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A photocrosslinkable poly(N,N′‐diethylacrylamide) copolymer allows for the photolithographic fabrication of hydrogel sheets with nonuniform crosslinking density and swelling ratio. Using this material system, different 3D shapes with nonzero Gaussian curvatureKare successfully programmed by prescribing a “metric” defined by in‐plane variations in swelling. However, this methodology does not control the direction of buckling adopted by each positive K feature, and therefore cannot controllably select between different isometric shapes defined by a single metric. Here, by introducing gradients in swelling through the thickness of the gel sheet by tuning the absorption of the UV‐light used for crosslinking, a preferential buckling direction is locally specified for each feature by the direction of UV exposure. By also controlling the strength of coupling between neighboring features, this is shown to be an effective method to program buckling direction of each unit within a canonical corrugated surface shape.

 

Self‐folding is a powerful approach to fabricate materials with complex 3D forms and advanced properties using planar patterning steps, but suffers from intrinsic limitations in robustness due to the highly bifurcated nature of configuration space around the flat state. Here, a simple mechanism is introduced to achieve robust self‐folding of microscale origami by separating actuation into two discrete steps using different thermally responsive hydrogels. First, the vertices are pre‐biased to move in the desired direction from the flat state by selectively swelling one of the two hydrogels at high temperature. Subsequently, the creases are folded toward their target angles by activating swelling of the second hydrogel upon cooling to room temperature. Since each vertex can be individually programmed to move upward or downward, it is possible to robustly select the desired branch even in multi‐vertex structures with reasonably high complexity. This strategy provides key new principles for designing shaping‐morphing materials that avoid undesired distractor states, expanding their potential applications in areas such as soft robotics, sensors, mechanical metamaterials, and deployable devices.

 

Assemblies of one-dimensional filaments appear in a wide range of physical systems: from biopolymer bundles, columnar liquid crystals, and superconductor vortex arrays; to familiar macroscopic materials, like ropes, cables, and textiles. Interactions between the constituent filaments in such systems are most sensitive to thedistance of closest approachbetween the central curves which approximate their configuration, subjecting these distinct assemblies to common geometric constraints. In this paper, we consider two distinct notions of constant spacing in multi-filament packings in${\mathbb{R}}^3$:equidistance, where the distance of closest approach is constant along the length of filament pairs; andisometry, where the distances of closest approach between all neighboring filaments are constant and equal. We show that, although any smooth curve in${\mathbb{R}}^3$permits one dimensional families of collinear equidistant curves belonging to a ruled surface, there are only two families of tangent fields with mutually equidistant integral curves in${\mathbb{R}}^3$. The relative shapes and configurations of curves in these families are highly constrained: they must be either (isometric) developable domains, which can bend, but not twist; or (non-isometric) constant-pitch helical bundles, which can twist, but not bend. Thus, filament textures that are simultaneously bent and twisted, such as twisted toroids of condensed DNA plasmids or wire ropes, are doubly frustrated: twist frustrates constant neighbor spacing in the cross-section, while non-equidistance requires additional longitudinal variations of spacing along the filaments. To illustrate the consequences of the failure of equidistance, we compare spacing in three ‘almost equidistant’ ansatzes for twisted toroidal bundles and use our formulation of equidistance to construct upper bounds on the growth of longitudinal variations of spacing with bundle thickness.

 

Modern fabrication tools have now provided a number of platforms for designing flat sheets that, by virtue of their nonuniform growth, can buckle and fold into target three-dimensional structures. Theoretically, there is an infinitude of growth patterns that can produce the same shape, yet almost nothing is understood about which of these many growth patterns is optimal from the point of view of experiment, and few can even be realized at all. Here, we ask the question: what is the optimal way to design isotropic growth patterns for a given target shape? We propose a computational algorithm to produce optimal growth patterns by introducing cuts into the target surfaces. Within this framework, we propose that the patterns requiring the fewest or shortest cuts produce the best approximations to the target shape at finite thickness. The results are tested by simulation on spherical surfaces, and new challenges are highlighted for surfaces with both positive and negative Gaussian curvatures.