What is a M?bius strip? - Trademark Inquiry Complete Network

What is a M?bius strip?

A brief explanation of the Mobius circle

Imagine a long strip of toilet paper, connect it end to end without sticking it together, and you will find that the original side is connected to its reverse side. For primary and middle school students, making the Mobius circle several times will help their understanding. Mobius Circle

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There is a story circulating in mathematics: Someone once proposed to use a rectangular piece of paper and stick it end to end to make a Make a paper circle, and then only allow one color to be painted on one side of the paper circle. Finally, paint the entire paper circle into one color without leaving any blank space. How should this paper circle be glued? If the paper circle made by sticking the end of the paper strips together has two sides, it is necessary to finish painting one side and then repaint the other side. This does not meet the requirements for painting. Can it be made with only one side and a closed curve as the boundary? Where is the paper circle? M?bius strip

Edit this paragraph The discovery of the Mobius strip

For such a seemingly simple problem, many scientists have conducted research over hundreds of years. After careful research, the results were unsuccessful. Later, the German mathematician Mobius became very interested in this. He concentrated on thinking and testing for a long time, but to no avail. One day, he was so confused by this problem that he went for a walk in the wild. The fresh air and cool wind made him feel relaxed and comfortable immediately, but in his mind he still only had the circle he had not yet found. The fat corn leaves turned into "green strips of paper" in his eyes. He couldn't help but squat down, fiddle with and observe. The leaves were bent and pulled down, and many of them were twisted into semi-circles. He tore off a piece at random and connected it into a circle along the natural twisting direction of the leaves. He was pleasantly surprised to find that this "green circle" was what he had dreamed of. That kind of circle. Mobius returned to the office, cut out a strip of paper, twisted one end of the paper 180°, and then glued the front and back of one end together, thus making a paper circle with only one side. After the circle was made, Mobius caught a small beetle and placed it on it to let it crawl. As a result, the little beetle crawled through all parts of the circle without crossing any borders. Mobius said excitedly: "Just little beetle, you have irrefutably proved that this circle has only one side." In this way, the Mobius circle was discovered. By doing a few simple experiments, you will find that the "Mobius Circle" has many surprising and interesting results. Make a circle and glue it. After making a circle, you will find that the entrance to the other side is blocked. This is the principle.

Experiment 1

If after cutting, Draw a line in the middle of a piece of paper and glue it to form a "Mobius circle". Then cut it along the line and divide the circle into two. It should be expected that you will get two circles. The strange thing is that after cutting, It's a big circle.

Experiment 2

If you draw two lines on a paper strip, divide the paper strip into three equal parts, glue it into a "Mobius Circle", and use scissors to cut along the drawn lines. Open, the scissors make two circles and return to the original starting point. Guess what is the result after cutting? Is it a big circle? Or three circles? None. What exactly is it? You can find out by doing this experiment yourself. You will be surprised to find that the paper tape is not divided into two, but a large and a small interlocking ring. What’s interesting is that the newly obtained longer paper circle is itself a double-sided curved surface. Although its two boundaries are not knotted, they are nested together. We can cut the above paper circle along the center line again, and this time it is really divided into two! What is obtained is two paper circles nested inside each other, and the original two borders are contained in the two paper circles respectively, but each paper circle itself is not knotted. Regarding the one-sidedness of the Mobius circle, we can intuitively understand it as follows. If we color the Mobius circle, the color pen always moves along the curved surface and does not cross its boundary. Finally, both sides of the Mobius circle can be evenly colored. Painted with color, it is impossible to distinguish what is the positive side and what is the negative side. It is different for cylindrical surfaces. It is impossible to color the other side without passing the boundary. Unilaterality is also called undirectability. Draw a small circle with each point on the curved surface as the center, and specify a direction for each small circle, which is called the accompanying Mobius circle. The direction of the center point of the single-sided curved surface can be made to make two adjacent points companion point in the same direction, the surface is said to be orientable, otherwise it is said to be non-orientable. The Mobius Circle is not orientable. The Mobius Circle also has even more bizarre characteristics. Some problems that cannot be solved on a flat surface are miraculously solved on the Mobius Circle.

For example, the "glove transposition problem" 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 much 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 Circle, then the solution is easy. The "glove displacement problem" tells us that if it is blocked on a distorted surface, left- and right-hand objects can be transformed through distortion. Let us spread the wings of our imagination and imagine that our space is curved like a Mobius circle at a certain edge of the universe. So, 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, the Mobius Circle has a very clear boundary. This seems to be a fly in the ointment. In 1882 AD, another German mathematician, Felix Klein (1849-1925), finally found a self-enclosed model with no obvious boundaries, which was later named "Klein" after him. bottle". This strange bottle can actually be seen as a pair of Mobius circles, glued along the border. The "Mobius Strip" was a bit mysterious and was not useful for a while, but people still made up some stories based on its characteristics. It is said that a thief stole something from an honest farmer and was caught on the spot. The thief was taken to the county government, and the county magistrate discovered that the thief was his son. So he wrote on the front of a piece of paper: The thief should be let go, and on the back of the paper he wrote: The farmer should be imprisoned. The magistrate handed the note to the deacon for him to handle. The clever deacon twisted the note and pinched the two ends together with his fingers. Then he announced to everyone that according to the order of the county magistrate, the peasants should be released and the thieves should be imprisoned. The county magistrate was furious and questioned the deacon. The deacon held the note in his hand and showed it to the county magistrate. Starting from the word "should", it was indeed correct. After carefully looking at the handwriting, there was no alteration. The county magistrate did not know the secret and had to admit that he was unlucky. The county magistrate knew that the deacon had tampered with the note, so he held a grudge and waited for an opportunity to retaliate. One day, another piece of paper was taken and the deacon was asked to blacken both sides of the paper in one stroke, otherwise he would be detained. The deacon calmly twisted the paper, glued the two ends, drew a pen on the paper ring, and then opened the two ends, only to see that the front and back of the paper were painted black. The magistrate's evil plan failed again. Such a story may not happen in reality, but this story well reflects the characteristics of the "Mobius Strip".

Edit this paragraph Mobius strip

There are three wonderful things

First, the Mobius strip only has one side. 2. If you cut along the middle of the Mobius strip, a ring will be formed that is twice as large as the original Mobius strip and has both positive and negative sides (numbered in this article as: Ring 0) instead of forming two Mobius rings or two other forms of rings. 3. If you cut along the middle of ring 0, two rings will be formed that are the same as ring 0 and have both front and back faces, and these two rings are nested together (in this article, we will The numbers are: ring 1 and ring 2). From now on, if you cut along the middle of ring 1 and ring 2 and all the rings generated by cutting along the middle of ring 1 and ring 2, two and There is no end to the rings with positive and negative sides that are the same as ring 0 space...and all the generated rings will be nested together and can never be separated, and they can never be independent without being connected to other rings. exist.

Six characteristics

The six characteristics of the Mobius strip 0 and all the generated rings: 1. The Mobius strip is made by inverting one end of the front and back sides. It is formed by turning 180 degrees and docking with the other end, so it unifies the front and back sides into one surface, but there is also a "twisting force", which we might as well call it "Mobius strip twisting force" 1 . 2. The generation of Mobius strip into ring 0 requires an "evolutionary fission" process. This "evolutionary fission" process decomposes the "Mobius strip twist" into "interconnected" or "connected" to separate them. It shows the four "twisting powers" of "twisting" in two directions: "spiral arc" downward and "spiral arc" upward.

The first and third of these four "twisting jins" transform the positive side into the negative side, while the second and fourth "twisting jin" transform the negative side into the positive side, or in other words , the first and third of these four "Nuan Jin" will transform the negative side into the positive side, while the second and fourth "Nuan Jin" will transform the positive side into the negative side, so that The generated ring 0 thus has two sides, "positive and negative". 3. The process of generating the Mobius strip into Ring 0 also gives Ring 0 the ability to transform into four "twisting elements" with different properties in the same direction due to mutual conversion. The "evolutionary fission" process decomposes the "Mobius twisting force" of the Mobius ring into the four "screwing points" in ring 0, and the "energy" of the "Mobius twisting force" is also generated The "energy" of these four "Ningjin" in Ring 0, but the "energy" of these four "Ningjin" in Ring 0 is twice the "energy" of "Mobius Ningjin", the newly generated The direction of "energy" that is 1 times that of "Mobius twisting force" is opposite to the direction of "energy" of the original "Mobius twisting force". 4. The process of generating the Mobius strip into ring 0 also makes the space of ring 0 double the space of the Mobius strip. 5. In the process of generating ring n and ring n+1 from ring 0, the "energy" of the four "twisting forces" in ring 0 will not increase, but from the "fission" of ring 0, each "fission" will Add a ring 0 space. 6. After ring 0 generates ring 1 and ring 2 and then "fissions" until ring n and ring n+1, all the generated rings n and ring n+1 will be nested together and can never be separated, never. It is impossible to exist independently without being connected to other rings.

Wonderful revelation

From the three wonderful features of the Mobius strip and the six characteristics of the Mobius strip, ring 0 and all generated rings, we Obtained wonderful revelations: 1. No matter where the Mobius strip is placed in the space and time of the universe, we will also find that the space outside the Mobius strip can only exist on one side. Therefore, any part of the space and time of the universe There is only one surface in space. If there is only one surface at any place in the universe's space-time, then we can think that any point in the universe's space-time is connected to other points, that is, the entire universe's space-time is connected, and any point is the center of the universe. The edge of the universe is the same as any matter in the space and time of the universe. It is also at the center of the universe and is also at the edge of the universe. 2: Any point in the space and time of the universe can generate an opposing yin and yang out of nothing through "fission". Regardless of whether the opposing yin and yang that are generated need a carrier to be presented, through the method of "fission", the opposing yin and yang that are created out of nothing requires a space that is twice as large as the original space to reflect this generation. , an opposing yin and yang. Three: As long as there is "fission", the original Mobius strip will no longer exist in its "original form", or in other words, the original Mobius strip no longer exists. To "restore" the original Mobius ring from a ring 0 that is created out of nothing and has an opposite yin and yang nature, it is necessary to resolve an opposite yin and yang side. 4. The process of generating the Mobius strip into Ring 0 also gives Ring 0 the ability to transform into four "twisting moments" with different properties in the same direction due to mutual conversion. We know that any affirmation should be a vector process (negation with a certain direction) that has a gap in the same direction, or can be said to be a non-absolute vector of negation of negation of negation of negation. 5. After generating ring 1 and ring 2 from ring 0 and then "fissioning" to ring n and ring n+1, all the generated rings n and ring n+1 will be nested together and can never be separated, and they will never be separated. It is impossible to exist independently without being connected to other rings. This shows that there is a universal law of connection between all things in the universe, and that any point or thing is connected to all other things in the universe and is inseparable and cannot be omitted. 6. There is no difference in the ultimate origin of all things in the universe. They all originate from a space with only one surface or a state without any surface. Therefore, it can also be said that everything in the universe comes from nothing, but shows differences in the process of evolution.

7. During the "fission" process in which the Mobius strip is generated into ring 0, new energy twice as much as the original "twisting energy" is generated out of thin air. That is to say, in the newly generated pair of yin and yang sexual relations The "fission" in the process does not follow the "principle of conservation of energy"; and the subsequent "fission" of all things in the universe can only increase the space-time of the universe and no longer generate new energy, and will inevitably Follow the "principle of conservation of energy". 8. Any point in the space and time of the universe can generate Yin and Yang for the first time by creating something out of nothing, and then use the newly generated Yin and Yang as the basis to generate the two substances of Yin and Yang for the first time. Three times... to eternity.

Edit this paragraph Mobius strips and Klein bottles

If we glue two Mobius strips along their only edge, you get Get a Klein bottle (of course don’t forget that we have to be in four-dimensional space to really be able to complete this bonding, otherwise we have to tear the paper a little). Similarly, if a Klein bottle is cut appropriately, we can get two Mobius strips. In addition to the Klein bottle we saw above, there is also a less well-known "figure 8" Klein bottle. It looks completely different from the surface above, but in four-dimensional space they are actually the same surface - a Klein bottle. In fact, it can be said that the Klein bottle is a three-dimensional Mobius strip. We know that if we draw a circle on a plane and put something inside the circle, if we take it out in two-dimensional space, we have to go beyond the circumference of the circle. But in three-dimensional space, it is easy to take it out and put it outside the circle without crossing the circumference. Projecting the trajectory of the object together with the original circle into a two-dimensional space is a "two-dimensional Klein bottle", that is, the Mobius strip (the Mobius strip here refers to the Mobius strip in the topological sense) bring). Imagine again that in our three-dimensional space it is impossible to remove the yolk from an egg without breaking the shell, but in four-dimensional space it is possible. If you project the trajectory of the egg yolk and the eggshell into a three-dimensional space, you will definitely see a Klein bottle. Attachment: A Klein bottle is broken in three-dimensional space. There must be at least one crack. If there are two cracks, it must be two partially connected Mobius strips. Similarly, there are n Mobius strips. Strips can also be combined into a Klein bottle with n cracks.

Edit this paragraph Application of Mobius Circle

Application of Mobius Circle in Mathematics

There is an important branch of mathematics called topology , mainly studies some characteristics and laws of geometric figures when they continuously change shape. The Mobius circle has become one of the most interesting one-sided problems in topology.

The application of Mobius Circle in real life

The concept of Mobius Circle has been widely used in architecture, art, and industrial production. Using the Mobius circle principle we can build overpasses and roads to avoid congestion of vehicles and pedestrians. Garbage recycling sign

1. In 1979, the famous American tire company BFGoodrich creatively made the conveyor belt into the shape of a Mobius loop. In this way, the entire conveyor belt ring was evenly distributed Power Architecture The logo

withstands wear and tear, avoiding damage to one side of the ordinary conveyor belt, fully doubling its lifespan. 2. Dot matrix printers rely on the printing needle to hit the ribbon to leave ink dots one by one on the paper. In order to make full use of the entire surface of the ribbon, the ribbon is often designed into a Mobius circle. 3. In the famous Kenny's Forest Amusement Park in Pittsburgh, USA, there is an "enhanced version" of the roller coaster - its track is a Mobius circle. Passengers race on both sides of the track. 4. The recurring geometric features of the Mobius circle contain eternal and infinite meaning, so it is often used in various logo designs. The trademark of the microprocessor manufacturer Power Architecture is a Mobius circle, and even the garbage collection logo is a variation of the Mobius circle.

Edit the geometry and topology structure of this paragraph

A method of using parametric equations to create a three-dimensional Mobius strip: M?bius strip depicted with Matlab

[1]x(u,v)=[1+v/2×cos（u/2）]cos(u)　y(u,v)=[1+v/2×cos（u/ 2)]sin(u)　z(u,v)=v/2×sin(u/2)　where 0≤u<2π and -1≤v≤1. .This system of equations can create a Mobius strip with a side length of 1 and a radius of 1. The location is the x-y plane and the center is (0, 0, 0). The parameter u wraps around the entire tape as v moves from one edge to the other. If expressed by polar coordinate equations (r, θ, z), an unbounded Mobius strip can be expressed as: log(r)sin(θ/2)=zcos(θ/2).

Edit this paragraph Introduction to Mobius (1790～1868)

Mobius, August Ferdinand, German mathematician and astronomer. Born on November 17, 1790 in Schulpfort near Naumburg, died on September 26, 1868 in Leipzig. In 1809, he entered the University of Leipzig to study law, and later switched to mathematics, physics and astronomy. He received his doctorate in 1814, became an associate professor in 1816, was elected as a corresponding member of the Berlin Academy of Sciences in 1829, and became a professor of astronomy and advanced mechanics at the University of Leipzig in 1844. Mobius's scientific contributions involve astronomy and mathematics. He led the establishment of the Leipzig University Observatory and served as its director. He was praised by astronomers for publishing "Calculations on Planetary Occultation". In addition, he also wrote "Principles of Astronomy" and "Fundamentals of Celestial Mechanics" and other astronomical works. In mathematics, Mobius developed algebraic methods of projective geometry. In his main work "Calculation of the Center of Gravity", he created the basic concept of algebraic projective geometry - homogeneous coordinates, independently of J. Plucker and others. In the same work he also revealed the relationship between the principle of duality and polarity, and gave a complete treatment of the concept of cross-ratio. Mobius's most famous mathematical discovery is the one-sided surface named after him - the Mobius strip. In addition, Mobius also made important contributions to other branches of mathematics such as topology and spherical trigonometry.

Edit this paragraph Art and Technology

The Mobius Strip has provided inspiration for many artists, such as the artist Maurits Cornelius Escher who used This structure appears in his woodcut paintings, the most famous of which is Mobius II. The painting shows some ants crawling on the Mobius strip. It also often appears in science fiction novels, such as Arthur C. Clarke's "The Wall of Darkness". Science fiction often imagines that our universe is a Mobius strip. The short story "A Station Called Mobius" by A.J. Deutsch creates a new route for a Boston subway station. The entire line twists in the Mobius way and trains that enter this line disappear. Another novel, "Star Trek: The Next Generation," also used the concept of Mobius strip space. There is a little poem that also describes the Mobius strip: Mathematicians assert that the Mobius strip has only one side. If you don’t believe it, please cut one to verify that the strip is still connected when separated. The Mobius strip is also used in industrial manufacturing. . A conveyor belt inspired by the Mobius strip could last longer because the entire belt could be better utilized, or it could be used to create magnetic tape that could carry twice the amount of information. There is a steel Mobius Strip sculpture located in the Smith Woods Museum of History and Technology in Washington, USA. Dutch architect Ben Van Berkel designed the famous Mobius House based on the Mobius Strip. In the Japanese comic "Doraemon", Doraemon has a prop that looks like a Mobius strip; in the story, as long as this ring is put on the doorknob, when people outside come in, they will see It's still outside. Ultraman Ace in Japan Chapter 23 "Reversal!" In "Zoffie's Appearance", the TAC team used the principle of the Mobius Belt to allow Beidou and Nan to enter a different dimension and destroy the Yabo people.

In the video game "Sonic Boy - Skateboard Meteor Story", the last level of the demon battle takes place on a track shaped like a Mobius strip. If you don't defeat the demon, you will keep sliding down the Mobius strip in an infinite loop. ..... The animated film Mobile Suit Gundam Char's Counterattack, released in Japan in 1988, uses the Mobius Strip as a metaphor for destiny: Humans are like ants walking on the Mobius Strip, forever escaping. If we don't get out of this vicious circle, we will keep repeating the same mistakes, and similar tragedies will keep happening. The theme song of the movie BEYOND THE TIME (メビウスのCosmos を日えて) also echoes this theme (Japanese メビウス means M?bius). Japan's Ultraman Mebius is also named after the Mobius Strip, and its transformation is the symbol of "infinity" and the cut Mobius Strip. Share it with your friends: iTieba Sina Weibo Tencent Weibo QQ Space Renren Douban MSN

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