On the foldability of rigid origami

Among the techniques of forming the geometry of a reconfigurable and programmable matter, origami (the art of paper folding) and kirigami (paper folding with cutting) are especially appealing, as they are effective tools for transforming simple two-dimensional materials into complex three-dimensional structures. Rigid origami is the underpinning kinematics and dynamics of origami and kirigami engineering, which considers possible path between two complex structures through continuous folding. Rigid origami serves as suitable scale-independent mathematical models for both structures found in nature (e.g., molecules, crystals, proteins, biological materials) and man-made structures (e.g., biomedical devices, reconfigurable metamaterials, self-folding robotics, stretchable electronics, fold-core deployable structures, architectures, large space elements), even in multiple forms of art. My research focus is on the unsolved theoretical challenges when establishing systematic theories for rigid origami. These challenges can be classified into the "forward problem'', which is the useful sufficient and necessary condition for a rigid origami with certain crease pattern to be foldable; and the "inverse problem'', which is to systematically design a rigid origami based on some targets, such as approximating a surface or following a given path, meanwhile balancing the freedom of approximation and stability of folding motion. This thesis will report results on four topics associated with the theoretical challenges mentioned above. (1) We set up the definitions of key concepts for origami and rigid origami. The major difficulties are how to describe the contact between different parts of a paper while preventing self-intersection, and how to describe rigid origami as a realization of an underlying graph. This modelling could be used to connect rigid origami and its cognate areas, such as the rigidity theory, graph theory, linkage folding, abstract algebra and computer science. (2) We develop the rigidity theory for rigid origami under the folding angle description, including the generic rigidity and local rigidity. Generic rigidity studies how the rigidity is determined from the underlying graph of a rigid origami. Local rigidity includes: first and second order rigidity, which are defined from local differential analysis on the consistency constraint; static rigidity and prestress stability, which are defined after finding the form of internal force and load. We show there is a hierarchical relation among these local rigidity with examples representing different levels. (3) For the forward problem, progress is made from a thorough analysis on the compatibility condition, then new mathematical tools are applied to extend the current solution set. We discover several new classes of large foldable quadrilateral meshes, which are presented symbolically with numeric examples. (4) For the inverse problem, new design examples are adapted from the foldable quadrilateral meshes discovered above. Further, new algorithms that are applicable for more design targets are developed. We propose several algorithms on approximating a target surface from a planar rigid origami, which can be folded continuously. The folding motion will stop at the desired shape due to the clashing of panels. A discussion on future work concludes this thesis.