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Liquid crystal elastomers and glasses can have significant shape changedetermined by their director patterns. Cones deformed from circular directorpatterns have non-trivial Gaussian curvature localised at tips, curvedinterfaces, and intersections of interfaces. We employ a generalised metriccompatibility condition to characterize two families of interfaces betweencircular director patterns -- hyperbolic and elliptical interfaces, and findthat the deformed interfaces are geometrically compatible. We focus onhyperbolic interfaces to design complex topographies and non-isometric origami,including n-fold intersections, symmetric and irregular tilings. The largedesign space of three-fold and four-fold tiling is utilized to quantitativelyinverse design an array of pixels to display target images. Taken together, ourfindings provide comprehensive design principles for the design of actuators,displays, and soft robotics in liquid crystal elastomers and glasses.

Evolving, complex topography from combining centers of Gaussian curvature