Although broadly admired for its aesthetic qualities, the art of origami is now being recognized also as a framework for mechanical metamaterial design. Working with the Miura-ori tessellation, we find that each unit cell of this crease pattern is mechanically bistable, and by switching between states, the compressive modulus of the overall structure can be rationally and reversibly tuned. By virtue of their interactions, these mechanically stable lattice defects also lead to emergent crystallographic structures such as vacancies, dislocations, and grain boundaries. Each of these structures comes from an arrangement of reversible folds, highlighting a connection between mechanical metamaterials and programmable matter. Given origami's scale-free geometric character, this framework for metamaterial design can be directly transferred to milli-, micro-, and nanometer-size systems.

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.

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.

We use numerical simulations to probe the dynamics of concentrated suspensions of spherical microswimmers interacting hydrodynamically. Previous work in the dilute limit predicted orientational instabilities of aligned suspensions for both pusher and puller swimmers, which we confirm computationally. Unlike previous work, we show that isotropic suspensions of spherical swimmers are also always unstable. Both types of initial conditions develop long-time polar order, of a nature which depends on the hydrodynamic signature of the swimmer but very weakly on the volume fraction up to very high volume fractions.

Origami and kirigami have emerged as potential tools for the design of mechanical metamaterials whose properties such as curvature, Poisson ratio, and existence of metastable states can be tuned using purely geometric criteria. A major obstacle to exploiting this property is the scarcity of tools to identify and program the flexibility of fold patterns. We exploit a recent connection between spring networks and quantum topological states to design origami with localized folding motions at boundaries and study them both experimentally and theoretically. These folding motions exist due to an underlying topological invariant rather than a local imbalance between constraints and degrees of freedom. We give a simple example of a quasi-1D folding pattern that realizes such topological states. We also demonstrate how to generalize these topological design principles to two dimensions. A striking consequence is that a domain wall between two topologically distinct, mechanically rigid structures is deformable even when constraints locally match the degrees of freedom. DOI: 10.1103/PhysRevLett.116.135501 Recent interest in origami mechanisms has been spurred by advances in fabrication and manufacturing [1][2][3], as well as a realization that folded structures can form the basis of mechanical metamaterials [4][5][6][7][8]. The ability to identify kinematic mechanisms-allowable folding motions of a crease pattern-is critical to the use of origami to design new deployable structures and mechanical metamaterials. For example, the mechanism in the celebrated Miura ori that allows it to furl and unfurl in a single motion [9,10] is also the primary determinant of the fold pattern's negative Poisson ratio [4,5]. Identifying these mechanisms becomes more challenging when the number of apparent constraints matches the number of degrees of freedom (DOF).When there is an exact balance between DOF and constraints in a periodic structure, the structure is marginally rigid [11,12]. In such a case, new mechanical properties such as nonlinear response to small perturbations emerge [13][14][15][16]. A recent realization is that the flexibility of such solids may be influenced by nontrivial topology in the phonon band structure [17,18]. Here, we show how to extend these topological ideas to origami and kirigami. We show that periodically folded sheets may exhibit distinct mechanical "phases" characterized by a topological invariant called the topological polarization, recently introduced by Kane and Lubensky [17] using a mapping of mechanically marginal structures to topological insulators [19]. The importance of this invariant has emerged in the study of the soft modes of spring networks [18], and the nonlinear mechanics of linkages [20] and buckling [21]. As in these examples, the phases in our origami and kirigami structures exhibit localized vibrational modes on certain boundaries, and transitions from between topological phases are characterized by the appearance of bulk modes that cost zero energy. These are the hallmarks of topol...