  We perform an accurate test of Ultrametricity in the aging dynamics of the
three dimensional Edwards-Anderson spin glass. Our method consists in
considering the evolution in parallel of two identical systems constrained to
have fixed overlap. This turns out to be a particularly efficient way to study
the geometrical relations between configurations at distant large times. Our
findings strongly hint towards dynamical ultrametricity in spin glasses, while
this is absent in simpler aging systems with domain growth dynamics. A recently
developed theory of linear response in glassy systems allows to infer that
dynamical ultrametricity implies the same property at the level of equilibrium
states.
