Why spheres are rarely used for packaging

Random close packing of spheres in three dimensions gives packing densities in the range 0. Compressing a random packing gives polyhedra with an average of For sphere packing inside a cube , see Goldberg , Schaer , Gensane , and Friedman. The results of Gensane improve those of Goldberg for , 12, and all from to except for and are almost certainly optimal.

Barlow, W. Conway, J. Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, Coxeter, H. New York: Wiley, pp. New York Acad. Critchlow, K. New York: Viking Press, Cundy, H.

Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub. Eppstein, D. Lagerungen in der Ebene, auf der Kugel und in Raum, 2nd ed. Berlin: Springer-Verlag, Friedman, E. Gardner, M. New York: Simon and Schuster, pp. New York: W. Norton, pp. Gauss, C. Gensane, T. For error-correcting codes, the centers of these spheres are our codewords. The radius will be chosen based on how much we think our transmission method might jiggle our message around when we send it.

If it could be 3 units, we pack spheres of radius 3. So we're looking for spheres in dimensional space whose radii are all some set distance away from each other. We just found a sphere-packing problem in its natural habitat.

Solving the sphere-packing problem in dimensions amounts to figuring out how many codewords we can pack into some defined region in dimensional space. The more efficiently we can pack codewords, the more distinct messages we can send using our dimensional message scheme. One might justifiably ask how important solving the sphere-packing problem is for finding error-correcting codes.

Is pretty good good enough? High-dimensional spheres get spiky. Hence it is all the more important to find the best sphere packings so we can get as much as possible out of or into?

Viazovska's breakthrough on sphere packing in eight dimensions—and the subsequent solution to the dimensional problem—did not have a practical effect on creating error-correcting codes using those dimensions. Researchers already knew the correct answer to within minute fractions of a percent, so anyone packing spheres in those dimensions was already using the right model.

But her work is a proof of concept that a new approach to sphere packing has the potential to make a difference in other dimensions and have a big impact on how we communicate with each other.

Alas, it doesn't. Knowing the densest packing in one dimension doesn't give you a clue of what it should be in the next. The graph below shows the density of the densest packings we know for dimensions 4 to 26, but these might not be the densest overall. The graph suggests, and this turns out to be true, that the sphere packing density decreases exponentially as the dimension increases.

When you're trying to find a number that attains some sort of a maximum, like being the highest packing density, but haven't got much luck, one approach is to lower your bar and look only for an upper bound : in our case a number you can prove the packing constant can't exceed. Various upper bounds for packing densities have been known for some time, but in Henry Cohn and Noam Elkies came up with a particularly interesting recipe for finding them in any dimension.

The recipe is hard to put into practice, so Cohn an Elkies were only able to approximate the associated upper bounds for dimensions up to The result is shown in the graphs below, together with the best-known values for the packing density, for dimensions 4 to 12 and 20 to The Cohn-Elkies upper bound blue and the density of the best-known packing green for dimensions 4 to 12 and 20 to What's striking here is that in dimensions 8 and 24 the two graphs appear to coincide.

If they really do coincide, then the densest packings we know in these dimensions are also the densest overall, and their density will be the sphere packing constant. But Cohn and Elkies weren't able to prove this: there remained the annoying possibility that denser packings exist, whose densities fall right into invisibly tiny gaps between the two graphs. Viazovska's work, which closed the gap for dimension 8 and was later extended with the help of Cohn, Abhinav Kumar, Stephen D.

Miller, and Danylo Radchenko to dimension 24, builds on the centre piece of Cohn and Elkies' work. If you forget about the actual spheres in a sphere packing and only consider their centres, you're left with a configuration of points. Rather than giving the coordinates of all the points, you can also characterise the configuration by the statistics of the distances between points: what is the smallest distance that occurs, how often does it occur, etc. The spheres in this eight-dimensional packing are centred on points whose coordinates are either all integers or all lie half way between two integers, and whose coordinates sum to an even number.

The radius of the spheres is. The E 8 lattice is related to the exceptional Lie group E 8. As the name suggests the group is an exceptional object in mathematics, so it's perhaps not surprising that it is connected to an exceptional sphere packing.

It's a fruitful approach that is often used in physics. It turns out that these statistics have to satisfy certain restrictions. If you want to have so many stars at this distance, and so many stars at that distance, and so many stars at another distance, it might not be possible to realise this in space.

Modular forms are functions that possess special symmetries like those in M. And although modular forms have been studied for centuries, mathematicians are still unlocking the deep secrets hidden inside their coefficients.

Sarnak calls them a gold mine. Eventually, Bondarenko and Radchenko moved on to other problems, but Viazovska pressed on alone. After two years of intense effort, she succeeded in coming up with the right auxiliary function for E 8 and proving that it is correct. After Viazovska posted her paper on March 14, she was startled by the surge of excitement it created among sphere-packing researchers.

That night, Cohn emailed to congratulate her, and as the two exchanged emails he asked if it might be possible to extend her method to the Leech lattice. The team posted its page paper online just a week after Viazovska had posted her first paper.

But the two proofs offer mathematicians both a sense of closure and a powerful new tool. This article was reprinted on Wired.