One of the most famous results in this area is Kepler's conjecture, which states that no arrangement of identical spheres can have a higher density than the face-centered cubic (FCC) lattice. This conjecture was first proposed in 1611 by Johannes Kepler, but it was not proven until 1998 by Thomas Hales.
The FCC lattice is a three-dimensional arrangement of spheres in which each sphere is surrounded by 12 other spheres. This arrangement has a density of about 74%, which means that about 26% of the space in the lattice is empty.
Kepler's conjecture is also true for other polyhedrons, such as cubes and octahedrons. However, the optimal packing arrangements for these polyhedrons are more complicated than the FCC lattice.
For example, the optimal packing arrangement for cubes is the body-centered cubic (BCC) lattice, in which each cube is surrounded by 8 other cubes. The BCC lattice has a density of about 68%, which means that about 32% of the space in the lattice is empty.
The optimal packing arrangement for octahedrons is the simple cubic (SC) lattice, in which each octahedron is surrounded by 6 other octahedrons. The SC lattice has a density of about 52%, which means that about 48% of the space in the lattice is empty.
Scientists and mathematicians are still studying the problem of packing polyhedrons into a box. There are many open questions in this area, such as the optimal packing arrangements for other polyhedrons and the densest arrangements possible for mixtures of different polyhedrons.
