A Twist in Mathematics
We are all quite familiar with the ‘Recycle’ logo. It seems like a single piece of paper folded thrice. But have you ever visualised what would happen if you straightened out that piece of paper?
You would get a one-sided object. Need a more practical, hands-on experiment? Take a strip of paper, give it one half-twist, and tape the two ends of the paper strip together. Now, draw a line along the vertical length of this strip. You will notice that the line seems to be present on “both” sides of the paper strip! Now try taking a pair of scissors and cutting the strip in half along this line you drew. You might expect two smaller endless strips, but you’ll be left with one long two-sided loop instead!
This object is what we call a Möbius Strip — a mathematical marvel that has stunned millions of artists, environmentalists, and mathematicians, among others, across time and space. It was discovered in 1858 by a German mathematician named August Möbius, who was working on the geometric theory of polyhedra, solid figures composed of vertices, edges, and flat faces.
A Knot-So-Ordinary Loop
The Möbius Strip has inspired innumerable things — the first being an entirely new mathematical field called “topology.” Topology is the study of objects whose geometrical properties and spatial relations remain unaffected when moved, bent, or stretched. An interesting example is the standard pair — a donut and a cup. They’re topologically the same because they both have only one hole. A pair of tangled earphones and a pair of untangled earphones are also topologically the same because they have simply been moved, bent, or twisted!
Another property of the Möbius Strip that has perplexed mathematicians is its non-orientability. Orientability is described as “a continuous choice of local orientation.” Mathematicians are unable to assign coordinates to the Möbius Strip, which means you can’t differentiate the top of a Möbius Strip from its bottom or its left side from its right side!
Do you wonder what would happen if the universe was non-orientable? Suppose a rocket with a few astronauts sets out into space, goes on a long journey, and then returns to orbit. If the universe were non-orientable, the astronauts would return in reverse, or well, as mirror images of themselves!
The Möbius Strip: Hidden in Plain Sight!
The Möbius Strip is more than just a fascinating mathematical theory. Although it has multiple applications in architecture, machinery, and many complex objects, there are numerous practical utilizations in the sphere of human imagination. Möbius strips are used in continuous-loop recording tapes, typewriter ribbons, and computer printer cartridges. Sandia Laboratories employed Möbius bands to create adaptable electronic resistors in the 1960s.
Möbius strips are used in conveyor belts because they allow the entire surface area of the belt to experience equal wear, allowing it to last longer. The Möbius strip has also been adapted for use in various artistic and cultural items. Möbius shapes have been seen in paintings, as well as earrings, necklaces, and other jewellery. The Möbius band is employed in hardware and popular iconography, but its mathematical and scientific curiosity has lasted for over a century.
Isaac Freund of Israel’s Bar-Ilan University proposed in 2005 that light’s polarisation may be distorted. The polarisation of light is a characteristic that explains how its electric field moves. In physics, polarisation is described as “the action of restricting the vibrations of a transverse wave, especially light, wholly or partially to one direction.” A transverse wave is one in which the oscillations of light flow perpendicular to the direction of its flow of energy. In simpler words, light can be manipulated into new shapes.
Peter Banzer of Germany’s Max-Planck-Institut für die Physik des Lichts tested Freund’s idea that light might be bent in 2015. Banzer’s team scattered two polarised green laser beams off a gold bead that was smaller than the wavelength of the light. The resulting conclusion created a polarisation pattern with three or five twists, giving it a Möbius-like shape.
The Primordial, Medieval, and Everything in Between
Surprisingly, the Möbius Strip has been heavily used as symbolism in the field of philosophy. The most important thing that the Möbius Strip represents is the idea of infinity.
The Greeks developed the notion of mathematical infinity around the sixth century B.C.E. Although it may have existed in previous civilizations, such as those of the Egyptians, Babylonians, and Chinese, most of these cultures focused on its application in daily life rather than the idea of infinity itself.
Interestingly, the Möbius strip can be seen in a Roman mosaic in Sentinum from the third century C.E. It showed Aion, a Hellenistic deity connected with time, standing within a Möbius strip embellished with zodiac signs. Thus, the connection of the Möbius Strip with infinity goes a long way back!
The Universe and its Artistic Prowess
The Möbius Strip also represents unity and non-duality. Numerous artists have used this concept in their artwork, the most notable being the Möbius Strip II — Red Ants by Maurits C. Escher, a Dutch graphic artist. It depicts red ants chasing each other on a Möbius Strip. He is also known for his other masterpieces, which cleverly use complex mathematical concepts. His other artworks include Möbius Strip I in 1961, which features abstract creatures chasing each other, and Horsemen in 1946, which portrays two groups of horses marching around the endless strip.
In some sense, all these creatures seem unified, running around the same loop endlessly.
While the Möbius strip represents unity, it can also portray futility and entrapment. After all, the creatures will be stuck in the same endless loop forever, all by themselves. This is also a metaphor for reality, where we are usually stuck in a loop. We are trapped by the routine, waking up daily and simply restarting the loop. You might seem like you’re making progress when, in reality, you’re constantly walking on a treadmill. In this case, the Möbius Strip represents hopelessness and a desire to break free from the rat race.
Finally, the last symbolism of the Möbius Strip, one that most of you are familiar with — a representation of the universe.
Space and time in the universe seem detached, just as the Möbius strip, but there is no separation since both comprise the cosmos. In actuality, all existing matter and space are considered as a whole.
Time travel to the past or future is prevalent in popular culture, although there is no evidence that it is conceivable. The Möbius strip was mentioned in the film Avengers: Endgame when a group of superheroes decided to go through time for a mission. In a metaphorical sense, they spoke of going back in time, comparable to the well-known experiment in which an ant returns to its initial position.
The Infinite Loop of Narrative
The Möbius Strip finds a use for itself even in the vast field of literature.
The continuum of crossing a Möbius strip exemplifies the nonlinear perception of time.
Gabriel Garcia Márquez’s “One Hundred Years of Solitude’’ is a notable literary example. The novel follows the Buendia family through cyclical patterns of behaviour and emotion, fusing history, memory, and prophecy. This core theme is shown by an exchange between two family members: “Ursula groaned. ‘Time moves on.’ ‘That’s how it goes,’ Aureliano acknowledged, ‘but not so much.’” Thus, in moments of bewilderment, Márquez’s Buendia family finds itself in a type of Möbius strip, repeating the same behaviours generation after generation despite witnessing the passage of time, technology, and politics in and outside Macondo.
However, certain characters appear to be in a state of being that frees them from the rigid temporal context of cultural conventions and familial history. This grants them the seemingly impossible ability to channel the pure “truth” of themselves and the events around them.
Read One Hundred Years of Solitude here: 100 YEARS OF SOLITUDE Gabriel Garcia Marquez (1967)
The past, present, and future are not placed in a linear sequence in “One Hundred Years of Solitude,” which means they are not neatly put in a straight line with the past at the rear, the future at the front, and the present in the middle. In the novel, time is arranged in a looping circle, and no one knows where it ends and where it begins.
Garcia Márquez deviated from the idea that time is ordered chronologically and used it to create another dimension to the narrative.
The End, but not of the Möbius Strip!
We’ve been traversing what feels like misshaped terrain and time during this year of solitude, bewilderment, and remoteness. However, unlike the residents of Márquez’s imaginary town of Macondo, which is “exiled from the memory of men,” we continue to recall and discover. Even as we mourn the loss of loved ones, scientific innovation is thriving and saving lives.
The Möbius strip has rich metaphorical and narrative ramifications: if you try to move ahead, you ring sideways; if you try to circle in, you end outside. An accurate allegory about relinquishing control, I think. Where will we be in 2024? We might question. Have we turned around after all this commotion just to find ourselves in the same situation we were in before? Or have we just started over?
