  Between space crystals and amorphous materials there exists a third class of
aperiodic structures which lack translational symmetry but reveal long-range
order. They are dubbed quasi-crystals and their formation, similarly as the
formation of space crystals, is related to spontaneous breaking of
translational symmetry of underlying Hamiltonians. Here, we investigate
spontaneous emergence of quasi-crystals in periodically driven systems. We
consider a quantum many-body system which is driven by a harmonically
oscillating force and show that interactions between particles result in
spontaneous self-reorganization of the motion of a quantum many-body system and
in the formation of a quasi-crystal structure in time.
