Magic, unreal - these are all epithets that can be awarded the ribbon of Mebius. One of the biggest mysteries of modernity. Perhaps it is the ribbon of Mebius that hides in himself the riddles of interaction of the entire existing in our universe. This figure has mysterious properties and quite real applications.

Mebius tape is one of the most unusual geometric shapes. Despite its unusual, it is easy to do at home.

Mebius tape is a three-dimensional unfulfilled figure with one border and side. It is unique and different from all other items that can meet in everyday life. Mebius tape is also called the leaf of the Mebius and the surface of the Mebius. It refers to topological objects, that is, the objects are continuous. Such objects studies the topology - science, exploring the continuity of the medium and space.

Interest is already the opening of the tape. Two mathematics unrelated to each other opened it in the same 1858. These openers were August Ferdinand Mebius and Johann Benedict Listing.

Conventionally distinguish between the ribbons in the folding method: clockwise and counterclockwise. They are also called right and left. But it is impossible to distinguish "on the eye".

Make such a figure extremely simple: you need to take the ABCD ribbon. Collapse it so as to connect the points a and d, in and s, glue the connected ends.

Some believe that this mysterious geometric shape is a mode of an inverted eight infinity, in fact it is incorrect. This symbol was introduced for use much earlier than the mebius tape was opened. But the similarity of the meaning of these figures is definitely there. Mystics are called Mebius tape with a symbol of dual perception of one. Mebius tape seems to speak of interpenetration, interconnectedness and infinity of everything in our world. No wonder, it is often used as emblems and trademarks. For example, international Symbol Recycling looks like a mebius tape. Mebius tape can also be a kind of illustration of some phenomena in nature, for example, a water cycle.

Mebius tape has characteristic properties, they do not change if the tape is compressed, lick or cut along.

These properties include:

• One-sidedness. If you take the ribbon of the Mebius and begin to paint in any place and direction, then gradually the whole figure will be painted entirely, while the figure will not need to turn the figure.
• Continuity. Each point of this figure can be connected to another point, while not going beyond the edges of the tape.
• Double-minded (or two-dimensional). The ribbon remains solid if you cut it along. From it will not work in this case two different figures.
• No focus. If you imagine that a person could go along this figure, then when returning to the point of starting the journey, he would turn into reflection. Traveling along the infinity sheet could continue forever.

If you take scissors and make a little bit at this mysterious surface, you will create additional unusual figures. If you cut it along, along the line removed from the edges at an equal distance, then the "Afghan ribbon" will be swirling. If the resulting tape is divided along, in the middle, then two ribbons are formed, interpenetrating each other. If you put a few stripes on each other and connect to the Mebius ribbon, if such a figure is deployed, the "Afghan ribbon" will turn out again.

If you cut the mebius tape with three or large numbers of semoligor, then rings are called paramro.

If you glue together two ribbons of the Mebius along the borders, then another amazing figure will be released - the bottle of Kleina, but it cannot be done in the usual three-dimensional space.

If you smooth out some vehicles of the Mesbius sheet, then the impossible triangle of Penrose will be released. This is a flat triangle-illusion when you look at it, it seems bulk.

The sheet of Mebius is an inexhaustible source for creativity writers, artists and sculptors. His mention is often found in fantastic and mystical literature. Artistic fictions were based on its properties about the emergence of the Universe, the arrangement of the afterlife, movement in time and space. The Lease of Mebius was mentioned in their works Arthur Clark, Vladislav Krapivin, Julio Cortasar, Haruki Murakami and many others.

A row of lithographs with the use of tape was created by the famous artist Esher. At his most famous work, ants crawl along the Lest Mebius.

Properties of Mebius tape will allow interesting tricks. Consider one of the most famous. Two ribbons of the Mebius from potash nitality are suspended, the magician touches the grilled cigarette to the midline of each of them. The flame broke out the first ribbon, and the second will turn into two, associated with each other. In the form of a ribbon of Mebius, a popular attraction "American slides" is made. Often use this geometric figure of jewelers when creating jewelry design.

Mebius ribbon is widely used in science and industry. It is a source for a variety of scientific research and hypotheses. There is, for example, the theory that DNA is part of the Mesbius sheet. Researchers in the field of genetics have already learned how to cut a single-chain DNA so as to get mebius ribbon from it. Physics suggest that optical laws are based on the properties of the Mobius sheet. For example, reflection in the mirror is a kind of movement in time by a similar trajectory. There is a scientific hypothesis that the Universe is a giant mebius tape.

At the beginning of the 20th century, Nikola Tesla invented a mebius resistor, which resists the flow of electricity without causing electromagnetic interference. It consists of two conductive surfaces that are twisted by 180 ° and form a mebius ribbon.

A ribbon conveyor strip (transporting machine of continuous action) is made in the form of a mebius tape. This surface allows you to increase the use of the tape, since its wear will occur evenly. Use the form of the Mebius tape and when recording on a continuous film.

Mebius leaf was used in matrix printers to extend the shelf life of the coloring tape.

Based on the mebius tape, an abrasive ring was created in sharpening mechanisms, automatic transmission works.

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

Mebius tape continues to cause persistent interest, not only in mathematicians and inventors, but also from ordinary people. She inspires art leaders to create mysterious works and fantastic theories. Experiments with this interesting figure - fascinating occupationfor both adult and child. Its properties found their use in science, technology and in everyday life. Mebius tape is an entertaining mathematical mysteryhiding the meaning of an idealistic understanding of the device of the universe, its impact on our life can be studied infinitely.

So he is the author of the amazing ribbon of the Mebius!
German Mathematician and Astronomer Theorient August Ferdinand Möbiius (1790-1868) - a student of the Great Gauss, a well-known Geometer, Professor of the University of Leipzig, director of the observatory. Long years of teaching, long years of work - usual life Professor.

And now it is necessary, it happened at the end of life! An amazing idea came ... It was the most significant event in his life! Unfortunately, he never had time to assess the significance of his invention. An article about the famous ribbon of Mebius was posthumously published.

How are Mebius tape (otherwise a sheet of mebius or mebius loop) mathematics?

In the language of mathematics is topological objectsimplest one-sided surface With the edge in the usual three-dimensional Euclidean space, where you can get from one point of this surface to any other, without crossing the edge.
Quite complex definition!

Therefore, it is more convenient to simply consider Mebius's ribbon closer. Take paper strip, twisted the strip in the half turn across (180 degrees) and glue the ends.

Another time "Mom would have stroked for such work"! But this time you are right! It should be a twisted ring.

We put in some place on the strip the point of the felt-tip pen. And now they drag along the whole of our tape the line until you meet your point again. You have not had to go through the edge anywhere - this is called a one-sided surface.

Look how the line you are read is interested in: it is inside the rings, then outside! And now measure the length of this line - from the point to the point.
Wonder?
It turns out to be twice as long as the initial paper strip!

So it should be, because you have a ribbon of Mebius in your hands! And the ribbon of the Mebius has only one side, and we again say - this is a one-sided surface with the edge.

And if it is for this feature to force crawling, not folding, ant, then you will receive a copy of the paintings by the artist Maurice Escher.
Poor ant on an endless road

And you can make two slightly different ribbons of the Mebius: one to twist the strip clockwise before gluing, and the other is counterclockwise. So the right and left ribbons of the Mebius differ.

And now Interesting surprises With Mebius Ribbon:

1. Cut the mebius car ribbon along the center line. Do not be afraid, it will not fall apart into two parts! The ribbon will turn into a long closed tape, twice the greatest than the initial one. Why does the mebius tape, with such a section, do not break into separate parts?
The cut did not touch the edge of the tape, so after cutting the edge (and therefore the entire paper strip) will remain a whole piece.

2. The mebius ribbon obtained after the first experience (twisted twice as large as the initial, i.e. 360 degrees) again cut down its central line.
What happens?
In your hands, now there are two identical, but the tapes of the Mebius are connected.

3. Make a new ribbon of the Mebius, but before gluing, turn it more than once, and three times (not 180 degrees, and at 540). Then cut it along the central line.

What happened?
You should get a closed tape, curled in trilisset knot. In a simple knot with three self-intersections.

4. If you make a mebius tape with an even large number of semoligitives before gluing, then unexpected and amazing figures, called paradromic rings.

5. If the Mebius ribbon is cut, not in the middle, and retreating from the edge to about a third of its widths, then two clutch tapes are obtained, one is a shorter mebius tape, and the other is a long mebius tape with two semolsters.

See how this can be done in practice:

Close to the mebius ribbon one-sided surface is bottle of Klein.
Interestingly, a cupin bottle can be obtained by gluing two mebius tapes along the edges. However, in the usual three-dimensional Euclidean space to do this without creating self-intersections, it is impossible.

There is another interesting object associated with the Mebius ribbon. it Mebius resistor.

In history, there are often cases when one idea comes into the head at the same time several inventors. It happened with the Mebius ribbon. In the same 1858, the idea of \u200b\u200btape came to another scientist - Johann Listing. He gave the name of science examining continuity - topology. And the championship in the opening of the topological object - the tape went by August to Mebius.

We are inconspicuously meeting the ribbon of Mebius in different devices: these are painting tapes in matrix printers, and belt transfers, grinding devices, belt connels and many others. In this case, the life of the product increases, because Decreased wear. And in continuous recording systems, the use of mebius tape allows you to double the recording time on one film.

The mysterious ribbon of Mebius always blinked the minds of writers, artists and sculptors.
The drawing of the Mebius tape is used in the chart. Alternate, for example, the emblem of the famous series of popular science books "QUANT" Library "or an international symbol of processing

Let's experiment: cut the strip from paper, make the ends of the ribbon, but not as usual, but with a turn of 180 degrees. We had a tape of Möbius.

German astronomer and mathematician Augustus Ferdinand Möbiius took one day paper ribbon., I turned one end to the half-turn (that is, 180 degrees), and then glued it with another end. Whether he did it from boredom, or for the sake of scientific interest - now it is already unknown. But it is known for certain that this is how the famous tape of Möbius appeared in the last century.

Tape properties Moebius

What is it famous? So that the surface of the tape Moebius has only one side. It is easy to check. Take a pencil and start painted with a ribbon in some direction. Soon you will return to the place where you started. And now you will look carefully: the whole tape turned out to be painted! But you did not turn it over to paint on the other side. Yes, and could not flip, even if they wanted very much. Because the surface of the Moebius tape - one-sided. Such is her curious property is observed.

We will work for scissors again: let's drive this tape and gently cut it along - exactly in the middle. "Well," you think, "there will be two separate rings ...".

But what is it? Instead of two rings, one turns out! And it is larger and thinner is initial, and twice twice. "This does not happen," you say. It happens.

What do you think will be with this figure if you cut it again? Maybe one whole, but the twisted strip of paper will come out again? Not. This time there are already two clutch rings.

These are such interesting metamorphoses of the Mebius ribbon. You can show your friends with these phenomena, giving them for tricks, whereas in fact you just demonstrate to them mathematical laws.

A simple strip of paper, but the twisted only once and glued into the ring, immediately turns into a mysterious migratory tape and acquires amazing properties. Such properties of surfaces and spaces studies a special section of mathematics - topology.
This science is so complicated that it does not pass at school. Only in institutions (and not in all!). But who knows, suddenly you will become a famous topologist with time and do not make one remarkable discovery. And perhaps, some intricate surface will be called your name!

Mebius tape in architecture

And where in real life Can I see the Moebius tape? Many architects in their projects are trying to use mysterious tape. So Belgian architect Vincent Callebo for the park in Taiwan developed a new building, which resembles a mebius tape.

The construction has the shape of a swallow socket and begins with a triangle, and then twisted into an ellipse. Inside the building can be admired by plants, art objects or just make a walk.

There are scientific knowledge and phenomena, which will bring the mystery and riddle in the usualness of our life.

Mebius tape belongs to the fullest. Modern mathematics remarkably describes with the help of formulas all its properties and features. But the usual people, weakly dismantled in toponymics and other geometric wisdoms, almost daily face objects made by her image and likeness, not even suspecting it.

What it is?

Mebius tape, which is also called a loop, surface or sheet, is an object of studying such mathematical discipline as the topology exploring general properties Figures persist in such continuous transformations as twisting, stretching, compression, bending and other non-violation of integrity. An amazing and unique feature of such a tape is that it has only one side and the edge and are not connected with its location in space. The metal sheet is topological, that is, a continuous object with the simplest one-sided surface with a boundary in a conventional Euclidean space (3-dimensional), where possible from one point of such a surface, without crossing the edge, to get into any other.

Who and when she opened it?

Such a difficult object, like a mebius tape, was and open quite unusual. First of all, we note that two mathematics, absolutely not related among themselves, discovered it at the same time - in 1858. Another one interesting fact is that both of these scientists in different time We were students of the same great mathematics - Johann Charles Friedrich Gauss. So, until 1858 it was believed that any surface is obliged to have two sides. However, Johann Benedict Listing and August Ferdinand Mebius discovered the geometric object, which only one party was only described, and describes its properties. The tape was named after Mebius, but the father-founder of the "rubber geometry" topologists consider listing and its work "Preliminary research on topology".

Properties

The mebius tape is inherent in the following properties that do not change when it is compressed, cut along or crushing:

1. The presence of one side. A. Mebios in its work "On the volume of polyhedra" described the geometric surface, then named then in his honor, having only one side. Check it is quite simple: we take a tape or a leaf of the Mebius and try to paint the inner side with one color, and the external one. It is not important that staining began in the direction and the direction, the whole figure will be painted in one color.

2. Continuity is expressed in the fact that any point of this geometric shape can be connected to any other point, without crossing the boundaries of the mebius surface.

3. Connectivity, or two-dimensionality, is that when cutting a tape along, several different figures will not work out of it, and it remains solid.

4. It does not have such an important property as orientation. This means that a person walking along this figure will return to the beginning of its path, but only in a mirror reflection of himself. Thus, the endless tape of the Mebius can lead to an eternal journey.

5. A special chromatic number showing what the maximum possible number of areas on the surface of the Mebius can be created so that any of them has a common border with all others. Mebius tape has a chromatic number - 6, but the paper ring is 5.

Scientific use

Today, the Mesbius leaf and its properties are widely used in science, serving the basis for building new hypotheses and theories, research and experimentation, creating new mechanisms and devices.

So, there is a hypothesis, according to which the Universe is a huge loop of Mebius. Indirectly evidenced by the theory of the relativity of Einstein, according to which even the flew straight straight can return to the same temporary and spatial point, from where it started.

Another theory considers DNA as part of the surface of the Mebius, which explains the difficulties with reading and deciphering the genetic code. Among other things, this structure gives a logical explanation of biological death - a spiral closed on itself leads to self-destruction of the object.

According to physicists, many optical laws are based on the properties of the Mesbius sheet. So, for example, a mirror reflection is a special transfer in time and a person sees his mirror twin in front of him.

Implementation in practice

In various industries of the Mebius tape, the application has found a long time ago. The great inventor of Nikola Tesla in the beginning of the century invented a mebius resistor consisting of two conductive surfaces twisted at 180 0, which can resist the stream of electric current without creating electromagnetic interference.

Based on the surface of the surface of the Mebius tape and its properties, many devices and devices were created. Its form is repeated when creating a strip of a ribbon conveyor and a coloring ribbon in printing devices, abrasive belts for sharpening tools and automatic transmission. This allows you to significantly increase the service life of their service, as wearing is more evenly.

Not so long ago, the amazing features of the Leaf of Mebius allowed to create a spring, which, unlike the usual, triggered in the opposite direction, does not change the direction of response. It is used in the steering stabilizer of the steering wheel, providing a refund of the steering wheel to its original position.

In addition, the Mebius tape sign is used in a variety of brands and logos. The most famous of them is an international symbol of recycling. It is affixed on the packages of goods or suitable for subsequent processing, or made from recycled resources.

Source of creative inspiration

Mebius tape and its properties have formed the basis of many artists, writers, sculptors and cinematographers. The most famous artist who used in such works as "Mebius II Tape (Red Ants)", "riders" and "knots", tape and its features - Mauritz Cornelis Escher.

Mebius sheets, or, as they are also called, the surface of minimal energy, became a source of inspiration for mathematical artists and sculptors, for example, Brent Collins or Max Bill. The most famous monument to Mebius tape is installed at the entrance to the Washington Museum of History and Technology.

Russian artists also did not remain aside from this topic and created their work. Sculptures "Mebius Tape" are installed in Moscow and Yekaterinburg.

Literature and topology

The unusual properties of the surfaces of the Mebius inspired many writers to create fantastic and surreal works. Mebius loop playing important role In the Roman R. Zelaznos "Doors in the sand" and serves as a means of moving through space and time for the main character of the novel "Nonoskop" B. Lamley.

It appears in the stories of the "wall of darkness" Arthur Clark, "on the ribbon of Mebius" M. Clifton and Leaf Mebius A. J. Deia. Based on the last director, Gustavo Moskurster was removed fantastic kinocartine "Mebius".

We do, do it yourself!

If you are interested in Mebius tape, how to make her model, you will tell you a small instruction:

1. For the manufacture of its model, you will need:

Sheet of ordinary paper;

Scissors;

Line.

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

3. The resulting paper strip is decaying on a flat surface. One end hold the hand, and the other turn 180 0 so that the strip twisted and the exhaust becomes the face.

4. glue the ends of the twisted strip as shown in the figure.

Mebius ribbon is ready.

5. Take a handle or marker and ribbon in the middle. Start drawing the track. If you did everything right, you will return to the same point where they started to draw a line.

In order to gain visual confirmation that Mebius tape is a one-way object, a pencil or handle, try to paint some kind of side. After a while you will see that they painted it completely.

Mebius tape (Möbius loop, Möbius leaf) - simple figure, but the mathematician would say that this is a two-dimensional surface with amazing properties: she has only one side and one edge, unlike an ordinary ring, which can be collapsed from the same strip as the Möbius tape, but he has There will be two sides and two edges. It is easy to make sure that if you draw a line in the middle of the tape, without taking the pencil from the paper until you return to the starting point. Surprisingly, but the fact: due to the half-edge 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: to get from one point of the tape of Möbius to any other, without moving through the edge.

Running on the ribbon Moebius

For a third-party observer, the journey along the Möbius ribbon is "running in a circle", full of surprises. He was visually depicted by a Dutch artist-schedule Mauritz Escher (1898-1972). In the film "Tape of Möbius II" in the role of running - ants. Following their movement, you can make an interesting discovery. By making one turn on the tape, each ant will turn out to be at the starting point, but already in the position of the antipode, - it will be "on the other side" a tape down head. And what will happen to a two-dimensional creature moving along the tape of Möbius? Bygoing the surface, it will turn into its mirror reflection (it is easy to imagine if you consider a transparent tape). To become myself, a two-dimensional creature will have to make another circle. So the ants need to go along the Möbius tape twice to return to the initial position.

Scientific curiosity or useful discovery

Ribe Möbius is often called mathematical curiosity. And its very appearance is attributed to the case. According to the legend, the tape came up with one German scientist when he saw an incorrectly tied neck scarf on the maid. It was a famous mathematician and astronomer, a student of Charles Friedrich Gauss. He described the one-sided surface with the only edge in 1858, but the article was not published during his life. In the same year, regardless of Möbius, a similar discovery was made by Johann Listing, another student Gauss.

Tape was still called in honor of Möbius. It has become one of the first objects of the topology - science that studies the most common properties of the figures, namely, which are preserved with continuous (without cuts and curlers) of transformations: stretching, squeezing, bending, twisting, etc. These transformations resemble deformations of rubber figures, Therefore, the topology is otherwise called "rubber geometry". Separate topological tasks decided in the XVIII century Leonard Euler. The beginning of a new region of mathematics laid the work of the listing "Preliminary research on topology" (1847) - the first systematic work on this science. He came up with the term "topology" (from the Greek words τόπος - place I. λόγος - Teaching).

Ribe Möbius could be considered a scientific curvoryer, the next fad of mathematicians, if she did not find practical applications and did not inspire people of art. She was once again depicted by artists, they put monuments of sculptors and dedicated their creations writers. This unusual surface faced as architects, designers, jewelers and even manufacturers of clothing and furniture. The inventors, designers, engineers were drew on it (for example, in the 1920s, audio and film ones in the form of Möbius tape were patented, allowing you to double the duration of the recording). But more often the fockers are dealing with this ribbon: they are attracted by unusual properties that are manifested when it is cutting it. So if you cut the Möbius ribbon along the midline, it will not break into two parts as you can expect. It will turn out a narrower and long bilateral tape, twice twice (similar shape has the design of the attraction "American slides"). But the "culinary focus": the cakes in the form of a tape of Möbius will seem tastier usual, because they can be smeared twice as much as the cream! In addition, there are interesting architectural projects of buildings performed "in the style of Möbius tape." While they exist only on paper, but, I want to believe, will certainly be implemented.

"Ambiguous" position

With its properties, Möbius ribbon in fact resembles an object from the casting card. And she herself, being an asymmetric figure, has a mirror twin. We will send stroll along the tape the imprint of the right foot and soon find that the left footprint will return home. Funny, really? And when only the "right" has time to become "left"? "Mount" in the tape a two-dimensional clock and make them make a complete turn. Looking at the clock, we will see that the arrows on the dial are moving at the same speed, but in reverse side! And what of the two directions of movement right?

While you think about the answer, I note that the mathematician would offer an elegant way out of this "ambiguous" position. It is necessary that, firstly, the clocks always showed the same time, and secondly, the arrows on the dial were in a position that would have survived under the mirror reflection, for example, it was vertically, forming an expanded angle.

Well, check the answer? In fact, on the Möbius tape, it is impossible to establish a certain direction of rotation. The same movement can be perceived as a turn clockwise, and as a rotation in the opposite direction. When the dot arbitrarily selected on the tape bypasses it, one direction continuously goes into another. At the same time, the "right" impossible is replaced by the "left". Two-dimensional creatures will not notice any changes. But they will see they will see the same creatures and, of course, we observe what is happening from another dimension. This is such an unpredictable, one-sided surface of Möbius.