  Liquid crystal elastomers (LCEs) can undergo large reversible contractions
along their nematic director upon heating or illumination. A spatially
patterned director within a flat LCE sheet thus encodes a pattern of
contraction on heating, which can morph the sheet into a curved shell, akin to
how a pattern of growth sculpts a developing organism. Here we consider,
theoretically, numerically and experimentally, patterns constructed from
regions of radial and circular director, which, in isolation, would form cones
and anticones. The resultant surfaces contain curved ridges with sharp V-shaped
cross-sections, associated with the boundaries between regions in the patterns.
Such ridges may be created in positively and negatively curved variants and,
since they bear Gauss curvature (quantified here via the Gauss-Bonnet theorem),
they cannot be flattened without energetically prohibitive stretch. Our
experiments and numerics highlight that, although such ridges cannot be
flattened isometrically, they can deform isometrically by trading the
(singular) curvature of the V angle against the (finite) curvature of the ridge
line. Furthermore, in finite thickness sheets, the sharp ridges are inevitably
non-isometrically blunted to relieve bend, resulting in a modest smearing out
of the encoded singular Gauss curvature. We close by discussing the use of such
features as actuating linear features, such as probes, tongues and limbs, and
highlighting the similarities between these patterns of shape change and those
found during the morphogenesis of several biological systems.
