We generate a large number of monodisperse disk packings in two dimensions via geometric-based packing algorithms including the relaxation algorithm and the Torquato–Jiao algorithm. Using the geometric-structure approach, a clear boundary of the geometrical feasible region in the order map is found which quite differs from that of the jammed region. For a certain packing density higher than 0.83, the crystalline degree varies in different packing samples. We find that the local hexatic order may increase in two fairly different ways as the system densifies. Therefore, two typical non-equilibrium jammed patterns, termed polycrystal and distorted crystal, are defined at high packing densities. Furthermore, their responses to isotropic compression are investigated using a compression–relaxation molecular dynamic protocol. The distorted crystal pattern is more stable than the polycrystal one with smaller displacements despite its low occurrence frequency. The results are helpful in understanding the structure and phase transition of disk packings.

We apply the ideal octahedron model and the relaxation algorithm in generating octahedron packings. The cubatic order parameter [P4]1">[P4]1, bond-orientational order metric Q6">Q6, and local cubatic order parameter P4local">P4local of the packings are calculated and their correlations with the packing density are investigated in the order maps. The border curve of packing density separates the geometrically feasible and infeasible regions in the order maps. Observing the transition phenomenon on the border curve, we propose the concept of the maximally dense random packing (MDRP) as the densest packing in the random state in which the particle positions and orientations are randomly distributed and there is no nontrivial spatial correlations among particles. The MDRP characterizes the onset of nontrivial spatial correlations among particles. A special packing with a density about 0.7 is found in the order maps and considered to be the MDRP of octahedra. The P4local">P4local is proposed as a new order parameter for octahedron packings, which measures the average order degree in the neighborhoods of particles. The [P4]1">[P4]1, Q6">Q6 and P4local">P4local evaluate the order degree of orientation, bond orientation and local structures, respectively and are applied simultaneously to measure the order degree of the octahedron packings. Their thresholds in the random state are determined by Monte Carlo simulations.