I wrote: << By the way, so far we've been speaking only about finite configurations. But there's another problem of at least theoretical interest: Does there exist a planar pattern (i.e., a map Z^2 ->{0,1}) that has every reasonable planar pattern as a descendant? Perhaps reasonable can be defined as those planar patterns each having arbitrarily long lines of ancestors. Or, what is a priori a different condition, maybe reasonable should mean those planar patterns each having at least one infinite line of ancestors.
Both of these options are impossible. First tile the plane by NxN squares for big enough N. By using either a small fixed finite periodic configuration, or a cemetery, in each NxN square, this shows the existence of uncountably many maps Z^2 -> {0,1} each having at least one infinite line of ancestors, which covers both questions. But any one map Z^2 -> {0,1} can only have countably many descendants. QED --Dan ________________________________________________________________________________________ It goes without saying that .