m bius strip/m bius band
What is Mobius circle?
Mobius circle (m? bius strip，M？ Bius band) is a kind of one-sided non-directional surface. Because a.f. Mobius (Auguste Ferdinand m? Bius， 1790- 1868)。 After fixing one end AB of a rectangular strip ABCD and twisting the other end DC for half a week, AB and CD are bonded together, and the obtained curved surface is Mobius circle.
The discovery of Mobius circle;
There is a story circulating in the field of mathematics: someone once suggested that a rectangular piece of paper be glued end to end into a paper circle, and then only one color is allowed to be painted on one side of the paper circle, and finally the whole paper circle is painted in one color without leaving any blank. How should this paper ring be glued? If the paper circle made of sticky tape has two sides, it is necessary to draw one side first and then the other side, which does not meet the requirements of painting. Can it be made into a paper circle with only one side and a closed curve as the boundary?
For such a seemingly simple problem, many scientists have done serious research for hundreds of years, and the results have not been successful. Later, the German mathematician Mobius became interested in this. He concentrated on thinking and experimented for a long time, but there was no result.
One day, he was confused by this question and went for a walk in the wild. The fresh air and cool wind made him feel relaxed and comfortable at once, but his mind still only had the circle that had not been found.
Pieces of fat corn leaves turned into "green notes" in his eyes, and he couldn't help squatting down, fiddling and observing. Leaves are bent and pulled down, and many are twisted into semicircles. He tore off a piece and butted it into a circle along the direction of natural distortion of the leaves. He was pleasantly surprised to find that this "green circle" is exactly the kind of circle he dreamed of.
Mobius went back to the office, cut out a piece of paper, twisted one end of the paper 180, and then glued the two ends together, thus making a paper circle with only one side.
After the circle was made, Mobius caught a small beetle and climbed on it. As a result, the little beetle crawled all over the circle without crossing any boundary. Mobius circle said excitedly, "beautiful little beetle, you irrefutably proved that this circle has only one side." This is how the Mobius circle was discovered.
Wonderful Mobius circle:
After doing a few simple experiments, we will find that the Mobius circle has many surprising and interesting results.
You form a circle and stick it. After a turn, you can find that the entrance on the other side is blocked. This is the principle.
If you draw a line in the middle of a cut piece of paper, glue it into a "Mobius circle", then cut it along this line and split the circle in two, you should get two circles. Strangely, it turned out to be a big circle after cutting.
If you draw two lines on a piece of paper, divide the paper into three equal parts, then glue it into a Mobius circle, cut it along the drawing line with scissors, and then return to the original starting point after the scissors are wound twice. Guess, what is the result after cutting? Is it a big circle? Or three laps? Neither. What is it? Just do the experiment yourself. You will be surprised to find that you didn't split the paper tape in two, but cut out twice the length of the paper circle.
Interestingly, the newly obtained long paper circle itself is a hyperboloid, and its two boundaries are not knotted, but nested together. We can cut the paper ring along the center line again, this time it really splits 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.
As for the unilateralism of Mobius circle, it can be intuitively understood that if Mobius circle is colored, then the colored pen always moves along the surface and does not cross its boundary. Finally, both sides of Mobius circle can be colored, but you can't tell what is the front and what is the back. It's different for cylindrical surfaces. It's impossible to paint without crossing the line. Unilateralism is also called unilateralism. Draw a small circle with all points on the surface except the edge as the center, and specify a direction for each small circle, which is called the direction accompanying the center point of the unilateral surface of Mobius circle. If two adjacent points can be accompanied by the same direction, the surface is said to be orientable, otherwise it is said to be non-orientable. Mobius circle is undirected.
Mobius circle has more bizarre characteristics. Some problems that could not be solved on the plane were actually solved in Mobius circle. For example, the "glove translocation problem" that ordinary space can't realize: 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 we move it to Mobius circle, it will be easy to solve.
"Glove Translocation Problem" tells us that left-handed and right-handed objects can be distorted when they are blocked on a twisted surface. Let's spread the wings of imagination and imagine that our space is at a certain edge of the universe, showing a Mobius-like bend. Then, one day, our interstellar astronauts will set off with the heart in the left chest and return to Earth with the heart in the right chest! Look, how amazing the Mobius circle is! However, Mobius circle has a very obvious boundary. This seems to be a fly in the ointment. In a.d. 1882, another german mathematician, Felix? Felix Klein (1849 ~ 1925) finally found a self-closed model with no obvious boundary, and later named it "Klein bottle" after him. This strange bottle can actually be regarded as a pair of Mobius rings glued together along the boundary.
Usually, the paper ring obtained by butting two ends of a piece of paper has two sides. You take a piece of paper, twist one end 180 degrees, and butt joint. So you point a point in the center of the paper tape with a pencil, and then draw a line and a circle along the paper tape from this point. These two points coincide, but they are not on the same plane. If you want to go back to the distant place, you must walk again. Mobius circle is actually a strange circle.
Application of Mobius circle:
There is an important branch of mathematics called "topology", which mainly studies some characteristics and laws of geometric figures when they constantly change shape. "Mobius circle" has become one of the most interesting one-sided problems in topology. The concept of Mobius circle has been widely used in architecture, art and industrial production. Using the principle of Mobius circle, we can build overpasses and roads to avoid traffic jams.