In A.D. 1858, the German mathematician Mobius (1790 ~ 1868) discovered that a piece of paper twisted by 180 was magically bonded at both ends.
Because ordinary paper has two sides (that is, a double-sided surface), one front and one back, and the two sides can be painted with different colors; And such a paper tape has only one side (that is, one side), and a bug can crawl all over it without crossing its edge!
We call this magical single-sided paper tape discovered by Mobius "Mobius tape".
Take a long piece of white paper, paint one side black, then turn one end over and stick it into a Mobius belt like the picture on the previous page. Now use scissors to cut along the center of the paper tape, as shown in the figure. You will be surprised to find that instead of splitting the paper tape in two, you cut a paper circle twice the length in the picture!
Interestingly, the newly obtained long paper circle itself is a double-sided surface, and its two boundaries are not knotted, but nested together! In order to let readers intuitively see this hard-to-imagine fact, we can cut the upper paper circle along the middle line again, and this time it is really split in two! What you get is two nested paper circles. Originally, the two boundaries were contained in two paper circles, but each paper circle itself was not knotted.
Mobius belt has more bizarre characteristics. Some problems that could not be solved on the plane were actually solved on Mobius!
For example, the problem of "glove translocation" that can't be realized in ordinary space: although people's left and right gloves are very similar, they are essentially different. We can't put the gloves on our left hand correctly on our right hand; You can't put the gloves on your right hand correctly on your left hand. No matter how you twist, the left-handed condom is always the left-handed condom, and the right-handed condom is always the right-handed condom! However, if you take it to Mobius, it will be easy to solve.
There are many gloves-like objects in nature. They have completely similar symmetrical parts, but one is left-handed and the other is right-handed. There are great differences between them.
"Mobius belt" has been applied in life and production. For example, the belt of belt-driven power machine can be made into the shape of "Mobius belt", so that the belt will not wear only one side. If the tape recorder is made into Mobius tape, there will be no positive or negative problems, and the tape has only one side.
Mobius belt is a topological graph. What is topology? Topology studies some properties of geometric figures, which remain unchanged when the figure is bent, enlarged, contracted or deformed at will, as long as the original different points do not overlap into the same point and no new points are generated during the deformation process. In other words, the condition of this transformation is that there is a one-to-one correspondence between the points of the original graph and the points of the transformed graph, and the adjacent points are also adjacent points. Such a transformation is called topological transformation. Topology has an image of rubber geometry. Because if the graphics are all made of rubber, many graphics can be topologically transformed. For example, a rubber band can be deformed into a circle or a square. But rubber bands cannot be converted from topology to Arabic numeral 8. Because the two points on the circle do not overlap, the circle will not become 8, and the "Mobius belt" just meets the above requirements.