The mathematical operations of objects that would otherwise seem simple can be surprisingly confusing. There is no better example of this than the Möbius belt.
It is a single-sided object that can be connected with tape by simply twisting a piece of paper. If you walk along the circle with your fingers, you'll eventually return to where you started, touching the entire surface of the circle on the journey. The Möbius band is one of the classical examples of topology.
One of these principles is non-directionality, that is, mathematicians cannot assign coordinates to an object, such as up and down or left and right. This principle has produced some interesting results, as scientists are not entirely sure whether the universe is directional.
This creates a confusing scenario: if a rocket carrying astronauts travels long enough in space and then returns, assuming the universe is unorientable, then all astronauts are likely to reverse.
In other words, when astronauts come back, they will become mirror images of their former selves, completely upside down. Their hearts will be on the right instead of on the left and they may be left-handed instead of right-handed. If one of the astronauts loses his right leg before flying, he loses his left leg on his return. This is what happens when you cross an unorientable surface, such as the Möbius belt.
Hopefully your mind is shaken – at least slightly – and we need to take a step back. What is the Möbius band? How could an object with such complex mathematical operations be made by simply twisting a piece of paper?
History of the Möbius belt
The Möbius belt was first discovered in 1858 by a German mathematician named August Möbius, who was working on geometric theory. Although Möbius is largely credited with the discovery (the belt got its name from him), it was almost simultaneously discovered by a mathematician named John Lister.
The strip itself is simply defined as a one-sided non-directional surface that is generated by adding a semi-torsional band. The Möbius band can be any band with an odd number of half-twists, which eventually leads to the Möbius belt having only one side and only one side.
Since its discovery, this one-sided strip has captivated artists and mathematicians alike. Even M.C. Escher fascinated him, creating his famous work "Möbius Comic Strips I and II".
The discovery of the Möbius band is also the basis for the formation of the field of mathematical topology, which studies the geometric properties of objects that remain unchanged when deformed or stretched. Topology is crucial for certain areas of mathematics and physics, such as differential equations and string theory.
For example, according to topology principles, a cup is actually a donut.
Practical use of the Möbius strip
The Möbius belt isn't just a great mathematical theory: it has some cool practical applications, both as a teaching aid to more complex objects, or in mechanics.
For example, since Möbius belts are physically single-sided, the use of Möbius belts in conveyor belts and other applications ensures that the belt itself does not suffer uneven wear throughout its entire life. NJ Wildberger, an associate professor at the School of Mathematics at the University of New South Wales in Australia, explained in a lecture series that the drive belts of machines are often twisted and "deliberately allow the belts to wear evenly on both sides." "The Möbius belt can also be seen in the building, such as the Five Forks Bridge in China.
Five forks bridge
People walk on the Wuchazi Bridge in Chengdu, Sichuan Province, China, which is designed according to the Möbius belt.
Dr. Edward Inglich Jr., a middle school math teacher and former optical engineer, said that when he first learned about the Möbius belt in elementary school, his teacher asked him to make one out of paper and cut the Möbius band along its long side, thus forming a longer Möbius belt with two complete twists.
"I think the curiosity and contact with the concept of two 'states' helped me when I encountered the up-and-down spin of electrons," he said, referring to his doctoral research. "It's not surprising to me to accept and understand various quantum mechanical concepts, because Möbius' comics made me realize this possibility." For many, the Möbius band is the first introduction to complex geometry and mathematics.
How do you create an Möbius band?
It is easy to do Möbius belt.
Creating an Möbius strip is very easy. Simply take a piece of paper, cut it into thin strips, and simply twist one end 180 degrees, or half screw. Then, take some tape, connect this end to the other, and make a half-twisted ring inside.
You can use your fingers along the sides of the strip, best adhering to the principles of this shape. You'll eventually wrap it around the shape and find your finger back where it started.
If you cut a long Möbius strip from the middle and follow it in a circle, you'll get a bigger ring, just this new Möbius band twisting more.
Cut along two third-grade lines, and that's what it looks like.