A family of weakly universal cellular automata in the hyperbolic plane with two states

In this paper, we construct a family of weakly universal cellular automaton for all grids {p, 3} of the hyperbolic plane for p ≥ 13. The scheme is general for p ≥ 17 and for 13 ≤ p < 17, we give such a cellular automaton for p = 13, which is enough. Also, an important property of this family is that the set of cells of the cellular automaton which are subject to changes is actually a planar set. The problem for p < 13 for a truly planar construction is still open. The best result, for p = 7, is four states and was obtained by the same author.