Packing deformable objects

If one pushes together identical deformable objects in 2D they tend to arrange themselves in a regular hexagonal array—and this configuration is known to minimize total boundary length. In 3D the arrangement one gets is typically not very regular—although as noted at various times since the 1600s individual objects often have pentagonal faces suggestive of dodecahedra. (The average number of faces for each object depends on the details of the random process used to pack them, but is typically around 14. Note that for a 3D Voronoi diagram with randomly placed points, the average number of faces for each region is 2 + 48 π^{2}/35 ≃ 15.5.) It was suggested by William Thomson (Kelvin) in 1887 that an array of 14-faced tetradecahedra on a bcc lattice might yield minimum total face area. But in 1993 Denis Weaire and Robert Phelan discovered a layered repetitive arrangement of 12- and 14-faced polyhedra (average 13.5) that yields 0.003 times less total area. It seems likely that there are polyhedra which fill space in a less regular way and yield still smaller total area. (Note that if the surfaces minimize area like soap films they are slightly curved in all these cases. See also pages 1007 and 1039.)