Mar 22, 2017 | By Tess

The “moving sofa problem” is something many of us may not have formally heard of, but most of us will have encountered at some point in our lives. Turns out, the issue of how to move a large sofa around a tight corner has not only been on movers’ minds, but also on those of mathematicians. And with the help of 3D printing, one mathematician is well on his way to figuring out an alternate solution to the tricky mathematical problem.

Romik's "ambidextrous moving sofa"

First off, you may be wondering what the “moving sofa problem” actually is. The question behind it is remarkably simple (or not so simple) and consists of asking “what is the largest sofa that can pivot around an L-shaped hallway corner?” For more than half a decade mathematicians have pondered over the question without having the means to a definitive answer, largely because it has been impossible to prove that a given sofa’s dimensions are the “largest.”

Dan Romik, a professor of math and the chair of the Department of Mathematics at UC Davis, explains: “It’s a surprisingly tough problem. It’s so simple you can explain it to a child in five minutes, but no one has found a proof yet.” With the help of 3D printing, however, Romik has not only attempted to tackle the “moving sofa problem” but has added a new element to it: a second corner.

Calling it the “ambidextrous moving sofa”, Romik was studying what kind of sofa could successfully be moved around both left and right 90-degree turns. The professor, who is an expert in combinatorics, decided to 3D print a model of a sofa and hallway to try out his experiments in a physical manner. “I’m excited by how 3D technology can be used in math. Having something you can move around with your hands can really help your intuition,” he explained.

Gerver sofa

To date, the largest sofa to fit around one 90-degree turn is a telephone shaped structure called the “Gerver sofa.” While playing around with the math behind the Gerver sofa, trying to make it 3D printer-friendly, Romik got so involved he began refining and extending Gerver’s initial ideas. Though he did not think much of his mathematical musings, considering the sofa problem to be a bit of a hobby, Romik eventually realized he could extend his methods to go to the next level: the two-turn hallway.

Given the challenge, Romik’s software generated a model of a dumbbell-shaped couch, featuring large symmetrical curves and a narrow middle. The ambidextrous sofa, though a breakthrough for the “moving sofa problem,” is still not a definitive proof, as someone could one day come along with a slightly larger sofa model.

Five versions of the "ambidextrous moving sofa"

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

Romik’s study, called “Differential equations and exact solutions in the moving sofa problem,” was recently published in the journal Experimental Mathematics.



Posted in 3D Printing Application



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polistra wrote at 4/25/2017 12:45:27 AM:

It's not a problem in the real world. Turn the sofa on end. It will pass through most doors that way. If the sofa is longer than 80 inches, slant it some. If it's still too wide, you can generally "L" the seat and back around one side of the doorjamb.

kb wrote at 3/25/2017 9:14:48 PM:

I would say that if all of the object's perimeter touches the walls at some point, doing a rotation of no more than the corner's angle, you have the largest shape possible that can go through.

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