  On the space of $\pm 1$ spin configurations on the 3d-square lattice, we
consider the \emph{shaken dynamics}, a parallel Markovian dynamics that can be
interpreted in terms of Probabilistic Cellular Automata, whose transition
probabilities are defined in terms of a pair ferromagnetic Ising-type
Hamiltonian with nearest neighbor interaction $J$, depending on an additional
parameter $q$, measuring the tendency of the system to remain locally in the
same state. We compute the stationary measure of the shaken dynamics and we
investigate its relation with the Gibbs measure for the Ising model. It turns
out that the two parameters $J$ and $q$ tune the geometry of the underlying
lattice. By a judicious use of perturbative methods we show rigorously that our
model exhibits a line of critical points in $J-q$ plane that separates the
ordered phase from the disordered one, and we perform numerical simulation to
determine the phase transition curve. Our method allows us to find in a unified
way the critical values of $J$ for Ising model with first neighbors
interaction, defined on a whole class of lattice, intermediate between the
two-dimensional hexagonal and the three-dimensional cubic one, such as, for
example, the tetrahedral lattice. Finally we estimate the critical exponents of
the magnetic susceptibility and show that our model captures a phase transition
in the geometry of the system at $q = 0$.
