... of perfect packing. The grain at depth n thenhas the freedom to point the ‘wrong’ way; we call such sites excitations, using the thermal analogy. The probability of observing an excitation at ... as the aspect ratio of a rectangular grain there, can take any value, rational or irrational, in this chapter.It can, in this chapter, equally represent the integer-value mismatch of rectangulargrains, ... shaken along the direction of gravity, the predominant dynamical response of the sandbox is known to be in the verticaldirection – recall the on- and off-lattice computer simulation results...