  Dualities are often supposed to be foundational, but they may come into
conflict with background independence, because a hidden fixed structures is
needed to define the duality transformation. This conflict can be eliminated by
extending a duality to a triality. This renders that fixed structure dynamical,
while unifying it with the dual variables.
  To illustrate this, we study matrix models with a cubic action, and show how
breaking its natural triality symmetry by imposing different compactifications
yields particle mechanics, string theory and Chern-Simons theory. These result
from compactifying, respectively, one, two and three dimensions. This may
explain the origin of Born's duality between position and momenta operators in
quantum theory, as well as some of the the dualities of string theory.
