  The spin system of the $2d$ Ising model having a hexagonal-lattice is
simulated using non-deterministic Cellular Automata. The method to implement
this program is outlined and our results show a good approximation to the exact
analytic solution. The phase transition in $2d$ Ising model is studied with a
$40\times40$ hexagonal-lattice with five different boundary conditions (bcs)
i.e., adiabatic, periodic, reflexive, fixed $+1$ and fixed $-1$ with random
orientation of spins as initial conditions in the absence of an external
applied magnetic field. The critical temperature below which the spontaneous
magnetization appears as well as other physical quantities such as the
magnetization, energy, specific heat, susceptibility and entropy with each of
the bcs are calculated. The phase transition occurs around $T^H_c$ = 1.5 which
approximates well with the result obtained from exact analytic solution by
Wannier and Houtappel. We compare the behavior of magnetisation per cell for
five different types of bcs by calculating the number of points close to the
line of zero magnetization for $T>T^H_c$. We find that the periodic, adiabatic
and reflexive bcs give closer approximation to the value of $T^H_c$ than fixed
$+1$ and fixed $-1$ bcs with all three initial conditions for lattice size less
than $50\times50$. However, for lattice size between $50\times50$ and
$200\times200$, fixed $+1$ bc and fixed $-1$ bc give closer approximation to
the $T^H_c$ with initial conditions in which all spins are in down
configuration and all spins are in up configuration respectively.
