It is well known that the best way to pack oranges in a (very large) box is to place them on a face-centered cubic lattice (also known as checkerboard), although this has been formally demonstrated only in 1998 (with difficulty). This talk focuses on what happens when the dimension or number of different spheres change. In particular, so-called compact packings (the term will be defined properly) seem good candidates to maximize density. In this tallk, we propose a survey of the known mathematical results and a discussion of possible applications in chemistry, including self-assembly of supercrystals.