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A Mathematician Just Cracked The Centuries-Old 'Sphere Problem' In Higher Dimensions

Because of the interwoven quilt that is mathematics, almost every profession is related to stacking spheres. For instance, it’s your job to distinct the signal from the noise in different pictures provided by spacecraft some billion miles away. Or maybe you are hired at the grocer and you have to pile fruit as efficiently as possible so that no free space is missed. Or maybe you’re a string theorist looking for hidden networks among fundamental constituents of the Universe. These all are, in a way related to stacking spheres, some of those spheres are just in other dimensions.

A tennis ball or an orange is a 3D object and if you’ve ever observed a pile of tennis balls, you would know that there’s always certain empty space among the balls. And if they’re just poured out of a bag, nearly 36 percent of the resultant pile will be air. This can be reduced to about 26 percent if you place the spheres very sensibly - called the 26 percent method - but mathematicians have recognized since the beginning of the 20th century that there could never be any other way to do better than that.

Mathematicians also work with the stuffing game in higher dimensions, where spheres have quite the same meaning, but the distances gain extra coordinates on top of the familiar three (x, y, and z, for instance) that we’re familiar with. And with adding more dimensions come more imaginable arrangements, so looking for the arrangements that minimalize the empty space has been puzzling.

But recently a mathematician named Maryna Viazovska claims to have finally solved this case, and for two particularly interesting cases.
She initiated with a higher dimensional sphere, an eight-dimensional spheres called E8. E8 is a somewhat like a higher-dimensional type of the 26 percent method, except that in eight dimensions, there’s enough space among the spheres that a new sphere can fit cozily between them.

Even though string theorists don’t actually use the spheres, they do use the configuration of E8 as a key element of the way that the distinct dimensions of string theory link to one another. String theory initiates with 26 dimensions, and has to bend them down into the three that we actually recognize and interfere with, and E8 has all of the properties required to accomplish that.

Many Mathematicians and physicists have debated that this couldn’t be just a coincidence - they believed that the dimensions would actually act in the most organized way imaginable, since any extra space would be tougher to explain.

And now it seems like they were right. Maryna Viazovska succeeded in showing that E8 leaves no empty space anywhere; it’s the most effective way of assembling eight-dimensional spheres together.

Maryna Viazovska didn’t stop there. After her research paper was published online at pre-print website, some of other mathematicians who desired crack the problem in 24 dimensions wrote to her. Viazovska told Quanta Magazine:

"I felt like, 'I am already tired and I deserve some rest. But I tried still to be useful."

But this time was different as the mathematicians were was confronting an arrangement called the Leech lattice, first presented by John Leech in the 1960s. Leech was actually working out to find ways to correct the errors or noise that would collect as signals traveled.

He discovered a way of ordering the data in 24 dimensions that made it very healthy for jobs like transmitting pictures of Jupiter from some half a billion miles away – which Voyagers 1 and 2 did a decade later with the help of the Leech lattice.

The Leech lattice assembles information like someone may organize spheres in those 24 dimensions, and researchers had again thought that looked like the most effective way to do it. Viazovska and her coworkers again verified the intuition accurate, with a research paper that published just a week after the E8 paper did.

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