First, imagine a long piece of paper, wrap it up and connect it end to end. Don't stick together, you will find that the original side is connected with its reverse side.
Related experiments
Experiment 1
If you draw a line in the middle of a cut piece of paper, glue it into a "Mobius belt", then cut it along this line and split the circle in two, you should get two circles. Strangely, after cutting, you will form a loop, twist the end of the paper tape twice and reassemble it (not Mobius tape).
Experiment 2
If you draw two lines on a piece of paper, divide the paper into three equal parts, then glue it into a Mobius belt, cut it along the drawing line with scissors, and then return to the original starting point after the scissors turn 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 the paper tape is not divided into two, but two buttons, one big and one small.
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.
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 it is moved to Mobius circle, it is easy to solve it in this space.
"The problem of glove displacement" tells us that objects tied to the left and right hands can be deformed by twisting if 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 1882, another German mathematician, Felix Klein (1849 ~ 1925), finally discovered a self-enclosed 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.
Experiment 3
How to turn two sides of a piece of paper into one side? The answer is Mobius circle.
A wonderful place
1. Mobius ring has only one side.
2. If it is cut along the middle of Mobius loop, it will form a loop twice as large as the original Mobius loop space, and the end of the paper tape will be wound four times and then reassembled (not Mobius tape, numbered as loop 0 in this paper), instead of forming two Mobius loops or two other forms of loops.
3. If you cut along the middle of ring 0, you will form two rings with the same spacing as ring 0, which are nested with each other (numbered as ring 1 and ring 2 in this paper), and then cut all the rings generated along the middle of ring 1 and ring 2 and along the middle of ring 1 and ring 2. Two rings with positive and negative sides will be formed, just like the space of ring 0, which has no end ... and all generated rings will be nested together, never separate, never exist independently, and will not contact with other rings.
trait
Six characteristics of Mobius ring 0 and all its generating rings;
1. Mobius ring is formed by turning one end of the front and back sides 180 degrees and butting with the other end, so it unifies the front and back sides into one surface, but there is also a "torsion force", which we might as well call "Mobius ring torsion force" 1 here.
2. The evolution from Mobius ring to ring 0 needs an evolutionary fission process, which decomposes the Mobius ring into four twisting directions: downward spiral arc and upward spiral arc. The first and third of these four twists turn heads into tails, while the second and fourth twists turn tails into tails. In other words, the first and third of these four twists turn tails into tails, while the second and fourth twists turn tails into tails.
Thirdly, the process from Mobius ring to ring 0 also makes ring 0 have four "twists" with different properties in the same direction due to mutual transformation. The evolutionary fission process decomposes the Mobius twist in Mobius ring into four twists in ring 0, which also produces the energy of Mobius twist, but the energy of the four twists in ring 0 is Mobius twist.
4. The process from Mobius ring to ring 0 also makes the space of ring 0 double that of Mobius ring.
5. In the process of generating ring n and ring n+ 1 from ring 0, the "energy" of the four "twists" in ring 0 will not increase, but from the perspective of "fission" of ring 0, the space of ring 0 will increase every time.
arouse
From the three wonders of Mobius ring and the six characteristics of Mobius ring, ring 0 and all generating rings, we get wonderful enlightenment:
First, no matter where the Mobius ring is placed in the cosmic space-time, we will also find that the space outside the Mobius ring can only have one face, so there is only one face outside any space in the cosmic space-time. If there is only one surface outside any space in cosmic space-time, then we can think that any point in cosmic space-time is connected with other points, that is, the whole cosmic space-time is connected, any point is the center and edge of the universe, and any matter in cosmic space-time is the same, all at the center and edge of the universe.
Two: At any point in the space-time of the universe, an opposite gender can be created out of nothing through "fission". Whether or not the generated heterosexuality needs a carrier to present it, the generated heterosexuality needs a space twice as large as the original space to reflect the generated heterosexuality through "fission".
Three: As long as there is fission, the original Mobius ring no longer exists as it is, or the original Mobius ring no longer exists. To "restore" a ring from scratch to the original Mobius ring, we must solve an opposite hermaphrodite.
Fourthly, the process from Mobius ring to ring 0 also makes ring 0 have four "twists" with different properties in the same direction due to mutual transformation. As we know, any affirmation should be a vector process of negation (negation with a certain direction), and there is no absoluteness of negation with the same direction, gap or negation.
5. After ring 1 and ring 2 are generated from ring 0 and "split" again until ring N and ring n+ 1, all generated rings N and ring n+ 1 will be nested together, never separate, never exist independently and never contact with other rings. This shows that there is a universal law of connection between all things in the universe, and any point or thing is connected with all other things in the universe, which is inseparable and indispensable.
6. There is no difference in the ultimate origin of everything in the universe, all of which originated from a space with only one face or a state without any face. So it can also be said that everything in the universe is from scratch, but it shows differences in the process of evolution.
7. In the process of "fission" of Mobius ring generating ring 0, new energy of 1 times the original "twisting force" is generated out of nothing, that is to say, the "fission" in the process of a pair of newly generated androgynous relations does not follow the "energy conservation principle"; The subsequent "fission" of everything in the universe can only increase the space-time of the universe and no longer produce new energy, and "fission" must follow the "principle of energy conservation"
Eight, any point in the space-time of the universe can generate yin and yang for the first time through the way of out of nothing, and then generate yin and yang for the first time on the basis of the newly generated yin and yang, the second time, the third time ... until eternity.
If we stick two Mobius bands together along their unique edges, you will get a Klein bottle (of course, don't forget, we will
Klein bottle
It is really possible to complete this bonding in four-dimensional space, otherwise the paper will be torn a little). Similarly, if we cut a Klein bottle properly, we can get two Mobius belts. In addition to the Klein bottle we saw above, there is also a little-known "8" Klein bottle. They look completely different from the surface above, but in four-dimensional space they are actually the same surface-Klein bottle. In fact, it can be said that Klein bottle is a three-dimensional Mobius belt. We know that drawing a circle on a plane, putting things in, and taking them out in two dimensions, you have to cross the circle. But in the three-dimensional space, it is easy to take it out and put it outside the circle without passing through it. Projecting the trajectory of an object together with the original circle into two-dimensional space is a "two-dimensional Klein bottle", that is, Mobius belt (Mobius belt here refers to Mobius belt in the topological sense). Imagine that in our three-dimensional space, it is impossible to take the yolk out of the egg without breaking the eggshell, but in our four-dimensional space, it is possible. Projecting the trajectory of egg yolk and eggshell into three-dimensional space, you can definitely see a Klein bottle. Attachment: Klein bottle is broken in three-dimensional space, and there must be at least one crack. If there are two cracks, it must be the Mobius belt with two parts connected. Similarly, n Mobius belts can be combined into a Klein bottle with n cracks.
Application in mathematics
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.
Application in practice
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.
1. 1979, the famous American tire company Baluchi creatively made the conveyor belt into the shape of Mobius circle, so that the entire conveyor belt circle was evenly distributed everywhere.
Bear the wear and tear, avoid the single-sided damage of ordinary conveyor belt, and prolong the service life by a whole time.
Second, the stylus printer hits the ribbon with a printing needle, leaving ink spots on the paper. In order to make full use of all the surfaces of the ribbon, the ribbon is usually designed as Mobius circle. 3. In the famous Kenny Forest Amusement Park in Pittsburgh, USA, there is an "enhanced version" roller coaster-its orbit is a Mobius circle. Passengers fly on both sides of the track.
Fourth, the geometric characteristics of Mobius circle contain eternal and infinite significance, which is often used in various logo designs. The trademark of Power Architecture, a microprocessor manufacturer, is a Mobius circle, and even the garbage collection sign is exchanged by Mobius circle.
Method of establishing three-dimensional Mobius region by using parametric equation;
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 belt with a side length of 1 and a radius of 1, which lies in the x-y plane and has a center of (0,0,0). When v moves from one side to the other, the parameter u surrounds the whole belt.
If expressed by the polar coordinate equation (r, θ, z), the borderless Mobius zone can be expressed as:
log(r)sin(θ/2)=zcos(θ/2).
Mobius belt has inspired many artists, such as the artist maurits cornelis escher, who used this structure in woodcuts. The most famous is the second generation Mobius belt, which shows some ants crawling on the Mobius belt.
Mobius, also known as Montbius, symbolizes infinity.
It often appears in science fiction, such as arthur clarke's The Wall of Darkness. Science fiction often imagines that our universe is a Mobius belt. A.J. deutsch's short story The Subway Station named Mobius created a new route for the Boston subway station. The whole route was distorted by Mobius belt, and all the trains entering this route disappeared. Another novel, Star Trek: The Next Generation, also uses the Mobius concept of space.
There is a little poem that also describes the Mobius belt:
Mathematicians assert that the Mobius belt has only one side. If you don't believe me, please cut a verification tape and attach it when you separate it.
Mobius belts are also used in industrial manufacturing. The conveyor belt inspired by Mobius belt can be used for a longer time because it can make better use of the whole belt, or it can be used to make magnetic tapes, which can carry twice as much information.
There is a steel Mobius sculpture in the Smith Forest History and Technology Museum in Washington, USA.
Dutch architect Ben van Becker designed the famous Mobius residence with the Mobius belt as a creative model.
In the Japanese comic "Doraemon Doraemon", Doraemon has a prop with Mobius appearance; In the story, as long as you put this ring on the doorknob, people outside will still see the outside when they come in.
Japan's Aesop Altman's 23rd sentence "Reverse! TAC team used Mobius belt principle in Zofi's debut, which made Beidou and Nannan enter another dimension and destroyed it.
In the video game "The Story of Sonic Boy-Skateboard Meteor", the devil's last battle was played on the shaped runway of Mobius. If you don't defeat the devil, you will never cycle down on Mobius. .....
1988 The animated film Xia Ya, which was released in Japan, takes the Mobius belt as a metaphor of fate: human beings are like ants walking on the Mobius belt, and they can never escape from this strange circle, repeating the same mistakes and similar tragedies. The theme song of the movie "Beyond Time" (メビスのをぇて) also echoes this theme (メビス is m & amp； OumlBius means).
Beyonce Altman, a Japanese dream, is also named after Mobius Belt, and its deformation is the symbol of infinity, that is, the cut Mobius Belt.