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One of the most intriguing mathematical concepts, the Möbius (also spelled Mobius or Moebius) strip is an infinite loop, featuring a one-sided surface without boundaries. It’s inspired various works of art, literature, technology, and even magic, making it an intriguing and versatile symbol. Here’s a closer look at the mysteries of this symbol and its significance today.
History of the Möbius Strip
Sometimes referred to as a twisted cylinder or a Möbius band, the Möbius strip was named after August Ferdinand Möbius, a theoretical astronomer and German mathematician who discovered it in 1858. He likely encountered the concept while he was working on the geometric theory of polyhedra, a three-dimensional object made of a polygon. The symbol had been explored independently a few months earlier by Johann Benedict Listing, another German mathematician, but he didn’t publish his work until 1861. This made August Mobius the first in the race and so the symbol was named after him.
The Möbius strip is created with a twisted strip of paper with joined ends. It’s one-sided, and only has a single continuous surface, which cannot be defined as inside or outside compared to a typical two-sided loop.
The Mysteries of the Möbius Strip
In an ordinary two-sided loop (with an inside and outside), an ant could crawl from the starting point and reach the ends only once, either on the top or the bottom—but not on both sides. In a one-sided Möbius strip, an ant has to crawl twice to return to where he started.
Most people become fascinated when the strip is split into halves. Typically, cutting an ordinary two-sided strip along the center will result in two strips of the same length. But in a one-sided Möbius strip, it will result in one strip twice as long as the first.
On the other hand, if a Möbius strip is cut lengthwise, dividing it in three equal parts, it will result in two intertwined rings—one shorter strip inside a longer strip.
Confused? It’s best to see this in action. This video very beautifully demonstrates these concepts.
Meaning and Symbolism of the Möbius Strip
Apart from theoretical mathematics, the Möbius strip has gained symbolic meaning in various works of art and philosophy. Here are some of figurative interpretations on the symbol:
- A Symbol of Infinity – In geometrical and artistic approaches, Möbius strip is depicted with one side and a never-ending path along its surface. It demonstrates infinity and endlessness.
- A Symbol of Unity and Non-Duality – The design of Möbius strip shows that the two sides, which are referred to as inside and outside, are joined together and became one side. Also, in various works of art, such as the Mobius Strip I, the creatures seem to chase one another, but they are unified in some sense, connected in an endless ribbon. This symbolizes unity and oneness and the concept that we’re all on the same path.
- A Representation of the Universe – Just like the Möbius strip, space and time in the universe seem to be unconnected, but there is no separation since both form the cosmos. In fact, all existing matter and space are considered as a whole. In pop culture, time travel to the past or future is common, even though there’s no evidence that it’s possible. The Möbius strip became a subject in Avengers: Endgame, when a team of superheroes planned to go back in time. Metaphorically speaking, they referred to returning to a point in time, which is similar to the known experiment of an ant returning to where it started.
- Futility and Entrapment – The strip can also convey the negative concept of futility and being trapped. While it might seem as though you’re getting somewhere and making progress, in reality, you’re in a loop, much like walking on a treadmill. This symbolizes a hopelessness, a rat race out of which most people never escape.
The Möbius Strip and Topology
The discovery of the Mobius strip led to new ways of studying the natural world, especially topology, a branch of mathematics that deals with the properties of a geometric object unaffected by deformations. The Mobius strip inspired the concept of the Klein bottle with one side, which cannot hold a liquid since there is no inside or outside.
The Concept in Ancient Mosaics
The concept of mathematical infinity began with the Greeks around 6th century B.C.E. While it might have been present in earlier civilizations of the Egyptians, the Babylonians, and the Chinese, most of these cultures dealt with its practicality in daily life—not the concept of infinity itself.
The Möbius strip was featured in a Roman mosaic in Sentinum, which can be dated back to the 3rd century C.E. It depicted Aion, a Hellenistic deity associated with time, standing inside a Möbius-like strip decorated with zodiac signs.
The Mobius in Modern Visual Arts
The Möbius strip has a visual appeal that attracts artists and sculptors. In 1935, Swiss sculptor Max Bill created the Endless Ribbon in Zurich. However, he wasn’t aware of the mathematical concept, as his creation was a result of finding a solution to a hanging sculpture. Eventually, he became an advocate of using mathematics as a framework of art.
The concept of the strip is also evident in works of Maurits C. Escher, a Dutch graphic artist who is famous for designing mathematically inspired prints, such as mezzotints, lithographs, and woodcuts. He created the Mobius Strip I in 1961, featuring a pair of abstract creatures chasing each other; and the Mobius Strip II – Red Ants in 1963, which depicts ants climbing the infinite ladder.
In 1946, he created the Horsemen, portraying two groups of horses marching around the strips endlessly. But according to a book To Infinity and Beyond: A Cultural History of the Infinite, the art isn’t a true Möbius strip, but something you can get when you split the strip into halves. In addition, the depiction itself connected the sides of the strip to let the two teams of horsemen meet.
Also, a triple-twist Möbius strip is featured on large stone sculptures by Keizo Ushio, a pioneer in geometrical sculpture in Japan. His split loop sculptures known as Oushi Zokei 540° Twists can be found at Bondi Beach, Australia and Tokiwa Park, Japan. His Möbius in Space depicts the strip in space, enclosed in a loop sculpture.
Uses of the Möbius Strip Today
From electrical components to conveyor belts and train tracks, the concept of the Möbius strip has many practical applications. It was used in typewriter ribbons and recording tapes too, and is commonly found on various packaging as a symbol for recycling.
In jewelry design, the motif is popular in earrings, necklaces, bracelets, and wedding rings. Some are designed with words inscribed on silver or gold, while others are studded with gemstones. The symbolism of the piece makes it an attractive design, especially as a gift for loved ones and friends. The symbol has also become a popular style for scarves in various materials and prints, as well as tattoos.
In literature and pop culture, the Möbius strip is often referenced to justify plots in science fiction such as Avengers: Endgame, A Subway Named Mobius, and The Wall of Darkness. There’s also a Mobius Chess, a game variant for 4 players, as well as LEGO sculptures and Mobius mazes.
Since its discovery, the Möbius strip has fascinated and inspired mathematicians and artists to design masterpieces beyond the space we live in. The Mobius strip has many practical applications in the fields of science and technology, as well as an inspiration in fashion, jewelry design, and pop culture.