  We analyze the geometrical background under which many Lie groups relevant to
particle physics are endowed with a (possibly multiple) hexagonal structure.
There are several groups appearing, either as special holonomy groups on the
compactification process from higher dimensions, or as dynamical string gauge
groups; this includes groups like SU(2),SU(3), G_2, Spin(7), SO(8) as well as
E_8 and SO(32). We emphasize also the relation of these hexagonal structures
with the octonion division algebra, as we expect as well eventually some role
for octonions in the interpretation of symmetries in High Energy Physics.
