Any fan of ‘Friends’ knows there’s only one way to get a couch around a tight corner: Pivot!
While this might not have worked out so well in that iconic scene, mathematicians have now revealed how Ross Geller could have moved his couch up the stairs.
Dr. Jineon Baek, a mathematician from Yonsei University, Korea, has found the largest possible bench that can be rotated around a 90-degree angle.
Unlike Ross’s sofa, that optimal design looks a lot like an old telephone receiver with a flat back, rounded corners and a semi-circular cutout in the front.
This design was first proposed by Joseph Gerber of Rutgers University in 1992, but until now no one has been able to prove that a larger bench was not possible.
In a 100-page proof, Dr. Baek now confirms that for a corridor one ‘unit’ wide, the largest bank you can get around the corner has an area of exactly 2.2195 units.
That means that if the Friend’s stairwell were 7 feet wide, the largest couch Ross could ever live in would be 4,439 square feet.
In addition to helping people move into small apartments, this solution also uncovers a 60-year-old mathematical puzzle.
While Ross Gellar may not have been able to get his couch up the stairs in this iconic Friend’s scene, mathematicians have now proven how he could have gotten his couch around the corner.
Dr. Jineon Baek, a mathematician from Yonsei University, Korea, has identified the ideal shape for a sofa that you have to move around a corner. This shape (shown) provides the largest seating area, depending on the size of the hallway
The moving bank problem was first proposed in 1966 by Austrian-Canadian mathematician Leo Moser.
This essentially laid out in mathematical terms a puzzle that almost everyone has tried to solve at least once in their lives.
The question is: for a hallway with a 90 degree bend, ignoring height, what is the largest bench you can get around the corner and how big is that bench?
While it may seem intuitive, this actually turns out to be a terribly difficult mathematical puzzle.
At its most basic, you could imagine pushing a square down the hall; that square could have a width and length equal to the size of the hallway.
This means that for a hallway with one arbitrary ‘unit’ across it, your square bench has an area of one unit.
That’s a good start, but anyone who has ever moved a couch will quickly see that you can still move a much larger couch.
For example, if you have a semicircular sofa with a radius as wide as the hallway, you can easily increase its area to 1.57 units.
This problem, known as the moving bank problem, was first proposed in 1966 and has baffled mathematicians ever since. The question is: for a 2D corridor one unit wide, what is the largest bench that you can make a 90 degree turn and how big is that bench? (stock image)
Mathematicians soon noticed that banks shaped like bananas or old telephone receivers could be even bigger.
In 1992, Professor Joseph Gerver proposed a design known as Gerver’s bench, a complex shape consisting of 18 curved parts.
For more than 30 years, no one had managed to find a shape that allowed a larger surface area, but on the other hand, no one could prove that a larger shape was not possible.
Now, after seven years of working on his test, Dr. Baek has finally succeeded in proving that Gerver’s bench is indeed the optimal shape.
Dr. Baek’s breakthrough came when he looked at a small number of sofa shapes and asked what qualities they all had in common.
These properties include a reasonably smooth outer edge, a mathematical property called balance that is similar to symmetry, and the ability to rotate a full 90 degrees around the corner.
Combining all these properties, Dr. Baek invented a new mathematical quantity called Q, which was closely related to surface area.
This turned the open question of how big a bank could be into a problem with one definitive answer.
In 1992, a mathematician named Joseph Gerver proposed “Gerver’s sofa,” a telephone-like shape that he believed was the largest possible sofa that you could still move around a 90-degree angle. Pictured: An illustration of what Gerver’s sofa might look like
More than 30 years later, Dr. Baek has finally proven that Gerver’s sofa (photo) is actually the largest possible sofa shape. Although his evidence needs to be verified, Dr. Baek is confident he will be proven right
By finding the highest possible value of Q, Dr. Baek was able to show which shape would fit that value.
And when he finished calculating the figures, the optimal bench shape turned out to be exactly the same as the shape that Professor Gerver had proposed thirty years ago.
Dr. Baek told me New Scientist: ‘I have spent a lot of time on this, without any publication so far.
“The fact that I can now say to the world that I have contributed something valuable to this problem is an affirmation.”
If Dr. Baek’s method is correct, it will prove mathematically that Professor Gerver was right when he said his bank had the largest possible shape.
Professor Gerver says: ‘I am of course very happy with this. I’m 75 years old and Baek can’t be older than 30.
“He has a lot more energy, stamina and surviving brain cells than I do, and I’m glad he’s taken over. “I’m also very glad I lived long enough to see him finish what I started.”
The results of Dr. Baek will have to be fully checked by other mathematicians before we can know for sure, but he remains confident that his results will prove correct.