It鈥檚 a tight squeeze. Mathematicians have proved that they know the best way to pack spheres in 8 and 24 dimensions 鈥� the first time this problem has been solved in a new dimension in almost 20 years.
鈥淚 think they鈥檙e fantastic results. I鈥檓 excited that this has been done at last,鈥� says at the University of Pittsburgh, Pennsylvania, who wrote the proof for the best way to pack spheres in 3 dimensions. 鈥淚 think it鈥檚 high time for us to escape the 3-dimensional ghetto and move on to 8 and 24 dimensions.鈥�
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The sphere-packing problem asks a deceptively simple question: What arrangement allows you to cram the most spheres into a limited volume? This is easy to describe, but profoundly difficult to prove.
In 1611, Johannes Kepler suggested that the best arrangement for stacking 3-dimensional spheres like cannonballs or oranges is a pyramid. But it took until 1998 for Hales to publish his proof 鈥� and it took another 16 years and computer assistance to formally verify it.
In the meantime, mathematicians have been gunning hard for higher dimensions. 鈥淚t turns out that every dimension is different,鈥� says at Microsoft Research New England in Cambridge, Massachusetts. 鈥淭hey have their own idiosyncrasies, and bizarre things happen in certain dimensions. This makes the problem enormously more subtle than you would expect.鈥�
Now, of Humboldt University of Berlin has proved that a uniquely useful grid called the , and almost immediately teamed up with Cohn and other researchers to prove that a related arrangement called the is best in 24 dimensions.
Magic functions
The fact that these were the next dimensions to fall is not a coincidence. For reasons mathematicians don鈥檛 quite understand, such lattices don鈥檛 show up in other dimensions. But they鈥檙e widely regarded as being the most efficient arrangements in the dimensions they apply to. 鈥淭hese are unbelievably good packings,鈥� Cohn says. 鈥淭he spheres in these dimensions fit perfectly, it works in ways that don鈥檛 happen in other dimensions.鈥�
In 2003, Cohn and his colleagues found a method for approximating how close these packings came to being the best answer. They also suggested that certain 鈥渕agic鈥� functions should exist that would prove outright that E8 and Leech were the best packings 鈥� if only they could find them.
鈥淭hat鈥檚 what Maryna did,鈥� Cohn says. 鈥淪he found the function in 8 dimensions by an incredibly clever method. She鈥檚 the real hero in this story.鈥�
Noisy signals
Unfortunately, it鈥檚 not obvious how to extend this proof to even more dimensions. But this isn鈥檛 just a mathematical game. The sphere-packing problem in 24 dimensions has applications in wireless communication, and has been used to communicate with .
It turns out that you can imagine sending signals over a noisy communication channel as being analogous to the sphere-packing problem. To send as much information as possible, you want to have many channels going at once. But you don鈥檛 want them to overlap, because that could introduce ambiguity and errors.
If you imagine each signal as a sphere, then the sphere-packing problem tells you how many channels you can have without overlap.
鈥淚f you grabbed someone on the street and said, 鈥楬ey, did you know mathematicians are studying spheres in high dimensions?鈥� They鈥檇 look at you like you were crazy,鈥� Cohn says. 鈥淏ut if you told them mathematicians are studying how to make your cell phone work better, it sounds a lot more reasonable.鈥�
Journal reference: ,
