Imagine holding a strip of paper. You give it a half-twist and then tape its ends together. The shape you’re now holding is the ticket to a world where surfaces have only one side and boundaries blur between inside and out. This is the realm of the Möbius Strip.
The Möbius Strip is one of the most intriguing mathematical structures we’ve encountered, a perfect blend of an ordinary shape with highly complex properties. It’s captivated amateurs and professional mathematicians for over a century. One of the most challenging puzzles is a deceptively simple question: How short and wide can a paper Möbius Strip get before it must tangle or pass within itself?
This question is far more subtle than it appears. The key constraint is the word “paper.” In geometry, this means the strip is “developable”—it can be made from a flat sheet without any stretching, tearing, or shrinking. The formal term is an isometric mapping, a transformation that preserves all distances and arc-lengths. You can’t just shrink a long, skinny band; the material itself forbids it. This rules out “origami monsters,” like folding a strip like an accordion into a tiny space. The strip must be smoothly embedded in 3D space.
Back in 1977, mathematicians Charles Weaver and Benjamin Halpern first dropped this brainteaser into the academic world. Mathematicians have been left frustrated ever since, trying to find the right answer. Now, Richard Schwartz, a mathematician from Brown University, claims he has finally solved the puzzle.
When a circle isn’t a circle anymore
The Möbius Strip has a “non-orientable” surface. In everyday terms, this means if you were an ant crawling on its surface, you wouldn’t be able to distinguish one side from another. If you take a pencil and draw a line along the center of the strip, the line will travel all the way around and return to its starting point without ever crossing an edge, revealing that the surface has only one continuous side. It’s quite mind-bending to see.
The German mathematicians August Ferdinand Möbius and Johann Benedict Listing independently discovered it in 1858. While Möbius got the naming rights, both men were drawn to its peculiar property: its unending surface.
This isn’t just some mathematical gimmick. Many engineers and scientists find the Möbius Strip fascinating for practical reasons. For instance, conveyor belts designed as a Möbius strip distribute wear and tear uniformly, lasting twice as long as conventional conveyor belts. In electronics, Möbius resistors are employed due to their unique electromagnetic properties.
Artists aren’t immune to the strip’s allure. M.C. Escher, the famed graphic artist, incorporated the Möbius Strip in his woodcut “Möbius Strip II,” where ants interlock and traverse the one-sided surface. Even the ubiquitous recycling symbol, found printed on the backs of aluminum cans and plastic bottles, is essentially a Möbius strip.
While the visual appeal of the strip is undeniable, its most significant impact has been in mathematics. Among its many contributions, the introduction of the Möbius Strip has revolutionized the field of topology, which studies the properties of objects that are preserved when moved, bent, stretched or twisted, without cutting or gluing parts together. A coffee mug and a doughnut are, for instance, topologically identical. Both objects have just one hole, which can be deformed through stretching and bending to create one or the other structure.
A breakthrough moment
But it’s not topology that intrigued Schwartz. He first heard about the minimum Möbius strip problem four years ago and has been hooked ever since. His efforts to untangle the Halpern-Weaver conjecture finally paid off. The mathematician reported the solution on the preprint server arXiv.org in August 2023.
His findings? The optimal Möbius strip must have an aspect ratio greater than √3 (about 1.73). In layman’s terms, a strip that is 1 centimeter wide must be more than 1.73 centimeters long — otherwise, the structure will inevitably collapse into itself.
Yet, the path to discovery wasn’t a straight line. Schwartz had to invent a new way to “see” the geometry hidden within the band. As he grappled with the problem, he employed various strategies over the years.
“The corrected calculation gave me the number that was the conjecture,” Schwartz told Scientific American. “I was gobsmacked… I spent, like, the next three days hardly sleeping, just writing this thing up.”
However, as is often the case in mathematics, solving a problem opens the door to solving another, more complex one. There is no limit, mathematically speaking, to how long a Möbius strip can be. But the next problem on Schwartz’s mind is finding the shortest strip of paper that can be used to make a Möbius strip with more twists.
A standard band has one half-twist. What about a band with three half-twists? This is the next frontier. In his paper, Schwartz notes that this is an active area of research. He and his collaborator, Brienne Brown, have been studying 3-twist bands and have identified two “candidate optimal models”. These are named the “crisscross” and the “cup,” both of which can be folded from a 1 x 3 strip of paper. This has led them to conjecture that for a 3-twist band [a Möbius strip with three half-twists (540°)], the aspect ratio must be greater than 3.
This opens up an infinite family of questions. What about 5-twist bands? Or 7-twist bands? What about “twisted cylinders,” which are made with an even number of half-twists (like two)?
Mathematics often pushes the boundaries of our understanding, nudging us to question the very fabric of reality. And in this fabric, the Möbius strip stands out as a mesmerizing thread, reminding us of the beauty that lies in endlessness and continuity.
The article was originally published on 13 September 2023 and has been edited to include more information.
Correction: An earlier version misstated the aspect ratio condition. Mathematician Richard Schwartz proved that a paper Möbius strip must have a length-to-width ratio greater than √3 (≈1.73). In practical terms, a strip that is 1 cm wide must be more than 1.73 cm long — not the other way around — to form a smooth Möbius strip without self-intersecting.
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