Imagine you are holding a strip of paper. You give it a half twist, then glue its ends together. The shape you hold now is the ticket to a world where surfaces are only one-sided and the lines blur between inside and outside. This is the kingdom of the Möbius strip.
The Möbius strip is one of the most intriguing mathematical structures we have encountered, a perfect blend of an ordinary shape with very complex properties. It has captivated amateurs and professional mathematicians for over a century. One of the hardest puzzles is a deceptively simple question: How short and wide can a paper Möbius strip be before it has to get tangled or run inside itself?
This question is much more subtle than it seems. The main constraint is the word “paper”. In geometry, this means that 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, thin strip; the material itself prohibits it. This rules out “origami monsters,” like folding a strip like an accordion into a tiny space. The tape should be integrated smoothly into the 3D space.
In 1977, mathematicians Charles Weaver and Benjamin Halpern first brought this puzzle to academia. Since then, mathematicians have been frustrated, trying to find the right answer. Now, Richard Schwartz, a mathematician at Brown University, claims to have finally solved the riddle.
When a circle is no longer a circle
The Möbius strip has a “non-orientable” surface. In everyday life, this means that if you were an ant crawling on its surface, you wouldn’t be able to tell one side from another. If you take a pencil and draw a line along the center of the strip, the line will go all the way around and back to where it started without ever crossing an edge, revealing that the surface has only one continuous side. It’s quite amazing to see.
German mathematicians August Ferdinand Möbius and Johann Benedict Listing discovered it independently in 1858. While Möbius obtained the naming rights, both men were attracted to its special property: its infinite surface area.
It’s not just a math gadget. Many engineers and scientists find the Möbius strip fascinating for practical reasons. For example, conveyor belts designed like a Möbius strip distribute wear evenly and last twice as long as conventional conveyor belts. In electronics, Möbius resistors are used because of their unique electromagnetic properties.
Artists are not immune to the strip’s charm. MC Escher, the famous graphic artist, incorporated the Möbius strip into his woodcut “Möbius Strip II”, where the ants fit together and cross the one-sided surface. Even the ubiquitous recycling symbol, printed on the back of aluminum cans and plastic bottles, is essentially a Möbius strip.
Although the visual appeal of the strip is undeniable, its most significant impact has been on mathematics. Among his many contributions, the introduction of the Möbius strip revolutionized the field of topology, which studies the properties of objects that are preserved when they are moved, folded, stretched, or twisted, without cutting or gluing parts together. A coffee cup and a donut are, for example, topologically identical. Both objects have only a single hole, which can be deformed by stretching and bending to create either structure.
A decisive moment
But it wasn’t the topology that intrigued Schwartz. He first heard about the minimum Möbius band problem four years ago and has been hooked ever since. His efforts to unravel the Halpern-Weaver conjecture ultimately bore fruit. The mathematician reported the solution on the preprint server arXiv.org in August 2023.
His discoveries? The optimal Möbius strip should have an aspect ratio greater than √3 (approximately 1.73). Simply put, a strip 1 centimeter wide must be more than 1.73 centimeters long, otherwise the structure will inevitably collapse on itself.
Yet the path to discovery was not a straight line. Schwartz had to invent a new way to “see” the geometry hidden in the group. To solve the problem, he has employed various strategies over the years.
“The corrected calculation gave me the number that was the guess,” Schwartz said. Scientific American. “I was stunned… I spent the next three days barely sleeping, writing this thing.”
However, as is often the case in mathematics, solving one problem opens the door to solving another, more complex one. There is no limit, mathematically speaking, to the length of a Möbius strip. But the next problem that worries Schwartz is finding the shortest strip of paper that can be used to create a Möbius strip with more twists.
A standard band has a half twist. How about a band with three half twists? This is the next frontier. In his article, Schwartz notes that this is an active area of research. He and his collaborator Brienne Brown studied 3-twist bands and identified two “optimal candidate models.” These are called “crisscross” and “cup”, both of which can be folded from a 1 x 3 strip of paper. This led them to conjecture that for a 3-twist strip (a three-half-twist (540°) Möbius strip), 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 of an even number of half twists (e.g. two)?
Mathematics often pushes the boundaries of our understanding, pushing us to question the very fabric of reality. And in this fabric, the Möbius strip stands out like a captivating thread, reminding us of the beauty that resides in infinity and continuity.
The article was originally published on September 13, 2023 and has been edited to include more information.
Correction: An earlier version incorrectly stated the aspect ratio condition. Mathematician Richard Schwartz proved that a strip of Möbius paper must have a length-to-width ratio greater than √3 (≈1.73). Concretely, a strip 1 cm wide must be more than 1.73 cm long – and not the other way around – to form a smooth Möbius strip without crossing.
