Cholesteric shells: two-dimensional blue fog and finite quasicrystals.

We study the phase behaviour of a quasi-two dimensional cholesteric liquid crystal shell. We characterise the topological phases arising close to the isotropic-cholesteric transition, and show that they differ in a fundamental way from those observed on a flat geometry. For spherical shells, we discover two types of quasi-two dimensional topological phases: finite quasicrystals and amorphous structures, both made up by mixtures of polygonal tessellations of half-skyrmions. These structures generically emerge instead of regular double twist lattices because of geometric frustration, which disallows a regular hexagonal tiling of curved space. For toroidal shells, the variations in the local curvature of the surface stabilises heterogeneous phases where cholesteric patterns coexist with hexagonal lattices of half-skyrmions. Quasicrystals, amorphous and heterogeneous structures could be sought experimentally by self assembling cholesteric shells on the surface of emulsion droplets.