There are scientific knowledge and phenomena that bring mystery and mystery to the everyday routine of our life. Ribbon Moebius refers to them fully. Modern mathematics perfectly describes all its properties and features using formulas. But ordinary people, poorly versed in toponymy and other geometrical intricacies, almost every day are confronted with objects made in her image and likeness, without even knowing it.
What it is?
The Mobius band, also called a loop, surface or sheet, is the object of study of such a mathematical discipline as topology, exploring the general properties of shapes that are preserved under such continuous transformations as twisting, stretching, compressing, 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 border in a conventional Euclidean space (3-dimensional), where it is possible to get from any other point from one point of such a surface without crossing the edges.
Who opened it and when?
Such a difficult object, like the Mobius strip, was and is opened rather unusual. First of all, we note that two mathematicians, completely unrelated to each other 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 Mobius discovered a geometric object that had only one side, and described its properties. The tape was named after Mobius, but topologists consider Listing and his work “Preliminary studies on topology” to be the founding father of “rubber geometry”.
The Mobius strip has the following properties that do not change when it is squeezed, cut along or creased:
1. The presence of one side. A. Mobius in his work “On the volume of polyhedra” described a geometric surface, then named after him, which has only one side. It’s quite simple to check: we take a ribbon or a Mobius strip and try to paint the inside with one color and the outside with another. It does not matter 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 of it, without crossing the boundaries of the Mobius surface.
3. Connectivity, or two-dimensionality, lies in the fact that when cutting the ribbon along, several different shapes will not come out of it, and it remains integral.
4. It lacks such an important property as orientation. This means that a person walking on this figure will return to the beginning of his path, but only in the mirror reflection of himself. Thus, the endless ribbon of Mobius can lead to eternal travel.
5. A special chromatic number indicating what the maximum possible number of areas on the Mobius surface can be created so that each of them has a common border with all others. The Mobius strip has a chromatic number - 6, but a paper ring is 5.
Today, the Möbius strip and its properties are widely used in science, serving as the 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 a huge Mobius loop. Einstein’s theory of relativity indirectly confirms this, according to which even a direct ship can return to the same temporal and spatial point from which it started.
Another theory considers DNA as part of the Möbius surface, which explains the difficulty in reading and decoding 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 an object.
According to physicists, many optical laws are based on the properties of the Mobius strip. So, for example, a mirror reflection is a special transfer in time and a person sees his mirror counterpart before him.
Implementation in practice
In various industries, the Mobius strip has been used for a long time. At the beginning of the century, the great inventor Nikola Tesla invented the Möbius resistor, consisting of two conducting surfaces twisted by 180 0, which can resist the flow of electric current without creating electromagnetic interference.
On the basis of studies of the surface of the Mobius strip and its properties, many devices and devices were created. Its shape is repeated when creating a strip of belt conveyor and ink ribbon in printing devices, abrasive belts for sharpening tools and automatic transfer. This allows you to significantly increase their service life, since wear occurs more evenly.
Not so long ago, the amazing features of the Mobius strip made it possible to create a spring, which, unlike the ordinary ones that operate in the opposite direction, does not change the direction of operation. It is used in the steering stabilizer of the steering wheel, ensuring the return of the steering wheel to its original position.
In addition, the Mobius strip badge is used in various trademarks and logos. The most famous of them is the international symbol of recycling. It is affixed to the packaging of goods or suitable for further processing, or made from recycled resources.
Source of creative inspiration
The Mobius strip and its properties formed the basis of the work of many artists, writers, sculptors and cinematographers. The most famous artist who used in such works as “The Ribbon of Moebius II (Red Ants)”, “The Riders” and “Knots”, the ribbon and its features is Maurits Cornelis Escher.
Mobius sheets, or, as they are also called, minimal energy surfaces, became a source of inspiration for mathematical artists and sculptors, for example, Brent Collins or Max Bill. The most famous monument of 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 works. Sculptures "Ribbon Moebius" installed in Moscow and Yekaterinburg.
Literature and Topology
The unusual properties of the Mobius surfaces have inspired many writers to create fantastic and surrealistic works. The Mobius loop plays an important role in the novel “Doors in the Sand” by R. Zelazny and serves as a means of moving through space and time for the main character of the novel “The Necroscope” B. Lumley.
She also appears in the stories The Wall of Darkness by Arthur Clarke, On the Mobius Tape by M. Clifton and The Mobius Sheet by A.J. Deutsch. Based on the last director Gustavo Mosquera was filmed a fantastic motion picture "Mobius".
We do it yourself!
If you are interested in the Mobius strip, how to make it a model, you will be prompted by a small instruction:
1. For the manufacture of its model will require:
- a sheet of plain paper;
2. Cut the strip from the sheet of paper so that its width is 5-6 times less than the length.
3. The resulting paper strip laid out on a flat surface. Hold one end with your hand, and turn the other end 180 ° 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. Ribbon Moebius ready.
5. Take a pen or marker and start drawing a path in the middle of the ribbon. If you did everything right, then return to the same point where you started to draw the line.
In order to get visual confirmation that the Mobius strip is a one-sided object, try to paint over any side of it with a pencil or pen. After a while you will see that you have painted it completely.