Self-assembled arrays of atomic chains on Si(111) represent a fascinating family of nanostructures with quasi-one-dimensional electronic properties. These surface reconstructions are stabilized by a variety of adsorbates ranging from alkali and alkaline earth metals to noble and rare earth metals. Combining the complementary strength of dynamical low-energy electron diffraction, scanning tunneling microscopy and angle-resolved photoemission spectroscopy, we recently showed that besides monovalent and divalent adsorbates, trivalent adsorbates are also able to stabilize silicon honeycomb chains. Consequently silicon honeycomb chains emerge as a most stable, universal building block shared by many atomic chain structures. We here present the systematics behind the self-assembly mechanism of these chain systems and relate the valence state of the adsorbate to the accessible symmetries of the chains.