  Ordered phases on curved substrates experience a complex interplay of
ordering and intrinsic curvature, commonly producing frustration and
singularities. This is an especially important issue in crystals as
ever-smaller scale materials are grown on real surfaces; eventually, surface
imperfections are on the same scale as the lattice constant. Here, we gain
insights into this general problem by studying two-dimensional smectic order on
substrates with highly localized intrinsic curvature, constructed from cones
and their intersections with planes. In doing so we take advantage of fully
tractable "paper and tape" constructions, allowing us to understand, in detail,
the induced cusps and singularities.
