sciencehabit writes with an article in Science about a new way to pack spheres into a cylinder. From the article: "One day, physicist Ho-Kei Chan of Trinity College Dublin was playing with steel ball bearings, trying to pack them into a little cylindrical tube in the most efficient way possible. It's a tricky problem that can take even a powerful computer a week to calculate. But after thinking about it for a while, Chan has figured out a way to simplify the math. The advance could help engineers pack all sorts of spheres more efficiently, from nano-sized buckyballs to Christmas tree ornaments. Another potential application is liquid crystal displays such as those used in televisions and computer monitors. If scientists could make liquid crystal molecules obey these rules, they could potentially create a whole new class of liquid crystals."
One caveat is that the diameter of the cylinder can be at most 2.7013 times as large as the diameter of the spheres being packed.