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| A universal cover such as the hexagon can cover up any shape. Photo: DVDP for Quanta Magazine |
Philip Gibbs is not a professional mathematician. So when he wanted a problem to chew on, he looked for one where even an amateur could make a difference. What he found was a challenge that could drive even the most exacting minds mad. In a paper completed earlier this year, Gibbs achieved a major advance on a 100-year-old question that hinges on the ability to accurately measure area down to the atomic scale.
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| Amateur mathematician Philip Gibbs reduced the size of the smallest known universal cover using techniques inspired by a compass and protractor. Photo: Courtesy of Philip Gibbs |
The problem was first proposed by Henri Lebesgue, a French mathematician, in a 1914 letter to his friend Julius Pál. Lebesgue asked: What is the shape with the smallest area that can completely cover a host of other shapes (which all share a certain trait in common)?
In the century since, Lebesgue’s “universal covering” problem has turned out to be a mousetrap: Progress, when it’s come at all, has been astonishingly incremental. Gibbs’ improvement is dramatic by comparison, though you still have to squint to see it.
Picture a dozen paper cutouts of different sizes and shapes lying on your floor. Now imagine being asked to design another shape that is just big enough to cover any of those dozen shapes. Through experimentation — by overlaying the shapes and rotating them — you could feel your way to a solution. But once you’d found a “universal” cover, how would you know if you’d found the smallest one? You could imagine returning to your cover throughout the day and finding places to trim a little more here or a little more there.
That is the spirit of Lebesgue’s universal covering problem. Instead of paper cutouts, it considers shapes where no two points are farther than one unit apart...
Over the next 80 years, two other mathematicians shaved slivers from Pál’s universal cover. In 1936 Roland Sprague removed a section near one of the corners; in 1992 H. C. Hansen removed two vanishingly small wedges from the lower right and left corners. Illustrations of Hansen’s area savings would convey something about the locations but inevitably mislead about the size: They had an area of 0.00000000004 units.
“You can’t really draw them in scale because they’d be atom-sized pieces,” said John Baez, a mathematician at the University of California, Riverside.
Baez lifted Lebesgue’s universal covering problem out of obscurity when he wrote about it in 2013 on his popular math blog. He confessed he was attracted to the problem the way you might be attracted to watching an insect drown...
Atomic Scissors
Early in his life, Gibbs thought he might become a scientist. He received an undergraduate degree in mathematics from the University of Cambridge and a Ph.D. in theoretical physics from the University of Glasgow. But he soon lost his enthusiasm for academic research and instead became a software engineer. He worked on systems for ship design, air traffic control and finance, before retiring in 2006...
In 2014 Gibbs ran computer simulations on 200 randomly generated shapes with diameter 1. Those simulations suggested he might be able to trim some area around the top corner of the previous smallest cover. He turned that lead into a proof that the new cover worked for all possible diameter-1 shapes. Gibbs sent the proof to Baez, who worked with one of his undergraduate students, Karine Bagdasaryan, to help Gibbs revise the proof into a more formal mathematical style.
The three of them posted the paper online in February 2015. It reduced the area of the smallest universal covering from 0.8441377 to 0.8441153 units. The savings — just 0.0000224 units — was almost one million times larger than the savings that Hansen had found in 1992.
Gibbs was confident he could do better. In a paper posted online in October, he lopped another relatively gargantuan slice from the universal cover, bringing its area down to 0.84409359 units...
In 2014 Gibbs ran computer simulations on 200 randomly generated shapes with diameter 1. Those simulations suggested he might be able to trim some area around the top corner of the previous smallest cover. He turned that lead into a proof that the new cover worked for all possible diameter-1 shapes. Gibbs sent the proof to Baez, who worked with one of his undergraduate students, Karine Bagdasaryan, to help Gibbs revise the proof into a more formal mathematical style.
The three of them posted the paper online in February 2015. It reduced the area of the smallest universal covering from 0.8441377 to 0.8441153 units. The savings — just 0.0000224 units — was almost one million times larger than the savings that Hansen had found in 1992.
Gibbs was confident he could do better. In a paper posted online in October, he lopped another relatively gargantuan slice from the universal cover, bringing its area down to 0.84409359 units.
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Source: Quanta Magazine
