Möbius strip — a mystery of our time

There are scientific knowledge and phenomena that add to the ordinariness of our life in mystery and Enigma. The Mobius strip applies to them in full. Modern mathematics is wonderful describes by means of formulas all of its properties and characteristics. But ordinary people, poorly versed in the toponymy and other geometric knowledge, it is almost daily confronted with items made in her image and likeness, without even knowing it.

What is it?The Mobius strip, also referred to as mesh, surface, or sheet, is the object of study in such mathematical disciplines such as topology, studies the General properties of figures, continued under these continuous transformations as torsion, tension, compression, bending and other not related to the violation of integrity. An amazing and unique feature of this tapes is that it has only one side and edge and nothing to do with its location in space. The Mobius strip is a topological, i.e. continuous object with the simplest one-sided surface with boundary in the usual Euclidean (3-dimensional), where possible from one point on a surface, without crossing the borders to get into any other.

When and who opened it?

Such a difficult object like a Mobius strip, was open and quite unusual. First, we note that two mathematicians, totally unrelated research, discovered it simultaneously in 1858. Another interesting fact is that both of these scientists at different times were students of the same great mathematician Johann Carl Friedrich Gauss. So, until 1858 it was considered 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 describe its properties. The tape was named after the Mobius, but the founding father of "rubber geometry" topologists think of a Listing and his work "a Preliminary study on topology".

Svojstvima Mobius has the following properties not changing when it is compressing, cutting or creasing along:





1. The presence of one side. A. möbius, in his work "About the volume of polyhedra" described geometric surface, then called in his honor, with just one side. To test for this is quite simple: take a tape or Mobius strip and try to paint the inside one color, and others. It does not matter in which location and direction was begun painting, the whole figure would be shaded one color.

2. The continuity is reflected in the fact that any point of this geometric shape can be connected to any other point without crossing the surface boundary Mobius.

3. Connectivity, or two-dimensionality, is that when you cut the tape along, it won't make a a number of different shapes, and it remains solid.

4. It is such an important property as the focus. This means that people walking on this figure, will return to the beginning of his career, but only in the mirror itself. Thus, an endless Mobius strip can lead to the eternal journey.

5. Special chromatic number showing what the maximum possible number of regions on the surface of Mobius, you can create so that any of them had a common border with all others. The möbius strip has chromatic number is 6, but the ring of paper – 5.

Scientific использование

Today, the Mobius strip and its properties are widely used in science as a basis for building new hypotheses and theories, research and experiments, the creation of new mechanisms and devices.

So, there is a hypothesis that the universe is a huge loop. Indirectly this is evidenced by the theory of relativity, according to which a melted directly, the ship may return to the same temporal and spatial point from which he started.

Another theory treats DNA as a part of the surface of a Mobius strip that explains the difficulty of reading and deciphering the genetic code. In addition, this structure gives a logical explanation of biological death – a closed on itself, the spiral leads to self-destruction of the object.

According to physicists, many optical laws are based on the properties of the Mobius strip. For example, the mirror reflection is a special transfer in time and see his mirror double.

Implementation on practical various industries möbius strip used for a long time. The great inventor Nikola Tesla in the beginning of the century invented the möbius resistor, consisting of two twisted 1,800 conductive surfaces that can resist the flow of electric current without creating electromagnetic interference.





Based on studies of the surface of a möbius strip and its properties, have created many devices and appliances. Its form is repeated when creating the band the belt conveyor and the ink ribbon in the printing device, abrasive belts for sharpening tools and an automatic transmission. This allows to considerably increase their service life, as wear occurs more evenly.

Not so long ago the amazing features of the Mobius strip is allowed to create a spring that, unlike a conventional actuated in the opposite direction, does not change the direction of operation. It is used in the stabilizer actuator of a steering wheel, providing the return steering wheel to original position.

In addition, the sign of the möbius strip is used in various trademarks and logos. The most famous of them is the international symbol of recycling. It is placed on product packaging or suitable for recycling or made from recycled resources.

The source of creative vdohnoveniye Mobius and its properties formed the basis of the creativity of many artists, writers, sculptors and filmmakers. The most famous artist that used in such his works as "möbius Strip II (Red ants)", "Riders" and "Nodes", the ribbon and its features — Maurits Cornelis Escher.




The möbius strip, or, as they are called, minimum energy surfaces were the source of inspiration for mathematical artists, e.g., 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 not remained aloof from this topic and created their works. The sculpture "Mobius Strip" is set in Moscow and Yekaterinburg.

Literature and topologically properties of the surfaces of the Mobius strip has inspired many writers to create fantastic and surreal works. The loop plays an important role in the novel by R. Zelazny "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 appears in the stories "the Wall of darkness" by Arthur C. Clarke, "the Mobius strip" by M. Clifton and "Moebius" by A. J. Deutsch. In explanation of the last Director Gustavo Mosquera was filmed a fantastic movie "Moebius".

Do with their own hands!If you are interested in the Mobius strip how to make a model of it will tell you a little guide:

1. For the manufacture of its model need to:

— sheet of plain paper;

— scissors;

— the range.

2. Cut a strip from a sheet of paper so that its width is 5-6 times less than the length.

3. Received a slip of paper spread on a flat surface. One end of the stick hand and rotated 1800 so that the strip is twisted and the wrong side was facing.

4. Glue the ends of the twisted band as shown in the figure.




The Mobius strip is ready.

5. Take a pen or marker and in the middle of the tape start to draw the track. If you did everything correctly, you will return to the same point where we started to draw the line.





In order to obtain visual confirmation that the Mobius strip — a one-sided object, a pencil or pen try to paint any side. After a while you will see that painted over it completely.published



Author: Svetlana Budarina

Source: www.syl.ru/article/172135/new_chto-takoe-lenta-mebiusa-lenta-mebiusa---zagadka-sovremennosti