The Mobius strip, also called a loop, surface, or sheet, is an object of study in a mathematical discipline such 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 … An amazing and unique feature of such a tape is that it has only one side and edge and has nothing to do with its location in space.
The 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, without crossing the edge, to get to any other.
Who opened it and when?
Such a complex object as the Mobius strip was and 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 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 Moebius, but topologists consider Listing and his work Preliminary Investigations in Topology to be the founding father of "rubber geometry".
Properties
The Mobius strip has the following properties that do not change when it is compressed, cut along or creased:
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1. Presence of one side. A. Mobius in his work "On the volume of polyhedra" described a geometric surface, named after him, which has only one side. It is quite simple to check this: we take a tape or Moebius strip and try to paint the inner side with one color, and the outer with another. It doesn't matter in what place and direction the painting was started, the whole shape will be painted over 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 of it without crossing the boundaries of the Mobius surface.
3. Connectivity, or two-dimensionality, means that when cutting the tape lengthwise, 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 along this figure will return to the beginning of his path, but only in a mirror image of himself. Thus, an endless Moebius strip can lead to an eternal journey.
5. A special chromatic number, showing the maximum possible number of regions on the Mobius surface, you can create so that any of them has a common border with all others. The Mobius strip has a chromatic number - 6, but a paper ring - 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, 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 views DNA as part of the Mobius 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 object's self-destruction.
According to physicists, many optical laws are based on the properties of the Moebius strip. So, for example, a mirror image is a special transfer in time and a person sees his mirror double in front of him.
Implementation in practice
The Mobius strip has found application in various industries for a long time. The great inventor Nikola Tesla at the beginning of the century invented the Mobius resistor, consisting of two conductive surfaces twisted by 1800, which can withstand 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 devices have been created. Its shape is repeated for the creation of conveyor belt and ink ribbon in printers, abrasive belts for sharpening tools and automatic transfer. This allows them to significantly increase their service life, since wear is more even.
Not so long ago, the amazing features of the Mobius strip made it possible to create a spring that, unlike conventional ones that work in the opposite direction, does not change the direction of operation. It is used in the stabilizer of the steering wheel drive, providing a return of the steering wheel to its original position.
In addition, the Mobius strip mark is used in a variety of brands and logos. The most famous of these is the international symbol for recycling. It is affixed to the packaging of goods either suitable for subsequent processing or made from recycled resources.
A source of creative inspiration
The Mobius strip and its properties formed the basis for the work of many artists, writers, sculptors and filmmakers. The most famous artist who used in such his works as "Mobius Tape II (Red Ants)", "Riders" and "Knots", the tape and its features - Maurits Cornelis Escher.
Mobius sheets, or as they are also called, minimal energy surfaces, 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 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 "Mobius strip" are installed in Moscow and Yekaterinburg.
Literature and topology
The unusual properties of Moebius surfaces have inspired many writers to create fantastic and surreal works. The Mobius loop plays an important role in the novel "Doors in the Sand" by R. Zelazny and serves as a means of movement through space and time for the protagonist of the novel "Necroscope" by B. Lumley.
It also appears in the stories “The Wall of Darkness” by Arthur Clarke, “On the Mobius Strip” by M. Clifton and “Mobius Leaf” by A. J. Deutsch. Based on the latter, the fantastic film "Mobius" was filmed by director Gustavo Mosquera.
We do it ourselves, with our own hands
If you are interested in the Mobius strip, a small instruction will tell you how to make its model:
1. To make her model you will need:
- a sheet of plain paper;
- scissors;
- ruler.
2. Cut off a strip from a sheet of paper so that its width is 5-6 times less than its length.
3. The resulting paper strip is laid out on a flat surface. We hold one end with our hand, and turn the other to 1800 so that the strip twists and the wrong side becomes the front side.
4. Glue the ends of the twisted strip as shown in the figure.
The Mobius strip is ready.
5. Take a pen or marker and start drawing a track in the middle of the tape. If you did everything correctly, then you will return to the same point from where you started drawing the line.
In order to get visual confirmation that the Mobius strip is a one-sided object, try to paint over one side of it with a pencil or pen. After a while, you will see that you have painted over it completely.