  We show how the trajectories of $d$-dimensional cellular automata (CA) can be
used to determine the ground states of $(d+1)$-dimensional classical spin
models, and we characterise their quantum phase transition, when in the
presence of a transverse magnetic field. For each of the 256 one-dimensional
elementary CA we explicitly construct the simplest local two-dimensional
classical spin model associated to the given CA, and we also describe this
method for $d>1$ through selected examples. We illustrate our general
observations with detailed studies of: (i) the $d=1$ CA Rule 150 and its $d=2$
four-body plaquette spin model, (ii) the $d=2$ CA whose associated model is the
$d=3$ square-pyramid plaquette model, and (iii) two counter-propagating $d=1$
Rule 60 CA that correspond to the two-dimensional Baxter-Wu spin model. For the
quantum spin models, we show that the connection to CAs implies a sensitivity
on the approach to the thermodynamic limit via finite size scaling for their
quantum phase transitions.
