We introduce an additive approach for the design of a class of transformable structures based on two-bar linkages (“scissor mechanisms”) joined at vertices to form a two dimensional lattice. Our discussion traces an underlying mathematical similarity between linkage mechanisms, origami, and kirigami and inspires our name for these structures: karigami. We show how to design karigami which unfold from a one dimensional collapsed state to two-dimensional surfaces of single and double curvature. Our algorithm for growing karigami structures is provably complete in providing the ability to explore the full space of possible mechanisms, and we use it to computationally design and physically realize a series of examples of varying complexity.