The Möbius strip, which is also called a loop, surface or leaf, is an object of study in 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 violations. . 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 to get from one point of such a surface without crossing the edges to any other.
Who opened it and when?
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 at different times were students of the same great mathematician - Johann Karl 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".
Properties
The Möbius strip has the following properties that do not change when it is compressed, cut along or crushed:
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 a 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 point 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 will not come out of it, and it remains whole.
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. Einstein's theory of relativity also indirectly testifies to this, 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, a mirror reflection is a special transfer in time and a person sees his mirror double in front of him.
Implementation 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 Mobius 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.
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 Moebius 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 located 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. Sculptures "Moebius Tape" installed in Moscow and Yekaterinburg.
Literature and topology
The unusual properties of Möbius surfaces 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 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. To make her model, you will need:
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 a visual confirmation that the Möbius strip is a one-sided object, try to paint over any of its sides with a pencil or pen. After a while, you will see that you have painted over it completely.published econet.ru
Mobius strip (Mobius loop, Mobius strip)- a simple-looking figure, but a mathematician would say that this is a two-dimensional surface with amazing properties: it has only one side and one edge, unlike the usual ring, which can be rolled up from the same strip as the Möbius strip, but it has There will be two sides and two edges. This is easy to verify 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 the half-turn of the strip, its upper and lower edges united 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 Möbius strip to any other without going over the edge.
Möbius strip running
For an outside observer, a journey along the Möbius strip is a “running in a circle”, full of surprises. He was vividly depicted by the Dutch graphic artist Maurits Escher (1898-1972). In the painting "Möbius Strip II" in the role of running - ants. Following their movement, you can make an interesting discovery. Having made one turn along the tape, each ant will be at the starting point, but already in the position of the antipode - visually it will be “on the other side” of the tape upside down. And what will happen to a two-dimensional creature moving along the Möbius strip? Bypassing the surface, it will turn into its mirror image (this is easy to imagine if we consider the tape transparent). In order to become itself, a two-dimensional being will have to make one more circle. So the ant needs to go twice along the Möbius strip in order to return to its initial 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, a German scientist invented the ribbon when he saw an incorrectly tied neckerchief on a maid. It was a famous mathematician and astronomer, a student of Carl Friedrich Gauss. He described a one-sided surface with a single edge in 1858, but the article was not published during his lifetime. In the same year, independently of Möbius, a similar discovery was made by Johann Listing, another student of Gauss.
The ribbon is 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 under continuous (without cuts and gluing) transformations: stretching, squeezing, bending, twisting, etc. These transformations resemble deformations of rubber figures, therefore topology is otherwise called "rubber geometry". Separate topological problems were solved back in the 18th century by Leonard Euler. The beginning of a new field of mathematics was laid by Listing's Preliminary Investigations 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 did not find practical application and did not inspire people of art. She was depicted more than once by artists, sculptors erected monuments to her and writers dedicated their creations. This unusual surface appealed to architects, designers, jewelers and even manufacturers of clothing and furniture. Inventors, designers, and engineers drew attention to it (for example, back in the 1920s, audio and film films in the form of a Möbius strip were patented, which made it possible to double the duration of the recording). But more often than others, conjurers deal with this tape: they are attracted by the unusual properties that appear when it is cut. So, if you cut the Möbius strip along the middle line, it will not fall into two parts, as you might expect. It will make a narrower and longer double-sided tape, twisted twice (the design of the Rollercoaster ride has a similar shape). And here's a "culinary trick": cakes in the form of a Mobius strip will seem tastier than usual 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 the Möbius strip. While 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 the Looking Glass. And she herself, being an asymmetric figure, has a mirror double. Let us send the imprint of the right foot for a walk along the tape and soon we will find that the imprint of the left foot will return home. Funny, right? And when did the “right” manage to become “left”? We “mount” a two-dimensional clock into the tape and make it complete 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 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 clock always shows the same time, and secondly, the hands on the dial should be in a position that would be preserved with a mirror image, for example, stand vertically, forming a developed angle.
Well, let's check the answer, shall we? In fact, on the Möbius strip it is impossible to set a certain direction of rotation. The same movement can be perceived as both a clockwise turn and a turn in the opposite direction. When a point arbitrarily chosen on the Möbius strip goes around it, one direction continuously changes into 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 such creatures and, of course, by us, who are watching what is happening from another dimension. This is such an unpredictable, one-sided Möbius surface.
