UC Davis Mathematician Uses 3D Printing to Deliver New Insight into Moving Sofa Problem

Like many people of my generation, I have moved many, many times in the last decade or so. I just moved again last month, and I’m hoping to stay in my current place of residence for a long time, maybe forever, not only because I love it but because moving is awful – especially when you’re dealing with multiple-floor apartment buildings with narrow halls, low ceilings, and no elevators. I did that several times, and never again. I still don’t know how my movers (i.e., family and friends to whom I offered food and/or beer) managed to get my bulky, heavy, horrible former couch, which is now presumably taking up well-deserved landfill space somewhere, around so many corners and up and down so many precariously steep and cramped staircases – I assume some sort of wizardry was involved.

While anyone who has moved into or out of an apartment building has likely spent some time pontificating on the physics involved in moving big couches around small corners, it turns out that mathematicians have been equally stumped. The “moving sofa problem” is a well-known one in the math world, and it started with a question asked by mathematician Leo Moser in 1966: “What is the shape of largest area in the plane that can be moved around a right-angled corner in a two-dimensional hallway of width 1?”

Dan Romik

In real-world terms, how big is the biggest couch that can be pivoted around a hallway corner without needing to be lifted or tilted? It doesn’t seem like it should be that difficult of a question to answer, although the idea of trying to move a couch around a corner without lifting and twisting it makes me break out in a cold sweat. The problem lies not only in finding the largest sofa that can perform such a task, but in proving that it’s the largest.

3D printing can’t help with a problem like that, can it? Come on, of course it can. Dan Romik, chair of the Department of Mathematics at UC Davis, is a bit obsessed with the moving sofa problem. He’s also really into 3D printing.

“I’m excited by how 3-D technology can be used in math,” said Romik. “Having something you can move around with your hands can really help your intuition.”

There goes the Gerver sofa!

In 1992, Joseph Gerver found what is generally accepted to be the biggest sofa to date that can be moved around a corner. The Gerver sofa, as it’s known, is roughly the shape of an old-fashioned telephone handset, or a slightly flattened rainbow, however you want to look at it. Romik was messing around with the data, trying to translate Gerver’s equations into something that could be translated into 3D printable form. After sitting on the problem for a while, he realized he could use his software and 3D printer to come up with an entirely new answer to the question.

“All this time I did not 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 went back to the program in April, I had a lightbulb flash. Maybe the methods I used for the Gerver sofa could be used for something else.”

That “something else” turned out to be a bow-shaped sofa that can go around not one, but two corners facing in opposite directions – a new twist in not only the moving sofa case, but its offshoot, the ambidextrous moving sofa problem (what’s the biggest couch that can go around a right AND left turn in the same hallway? Again, cold sweat). Romik’s software generated the shape, and Romik 3D printed it, along with a hallway shape, to physically demonstrate how it worked and give some extra padding to his theory. He has included several CAD files at the bottom of his webpage if you’d like to print some sofas and hallways of varying shapes and slide them around, too.

Romik’s new sofa design

While the moving sofa problem hasn’t been solved, per se, Romik’s discovery is still an interesting mathematical development.

“Although the moving sofa problem may appear abstract, the solution involves new mathematical techniques that can pave the way to more complex ideas,” Romik said. “There’s still lots to discover in math.”

Romik published his study in a paper entitled “Differential Equations and Exact Solutions in the Moving Sofa Problem,” which you can access here. It was also published in the journal Experimental Mathematics. Discuss in the Moving Sofa forum at 3DPB.com.