In the world of mathematics , two things always hold rightful : firstly , some of the most stubborn and complex problem often have astonishingly existent - world applications ; and second , for multitude who spend all their metre actuallyinthat real existence , those problem can seem … well , passably silly .
Take , for object lesson , the “ lounge job ” : a conundrum that has both stumped mathematicians for decade , and also been “ puzzle out ” by just about anybody who ’s ever go business firm in their life . It ’s a interrogation of how to move a curved sofa around a 90 - level recess – yes , just like in that one episode ofFriendsyou’re all now quote .
The math of moving
Technically , the sofa problemis this : What is the region of largest area which can be go around a right - angled quoin in a corridor of breadth one ? It was first officially statedin 1966by the Austrian - Canadian mathematician Leo Moser – though it had been a issue of conversation around the numerical urine coolers for many years before that – and until now , never once and for all solved .
Now , you ’ll notice that there ’s no mention of the eponymous settee in this formulation , and indeed the first musical composition of “ furniture ” evoke as a solution was actually a “ forte-piano ” . Nevertheless , the “ couch ” nomenclature soon took off , mostly because – well , seem at it :
It was the determination of a lower bound that gave rise to the iconic sofa shape : in a 1968 treatise named , we kid you not , On the enfeeblement of numerical acquisition by ' Modern Mathematics ' and by similar soft noetic trash in schools and university , John Hammersley show with some relatively elementary calculus that this shape apply an area of ( π/2 ) + ( 2 / π ) – some 2.2074 .
The Gerver sofa, showing each of the 18 segments making up its boundary.Image Credit: TilmannR viaWikimedia Commons, Public Domain.
In fact , he choke further . In the same paper , he proved that an upper bind on the surface area was yield by 2√2 – roughly 2.8284 . It had only been a twain of years , but the sofa trouble was already on its way to a solution : the exact figure had n’t been nailed down yet , but mathematicians knew it had to be between these two values . sure as shooting it would n’t take much more body of work to observe the true answer ?
Fast forward 25 years , though , and Hammersley ’s bounds were still the good we had . That was , until Rutgers mathematician Joseph Gerver stepped up to the plate , offer a sofa constructed from 18analytically smoothconnected breaking ball sections . The “ Gerver lounge ” , as it became known , increased the lower bound to 2.2195 .
It would be another quarter - century again before the chain of mountains of possible solutions would be pared down even further : in 2018 , mathematicians Yoav Kallus and Dan Romik used a computer - assisted proof to plane the upper bond down to 2.37 .
It was a large improvement on Hammersley ’s original bounce – but that exact solution was still evading seizure .
Baek in the game
It would have been around the same fourth dimension as Kallus and Romik were working on their solution that Jineon Baek , a postdoctoral researcher at Yonsei University in Seoul , Korea , first begin thinking about the sofa problem . Now , seven long time subsequently , he compute he ’s check it in a proof that has yet to be equal - refresh .
“ I consecrate a lot of time to this , without any publication so far , ” he toldNew Scientist . “ The fact that now I can say to the world that I committed something worthful to this problem is validating . ”
For a question so easily stated and opine , Baek ’s proof was no lowly project . cross more than 100 varlet , it does far more than only brute forcefulness the problem or continuously shave off ever - smaller gash of area . Rather , it is , Romik told New Scientist , a “ wonderful growth ” .
“ I get it on I could never have done this , ” Romik say . “ I do n’t have a impression of regret , or like , how could I pretermit this , because it ’s clear it ’s just not the sort of cerebration that I call up I would have been able to . [ Baek ] was just come at it from a completely dissimilar centering . ”
Without cause into the nitty - gritty , the proof goes like this : first , Baek said that the optimal lounge , whatever it turn out to be , had to have three specific attribute – it had to be flat , balanced , and have a rotation slant π/2 . Again , these are quite expert to delineate , but essentially it churn down to this : the “ sofa ” we ’ve been using so far is pretty much the ripe shape already .
second , Baek set about proving a term on how this sofa would move around the corner – a small thing , but essential for completing the concluding stride : defining the upper bounce for the area of this couch , and then showing that it was equal to Gerver ’s lower bound .
That ’s right : after 32 years , it turns out Gerver was right all along .
“ I am of path very glad about all of this , ” Gerver severalize New Scientist . “ I am 75 years old , and Baek ca n’t be more than 30 . He has a peck more energy , stamen and pull round brain jail cell than I do , and I am glad that he pick up the baton . I am also very happy that I lived long enough to see him finish what I begin . ”
Put your feet up
So , is the sofa problem now consummate ? Well , technically , it remain to be seen . As with all mathematical proof , it necessitate to be equal - reexamine for truth – a process that Baek is quietly bright for .
“ I ca n’t say that I ’m confident 100 per cent , because we are humanity , we make errors , ” he told New Scientist . “ But still , I did my effective to be as surefooted as I can . ”
But if your hope of solving the sofa job yourself have been dart by this word , take heart : since Baek define his sofa so strictly , you could always choose a unlike shape for your own .
It might not make quite as honest a lounge for your aliveness room of course , but there ’s really nothing stopping you from go … Baek to the draft board , you might say .
The proof can be foundon the ArXiv preprint host .
