# 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 arXiv.org, 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.

**This post was written by Umer Abrar. To contact the author of this post, write to mirzavadoodulbaig@gmail.com or add/follow him on facebook :**

A Mathematician Just Cracked The Centuries-Old 'Sphere Problem' In Higher Dimensions
Reviewed by Umer Abrar
on
4/11/2016
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