Packing problems have a long history in physics and mathematics. Often hundreds of years or more elapse between the statement of a particular packing problem and the proof that the best packing has been found. For example, the best way to pack spheres has been known, undoubtedly, for millennia. It is as a face-centered cubic lattice – much the way grocers stack oranges. The proof, however, was only in 1999. Packing problems are important in understand how molecules and particles of different shapes arrange themselves in space, and the various shapes of molecules and particles have inspired a search for the densest packings of many different shapes.
What is the densest way to pack squares into a larger square? It is a problem that is deceptively simple to state, yet subtle and difficult to solve. We have simplified this problem further by asking for the densest packing of squares into a larger square with periodic boundary conditions, in which the sides of a square are identified. Even in this packing problem, exact results are sparse. On the one hand, the addition of translation symmetry simplifies the arrangements of squares, yet numerical analysis indicates that the densest packings can be still quite unpredictable. See, for example, the densest packing of 23 squares that we found.
D. Blair, C.D. Santangelo and J. Machta, “Packing squares in a torus,” accepted to J. Stat, [ARXIV] (2011).
Want to try your hand at packing regular polygons? Visit http://donblair.org/grains6a/