A trapezoid was key to discovering the elusive answer to a riddle about Möbius strips
Möbius strips are curious mathematical objects. To construct one of these single-sided surfaces, take a strip of paper, twist it once and then tape the ends together. Making one of these beauties is so simple that even young children can do it, yet the shapes’ properties are complex enough to capture mathematicians’ enduring interest.
Schwartz first learned about the problem about four years ago, when Sergei Tabachnikov, a mathematician at Pennsylvania State University, mentioned it to him, and Schwartz read a chapter on the subject in a book Tabachnikov and Fuchs had written. “I read the chapter, and I was hooked,” he says. Now his interest has paid off with a solution to the problem at last. In a preprint paper posted on arXiv.org on August 24, Schwartz proved the Halpern-Weaver conjecture.
Before arriving at the successful strategy, however, Schwartz tried other tactics on and off again over a few years. He recently decided to revisit the problem because of a nagging sensation that the approach he had used in a 2021 paper should have worked. In his earlier work, Schwartz identified two straight lines that are parallel to each other and also in the same plane, forming a T-pattern on every Möbius strip. “It is not at all obvious that these things exist,” Schwartz says. Showing that they do was the first part of proving the lemma, however.
During those experiments, Schwartz cut open a Möbius band and realized, “Oh, my God, it’s not the parallelogram. It’s a trapezoid.” Discovering his mistake, Schwartz was first annoyed but then driven to use the new information to rerun other calculations. “The corrected calculation gave me the number that was the conjecture,” he says. “I was gobsmacked.... I spent, like, the next three days hardly sleeping, just writing this thing up.
United States Latest News, United States Headlines
Similar News:You can also read news stories similar to this one that we have collected from other news sources.
Santa Rosa police arrest 20-year-old on suspicion of threats against 13-year-oldSanta Rosa police arrested a second suspect who allegedly threatened to shoot a Slater Middle School student on Sept. 3.
Read more »
Amber Alert for 2-month-old boy, 1-year-old girl issued by Florida authoritiesAccording to authorities, the 2-month-old boy and 1-year-old girl were last seen in Lake Worth, Florida.
Read more »
30-Year-Old Civil War Movie Ages Well With Near Perfect Historical Depiction Of Battle Scenes1993's Gettysburg gets a seal of approval.
Read more »
13-year-old bicyclist in critical condition after struck by police vehicle in Suffolk CountyAfter colliding with the bicyclist, the police officer struck another vehicle stopped in traffic, a tree and two parked cars.
Read more »