The structure of a bird's feather under a microscope. Abstract of GCD in the senior group Topic: “Experiments and experiments with bird feathers

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 may be found 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, they do not change if the tape 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 equidistant 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 Mobius strip with three or large quantities 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. Artistic fictions about the origin of the Universe, the structure of afterlife, 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. She is the source for many scientific research 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. Eat 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.

Band belt conveyor(continuous transport machine) is made in the form 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 Mobius 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.

30.07.11 Perhaps the very first unusual figure was invented in the middle of the 19th century by Augustus 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 a Mobius strip. However, let's do it simpler: let's look at the Möbius strip depicted in the well-known drawing by Maurice Escher.

Having made a circle, the ant runs back to the same place from where it started moving, but at the same time finds itself with opposite side flat tape! Naturally, after running one more lap, he will return to the starting point. (Of course, this assumes 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 new classification curves and surfaces, established general concept projective transformation, studied correlative transformations. For the first time he established the existence of one-sided surfaces.

Rumor has it that Möbius came up with the idea for this unusual geometric figure when he saw a maid who had tied her hair incorrectly. neckerchief. Well, maybe, maybe! After all, Isaac Newton also delayed 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 all the Möbius strip, was simultaneously and independently constructed in the same 1858 by another German mathematician Johann Benedict 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 a 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 Möbius strip from a wide strip of paper and try cutting 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 again see a Möbius strip, albeit more “screwed” (right picture). 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 Mobius strip to the surprise of the audience call the resulting figure for some reason “Afghan strip.” But don't think that the wonders of the Möbius strip end there. What happens if you turn the strip several times before gluing it?

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

Well, what happens when you turn the tape twice before gluing? It turns out that in this case the result is simply a “twisted” ring. But if you turn the tape before gluing it again in the same direction. Then again you will 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 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 meet”, then this means that the glued tape will have two surfaces - one white and the other gray , i.e. a ring with a helical forming tape will be formed. If at the junction during gluing the gray side meets the white side, then after gluing we will get a Möbius strip, although also an intricate one. It will have only one surface: after all, Escher’s 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.

Let's tightly connect two ordinary flat rings and let the Escher ant crawl along the outer surface of the left ring. When it crawls to the junction of the rings, it can move to the inner surface of the second ring. If you 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 you build a chain of flat rings or a chain of Möbius strips in this way, then these properties will be preserved.

You can continue interesting experiments with the Möbius strip. Make a blank from a sheet of paper as shown in the figure. Cut along the lines, and then roll each of the resulting strips, not separated from the main part, into a Möbius strip. The result will be a kind of multi-story structure.

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

An ant would have plenty of time to travel around this “Möbius bush”! Of course, you can come up with a lot of similar 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 are twisted. Of course, it’s not without some little tricks: you can get from the outside to the inside via an “escalator” in the center of the ring.

A “leaky” ring with the properties of a Möbius strip.

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

There is scientific knowledge and phenomena that bring mystery and mystery into the everyday life of our lives.

The Mobius strip applies to them fully. Modern mathematics wonderfully describes all its properties and features using formulas. And here ordinary people, poorly versed in toponymy and other geometric wisdom, almost every day they encounter objects made in its image and likeness, without even knowing it.

What it is?

A Möbius strip, also called a loop, surface, or sheet, is an object of study in the mathematical discipline of topology, which studies general properties figures that are preserved during such continuous transformations as twisting, stretching, compression, bending and others not associated with 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.

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 simultaneously - in 1858. One more 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 strip was named after Möbius, but topologists consider Listing and his work “Preliminary Studies in Topology” to be the founding father of “rubber geometry.”

Properties

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

1. The presence of one side. A. Mobius in his work “On the Volume of Polyhedra” described a geometric surface, later named in his honor, with only one side. Checking this is quite simple: take a Mobius strip or strip and try to paint over it inner side one color, and the outer one - 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, it will not turn out to be several different figures, and it remains intact.

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

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

Scientific use

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

Thus, there is a hypothesis according to which the Universe is a huge Mobius loop. This is indirectly evidenced by Einstein's theory of relativity, according to which even a ship that has flown straight can return to the same temporal and spatial point from which it started.

Another theory sees DNA as part of the Mobius surface, which explains the difficulty in reading and deciphering 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. For example, mirror reflection- this 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. Great Inventor Nikola Tesla at the beginning of the century invented the Mobius resistor, consisting of two conductive surfaces twisted by 180°, 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 trademarks 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 “Möbius 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. Mobius loop plays important role in R. Zelazny’s novel “Doors in the Sand” and serves as a means of moving 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 180 0 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.

Do you know what information you can get about a product based only on its packaging? Even if everything is written on it using hieroglyphs. It's okay if you don't know the meaning of any of them. You will still understand the pictograms. They are there for this reason, so that the information can be read and understood in all corners globe.

So, if you see a glass on the box, this means that there is a fragile product inside, and if there is a raging flame on the pictogram, then the contents of the box are flammable.

What do these signs mean?

This pictogram depicts the famous Moebius strip or loop. It represents a kind of mathematical paradox, since it is a one-sided surface. Yes, yes - she only has one side. You can see this for yourself if you pick it up. Making a Mobius loop is simple - take a strip of paper about 30 cm long and 1.5 cm wide.

Rotate one end 180 degrees and glue it to the other. To make sure that it really has one side, place a pencil exactly in the middle of the tape and draw a line without lifting it from the paper. After a while you will run into the beginning of your own line. You didn’t turn the paper over, you didn’t lift the pencil from it, but the line connected, therefore, the Mobius loop really only has one side, and your eyes are simply deceiving you. In general, it is very interesting to explore it. Try cutting it along a pencil line - you will get interconnected rings.

But this excursion into the jungle of mathematical paradoxes does not at all explain what the Mobius loop does on the packaging. This sign means that the packaging itself is made from a material that can be recycled. If there are numbers from 1 to 7 inside the pictogram, then they mean the name of the material from which the packaging is made. In ascending order of numbers, they mean: polyethylene terephthalate, high-density polyethylene, PVC, polypropylene, polystyrene or other plastic. Sometimes, instead of letters, capital Latin letters can be used, which mean the same thing.

It may also happen that instead of letters or just numbers from 1 to 7, some percentage value will be indicated inside the loop, or underneath it. In this case, the Mobius loop tells how much recycled material is already contained in this package. Why was this particular drawing chosen? This is easy to explain. The arrows mean that the manufacturing and processing cycle turns on itself, that is, it is closed.

In fact, the placement of this sign is not regulated by any legal requirements and is placed solely at the request of the manufacturer. But in light of the fact that they are now fighting for the environment at an accelerated pace, almost all packaging materials used in industry are subject to recycling. So don’t be surprised if you see a Moebius loop on Tetra-Pak packaging or plastic bottle. They have actually already learned how to recycle them, despite the fact that previously they were considered unsuitable for recycling.