This is the biggest sofa you can fit around a corner, according to a mathematician
If you have already had trouble tightening a sofa in a tight corner while passing through a new apartment, you will probably see that the problem of pure mathematics known as “sofa problem” is incredibly relatable.
The question seeks to find a maximum value for the area of ​​a sofa that can slide around a 90 degree corner in a given corridor. The mathematicians had long suspected that the answer lies in a form called Sofa of Gerver. Now, a postdoctoral researcher in South Korea may have finally provided definitive proof that they were correct.
What is Gerver’s sofa?
While numerous sadly eyes with sad eyes have probably thought about it over the years, the question has been officially asked in 1966 by the mathematician Leo Moser. In 1992, Joseph L. Gerver demonstrated the construction of a form that has since been called the Gerver sofa, claiming that it provided the maximum possible area for a form that can move in a corner. The mathematicians suspected for a long time that he was right, but no one could prove it in a conclusive manner.
The mathematician not forward Jineon Baek, whose 119-page epic paper on the problem was subjected to the pre-print server Arxiv. The document concludes that the Gerver conjecture was correct: the Gerver sofa, which has an area of ​​2,2195 units (assuming that the corridor is a large unit) is indeed the best we can do.
The first thing you will notice on the Gerver’s sofa is that, well, it does not look much like that of your living room. Baek, a postdoctoral student at Yonsei University in South Korea, says Popular science That the term “sofa” here is more a cute nickname for a “theoretical form” which he describes as “as an old phone”.
So why is the Gerver sofa in the shape of this? The simple answer is that the shape maximizes its area, while sliding continuously in a corner. The large cut in the middle allows him to rotate at the corner of the street, and the curves of each opposite corner allow him to slide along the walls.
Baek explains that Gerver has built the shape assuming that the sofa must touch the wall at all times. “THE [points of contact between wall and sofa] Make curves that trace the limit of the sofa, ”he says.
Optimizing these curves – building them in a way that maximizes the area of ​​the form – the Gerver sofa. The form itself is in fact extremely complex, which is part of the reason why the problem has proven so difficult to break over the years.
In addition to Gerver and Baek, other mathematicians have worked on this problem over the years, and the collective work body had already established the minimum and maximum possible zones of the sofa. The lower terminal was established by Gerver itself, while the upper limit – 2.37 – was demonstrated in a 2017 article by Yoav Kallus and Dan Romik.
In other words, mathematicians knew that the maximum area of ​​the sofa was somewhere between 2,2195 and 2.37, but not exactly where. Baek says that his paper answers this question, proving that the lower terminal is actually as large as the sofa can get it.
[ Related: Mathematician solves algebra’s oldest problem.]
The proof of the sofa
Baek explains that his proof has three steps. The first confirmed that the optimal shape to maximize the sofa area was indeed the traditional phone form of the Gerver sofa. The second was exactly what this form should look like. And the third was to establish a higher limit on the area of ​​this form.
The third step was the most delicate, because it is not easy to calculate the area of ​​the Gerver sofa. The nature of the form means that there is no simple formula to determine its area
“THE [shape of the] The original sofa can change arbitrarily … It can also consist of saying, 100 different curves “, explains Baek.” And you cannot even control the number of different curves [needed]. SO [its area] does not have a concrete formula. »»
To get around this problem, Baek has built another form which was essentially a simplified version of the Gerver sofa, showing that this form must lock the whole sofa. In other words, if you took a gerver sofa of a certain length and width, it would always be integrated into a simplified sofa of the same dimensions.
Since the simplified form area was simple to calculate, and it always locked the shape of the sofa, finding a way to optimize the simplified shape would put a higher limit on the gerver sofa.
Baek therefore decided to find how large the simplified form could be and its optimal shape. The answer turned out to be the same shape and the same size as the Gerver sofa. This result means that the lower and upper limits on the optimal sofa were the same, so the Gerver sofa is the largest possible sofa that could adapt in a corner.
“I used convex optimization and geometry to really optimize [the simple shape’s area]And the optimal solution was Gerver’s sofa, completing the proof, ”explains Baek.
Sofa, so good
Although the answer will not necessarily help you maneuver a old simple rectangular sofa in a tight corner, it has potential applications of the real world. Baek explains that the problem unites the field planning fields (studying how to move stitch objects in the most effective way possible) and an optimization of the area (a field of pure mathematics which study how to maximize the area of ​​a given form).
That said, Baek points out that, like many problems studied in mathematical research, the problem of the sofa was born from the curiosity and desire of mathematicians to extend their knowledge. “Like many pure mathematics results, it is … likely that the result will not be used in real life.”
However, Baek says he is looking for a new piece of furniture to celebrate the resolution of this mathematical problem of almost 60 years.
“I always want a physical gerver sofa at the corner of my desktop I can really sit!”
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