Möbius strips are fun geometric shapes that only have one side . Take a strip of paper – it ’s got a front and a back . Now twist it and glue the two myopic edges together . Suddenly there is no front or back . You could draw a line across its whole airfoil without having to purloin the pencil from the paper . Forty - six years ago mathematicians paint a picture the minimum size for such a strip show but they could n’t prove it . Now , someone ultimately has .
Since the creation of the funnies by August Ferdinand Möbius and Johann Benedict Listing , its simplicity in form and visualize it had to be balance with themathematicalcomplexity of such a shape . It is not surprising that in 1977 , Charles Sidney Weaver and Benjamin Rigler Halpern created the Halpern - Weaver Conjecture , which stated the minimal ratio between the breadth of the funnies and its length . They suggested that for a strip with a breadth of 1 centimetre ( 0.39 inches ) , the duration had to be at least the square etymon of 3 centimeters ( about 1.73 centimeters or 0.68 inches ) .
For unruffled Möbius strip show that are “ embedded ” , mean they do n’t intersect with each other , the conjecture had no solution . If the strip can go through itself , it is a much easier problem to solve , Brown University ’s mathematician Richard Evan Schwartz proposed in 2020 – but he had made a misunderstanding . In a paper posted as a preprint – meaning it is yet to be subjected topeer review – Schwartz corrected the error and found the right solution for the speculation .
The solution comes from a flowering glume in hisprevious newspaper . A crucial concept is that on the surface of the Möbius strips exist straight lines move through every point and ending at the limit . To prove the first part of the lemma , he need to prove that there were perpendicular lines to those straight lines existing in the same plane . And he did .
“ It is not at all obvious that these thing survive , ” Schwartz said toScientific American .
The next step was to slice up the Möbius strips and understand what form of shapes they were mold . The idea was to simplify the problem by flattening the strip onto a plane . In the original paper , Schwartz think that a chopped strip would look like a parallelogram , but it turned out to be a unlike quadrilateral – atrapezoid .
“ The corrected figuring gave me the bit that was the conjecture , ” he said . “ I was gobsmacked … I pass , like , the next three days hardly sleep , just writing this matter up . ”
The preprint has been posted toArXiv .
[ H / T : Scientific American ]
