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Rigid origami, with applications ranging from nano-robots to unfolding solar sails in space, describes when a material is folded along straight crease line segments while keeping the regions between the creases planar. Prior work has found explicit equations for the folding angles of a flat-foldable degree-4 origami vertex and some cases of degree-6 vertices. We extend this work to generalized symmetries of the degree-6 vertex where all sector angles equal 60 ∘ . We enumerate the different viable rigid folding modes of these degree-6 crease patterns and then use second-order Taylor expansions and prior rigid folding techniques to find algebraic folding angle relationships between the creases. This allows us to explicitly compute the configuration space of these degree-6 vertices, and in the process we uncover new explanations for the effectiveness of Weierstrass substitutions in modelling rigid origami. These results expand the toolbox of rigid origami mechanisms that engineers and materials scientists may use in origami-inspired designs. 

We introduce basic, but heretofore generally unexplored, problems in computational origami that are similar in style to classic problems from discrete and computational geometry. We consider the problems of folding each corner of a polygon P to a point p and folding each edge of a polygon P onto a line segment L that connects two boundary points of P and compute the number of edges of the polygon containing p or L limited by crease lines and boundary edges. 

Given a locally flat-foldable origami crease pattern $G=(V,E)$ (a straight-line drawing of a planar graph on the plane) with a mountain-valley (MV) assignment $\mu:E\to\{-1,1\}$ indicating which creases in $E$ bend convexly (mountain) or concavely (valley), we may \emph{flip} a face $F$ of $G$ to create a new MV assignment $\mu_F$ which equals $\mu$ except for all creases $e$ bordering $F$, where we have $\mu_F(e)=-\mu(e)$. In this paper we explore the configuration space of face flips that preserve local flat-foldability of the MV assignment for a variety of crease patterns $G$ that are tilings of the plane. We prove examples where $\mu_F$ results in a MV assignment that is either never, sometimes, or always locally flat-foldable, for various choices of $F$. We also consider the problem of finding, given two locally flat-foldable MV assignments $\mu_1$ and $\mu_2$ of a given crease pattern $G$, a minimal sequence of face flips to turn $\mu_1$ into $\mu_2$. We find polynomial-time algorithms for this in the cases where $G$ is either a square grid or the Miura-ori, and show that this problem is NP-complete in the case where $G$ is the triangle lattice. 

In this paper, we show that the rigid-foldability of a given crease pattern using all creases is weakly NP-hard by a reduction from the partition problem, and that rigid-foldability with optional creases is NP-hard by a reduction from the 1-in-3 SAT problem. Unlike flat-foldabilty of origami or flexibility of other kinematic linkages, whose complexity originates in the complexity of the layer ordering and possible self-intersection of the material, rigid foldabilltiy from a planar state is hard even though there is no potential self-intersection. In fact, the complexity comes from the combinatorial behavior of the different possible rigid folding configurations at each vertex. The results underpin the fact that it is harder to fold from an unfolded sheet of paper than to unfold a folded state back to a plane, frequently encountered problem when realizing folding-based systems such as self-folding matters and reconfigurable robots. 

A variety of electronic phases in solid-state systems can be understood by abstracting away microscopic details and refocusing on how Fermi surface topology interacts with band structure to define available electron states1. In fact, topological concepts are broadly applicable to non-electronic materials and can be used to understand a variety of seemingly unrelated phenomena2,3,4,5,6. Here, we apply topological principles to origami-inspired mechanical metamaterials7,8,9,10,11,12, and demonstrate how to guide bulk kinematics by tailoring the crease configuration-space topology. Specifically, we show that by simply changing the crease angles, we modify the configuration-space topology, and drive origami structures to dramatically change their kinematics from being smoothly and continuously deformable to mechanically bistable and rigid. In addition, we examine how a topologically disjointed configuration space can be used to constrain the locally accessible deformations of a single folded sheet. While analyses of origami structures are typically dependent on the energetics of constitutive relations11,12,13,14, the topological abstractions introduced here are a separate and independent consideration that we use to analyse, understand and design these metamaterials.