Easy to state and hard to solve problems make mathematics interesting. Packing problems are one such type. In fact there is a very nice Wikipedia article on this topic:
Packing problems involve attempting to pack objects together into containers. The goal is to either pack a single container as densely as possible or pack all objects using as few containers as possible.
I came across this problem a couple of years ago, while watching the following TED Talk by Eduardo Sáenz de Cabezón:
How you can chop 3d space into cells of equal volume with the minimum surface area per cell?
The packing problem for higher dimensions was in news last year, since a problem about “Densest Packing Problem in Dimensions 8 and 24” was solved by a young mathematician (Maryna Viazovska). The mathematics involved in the solution is very advanced but we can start gaining knowledge from this book:
Recently, while reading Matt Parker’s book, I discovered a wonderful website called Packomania by Eckard Specht (Otto-von-Guericke-Universität Magdeburg) containing data about packing problems in 2D and 3D. (also checkout his Math4u.de website, it’s a good reference for elementary triangle geometry and inequalities problems.)
No arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements.
using HOL Light proof assistant.
The proof was a huge collaboration and was called Flyspeck Project. The details about this proof are available in this book:
NOTE: You can construct your own Truncated octahedron and Weaire-Phelan polyhedra by printing the nets available at Matt Parker’s website.