  One of the most important problems in contemporary science, and especially in
biology, is to reveal mechanisms of pattern formation. On the level of
biological tissues, patterns form due to interactions between cells. These
interactions can be long-range if mediated by diffusive molecules or
short-range when associated with cell-to-cell contact sites. Mathematical
studies of long-range interactions in-volve models based on differential
equations while short-range interactions are modelled using discrete type
models. In this paper, we use cellular automata (CA) technique to study
formation of patterns due to short-range interactions. Namely, we use von
Neumann cellular automata represented by a finite set of lattices whose states
evolve according to transition rules. Lattices can be considered as
representing biological cells (which, in the simplest case, can only be in one
of the two different states) while the transition rules define changes in their
states due to the cell-to-cell contact interactions. In this model, we identify
rules result-ing in the formation of stationary periodic patterns. In our
analysis, we distin-guish rules which do not destroy preset patterns and those
which cause pattern formation from random initial conditions. Also, we check
whether the forming patterns are resistant to noise and analyze the time frame
for their formation. Transition rules which allow formation of stationary
periodic patterns are then discussed in terms of pattern formation in biology.
