  Programming the non-uniform growth of a responsive polymer gel has emerged as
a powerful tool to shape sheets into prescribed three dimensional shapes. We
demonstrate that shapes with zero Gaussian curvature, except at singularities,
produced by the growth-induced buckling of a thin elastic sheet are the same as
those produced by the Volterra construction of topological defects in which
edges of an intrinsically flat surface are identified. With this connection, we
study the problem of choosing an optimal pattern of growth for a prescribed
developable surface, finding a fundamental trade-off between optimal design and
the accuracy of the resulting shape which can be quantified by the length along
which an edge should be identified.
