*Hobby 3-D Printing Leads to New Insights into Moving Sofa Problem*

*Hobby 3-D Printing Leads to New Insights into Moving Sofa Problem*

*By Becky Oskin*

Most of us have struggled with the mathematical puzzle known as the “moving sofa problem.” It poses a deceptively simple question: What is the largest sofa that can pivot around an L-shaped hallway corner?

A mover will tell you to just stand the sofa on end. But imagine the sofa is impossible to lift, squish or tilt. Although it still seems easy to solve, the moving sofa problem has stymied math sleuths for more than 50 years. That’s because the challenge for mathematicians is both finding the largest sofa and *proving* it to be the largest. Without a proof, it’s always possible someone will come along with a better solution.

“It’s a surprisingly tough problem,” said math professor Dan Romik, chair of the Department of Mathematics at UC Davis. “It’s so simple you can explain it to a child in five minutes, but no one has found a proof yet.

The largest area that will fit around a corner is called the “sofa constant” (yes, really). It is measured in units where one unit corresponds to the width of the hallway.

Inspired by his passion for 3-D printing, Romik recently tackled a twist on the sofa problem called the ambidextrous moving sofa. In this scenario, the sofa must maneuver around both left and right 90-degree turns. His findings are published online and appear in the journal *Experimental Mathematics*.

**Eureka Moment**

Romik, who specializes in combinatorics, enjoys pondering tough questions about shapes and structures. But it was a hobby that sparked Romik’s interest in the moving sofa problem—he wanted to 3-D print a sofa and hallway. “I’m excited by how 3-D technology can be used in math,” said Romik, who has a 3-D printer at home. “Having something you can move around with your hands can really help your intuition.”

The Gerver sofa—which resembles an old telephone handset—is the biggest sofa found to date for a one-turn hallway. As Romik tinkered with translating Gerver’s equations into something a 3-D printer can understand, he became engrossed in the mathematics underlying Gerver’s solution. Romik ended up devoting several months to developing new equations and writing computer code that refined and extended Gerver’s ideas. “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.”

Romik decided to tackle the problem of a hallway with two turns. When tasked with fitting a sofa through the hallway corners, Romik’s software spit out a shape resembling a dumbbell, with symmetrical curves joined by a narrow center. “I remember sitting in a café when I saw this new shape for the first time,” Romik said. “It was such a beautiful moment.”

**Finding Symmetry**

Like the Gerver sofa, Romik’s ambidextrous sofa is still only a best guess. But Romik’s findings show the question can still lead to new mathematical insights. “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.”

## More information

Dan Romik’s page on the moving sofa problem (with animations and 3-D printer files)

Video: Dan Romik explains the Moving Sofa Problem for Numberphile

*Becky Oskin writes for the Division of Mathematical and Physical Sciences, College of Letters and Science. Follow her on Twitter @beckyoskin.*

Couldn’t you just rotate the Gerber sofa around it’s axis 180 degrees so it could handle a two corner (or x corner) scenario? The only constraint would be the length of the straight section needing to be some value to allow a full turn to be completed before starting the next.

The Shephard Piano and the Conway Car have been known for decades. How much of an improvement is this on Shephard?

It’s hard to believe the best math guys or some engineers can’t figure this out. Couldn’t a computer just run different shapes and simulate it until it finds the optimal one?

@Joe It’s not about finding the best shape. It’s about finding it and proving that it is the optimal one.