There are scientific knowledge and phenomena that bring to the everyday life of our lives a mystery and a riddle. Mobius tape refers to them in full. Modern mathematics describes remarkably with the help of formulas all its properties and features. But ordinary people, poorly versed in toponymy and other geometric meanings, almost daily encounter objects made in its image and likeness, even without realizing it.
What it is?
The Mobius band, also called a loop, surface, or sheet, is the object of studying a mathematical discipline such as a topology that examines the general properties of figures that persist under continuous transformations such as twisting, stretching, squeezing, bending, and other unrelated integrity . The surprising 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 Mobius strip is a 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 surface, without crossing the edges, to get into any other.
Who and when did it open?
Such a complex object, like the Mobius tape, was and is quite unusual. First of all, we note that two mathematicians, absolutely unrelated to each other in research, discovered it simultaneously - in 1858. Another interesting fact is that both these scientists at different times were pupils 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 Augustus Ferdinand Moebius discovered a geometric object that had only one side and described its properties. The tape was named in honor of Moebius, but the founding father of "rubber geometry" topologists believe Listing and his work "Preliminary research on topology."
The following properties are inherent to the Moebius tape: they do not change when it is squeezed, cut along or crumpled:
1. The presence of one side. A. Mobius in his work "On the volume of polyhedra" described a geometric surface, named after him in his honor, which has only one side. Check it is quite simple: take a tape or a sheet of Moebius and try to paint the inside with one color, and the outer one with the other. It is not important, in which place and direction the painting was started, the whole figure will be shaded 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 boundary of the Möbius surface.
3. Connectivity, or two-dimensionality, consists in the fact that when cutting a tape along, it will not produce several different figures, and it remains intact.
4. It lacks such an important property as orientation. This means that the person walking along this figure will return to the beginning of his journey, but only in the mirror image of himself. Thus, the endless ribbon of Moebius can lead to an eternal journey.
5. A special chromatic number showing how many possible areas on the Möbius surface can be created so that either of them has a common boundary with all the others. Moebius tape has a chromatic number - 6, but a ring of paper - 5.
Today, the Möbius sheet and its properties are widely used in science, serving as a basis for constructing new hypotheses and theories, conducting research and experiments, creating new mechanisms and devices.
So, there is a hypothesis that the Universe is Mobius's huge loop. Indirectly, this is evidenced by the theory of relativity of Einstein, according to which even a ship that has flown directly can return to the same time and space point, whence it started.
Another theory considers DNA as part of the Moebius 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 self-contained spiral leads to self-destruction of the object.
According to physicists, many optical laws are based on the properties of the Moebius sheet. So, for example, mirror reflection is a special transfer in time and a person sees before himself his mirror double.
Implementation in practice
In various industries, the Möbius band has been used for a long time. The great inventor Nikola Tesla at the beginning of the century invented a Möbius resistor, consisting of two conductive surfaces twisted at 180 degrees, which can withstand the flow of electric current without the creation of electromagnetic interference.
Based on studies of the surface of the Möbius band and its properties, many devices and instruments have been created. Its shape is repeated when creating a band of belt conveyor and ink ribbon in printing devices, abrasive belts for sharpening tools and automatic transmission. This makes it possible to significantly increase their service life, since wear occurs more evenly.
Not so long ago, the amazing features of the Moebius leaf allowed the creation of a spring, which, unlike the usual ones that operate in the opposite direction, does not change the direction of operation. It is used in the steering wheel steering stabilizer, ensuring the steering wheel returns to its original position.
In addition, the Mobius ribbon sign is used in a variety of trademarks and logos. The most famous of them is the international symbol of secondary processing. It is placed on packages of goods either suitable for subsequent processing, or made from recycled resources.
Source of creative inspiration
Moebius tape and its properties formed the basis for the work of many artists, writers, sculptors and filmmakers. The most famous artist, who used in such works as "Mobius II Tape (Red Ants)", "Horsemen" and "Knots", ribbon and its features - Maurits Cornelis Escher.
Möbius sheets, or, as they are also called, surfaces of minimal energy, have become a source of inspiration for mathematical artists and sculptors, for example, Brent Collins or Max Bill. The most famous monument to the Mobius Ribbon is set 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 "Mobius tape" are installed in Moscow and Yekaterinburg.
Literature and topology
The unusual properties of Möbius surfaces inspired many writers to create fantastic and surrealistic works. The Möbius loop plays an important role in R. Zelyazny'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" B. Lumley.
She also depicts in the stories "The Wall of Darkness" by Arthur Clarke, "On the Moebius Band" by M. Clifton and "List Moebius" by AJ Deutsch. Based on the latter's motives, Gustavo Mosquera was shot by a fantastic film called Mobius.
We do it ourselves, with our own hands!
If you are interested in the Mobius tape, 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;
2. Cut the strip from the sheet of paper so that its width is 5-6 times smaller than the length.
3. The resulting strip of paper is laid out on a flat surface. One end is held by hand, and the other is rotated by 180 ° so that the strip is twisted and the infernal becomes the face.
4. Bond the ends of the twisted strip as shown in the picture. Mobius tape is ready.
5. Take a pen or marker and in the middle of the tape, start drawing the track. If you did everything correctly, then return to the same point where you started drawing the line.
In order to get a visual confirmation of the fact that the Möbius tape is a one-sided object, try using a pencil or a pen to paint some of its side. After a while you will see that you have painted it completely.