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 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 and solved them in specific instances.

For origami,  following early work that showed how these folded structures can arise via spontaneous self-organizing instabilities, we showed how to approximate complex doubly-curved surfaces using computational algorithms , and then built physical models to realize the results.  However, these algorithms require the solution of non-convex optimization problems that requires good initial guesses that are hard to intuit. Inspired by the observation that all origami artists fold physical models inward from the free edges (thus taking advantage of the free-boundary-induced softness), we switched to qualitatively different perspective and designed (provably complete) additive algorithms to grow origami designs and provide a constructive algorithm to approximate any shape to any degree of accuracy.

For kirigami, we can follow an analogous procedure and showed how to approximate a given planar shape from a compact kirigamized sheet that can be deployed by articulated joints connecting 4-coordinated quads, and then showed how to create doubly-curved surfaces similarly, “hiding” the curvature in the gaps between the quads. We then constructed physical models to realize our computational results. Again, just as in origami, these algorithms require the solution of  non-convex optimization problems that requires good initial guesses that are hard to intuit. Inspired by our additive algorithms for origami, we showed that it is possible to create (provably complete) additive algorithms to grow planar and 3-dimensional kirigami designs and provide a constructive algorithm to approximate any shape to any degree of accuracy.

Together with our work on algorithms for 4d printing, that were shown to easy to be realized physically, this completes a program to answer the simple question: can we solve the inverse problem of creating physical geometries of arbitrary complexity, using 2d surfaces? Our use of either configurational degrees of freedom (e.g. using either the geometry of folding, or the topology of cutting) or accretionary degrees of freedom (e.g. using 3d printing of filaments or the growth of surfaces) answers this in the affirmative with provably convergent constructive algorithms that have begun to be used widely following our first demonstrations.

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.  In this vein, we have shown that we can create origami memories that show interesting rigidity percolation transitions, along with similar transitions for kirigami structures. We have also shown how to generalize kirigami to every one of the 17 members of the wallpaper group, and also shown how to circle the square or create multiple closed, compact states with kirigami, and analyzed the mechanical behavior of random kirigami structures with implications for reconfigurable windows, cryptography, and soft robotics. What is next? Perhaps conductors, diodes, amplifiers, and   even computers!