After 50 years of research, Richard Schwartz finally solves the Möbius mystery, shedding light on its unique properties and shape.
Individuals who like mathematics will probably be familiar with Möbius strips. According to IFL Science, all one has to do to make one is take a small strip of paper and stick the two short edges together. What is interesting about this shape is that a person could draw a line across the whole surface without taking the pencil away from the paper. Mathematicians attempted to come up with the minimum size for such a strip forty-six years ago but could not find it.
A trapezoid was key to discovering the elusive answer to a riddle about Möbius strips https://t.co/sTiXAptrFz— Scientific American (@sciam) September 15, 2023
Fortunately, someone has discovered the minimum size required for achieving the unique shape that a Möbius strip has. The strip was created by August Ferdinand Möbius and Johann Benedict Listing. Despite the simple nature of making this strip, it came to be associated with many mathematical complications because of its shape. Charles Sidney Weaver and Benjamin Rigler Halpern came up with the Halpern-Weaver Conjecture, which gave the minimal ratio between the width of the strip and its length.
They came up with the fact that for a strip having a width of 1 centimeter (0.39 inches), the length would at least need to be the square root of 3 centimeters. However, the conjecture could not find a solution for smooth Möbius strips that were "embedded" which meant that they didn't intersect with each other. Brown University's mathematician Richard Evan Schwartz came up with a solution in 2020 by saying that it would be a much easier problem to solve if the strip could go through itself.
He did make a mistake in his initial paper that was posted as a preprint and managed to find the right solution for the conjecture. Schwartz came up with a solution with a "lemma" from his previous paper. He utilized the core concept of a Möbius strip. It detailed the existence of straight lines going through every point of the shape and reaching the boundaries. The first part involved proving this concept, which he successfully did.
He spoke to Scientific American, reflecting on his discovery, "It is not at all obvious that these things exist." He also revealed how he came to be interested in the problem four years back when Sergei Tabachnikov, a mathematician at Pennsylvania State University, told him about it. Once he confirmed the existence of these lines, he had to slice open a Möbius band at an angle along a line segment that went across the width of the band and examine the shape that came about.
Schwartz identified it as a parallelogram, but it actually turned out to be a trapezoid. He spoke about the discovery, "Oh, my God, it's not the parallelogram. It's a trapezoid." The discovery excited him so much that he couldn't sleep for the next three days and all he did was wrote about it. His excitement was understandable, considering just how long the problem had plagued mathematicians.
Tabachnikov spoke about his fellow mathematician's achievement saying, "It takes courage to try to solve a problem that remained open for a long time. It is a characteristic of Richard Schwartz's approach to mathematics: He likes attacking problems that are relatively easy to state and that are known to be hard. And typically, he sees new aspects of these problems that the previous researchers didn't notice."