  We consider nematic liquid crystals in a bounded, convex polyhedron described
by a director field n(r) subject to tangent boundary conditions. We derive
lower bounds for the one-constant elastic energy in terms of topological
invariants. For a right rectangular prism and a large class of topologies, we
derive upper bounds by introducing test configurations constructed from local
conformal solutions of the Euler-Lagrange equation. The ratio of the upper and
lower bounds depends only on the aspect ratios of the prism. As the aspect
ratio is varied, the minimum-energy conformal state undergoes a sharp
transition from being smooth to having singularities on the edges.
