Anika Patel
Since its creation by August Ferdinand Möbius and Johann Benedict Listing, the Möbius strip has captivated mathematicians with its simplicity in construction and visualization, while posing complex mathematical challenges. In 1977, Charles Sidney Weaver and Benjamin Rigler Halpern introduced the Halpern-Weaver Conjecture, which aimed to determine the minimum ratio between the strip’s width and length. According to their conjecture, a strip with a width of 1 centimeter (0.39 inches) should have a length of at least the square root of 3 centimeters (approximately 1.73 centimeters or 0.68 inches).
For Möbius strips that are “embedded” and do not intersect with each other, the conjecture remained unsolved. However, Richard Evan Schwartz, a mathematician from Brown University, proposed in 2020 that if the strip could pass through itself, the problem would be easier to solve. Unfortunately, Schwartz initially made an error in his calculations. In a preprint paper, which has yet to undergo peer review, Schwartz corrected his mistake and arrived at the correct solution for the conjecture.
The key to Schwartz’s solution lies in a lemma from his previous paper. He discovered that straight lines exist on the surface of Möbius strips, passing through every point and reaching the boundaries. To prove this, he needed to establish the existence of perpendicular lines to these straight lines within the same plane. And he succeeded.
“It is not at all obvious that these things exist,” Schwartz explained to Scientific American.
The next step involved slicing the Möbius strips to better understand the shapes they formed. By flattening the strip onto a plane, Schwartz aimed to simplify the problem. Initially, he believed that a sliced strip would resemble a parallelogram, but it turned out to be a different quadrilateral – a trapezoid.
“The corrected calculation confirmed the conjecture,” Schwartz exclaimed. “I was amazed… I spent the next three days barely sleeping, just documenting my findings.”
The preprint of Schwartz’s paper has been shared on ArXiv.
[H/T: Scientific American]
