Darwin’s finches are a classic example of adaptive radiation, exemplified by their adaptive and functional beak morphologies. To quantify their form, we carry out a morphometric analysis of the three-dimensional beak shapes of all of Darwin’s finches and find that they can be fit by a transverse parabolic shape with a curvature that increases linearly from the base toward the tip of the beak. The morphological variation of beak orientation, aspect ratios, and curvatures allows us to quantify beak function in terms of the elementary theory of machines, consistent with the dietary variations across finches. Finally, to explain the origin of the evolutionary morphometry and the developmental morphogenesis of the finch beak, we propose an experimentally motivated growth law at the cellular level that simplifies to a variant of curvature-driven flow at the tissue level and captures the range of observed beak shapes in terms of a simple morphospace. Altogether, our study illuminates how a minimal combination of geometry and dynamics allows for functional form to develop and evolve.

When a thin stream of aqueous sodium alginate is extruded into a reacting calcium chloride bath, it polymerizes into a soft elastic tube that spontaneously forms helical coils due to the ambient fluid drag. We quantify the onset of this drag-induced instability and its nonlinear evolution using experiments, and explain the results using a combination of scaling, theory and simulations. By co-extruding a second (internal) liquid within the aqueous sodium alginate jet and varying the rates of co-extrusion of the two liquids, as well as the diameter of the jet, we show that we can tune the local composition of the composite filament and the nature of the ensuing instabilities to create soft filaments of variable relative buoyancy, shape and mechanical properties. All together, by harnessing the fundamental varicose (jetting) and sinuous (buckling) instabilities associated with the extrusion of a jelling filament, we show that it is possible to print complex three-dimensional filamentous structures in the ambient fluid.

Inspired by the quest for shape-shifting structures in a range of applications, we show how to create morphable structural materials using a neutrally stable unit cell as a building block. This unit cell is a self-stressed hinged structure with a one-parameter family of morphing motions that are all energetically equivalent. However, unlike kinematic mechanisms, the unit cell is not infinitely floppy and instead exhibits a tunable mechanical response akin to that of an ideal rigid-plastic material. Theory and simulations allow us to explore the properties of planar and spatial assemblies of neutrally stable elements, and solve the inverse problem of designing assemblies that can morph from one given shape into another. Simple experimental prototypes of these assemblies corroborate our theoretical results and show that the addition of switchable hinges allows us to create load-bearing structures. Altogether, totimorphs pave the way for structural materials whose geometry and deformation response can be controlled independently and at multiple scales.

Kirigami involves cutting a flat, thin sheet that allows it to morph from a closed, compact configuration into an open deployed structure via coordinated rotations of the internal tiles. By recognizing and generalizing the geometric constraints that enable this art form, we propose a design framework for compact reconfigurable kirigami patterns, which can morph from a closed and compact configuration into a deployed state conforming to any prescribed target shape, and subsequently be contracted into a different closed and compact configuration. We further establish a condition for producing kirigami patterns and mechanisms which are reconfigurable and rigid deployable allowing us to connect the compact states via a zero-energy family of deployed states. All together, our inverse design framework lays out a path for the creation of shape-morphing material structures.

Singularities are common in diverse physical systems1 and lead to universal structures. This universality suggests that they should also naturally arise in biological systems, where active growth, autonomous motion, kinesis and taxis focus deformations in spacetime, as exemplified in the morpho- genetic processes determining biological size and shape4. A familiar example of a morphogenetic singularity is seen in the humble apple, which forms in the neighbourhood of the stalk as the apple grows. Here we study the geometry and morpho- genesis of the cusp of an apple by combining observations of fruit growth with a simple theory, finite element simulations and controlled swelling experiments using a physical gel sim- ulacrum. Our observations show that the axisymmetric cusp develops into a self-similar form, which can be understood in terms of a mechanical theory for the inhomogeneous growth of a soft sphere. Physical experiments using local inhibition in swelling gels corroborate our theoretical predictions. These experiments further show that axisymmetric cusps can lose stability and become lobed. We use simulations to show that the number of cuspidal lobes depends on the ratio of the size of the stalk to the size of the sphere, as well as the amplitude and periodicity of perturbations that mimic the role of fruit anatomy, consistent with observations of multi-cusped fruits.

Interactions with animal pollinators have helped shape the stunning diversity of flower morphologies across the angiosperms. A common evolutionary consequence of these interactions is that some flowers have converged on suites of traits, or pollination syndromes, that attract and reward specific pollinator groups. Determining the genetic basis of these floral pollination syndromes can help us understand the processes that contributed to the diversification of the angiosperms. Here, we characterize the genetic architecture of a bee-to-hummingbird pollination shift in Aquilegia (columbine) using QTL mapping of 17 floral traits encompassing color, nectar composition, and organ morphology. In this system, we find that the genetic architectures underlying differences in floral color are quite complex, and we identify several likely candidate genes involved in anthocyanin and carotenoid floral pigmentation. Most morphological and nectar traits also have complex genetic underpinnings; however, one of the key floral morphological phenotypes, nectar spur curvature, is shaped by a single locus of large effect.

Kirigami, the art of paper cutting, has become a paradigm for mechanical metamaterials in recent years. The basic building blocks of any kirigami structures are repetitive deployable patterns that derive inspiration from geometric art forms and simple planar tilings. Here, we complement these approaches by directly linking kirigami patterns to the symmetry associated with the set of 17 repeating patterns that fully characterize the space of periodic tilings of the plane. We start by showing how to construct deployable kirigami patterns using any of the wallpaper groups, and then design symmetry-preserving cut patterns to achieve arbitrary size changes via deployment. We further prove that different symmetry changes can be achieved by controlling the shape and connectivity of the tiles and connect these results to the underlying kirigami-based lattice structures. All together, our work provides a systematic approach for creating a broad range of kirigami-based deployable structures with any prescribed size and symmetry properties.

The self-assembly of peptides and proteins into amyloid fibrils plays a causative role in a wide range of increasingly common and currently incurable diseases. The molecular mechanisms underlying this process have recently been discovered, prompting the development of drugs that inhibit specific reaction steps as possible treatments for some of these disorders. A crucial part of treatment design is to determine how much drug to give and when to give it, informed by its efficacy and intrinsic toxicity. Since amyloid formation does not proceed at the same pace in different individuals, it is also important that treatment design is informed by local measurements of the extent of protein aggregation. Here, we use stochastic optimal control theory to determine treatment regimens for inhibitory drugs targeting several key reaction steps in protein aggregation, explicitly taking into account variability in the reaction kinetics. We demonstrate how these regimens may be updated “on the fly” as new measurements of the protein aggregate concentration become available, in principle, enabling treatments to be tailored to the individual. We find that treatment timing, duration, and drug dosage all depend strongly on the particular reaction step being targeted. Moreover, for some kinds of inhibitory drugs, the optimal regimen exhibits high sensitivity to stochastic fluctuations. Feedback controls tailored to the individual may therefore substantially increase the effectiveness of future treatments.

Inspired by the allure of additive fabrication, we pose the problem of origami design from a different perspective: How can we grow a folded surface in three dimensions from a seed so that it is guaranteed to be isometric to the plane? We solve this problem in two steps: by first identifying the geometric conditions for the compatible completion of two separate folds into a single developable fourfold vertex, and then showing how this foundation allows us to grow a geometrically compatible front at the boundary of a given folded seed. This yields a complete marching, or additive, algorithm for the inverse design of the complete space of developable quad origami patterns that can be folded from flat sheets. We illustrate the flexibility of our approach by growing ordered, disordered, straight, and curved-folded origami and fitting surfaces of given curvature with folded approximants. Overall, our simple shift in perspective from a global search to a local rule has the potential to transform origami-based metastructure design.