Symmetry, shape, and order

Packing problems have been of great interest in many diverse contexts for many centuries. The optimal packing of identical objects has been often invoked to understand the nature of low-temperature phases of matter. In celebrated work, Kepler conjectured that the densest packing of spheres is realized by stacking variants of the face-centered-cubic lattice and has a packing fraction of pi/(3 root 2) similar to 0.7405. Much more recently, an unusually high-density packing of approximate to 0.770732 was achieved for congruent ellipsoids. Such studies are relevant for understanding the structure of crystals, glasses, the storage and jamming of granular materials, ceramics, and the assembly of viral capsid structures. Here, we carry out analytical studies of the stacking of close-packed planar layers of systems made up of truncated cones possessing uniaxial symmetry. We present examples of high-density packing whose order is characterized by a broken symmetry arising from the shape of the constituent objects. We find a biaxial arrangement of solid cones with a packing fraction of pi/4. For truncated cones, there are two distinct regimes, characterized by different packing arrangements, depending on the ratio c of the base radii of the truncated cones with a transition at C* = root 2- 1.