Science

The Timeless Journey of the Möbius Strip


If you were to trace both “sides” of a Möbius strip, you would never have to lift your finger. A single-sided surface with no boundaries, the strip is an artist’s reverie and a mathematician’s feat. A typical thought experiment to demonstrate how the three-dimensional strip operates involves imagining an ant on an adventure. Picture the insect traversing the Möbius band. One apparent loop would land the ant not where it started but upside down, only halfway through a full circuit. After two loops, the ant would be back at the beginning—but dizzy.

The figurative and narrative implications of the Möbius strip are rich: when you try to go forward, you ring sideways, when you try to circle in, you find yourself outside. It’s an apt allegory for losing control. We might ask ourselves after 2020, where are we? Have we spun around after so much chaos, and found our position stagnated, back where we started? Or are we at a new beginning?

The continuum of crossing a Möbius strip is emblematic of how we experience time in a nonlinear way. Artists and authors explore this phenomenon as well. One salient literary example is Gabriel García Márquez’s One Hundred Years of Solitude. Melding history, memory, and prophecy, the novel follows the Buendía family through cyclical patterns of behavior and emotion. An exchange between two family members illustrates this central theme: “Úrsula sighed. ‘Time passes.’ ‘That’s how it goes,’ Aureliano admitted, ‘but not so much.’”

The unorientable quality of the Möbius strip is perhaps its most distinctive. Orientability can be defined as “a continuous choice of local orientation.” A more colloquial explanation: “a space is orientable if you can choose ‘inward’ and ‘outward’ or ‘up’ and ‘down’ directions at every point on the surface that are compatible: you will never accidentally end up at the same point but with ‘up’ flipped to ‘down.’”

The Möbius strip was independently discovered by two German mathematicians in 1858. August Ferdinand Möbius was a mathematician and theoretical astronomer (and also the first to introduce “homogenous coordinates” into “projective geometry”). Johann Benedict Listing, a younger mathematician, coined the term “topology” for the study of surfaces, and in conducting that research, independently determined the properties of the Möbius strip. Though perhaps too neat a metaphor, it’s interesting to note that these two men arrived at the same conclusion, from different directions, at the same time.

You can make a model of the Möbius strip with just a rectangular piece of paper: give it an odd number of half-twists, then tape the ends back together. But the strip was long thought never to occur in the natural world. Because it has never been observed in our organic environment, it is sometimes referred to as an “impossible shape.” Practical applications of it abound in the world of human invention, however. For instance, Möbius strips are used in continuous-loop recording tapes, typewriter ribbons and computer print cartridges. In the 1960s, Sandia Laboratories also used Möbius bands in the design of adaptable electronic resistors. Conveyor belts use Möbius strips because they allow the entire surface area of the belt to receive an equal amount of wear, which makes it last longer.

The Möbius strip has also been tailored to various artistic and cultural products. Paintings have displayed Möbius shapes, as have earrings, necklaces and other pieces of jewelry. The green, three-arrowed universal sign for recycling also composes the Möbius band. There’s a depth to the image that reminds you to reduce, reuse and recycle. It is not just a circular action; it’s dynamic. The symbol seeks to represent the three interdependent aspects of a sustainable loop: the collection of materials to be recycled, the manufacturing of recycled materials into new products, and the purchase and use of the products made from recycled materials. Each arrow pleats and pivots itself, as all three arrows pursue and power one another.

The Möbius band is used in hardware and popular imagery, but the mathematical and scientific fascination with the Möbius strip has also endured for over a century. In 2005, Isaac Freund at Bar-Ilan University in Israel, proposed that light’s polarization could be twisted. Light’s polarization is a property that describes how its electric field moves. Polarization in physics is defined as “the action of restricting the vibrations of a transverse wave, especially light, wholly or partially to one direction.” A transverse wave means that light’s oscillations travel perpendicular to the direction in which its energy moves. Scientists can confine light to certain planes depending on its orientation in space. In other words, light can be coaxed into new shapes.

In 2015, Peter Banzer of the Max Planck Institute for the Science of Light in Erlangen Germany tested Freund’s hypothesis that light could be twisted. According to Katherine Kornei, writing at New Scientist, “Banzer’s team scattered two [polarized] green laser beams off a gold bead that was smaller than the wavelength of the light. The resulting inference introduced a polarization pattern with either three or five twists, giving it a Möbius-like structure.”

When I was in high school, my English teacher, Ms. Mulvihill, gave each of our classmates an individual question to answer for a final assignment. One of my peers was asked, “What is the shape of time in One Hundred Years of Solitude?” He responded with a story: his family used to have a dog that would chase his tail. The dog drove himself crazy, running in circles, teeth bared, hunting his own hind legs. One day, racing after his tail, he bit it off. That’s the shape of time in One Hundred Years of Solitude.

In this year of solitude, disorientation and distance, we have been navigating what feels like misshapen terrain and time. But unlike the characters in Macondo, Márquez’s fictional town that is “exiled from the memory of men,” we continue to remember and discover. Even as we mourn the people we have lost, scientific innovation is thriving, and salving.

Darkness can seem like an endless, unbounded surface. We can walk it alone; we can walk it together. But we now know something else can be an infinite curl, a dizzying ground, an effervescent hypothesis confirmed: light. It only needs a little nudging.

 

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