New twist on sofa problem that stumped mathematicians and furniture movers
The problem with the movable sofa is asking what is the largest shape that can move around a right angle curve. UC Davis’ mathematician Dan Romik has extended this problem to a two-turn hallway, showing that a bikini top-shaped sofa is the largest that has been found that can move around in such a hallway. Photo credit: Dan Romik, UC Davis
Most of us struggle with the math puzzle known as the “movable sofa problem”. A deceptively simple question arises: what is the largest sofa that can rotate around an L-shaped hallway corner?
A mover will tell you to just finish the sofa. But imagine if the sofa cannot be lifted, squashed or tilted. While it still seems easy to solve, the moving sofa problem has been hampering math problems for more than 50 years. That’s because the challenge for mathematicians is to find the biggest sofa and prove it to be the biggest. Without proof, it is always possible that someone will come up with a better solution.
“It’s a surprisingly difficult problem,” said math professor Dan Romik, chair of the math department at UC Davis. “It’s so easy you can explain it to a kid in five minutes, but no one has found any evidence yet.
The largest area that fits around a corner is known as the “sofa constant” (yes, really). It is measured in units, with one unit being the width of the hallway.
Inspired by his passion for 3D printing, Romik recently tackled a twist on the sofa problem, the ambidextrous sofa. In this scenario, the sofa has to maneuver both the left and right 90 degree rotations. His results are published online and published in the journal Experimental Mathematics.
The Gerver sofa is the largest found that fits in one turn. It has a “sofa constant” of 2.22 units, with one unit representing the width of the hallway. Photo credit: Dan Romik / UC Davis
Romik, who specializes in combinatorics, likes to think about difficult questions about shapes and structures. But it was a hobby that sparked Romik’s interest in the moving sofa problem – he wanted to 3D print a sofa and hallway. “I’m excited about how 3D technology can be used in math,” said Romik, who has a 3D printer at home. “Having something that you can move with your hands can really help your intuition.”
The Gerver sofa, which resembles an old telephone receiver, is the largest sofa ever found for a one-turn hallway. When Romik was tinkering with translating Gerver’s equations into something a 3D printer could understand, he immersed himself in the math that underlies Gerver’s solution. Romik devoted several months to developing new equations and writing computer code that refined and expanded Gerver’s ideas. “The whole time I didn’t think I was doing research. I was just playing around,” he said. “Then in January 2016 I had to put this aside for a few months. When I returned to the program in April, I had a lightbulb flash. Perhaps the methods I used on the Gerver sofa could be used for something else. “”
Romik decided to address the problem of a hallway with two curves. When tasked with assembling a sofa through the hallway corners, Romik’s software spat out a shape that resembled a bikini top, with symmetrical curves connected by a narrow center. “I remember sitting in a cafe when I first saw this new shape,” said Romik. “It was such a nice moment.”
Like the Gerver sofa, Romik’s two-handed sofa is only a good guess. However, Romik’s results show that the question can still lead to new mathematical knowledge. “Although the moving sofa problem seems abstract, the solution involves new mathematical techniques that can pave the way for more complex ideas,” said Romik. “There’s still a lot to discover in math.”
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Dan Romik, Differential Equations and Exact Solutions in the Problem of the Movable Sofa, Experimental Mathematics (2017). DOI: 10.1080 / 10586458.2016.1270858
Quote: A new variant of the sofa problem that baffled mathematicians and furniture manufacturers (2017, March 20) was found on December 30, 2020 from https://phys.org/news/2017-03-sofa-problem-stumped-mathematicians- furniture.html
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