"Secrets of the Möbius strip". Mobius strip and its surprises. Science Toys Moebius Fence

Magical, unreal - these are all the epithets that can be awarded to the Möbius strip. One of the biggest mysteries of our time. Perhaps it is the Möbius 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 unusual geometric shapes. Despite its unusualness, 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 items that can be found in Everyday life. The Möbius strip is also called the Möbius strip and the 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 very opening of the tape is of interest. Two unrelated mathematicians discovered it in the same 1858. These discoverers were August Ferdinand Möbius and Johann Benedict Listing.

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

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

Some believe that this mysterious geometric figure is a prototype of an inverted figure-eight-infinity, 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 Möbius 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. Not without reason, it is often used as emblems and trademarks. For example, the international symbol for recycling looks like a Möbius strip. The Möbius strip can also be a kind of illustration of some phenomena in nature, for example, the water cycle.

The Möbius strip has characteristic properties; they do not change if the strip is compressed, crumpled or cut along.

These properties include:

• One-sidedness. If you take a Mobius strip and start painting in any of its places and directions, then gradually the whole figure will be painted over entirely, and the figure will not need to be turned over.
• Continuity. Each point of this figure can be connected to its other point, while never going beyond the edges of the tape.
• Biconnectivity (or two-dimensionality). The tape stays intact when cut lengthwise. In this case, two different figures will not work out of it.
• Lack of orientation. If we imagine that a person could walk along this figure, then upon returning to the starting point of the journey, he would turn into his own reflection. The journey through the leaf of infinity could go on forever.

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

If we cut the Möbius strip with three or large quantities half-turns, then you get rings called paradromic.

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

If we smooth out some of the edges of the Möbius strip, we get an impossible Penrose triangle. This is a flat illusion triangle, when you look at it, it seems voluminous.

The Möbius strip is an inexhaustible source for the creativity of writers, artists and sculptors. His mention is often found in fantastic and mystical literature. Artistic fictions about the origin of the Universe, the structure of the afterlife, movement in time and space were based on its properties. The Möbius strip was mentioned in their works by Arthur C. Clarke, Vladislav Krapivin, Julio Cortazar, Haruki Murakami and many others.

The famous artist Escher created a number of lithographs using ribbon. In his most famous work, ants crawl on a Möbius strip.

The properties of the Möbius strip will allow you to show interesting tricks. Consider one of the most famous. Two Möbius bands of potassium nitrate are suspended, the magician touches a lit cigarette to the middle line of each of them. The blazing flame will lengthen the first ribbon, and turn the second into two connected to each other. The popular rollercoaster ride is made in the form of a Mobius strip. Jewelers often use this geometric figure when creating jewelry designs.

The Möbius strip is 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. Researchers in the field of genetics have already learned how to cut single-stranded DNA in such a way as to obtain a Möbius strip from it. Physicists say that optical laws are based on the properties of the Möbius strip. For example, a 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 Möbius strip.

In the early 20th century, Nikola Tesla invented the Möbius resistor, which opposes 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 a belt conveyor (transporting machine of continuous action) is made in the form of a Möbius strip. Such a surface allows you to increase the life of the tape, since its wear will occur evenly. Use the form of the Möbius strip and when recording on a continuous film.

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

On the basis of the Möbius strip, an abrasive ring was created in sharpening mechanisms, an automatic transmission works.

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

The Möbius strip continues to arouse strong 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 everyday life. Möbius strip is entertaining mathematical riddle, which hides the meaning of an idealistic understanding of the structure of the Universe, its impact on our lives can be studied endlessly.

30.07.11 Perhaps the very first unusual figure was invented in the middle of the 19th century by August Möbius. It was the so-called "Möbius strip", or "Möbius strip" - a very simple and at the same time very strange design.

It's easy to see that this figure has only one surface!

Imagine that, for example, an ant is running along the Möbius strip. However, let's do it easier: look at the Möbius strip depicted in the well-known drawing by Maurice Escher.

Having made a circle, the ant resorts to the same place from which it started moving, but at the same time it turns out to be on the opposite side of the flat tape! Naturally, after running one more lap, he will return to the starting point. (Of course, it is assumed that the ant cannot get over the edge of the tape.)

August Ferdinand Möbius (1790 - 1868)

German geometer and astronomer, professor at the University of Leipzig. Basic works on geometry. For the first time he introduced a coordinate system and analytical methods of research into projective geometry, received a new classification of curves and surfaces, established general concept projective transformation, studied correlative transformations. He was the first to establish the existence of one-sided surfaces.

Rumor has it that Mobius got the idea for this unusual geometric figure when he saw a maid incorrectly tying her neckerchief. Well, what, maybe, maybe! After all, Isaac Newton also pulled with the discovery of the universal law of gravity, until an apple fell on his head.
In fairness, it should be noted that the figure itself, called by everyone the Möbius strip, was simultaneously and independently built in the same 1858 by another German mathematician Johann Benedikt Listing (1808-1882), who, by the way, put into mathematical use the term "topology ".

The Möbius strip immediately attracted the attention of mathematicians. One of the interesting problems is the following: how long (for a given width) should the strip be so that it can be folded into a Möbius strip? A very important practical question, isn't it?

But the matter is not limited to a simple "classical" Möbius strip. Glue a Mobius strip from a wide strip of paper and try to cut it along the middle line. The initial cutting phase is shown in the left figure. And when you cut this ring to the end, then ... you will see the Möbius strip again, however, more “screwed” (right figure). But the ant, having started to crawl, will again run along both sides of the strip and return to the starting point.

By the way, magicians who cut the Möbius strip to the surprise of the audience call the resulting figure for some reason “Afghan ribbon”. But do not think that the miracles with the Möbius strip ended there. And what happens if the strip is rotated several times before gluing?

It all depends on how twisted the tape is. With one twist from a simple ring, we go to a Möbius strip.

Well, what happens when you double-turn the tape before gluing? It turns out that in this case we get just a "twisted" ring. But if the tape is turned before gluing again in the same direction. Then again you get a Möbius strip, but already “twisted”!

For the convenience of explaining the essence of the operations performed, a tape was chosen, one side of which is white and the other side is gray. Then it is completely clear that no matter how many times we twist the tape, if it turns out that at the junction “sides with the same color met, this means that the glued tape will have two surfaces - one white and the other gray , i.e. a ring with a helical generatrix will be formed. If at the junction, when gluing, the gray side “meets with the white one, then after gluing we will get a Möbius strip, although also intricate. It will have only one surface: after all, the Escher ant, running along the white side, eventually reaches the border, where the gray side begins and continues to run along it.

The properties of chains formed by flat rings and Möbius strips are also interesting.

We will tightly connect two ordinary flat rings and let the Escher ant crawl along the outer surface of the left ring. When he crawls to the junction of the rings, he can move to the inner surface of the second ring. If we launch the second ant onto the inner surface of the left ring, then it can move to the outer surface of the right ring. In other words, these two ants will never meet - each will crawl on its own surface.

It is clear that if a chain of flat rings or a chain of Möbius strips is constructed in this way, then these properties will be preserved.

With the Möbius strip, interesting experiments can be continued further. Make a blank from a sheet of paper, as shown in the figure. Cut along the lines, and then fold each of the resulting stripes, not separated from the main part, into a Möbius strip. It turns out a sort of multi-storey structure.

Of course, the figure shows a schematic representation of the resulting structure. A real "fractal" figure of this type looks much more intricate.

Here on such a “Möbius bush” an ant would travel a lot! Of course, there are a lot of such multi-tiered and nested Möbius strips.

In conclusion, we give another example of a figure that has the properties of a Möbius strip and at the same time none of the sides is twisted. Of course, things could not do without little tricks: you can get from the outside to the inside along the “escalator” in the center of the ring.

A "leaky" ring with the properties of a Möbius strip.

It is very easy to make a ring of this kind even with two escalators, which will allow the ant to make a complete cycle without ever visiting the same point (unless, of course, it does not make loops, but only moves forward).

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, in which an ant can crawl to any place without climbing over the edge, was given by Möbius in 1858.

M. Escher "Möbius strip II" "Transition" through the Möbius 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 gave enough money not to think about them, and left time for one's own reflections. And Möbius became one of the greatest geometers of the 19th century.

At the age of 68, Mobius managed to make a discovery of striking beauty. This is the discovery of one-sided surfaces, one of which is a Möbius strip (or strip). Mobius came up with the ribbon while watching a maid put her handkerchief on the wrong way around her neck.

M. Escher "Möbius strip"

Let's make a Möbius strip: take a paper strip - a long narrow rectangle ABCD (convenient dimensions: length 30 cm, width 3 cm). Twisting one end of the strip 180º, glue a ring out of 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 of the ends of the strip in half a turn and connecting it with the other end to form a closed shape. If you start drawing a line with a pencil on the surface of the tape, then the line will go deep into the figure and pass under the starting point of the line, as if 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 the Möbius strip has only one side and one boundary.

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

The Möbius strip will surprise you 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, knitted closed strip. Cut the tape obtained after the first cut along the center line again. Before the last scissor squeeze try to guess what will happen?

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

And now let's try to make such a model: cut a slot in the ABCD strip and thread one end through it. Turning half a turn, glue, as shown in the figure.

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

The mysterious and famous Möbius strip, which appeared in 1858, excited artists and sculptors. Many drawings with images of the Möbius strip were left by the famous Dutch artist Maurice Escher (see the article Mathematical Art by M.C. Escher).

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

Romance with stone. Mobius' sling. S. Karpikov Monument to the Möbius strip in Moscow. A. Nalich

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

Minsk. A public garden near the Central Scientific Library named after Yakub Kolas.

Architectural solutions using the Mobius 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 helix itself is also a fragment of the Möbius strip.

The international symbol for recycling is the Möbius strip..

The Möbius strip is also recurring 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 generalized Möbius strip. Also, the Möbius ring is constantly mentioned in the works of the Ural writer Vladislav Krapivin, the cycle “In the depths of the Great Crystal” (for example, “Outpost on the Anchor Field. Tale”). In A.J. Deitch's short story "Möbius Strip", the Boston subway builds a new line whose route becomes so confusing that it becomes a Möbius strip, and trains begin to disappear on the line. Based on the story, the fantastic film "Mobius" directed by Gustavo Mosquera was shot. Also, the idea of ​​the Mobius strip is used in the story of M. Clifton "On the Mobius strip". The flow 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 Möbius ring: two storylines - "boys" and "girls" - are intertwined, flow into each other, but do not intersect."

There are scientific knowledge and phenomena that bring mystery and mystery into the everyday life of our life. The Möbius strip applies to them fully. Modern mathematics remarkably describes with the help of formulas all its properties and features.

But ordinary people, who are poorly versed in toponymy and other geometric intricacies, almost daily encounter objects made in her image and likeness, without even knowing it. What is it? Mobius strip, which is also called a loop, surface or sheet, this is the object of study of such a mathematical discipline as topology, which studies the general properties of figures that are preserved under such continuous transformations as twisting, stretching, compression, bending, and others not related to integrity violation. An amazing and unique feature of such a tape is that it has only one side and edge and is in no way connected with its location in space. The Möbius strip is topological, that is, a continuous object with the simplest one-sided surface with a boundary in the usual Euclidean space (3-dimensional), where it is possible from one point of such a surface, without crossing the edges, to get into any other. Who and when discovered it?

Such a complex object as the Möbius strip was and was discovered in a rather unusual way.

First of all, we note that two mathematicians, absolutely unrelated in research, discovered it at the same time - in 1858.

Another interesting fact is that both of these scientists in different time 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 tape was named after Möbius, but topologists consider Listing and his work “Preliminary Studies in Topology” to be the founding father of “rubber geometry”.

1. The presence of one side. A. Möbius in his work "On the Volume of Polyhedra" described a geometric surface, then named after him, having only one side. Checking this is quite simple: we take a strip or Möbius strip and try to paint over the inside with one color, and the outside with another. It does not matter in what place and direction the coloring was started, the whole figure will be painted over with one color.

2. Continuity is expressed in the fact that any point of this geometric figure can be connected to any other of its points without crossing the boundaries of the Möbius surface.

3. Connectivity, or two-dimensionality, lies in the fact that when cutting the tape along, several different figures and it stays intact.

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

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

Scientific use

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

So, there is a hypothesis according to which the Universe is a huge Mobius loop. This is indirectly evidenced by Einstein's theory of relativity, according to which even a ship flying straight can return to the same time and space point from which it started.

Another theory sees DNA as part of the Möbius surface, which explains the difficulty in reading and deciphering the genetic code. Among other things, such a structure provides a logical explanation for biological death - a spiral closed on itself leads to the self-destruction of the object.

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

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

On the basis of studies of the surface of the Möbius strip and its properties, many devices and devices were created. Its shape is repeated in the creation of a conveyor belt strip and an ink ribbon in printing devices, abrasive belts for sharpening tools and automatic transmission. This allows you to significantly increase their service life, as wear occurs more evenly.

Not so long ago, the amazing features of the Möbius strip made it possible to create a spring that, unlike conventional ones 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 trademarks and logos. The most famous of them is the international symbol of recycling. It is affixed to the packaging of goods that are either recyclable or made from recycled resources.

A source of creative inspiration The Möbius strip and its properties formed the basis of the work of many artists, writers, sculptors and filmmakers. The most famous artist who used the ribbon and its features in such works as Mobius Ribbon II (Red Ants), Horsemen and Knots is Maurits Cornelis Escher.

Möbius strips, or, as they are also called, minimum energy surfaces, became a source of inspiration for mathematical artists and sculptors, such as Brent Collins or 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 Mobius surfaces inspired many writers to create fantastic and surrealistic works. The Mobius loop plays an important role in R. Zelazny's novel "Doors in the Sand" and serves as a means of moving through space and time for the protagonist of the novel "Necroscope" B. Lumley.

It also appears in the stories "The Wall of Darkness" by Arthur C. Clarke, "On the Mobius Strip" by M. Clifton and "The Mobius Leaf" by A. J. Deitch. Based on the latter, directed by Gustavo Mosquera, the fantastic film "Mobius" was shot.

We do it ourselves, with our own hands!

If you are interested in the Möbius strip, how to make its model, you will be prompted by a small instruction: 1. For the manufacture of its model will require:

A sheet of plain paper;

Scissors;

Ruler.

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

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

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

The Möbius strip is ready.

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

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

Source econet.ru

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

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

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

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

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

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

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

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

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

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