Considered as a direct infinitesimal structure, the Killing root space diagram A₃ concretely realizes the SU(3) Lie algebra in ordinary three-dimensional Euclidean space, both as the ground vector system of “straight lines of length equaling zero” which  from the mutual center radiate these “rectilinear generatrices” of the “infinitely small sphere” as “image-points” onto the inner surface of this, and so marking there the vertex arrangement of the tetrahedral-octahedral honeycomb lattice which can be outlined as a cellular automaton by iteration of the thus hybridized “grain of nature”. The present report focuses upon this compound automaton cell and shows that it is an effective template and incubator of the elementary particle symmetries and transformations, including the internal ”hot nuclear matter” events and among them also the charm and bottom flavors, the Tau and the gauge vector bosons.