Fundamental conical defects: The d-cone, its e-cone, and its p-cone

Truncated cones from indenting a clamped disk

When a simply supported thin disk is indented by a centrally applied point force, it buckles out-of-plane to form a shape dominated by two conical portions: a uniform region indenting against the support, interrupted by a smaller elevated portion detached from the support, altogether known as a "developable cone" or d-Cone. If a central circular region of the disk is clamped instead, then the buckling complexion changes markedly: The indenting region is interspersed with several detached and elevated cones, now "truncated," where their number depends on the clamping extent as well as the radius of the circular simple support. Studies of d-Cone kinematics often consider its shape as an analogous vertex, which forms by folding along hinge lines separating triangular facets. We extend this methodology by, first, showing that each truncated cone, or "t-Cone," operates as a pair of connected d-Cone vertices that fold synchronously and that their number, viz. distribution, around the indented disk stems from optimal "packaging" of the folded shape in the annular space between the clamping edge and support; furthermore, because our analysis presumes a geometrically dominant character, it captures the "saturated," i.e., final number of t-Cones, in experiments from a recent study. Our predictions agree rather well.

Curved crease origami and topological singularities enable hyperextensibility of L. olor

Fundamental limits of cellular deformations, such as hyperextension of a living cell, remain poorly understood. Here, we describe how the single-celled protist Lacrymaria olor, a 40-micrometer cell, is capable of reversibly and repeatably extending its necklike protrusion up to 1200 micrometers in 30 seconds. We discovered a layered cortical cytoskeleton and membrane architecture that enables hyperextensions through the folding and unfolding of cellular-scale origami. Physical models of this curved crease origami display topological singularities, including traveling developable cones and cytoskeletal twisted domain walls, which provide geometric control of hyperextension. Our work unravels how cell geometry encodes behavior in single cells and provides inspiration for geometric control in microrobotics and deployable architectures.

Proportions of conical motifs: Optimal packing via the spherical image

We present a scheme for calculating the shape of two well-known conical motifs: the d-Cone and the e-Cone. Each begins as a thin, flat disk, before buckling during loading into a deformed shape with distinctive, asymmetrical conical features and a localised apex. Various deformed equilibrium models rightly assume a developable shape, with a particular focus on determining how much of the disk detaches from how it is supported during buckling; they are, nevertheless, extensively curated analytically, and must confront (some, ingeniously) the question of singular, viz., infinite properties at the conical apex. In this study, we find an approximate description of shape that reveals the extent of detachment, from an analogous mobile vertex that packages optimally according to its constraints. To this end, we further develop the usage of Gauss's Mapping and the associated spherical image, which has been used previously, but only to confirm known properties of deformed shape. Despite the simplicity of our approach, remarkably good predictions are availed, perhaps because such problems of extreme deformation are geometrically (rather than equilibrium) dominated.

Connecting the branches of multistable non-Euclidean origami by crease stretching

Non-Euclidean origami is a promising technique for designing multistable deployable structures folded from nonplanar developable surfaces. The impossibility of flat foldability inherent to non-Euclidean origami results in two disconnected solution branches each with the same angular deficiency but opposite handedness. We show that these regions can be connected via "crease stretching," wherein the creases exhibit extensibility in addition to torsional stiffness. We further reveal that crease stretching acts as an energy storage method capable of passive deployment and control. Specifically, we show that in a Miura-Ori system with a single stretchable crease, this is achieved via two unique, easy to realize, equilibrium folding pathways for a certain wide set of parameters. In particular, we demonstrate that this connection mostly preserves the stable states of the non-Euclidean system, while resulting in a third stable state enabled only by the interaction of crease torsion and stretching. Finally, we show that this simplified model can be used as an efficient and robust tool for inverse design of multistable origami based on closed-form predictions that yield the system parameters required to attain multiple, desired stable shapes. This facilitates the implementation of multistable origami for applications in architecture materials, robotics, and deployable structures.

Curved crease origami and topological singularities at a cellular scale enable hyper-extensibility of Lacrymaria olor

Eukaryotic cells undergo dramatic morphological changes during cell division, phagocytosis and motility. Fundamental limits of cellular morphodynamics such as how fast or how much cellular shapes can change without harm to a living cell remain poorly understood. Here we describe hyper-extensibility in the single-celled protist Lacrymaria olor, a 40 μm cell which is capable of reversible and repeatable extensions (neck-like protrusions) up to 1500 μm in 30 seconds. We discover that a unique and intricate organization of cortical cytoskeleton and membrane enables these hyper-extensions that can be described as the first cellular scale curved crease origami. Furthermore, we show how these topological singularities including d-cones and twisted domain walls provide a geometrical control mechanism for the deployment of membrane and microtubule sheets as they repeatably spool thousands of time from the cell body. We lastly build physical origami models to understand how these topological singularities provide a mechanism for the cell to control the hyper-extensile deployable structure. This new geometrical motif where a cell employs curved crease origami to perform a physiological function has wide ranging implications in understanding cellular morphodynamics and direct applications in deployable micro-robotics.

Indentation of an elastic disk on a circular supporting ring

Thin elastic two-dimensional systems under compressive stresses may relieve part of their stretching energy by developing out-of-plane undulations. We investigate experimentally and theoretically the indentation of an elastic disk supported by a circular ring and show that compressive stresses are relieved via two different routes: either developing buckles which are spread over the system or developing a d-cone where deformation is concentrated in a subregion of the system. We characterize the indentation threshold for wrinkles or d-cone existence as a function of aspect ratio.

Large-displacement symmetrical foldable cones

A foldable cone, or f-cone, is a discrete vertex folded from flat with curved faces, and is an exemplar of nonrigid origami. We determine the exact equilibrium shape of a well-folded symmetrical f-cone by geometrical considerations alone, by treating as cones intersecting along original fold lines expressing equal fold angles. When moderate displacements prevail, the shape is found alternatively from the spherical image of the vertex, from deploying Gauss's mapping. The analytical working is much less compared to direct linearization of the exact solution framework; expressions for shape parameters are availed in closed form, and some of their variation is accurately retained compared to the exact case.

The Biology of Anastomotic Healing-the Unknown Overwhelms the Known

BACKGROUND

Anastomotic complications are among the most devastating consequences of gastrointestinal surgery. Despite its high morbidity, the factors responsible for anastomotic regeneration following surgical construction remain poorly understood. The aim of this review is to provide an overview of the typical and atypical factors that have been implicated in anastomotic healing.

METHODS

A review and analysis of select literature on anastomotic healing was performed.

RESULTS

The healing of an anastomotic wound mirrors the phases of cutaneous wound healing- inflammation, proliferation, and remodeling. The evidence supporting much of the traditional dogma for optimal anastomotic healing (ischemia, tension, nutrition) is sparse. More recent research has implicated atypical factors that influence anastomotic healing, including the microbiome, the mesentery, and geometry. As technology evolves, endoscopic approaches may improve anastomotic healing and in some cases may eliminate the anastomosis altogether.

DISCUSSION

Much remains unknown regarding the mechanisms of anastomotic healing, and research should focus on elucidating the dynamics of healing at a molecular level. Doing so may help facilitate the transition from traditional surgical dogma to evidence-based medicine in the operating room.

Foldable cones as a framework for nonrigid origami

The study of origami-based mechanical metamaterials usually focuses on the kinematics of deployable structures made of an assembly of rigid flat plates connected by hinges. When the elastic response of each panel is taken into account, novel behaviors take place, as in the case of foldable cones (f-cones): circular sheets decorated by radial creases around which they can fold. These structures exhibit bistability, in the sense that they can snap through from one metastable configuration to another. In this work, we study the elastic behavior of isometric f-cones for any deflection and crease mechanics, which introduce nonlinear corrections to a linear model studied previously. Furthermore, we test the inextensibility hypothesis by means of a continuous numerical model that includes both the extended nature of the creases, stretching and bending deformations of the panels. The results show that this phase-field-like model could become an efficient numerical tool for the study of realistic origami structures.

Bioinspired 3D structures with programmable morphologies and motions

Living organisms use spatially controlled expansion and contraction of soft tissues to achieve complex three-dimensional (3D) morphologies and movements and thereby functions. However, replicating such features in man-made materials remains a challenge. Here we report an approach that encodes 2D hydrogels with spatially and temporally controlled growth (expansion and contraction) to create 3D structures with programmed morphologies and motions. This approach uses temperature-responsive hydrogels with locally programmable degrees and rates of swelling and shrinking. This method simultaneously prints multiple 3D structures with custom design from a single precursor in a one-step process within 60 s. We suggest simple yet versatile design rules for creating complex 3D structures and a theoretical model for predicting their motions. We reveal that the spatially nonuniform rates of swelling and shrinking of growth-induced 3D structures determine their dynamic shape changes. We demonstrate shape-morphing 3D structures with diverse morphologies, including bioinspired structures with programmed sequential motions.

k-cones and kirigami metamaterials

We are inspired by the tensile buckling of a thin sheet with a slit to create a foldable planar metamaterial. The buckled shape comprises two pairs of identical e-cones connected to the slit, which we refer to as a k-cone. We approximate this shape as discrete vertices that can be folded out of plane as the slit is pulled apart. We determine their kinematics and we calculate generic shape properties using a simple elastic model of the folded shape. We then show how the folded sheet may be tessellated as a unit cell within a larger sheet, which may be constructed a priori by cutting and folding the latter in a regular way, in order to form a planar kirigami structure with a single degree of freedom.