  An arithmetical discrete plane is said to have critical connecting thickness
if its thickness is equal to the infimum of the set of values that preserve its
$2$-connectedness. This infimum thickness can be computed thanks to the fully
subtractive algorithm. This multidimensional continued fraction algorithm
consists, in its linear form, in subtracting the smallest entry to the other
ones. We provide a characterization of the discrete planes with critical
thickness that have zero intercept and that are $2$-connected. Our tools rely
on the notion of dual substitution which is a geometric version of the usual
notion of substitution acting on words. We associate with the fully subtractive
algorithm a set of substitutions whose incidence matrix is provided by the
matrices of the algorithm, and prove that their geometric counterparts generate
arithmetic discrete planes.
