In 1858 AD, German mathematicians Mobius (1790-1868) and John Listin discovered that after twisting a piece of paper 180°, the two ends were then glued together. Paper tape circles have magical properties. Ordinary paper tape has two sides (i.e., double-sided curved surface), one front and one back, and the two sides can be painted in different colors; while such paper tape has only one side (i.e., one-sided curved surface), and a small insect can Climb the entire surface without stepping over its edges. This paper strip is called a "Mobius strip". (That is to say, it has only one curved surface)

Take a long white strip of paper, paint one side black, then turn one end over and glue it to form a M?bius strip. Use scissors to cut the paper strip down the center. Not only did the paper strip not split into two, but a paper circle twice as long was cut out. The newly obtained longer paper circle is itself a double-sided curved surface. Although its two boundaries are not knotted, they are nested together. Cut the above paper circle again along the center line. This time it is really divided into two. What you get is two paper circles nested inside each other, and the original two borders are included in the two paper circles respectively. Among them, it’s just that each paper circle is not knotted.

The M?bius strip has even more bizarre properties. Some problems that cannot be solved on a flat surface are miraculously solved on the M?bius strip. For example, the "glove translocation" problem that cannot be realized in ordinary space: although the gloves on the left and right hands of a person are very similar, they are essentially different. We cannot fit the left-hand glove onto the right hand; nor can we fit the right-hand glove onto the left hand. No matter how you twist and turn, the left glove will always be the left glove, and the right glove will always be the right glove! However, if you move it to the Mobius Belt, then solving it is easy.

There are many objects in nature that are similar to gloves. They have exactly the same symmetrical parts, but one is left-handed and the other is right-handed. There is a huge difference between them.

You can use parametric equations to create a three-dimensional M?bius strip

This set of equations can create a M?bius strip with a side length of 1 and a radius of 1, located at

x-y

The center is (0, 0, 0). The parameter

u

wraps around the entire tape as

v

moves from one edge to the other.

Topologically speaking, the M?bius strip can be defined as a matrix ×, with the edges determined by (x, 0) ~ (1-x, 1) when 0 ≤ x ≤ 1.

A M?bius strip is a two-dimensional compact manifold (that is, a bounded surface) that can be embedded in a three-dimensional or higher-dimensional manifold. It is a non-directional standard paradigm and can be viewed as RP#RP. It is also one of the examples of describing fiber bundles mathematically. In particular, it is a nontrivial bundle on a circle S with one fiber unit interval, I = . Looking only at the edge of the M?bius strip gives a non-trivial distribution of two points on S (or Z2).