A conjecture concerning the symmetries of planar nets and the hard disk freezing transition

We examine the conjecture that in a 2D system of hard disks the packing fraction at which the continuous transition from the ordered 2D solid to the hexatic phase occurs, and that at which the very weak first-order or continuous transition from the hexatic to the fluid phase occurs, can be correlated with the packing fractions of patterned networks (tessellations) of disk positions that span the 2D space. We identify three tessellations that have less than close packed density, span 2D space, and have percolated continuity of disk-disk contact. One has a packing fraction of eta = 0.729, very slightly larger than the estimated packing fraction at the ordered solid-to-hexatic transition, eta = 0.723, and the other two have packing fractions of approximately 0.680, slightly smaller than that identified as the upper end of the stability range of the liquid phase, eta = 0.699. The region 0.680 < eta < 0.729 is identified with the hexatic domain. The end points of this region can be placed in correspondence with nets for which the defining unit structures are regular polygons, but not the hexatic domain, in which there are randomly dispersed clusters that need not be regular polygons. The densities at which the percolated tessellations span the 2D space are regarded as special points along the density axis. We suggest that the possibility of forming different symmetry nets with sensibly the same packing fraction is a geometric analogue of a bifurcation condition that divides the configuration space into qualitatively different domains, and that the onset and end of the hexatic region are correlated with such divisions of the configuration space.