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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.

Liquid crystal elastomers and glasses can have significant shape change determined by their director patterns. Cones deformed from circular director patterns have nontrivial Gaussian curvature localized at tips, curved interfaces, and intersections of interfaces. We employ a generalized metric compatibility condition to characterize two families of interfaces between circular director patterns, hyperbolic and elliptical interfaces, and find that the deformed interfaces are geometrically compatible. We focus on hyperbolic interfaces to design complex topographies and nonisometric origami, including n-fold intersections, symmetric and irregular tilings. The large design space of threefold and fourfold tiling is utilized to quantitatively inverse design an array of pixels to display target images. Taken together, our findings provide comprehensive design principles for the design of actuators, displays, and soft robotics in liquid crystal elastomers and glasses.

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