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Mobius' Mathematical Explanation
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. 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.