  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.
