 of the famous ones is Mobius Strip II, which shows ants crawling on the surface of a Mobius strip.

The Möbius strip is the emblem of the series of popular science books in the “Quantum” Library series. It also appears regularly in science fiction, such as 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 the 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 Möbius strip is used in M. Clifton’s story “On the Möbius Strip.”

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

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 Mobius strip.

The flow of the novel “Echo” by the modern Russian writer Alexei Shepelev 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.”

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

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

In 1987, Soviet jazz pianist Leonid Chizhik recorded the album “Mobius Strip,” which included the composition of the same name.

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

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

Also above the entrance to the Institute of Central Economics and Mathematics of the Russian Academy of Sciences there is a mosaic high relief “Mobius Strip” by the architect Leonid Pavlov in collaboration with the artists E. A. Zharenova and V. K. Vasiltsov (1976)

Architectural solutions using the Moebius strip idea:

Jewelry in the form of a Mobius strip:

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

A device called a Möbius resistor is a recently 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 matrix printers, the ink ribbon also had the form of a Möbius strip to increase shelf life.

Magical, unreal - these are all the adjectives that can be used to describe a Mobius strip. One of the biggest mysteries of our time. Perhaps it is the Mobius strip that hides the mysteries of the interaction of everything that exists in our Universe. This figure has mysterious properties and very real applications.

The Möbius strip is one of the most extraordinary geometric figures. Despite its unusual nature, it is easy to make at home.

A Möbius strip is a three-dimensional non-orientable figure with one boundary and one side. This makes it unique and different from all other objects that can be encountered in everyday life. A Möbius strip is also called a Möbius strip and a Möbius surface. It refers to topological objects, that is, continuous objects. Such objects are studied by topology - a science that studies the continuity of the environment and space.

The opening of the tape itself arouses interest. Two unrelated mathematicians discovered it in the same year, 1858. These discoverers were August Ferdinand Möbius and Johann Benedict Listing.

Ribbons are conventionally distinguished by the method of folding: clockwise and counterclockwise. They are also called right and left. But it is impossible to distinguish the type of tape by eye.

Making such a figure is extremely simple: you need to take ABCD tape. Fold it so as to connect points A and D, B and C, and glue the connected ends.

Some believe that this mysterious geometric figure is a prototype of an inverted figure eight-infinity, but in fact this is not true. This symbol was introduced for use long before the Möbius strip was discovered. But there is definitely a similarity in the meaning of these figures. Mystics call the Mobius strip a symbol of the dual perception of the one. The Mobius strip seems to speak of the interpenetration, interconnectedness and infinity of everything in our world. No wonder it is often used as emblems and trademarks. For example, the international symbol for recycling looks like a Mobius strip. The Mobius strip can also be a unique illustration of certain natural phenomena, for example, the water cycle.

The Möbius strip has characteristic properties that do not change if the strip is compressed, crumpled or cut lengthwise.

These properties include:

One-sidedness. If you take a Mobius strip and start painting in any place and direction, then gradually the entire figure will be painted over entirely, without the need to turn the figure over.
Continuity. Each point of this figure can be connected to another point without ever going beyond the edges of the tape.
Biconnectivity (or two-dimensionality). The tape remains intact if you cut it lengthwise. In this case, it will not produce two different figures.
Lack of orientation. If we imagine that a person could follow this figure, then when returning to the starting point of the journey, he would turn into his own reflection. The journey along the sheet of infinity could go on forever.

If you take scissors and do a little magic on this mysterious surface, you will be able to create additional unusual shapes. If you cut it lengthwise, along a line equally distant from the edges, you will get a twisted “Afghan Ribbon”. If the resulting tape is divided lengthwise, in the middle, then two tapes are formed, interpenetrating each other. If you put several strips on top of each other and connect them into a Mobius strip, then if you unfold such a figure, you will again get an “Afghan strip”.

If you cut a Möbius strip with three or more half-turns, you get rings called paradromic rings.

If you glue two Mobius strips together along the boundaries, you will get another amazing figure - a Klein bottle, but it cannot be made in ordinary three-dimensional space.

If you smooth out some of the edges of the Mobius strip, you will get an impossible Penrose triangle. This is a flat triangle illusion; when you look at it, it seems three-dimensional.

The Möbius strip is an inexhaustible source for the creativity of writers, artists and sculptors. Its mention is often found in fantasy and mystical literature. Its properties were the basis for artistic fiction about the origin of the Universe, the structure of the afterlife, and movement in time and space. The Möbius strip was mentioned in their works by Arthur Clarke, Vladislav Krapivin, Julio Cortazar, Haruki Murakami and many others.

The famous artist Escher created a number of lithographs using tape. In his most famous work, ants crawl along a Mobius strip.

The properties of the Mobius strip will allow you to show interesting tricks. Let's look at one of the most famous. Two Möbius strips made of potassium nitrate are suspended, and the magician touches a lit cigarette to the midline of each of them. The flaming flame will lengthen the first ribbon, and turn the second into two connected to each other. The popular roller coaster ride is made in the shape of a Mobius strip. Jewelers often use this geometric figure when creating jewelry designs.

Mobius strips are widely used in science and industry. It is the source for many scientific studies and hypotheses. There is, for example, a theory that DNA is part of a Mobius strip. Genetics researchers have already learned how to cut single-stranded DNA to create a Möbius strip. Physicists say that optical laws are based on the properties of the Mobius strip. For example, reflection in a mirror is a kind of movement in time along a similar trajectory. There is a scientific hypothesis that the Universe is a giant Mobius strip.

In the early 20th century, Nikola Tesla invented the Möbius resistor, which resists the flow of electricity without causing electromagnetic interference. It consists of two conductive surfaces that are twisted 180° to form a Möbius strip.

The strip of the conveyor belt (continuous transport machine) is made in the shape of a Mobius strip. This surface allows you to increase the life of the tape, since its wear will occur evenly. The Moebius strip form is also used when recording on continuous film.

The Mobius strip was used in dot matrix printers to extend the shelf life of the ink ribbon.

An abrasive ring in sharpening mechanisms is created on the basis of a Moebius strip, and automatic transmission operates.

Currently, many inventors use the properties of this tape to conduct experiments and create new devices.

The Mobius strip continues to arouse persistent interest, not only among mathematicians and inventors, but also among ordinary people. She inspires artists to create mysterious works and fantastic theories. Experimenting with this interesting figure is a fascinating activity for both adults and children. Its properties have found their application in science, technology and in everyday life. The Mobius strip is an entertaining mathematical riddle that hides the meaning of an idealistic understanding of the structure of the Universe; its impact on our lives can be studied endlessly.

Möbius strip (Möbius loop, Möbius strip)- a simple-looking figure, but a mathematician would say that it is a two-dimensional surface with amazing properties: it has only one side and one edge, unlike an ordinary ring, which can be rolled up from the same strip as a Möbius strip, but it has there will be two sides and two edges. You can easily verify this if you draw a line in the middle of the tape, without lifting the pencil from the paper until you return to the starting point. Surprisingly, but true: due to a half-turn of the strip, its upper and lower edges merged into one continuous line, and the two sides turned into a single whole and became one side. And here is the result: you can get from one point of the Mobius strip to any other without going over the edge.

Running on a Mobius strip

For an outside observer, a journey along a Mobius strip is a “running in a circle”, full of surprises. It was clearly depicted by the Dutch graphic artist Maurits Escher (1898-1972). In the painting “The Mobius Strip II” the ants are running. By following their movement, you can make an interesting discovery. Having made one revolution along the tape, each ant will be at the starting point, but already in the antipode position - visually it will be “on the other side” of the tape upside down. What happens to a two-dimensional creature moving along a Mobius strip? Having gone around the surface, it will turn into its mirror image (this is easy to imagine if you consider the tape transparent). To become itself, a two-dimensional being will have to make one more circle. So the ant needs to walk along the Möbius strip twice to return to its starting position.

Scientific curiosity or useful discovery

The Möbius strip is often called a mathematical curiosity. And its very appearance is attributed to chance. According to legend, the ribbon was invented by a German scientist when he saw an incorrectly tied neckerchief on a maid. He was a famous mathematician and astronomer, a student of Carl Friedrich Gauss. He described a one-sided surface with a single edge back in 1858, but the paper was not published during his lifetime. In the same year, independently of Mobius, a similar discovery was made by Johann Listing, another student of Gauss.

The tape was still named after Möbius. It became one of the first objects of topology - a science that studies the most general properties of figures, namely those that are preserved during continuous (without cuts or gluing) transformations: stretching, squeezing, bending, twisting, etc. These transformations resemble the deformations of figures made of rubber, Therefore, the topology is otherwise called “rubber geometry”. Some topological problems were solved by Leonhard Euler back in the 18th century. The beginning of a new field of mathematics was laid by Listing’s work “Preliminary Studies in Topology” (1847), the first systematic work on this science. He also coined the term “topology” (from the Greek words τόπος - place and λόγος - teaching).

The Möbius strip could be considered a scientific curiosity, another whim of mathematicians, if it had not found practical application and did not inspire artists. Artists have depicted her more than once, sculptors erected monuments to her and writers dedicated their creations to her. This unusual surface has attracted the attention of architects, designers, jewelers and even clothing and furniture manufacturers. Inventors, designers, and engineers paid attention to it (for example, back in the 1920s, audio and film tapes in the form of a Möbius strip were patented, allowing the recording duration to be doubled). But magicians deal with this strip more often than others: they are attracted by the unusual properties that appear when it is cut. So, if you cut a Möbius strip along the middle line, it will not break into two parts, as you might expect. It will make a narrower and longer double-sided tape, twisted twice (the design of the roller coaster ride has a similar shape). Here’s a “culinary trick”: cakes in the shape of a Mobius strip will seem tastier than regular ones, because you can spread twice as much cream on them! In addition, there are interesting architectural designs of buildings made “in the style of a Möbius strip.” For now they exist only on paper, but, I want to believe, they will certainly be implemented.

"Ambiguous" position

With its properties, the Möbius strip actually resembles an object from Through the Looking Glass. And she herself, being an asymmetrical figure, has a mirror double. Let us send the print of the right foot for a walk along the tape and soon we will find that the print of the left foot will return home. It's funny, isn't it? And when did the “right” manage to become the “left”? Let’s “mount” a two-dimensional clock into the tape and force it to make a full revolution along it. Looking at the clock, we will see that the hands on the dial are moving at the same speed, but in the opposite direction! And which of the two directions of movement is correct?

While you are thinking about the answer, I note that a mathematician would offer an elegant way out of even this “ambiguous” situation. It is necessary that, firstly, the watch always shows the same time, and secondly, the hands on the dial should be in a position that would be preserved in a mirror reflection, for example, stand vertically, forming a reversed angle.

Well, let's check the answer? In fact, it is impossible to set a specific direction of rotation on a Möbius strip. The same movement can be perceived as both a clockwise turn and a turn in the opposite direction. When a point randomly selected on the Möbius strip goes around it, one direction continuously changes to another. At the same time, “right” is subtly replaced by “left”. A two-dimensional being will not notice any changes in itself. But they will be seen by other similar creatures and, of course, by us, who are watching what is happening from another dimension. This is such an unpredictable, one-sided Möbius surface.

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