  We investigate a stochastic model describing a column of grains in the
jamming limit, in the presence of a low vibrational intensity. The key control
parameter of the model, $\epsilon$, is a representation of granular shape,
related to the reduced void space. Regularity and irregularity in grain shapes,
respectively corresponding to rational and irrational values of $\epsilon$, are
shown to be centrally important in determining the statics and dynamics of the
compaction process.
