Art and artists have inspired mathematics and science for aeons, prefiguring many ideas and artifacts at an intuitive level long before it was realized that there was even a scientific question to which art had already provided an answer. This can be seen in many instances in perspective painting, chiarascuro,  trompe l’oeil, origami and its cousins, and a range of crafts such as glass-blowing, weaving, knitting, crocheting etc.   We have worked on a small subset of these questions from both a predictive and a design perspective, focusing mostly on the paper arts.

Origami is an art form that probably originated with the invention of paper in China, but was refined in Japan. Kirigami is a cousin of origami that replaces folds by cuts in a thin sheet and enables articulated deployment that leads to even more remarkable shapes. The ability to create complex origami and kirigami structures depends on folding (cutting) thin sheets along creases, a natural consequence of the large-scale separation between the thickness and the size of the sheet. This allows origami (kirigami) patterns to be preserved across many length scales: the same pattern can be used to design human architectures and graphene gadgets.  Artists have an intuitive way to work with these forms, but can mathematics help quantify the design principles behind these art forms?  How can we design the number, size and location of folds (cuts) on a sheet of paper that will enable us to fold (cut) it into a given shape? And can engineering help realize and (mass) produce the shapes and functionalize them?

Inspired by the mathematical analogies between the two art forms that manipulate geometrical and topological degrees of freedom, we have posed the inverse problem of origami and kirigami as  constrained optimization problems or evolutionary algorithms and solved them in specific instances. Some questions of continuing interest in these areas include the quest for impossibility theorems  (or their converse),  the physical behavior of these unusual auxetic meta-materials, the Nash-Kuiper-Gromov convex integration program and its generalizations in terms of the smoothness requirements for the class of feasible solutions, and the exploration of floppy origami and floppy kirigami as mechanical memories, conductors, diodes, amplifiers, and perhaps even computers.