Consider the ideal I ⊆ K[x, y, z] corresponding to six points of P 2 . We study the limiting behaviour of the symbolic generic initial system, {gin(I (m) }m of I obtained by taking the reverse lexicographic generic initial ideals of the uniform fat point ideals I (m) . The main result of this paper is a theorem describing the limiting shape of {gin(I (m) }m for each of the eleven possible configuration types of six points. m P gin(I (m) ) , where P gin(I (m) )