  We systematically study phase transformations from one helical structure to
another. Motivated in part by recent work that relates the presence of
compatible interfaces with properties such as the hysteresis and reversibility
of a phase transformation [35, 33, 12, 28], we give necessary and sufficient
conditions on the structural parameters of two helical phases such that they
are compatible. We show that, locally, four types of compatible interface are
possible: vertical, horizontal, helical and elliptical. We discuss the mobility
of these interfaces and give examples of systems of interfaces that are mobile
and could be used to fully transform a helical structure from one phase to
another.
  These results provide a basis for the tuning of helical structural parameters
so as to achieve compatibility of phases. In the case of transformations in
crystals, this kind of tuning has led to materials with exceptionally low
hysteresis and dramatically improved resistance to transformational fatigue.
Compatible helical transformations with low hysteresis and fatigue resistance
would exhibit an unusual shape memory effect involving both twist and
extension, and may have potential applications as new artificial muscles and
actuators.
