  Many solid materials and liquid crystals exhibit geometric frustration,
meaning that they have an ideal local structure that cannot fill up space. For
that reason, the global phase must be a compromise between the ideal local
structure and geometric constraints. As an explicit example of geometric
frustration, we consider a chiral liquid crystal confined in a long cylinder
with free boundaries. When the radius of the tube is sufficiently small, the
director field forms a double-twist configuration, which is the ideal local
structure. However, when the radius becomes larger (compared with the natural
twist of the liquid crystal), the double-twist structure cannot fill space, and
hence the director field must transform into some other chiral structure that
can fill space. This space-filling structure may be either (1) a cholesteric
phase with single twist, or (2) a set of double-twist regions separated by a
disclination, which can be regarded as the beginning of a blue phase. We
investigate these structures using theory and simulations, and show how the
relative free energies depend on the system size, the natural twist, and the
disclination energy. As another example, we also study a cholesteric liquid
crystal confined between two infinite parallel plates with free boundaries.
