Explicit kinematic equations for degree-4 rigid origami vertices, Euclidean and non-Euclidean.

We derive algebraic equations for the folding angle relationships in completely general degree-4 rigid-foldable origami vertices, including both Euclidean (developable) and non-Euclidean cases. These equations in turn lead to elegant equations for the general developable degree-4 case. We compare our equations to previous results in the literature and provide two examples of how the equations can be used: in analyzing a family of square twist pouches with discrete configuration spaces, and for proving that a folding table design made with hyperbolic vertices has a single folding mode.

Rigid folding equations of degree-6 origami vertices.

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.

Origami structures with a critical transition to bistability arising from hidden degrees of freedom.

Origami is used beyond purely aesthetic pursuits to design responsive and customizable mechanical metamaterials. However, a generalized physical understanding of origami remains elusive, owing to the challenge of determining whether local kinematic constraints are globally compatible and to an incomplete understanding of how the folded sheet's material properties contribute to the overall mechanical response. Here, we show that the traditional square twist, whose crease pattern has zero degrees of freedom (DOF) and therefore should not be foldable, can nevertheless be folded by accessing bending deformations that are not explicit in the crease pattern. These hidden bending DOF are separated from the crease DOF by an energy gap that gives rise to a geometrically driven critical bifurcation between mono- and bistability. Noting its potential utility for fabricating mechanical switches, we use a temperature-responsive polymer-gel version of the square twist to demonstrate hysteretic folding dynamics at the sub-millimetre scale.

Programming reversibly self-folding origami with micropatterned photo-crosslinkable polymer trilayers.

Self-folding microscale origami patterns are demonstrated in polymer films with control over mountain/valley assignments and fold angles using trilayers of photo-crosslinkable copolymers with a temperature-sensitive hydrogel as the middle layer. The characteristic size scale of the folds W = 30 µm and figure of merit A/ W (2) ≈ 5000, demonstrated here represent substantial advances in the fabrication of self-folding origami.