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

Nonuniform growth and topological defects in the shaping of elastic sheets