  We study the Hard Core Model on the graphs ${\rm {\bf \scriptstyle G}}$
obtained from Archimedean tilings i.e. configurations in $\scriptstyle
\{0,1\}^{{\rm {\bf G}}}$ with the nearest neighbor 1's forbidden. Our
particular aim in choosing these graphs is to obtain insight to the geometry of
the densest packings in a uniform discrete set-up. We establish density bounds,
optimal configurations reaching them in all cases, and introduce a
probabilistic cellular automaton that generates the legal configurations. Its
rule involves a parameter which can be naturally characterized as packing
pressure. It can have a critical value but from packing point of view just as
interesting are the noncritical cases. These phenomena are related to the
exponential size of the set of densest packings and more specifically whether
these packings are maximally symmetric, simple laminated or essentially random
packings.
