On Aug. 24, Richard Schwartz, professor of mathematics, published a preprint paper to arXiv.org that proved a conjecture that has stumped mathematicians for nearly 50 years: What is the shortest strip of paper needed to create a Möbius strip?
This question was first answered in 1977, when a paper concluded that the Möbius strip must have an aspect ratio of one to the square root of 3, which is roughly 1.73. This means that if a piece of paper is one inch wide, the length must be greater than 1.73 inches in order to form a Möbius strip. This is commonly known as the Halpern-Weaver conjecture, which remained unproven — until Schwartz.
Möbius strips have captured the interest of mathematicians and the public alike due to their special geometric properties — they only have a single side. They were first discovered in 1858 by August Ferdinand Möbius and Johann Benedict Listing. Making a Möbius strip at home is relatively easy: take a strip of paper, twist it once and tape its ends together.
“They’re extremely simple and accessible objects, which is not always the case with math research,” Schwartz said. “It’s the kind of thing that you could explain … to an eight-year-old.”
Kyne Santos, a drag queen who has amassed 1.5 million TikTok followers for posting educational math content, is no stranger to Möbius strips. In July 2021, she posted a video to the platform that has since received more than 14 million views, explaining the properties of Möbius strips.
“When you flip that paper around and create (a Möbius strip), you stumble upon this mathematical object that is — on one hand — really simple at first glance but actually a lot more interesting than meets the eye,” Santos said in an interview with The Herald.
Two types of Möbius strips dominate mathematicians’ interest: immersed and embedded. “Embedded are the kind you can actually make,” Schwartz said. Immersed, on the other hand, can intersect with itself, requiring different lengths to form a Möbius strip.
“If you have to walk to your bathroom and you’re allowed to go through the walls, you don't have to take as many turns probably — you can probably just go straight there,” he said. “So, the embedded ones have to be longer in order to make them because it has to get out of its own way.”
Schwartz was first introduced to the problem by a friend and colleague who had written an article on the topic. At the same time, he was thinking about another problem called the Square Peg conjecture, which gave him insight into how to solve the Halpern-Weaver conjecture.
Specifically, Schwartz applied his knowledge of T patterns — which he described as a “special arrangement of four points on a loop.” On Möbius strips, the T patterns are a “special arrangement of two line segments,” a property which eventually helped Schwartz to solve the proof.
The road to the solution did not come without bumps, however. In 2021, when Schwartz first attempted the problem, a mistake in his proof resulted in a parallelogram instead of a trapezoid.
“It was just some stupid error (that) got baked into my brain, and I never checked it until three years later,” he recalled.
When Schwartz realized his mistake, he issued a correction to the journal on his 2021 paper with new calculations in line with his realization. “I did it out, and it just solved the whole conjecture,” Schwartz said. “Then I realized — now it’s a different ballgame.”
He spent multiple days writing a new paper, which has since been uploaded to the open-access site arXiv.org and sent to a journal for potential publication. Publication and peer review is a lengthy process, but Schwartz is optimistic about the paper’s future, highlighting that his solution is relatively short — only 13 pages without references — and proves an old conjecture.
Jeffrey Hoffstein, professor of mathematics and department chair, highlighted the uniqueness of the Halpern-Weaver conjecture and the rarity of a proof like Schwartz’s.
“It is very unusual to have a math problem that is so easy to state, and so hard to prove,” he wrote in an email to The Herald. “In fact, most problems that fit that description are either incredibly hard to prove (like Fermat’s Last Theorem, which remained unsolved for hundreds of years), or have never been proved, like Goldbach’s conjecture.”
Santos hopes that the proof — which she uploaded two TikTok videos about — helps underscore the creativity associated with math that often goes unnoticed. “People have a misconception about math that it’s all about getting the right answer,” she said. “When you’re an actual mathematician, there’s no right or wrong way to go about it — it’s a matter of problem-solving.”
Ryan Doherty is a senior staff writer covering faculty, higher education and science & research. He is a sophomore concentrating in chemistry and history who likes to partially complete crosswords in his free time.