  We have recently devised organizing principles to obtain maximally dense
packings of the Platonic and Archimedean solids, and certain smoothly-shaped
convex nonspherical particles [Torquato and Jiao, Phys. Rev. E 81, 041310
(2010)]. Here we generalize them in order to guide one to ascertain the densest
packings of other convex nonspherical particles as well as concave shapes. Our
generalized organizing principles are explicitly stated as four distinct
propositions. We apply and test all of these organizing principles to the most
comprehensive set of both convex and concave particle shapes to date, including
Catalan solids, prisms, antiprisms, cylinders, dimers of spheres and various
concave polyhedra. We demonstrate that all of the densest known packings
associated with this wide spectrum of nonspherical particles are consistent
with our propositions. Among other applications, our general organizing
principles enable us to construct analytically the densest known packings of
certain convex nonspherical particles, including spherocylinders, "lens-shaped"
particles, square pyramids and rhombic pyramids. Moreover, we show how to apply
these principles to infer the high-density equilibrium crystalline phases of
hard convex and concave particles. We also discuss the unique packing
attributes of maximally random jammed packings of nonspherical particles.
