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.

Deployable motion of rotational sliceforms.

A rotational sliceform (RS) forms a stiff, ringlike array of intersecting planar slices. Removing a few slices and disconnecting the ends of an RS enables the incomplete array to be collapsed scissorlike into a compact stack; it can be expanded smoothly as far as the original incomplete configuration, but not beyond. Its structured architecture, coupled to apparent mechanistic motion and a natural self-locking ability, expresses equivalently a novel deployable metamaterial, and we set out to determine its natural limits of motion for symmetrical and asymmetrical RS architectures. We first reconceptualize the RS as an array of plane-faced pyramidal cells bounded by rigid slices of zero thickness. The minimum articulation range from all cells is shown to set an upper bound on the range of motion of an incomplete RS, specifically, that symmetrical architectures can collapse fully while asymmetrical cannot and that expansion always stops at the design configuration. We also find that planar rotation of slices is not possible without distorting the original intersections. Each slice is then permitted to kink out-of-plane while preserving the initial geometry of each cell, in order to marshal compatible rotations of now compliant slices. Our analysis then reliably captures the deployment features: the minimum collapsed state, the degree of slice deformation as they rotate, and the limit of expansion.

Spherical images and inextensible curved folding.

In their study, Duncan and Duncan [Proc. R. Soc. London A 383, 191 (1982)1364-502110.1098/rspa.1982.0126] calculate the shape of an inextensible surface folded in two about a general curve. They find the analytical relationships between pairs of generators linked across the fold curve, the shape of the original path, and the fold angle variation along it. They present two special cases of generator layouts for which the fold angle is uniform or the folded curve remains planar, for simplifying practical folding in sheet-metal processes. We verify their special cases by a graphical treatment according to a method of Gauss. We replace the fold curve by a piecewise linear path, which connects vertices of intersecting pairs of hinge lines. Inspired by the d-cone analysis by Farmer and Calladine [Int. J. Mech. Sci. 47, 509 (2005)IMSCAW0020-740310.1016/j.ijmecsci.2005.02.013], we construct the spherical images for developable folding of successive vertices: the operating conditions of the special cases in Duncan and Duncan are then revealed straightforwardly by the geometric relationships between the images. Our approach may be used to synthesize folding patterns for novel deployable and shape-changing surfaces without need of complex calculation.

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.

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

We consider well-known surface disclinations by cutting, joining, and folding pieces of paper card. The resulting shapes have a discrete, folded vertex whose geometry is described easily by Gauss's mapping, in particular, we can relate the degree of angular excess, or deficit, to the size of fold line rotations by the area enclosed by the vector diagram of these rotations. This is well known for the case of a so-called "d-cone" of zero angular deficit, and we formulate the same for a general disclination. This method allows us to observe kinematic properties in a meaningful way without needing to consider equilibrium. Importantly, the simple vector nature of our analysis shows that some disclinations are primitive; and that other types, such as d-cones, are amalgamations of them.

Inverted cones and their elastic creases.

We study the elastic inversion of a right circular cone, in particular, the uniform shape of the narrow crease that divides its upright and inverted parts. Our methodology considers a cylindrical shell analogy for simplicity where the crease is the boundary layer deformation. Solution of its governing equation of deformation requires careful crafting of the underlying assumptions and boundary conditions in order to reveal an expression for the crease shape in closed form. We can then define the characteristic width of crease exactly, which is compared to a geometrically nonlinear, large displacement finite element analysis. This width is shown to be accurately predicted for shallow and steep cones, which imparts confidence to our original assumptions. Using the shape of crease, we compute the strain energy stored in the inverted cone, in order to derive an expression for the applied force of inversion by a simple energy method. Again, our predictions match finite element data very well. This study may complement other studies of creases traditionally formed in a less controlled manner, for example, during crumpling of lightweight sheets.

Absence of branches from xylan in Arabidopsis gux mutants reveals potential for simplification of lignocellulosic biomass.

As one of the most abundant polysaccharides on Earth, xylan will provide more than a third of the sugars for lignocellulosic biofuel production when using grass or hardwood feedstocks. Xylan is characterized by a linear ß(1,4)-linked backbone of xylosyl residues substituted by glucuronic acid, 4-O-methylglucuronic acid or arabinose, depending on plant species and cell types. The biological role of these decorations is unclear, but they have a major influence on the properties of the polysaccharide. Despite the recent isolation of several mutants with reduced backbone, the mechanisms of xylan synthesis and substitution are unclear. We identified two Golgi-localized putative glycosyltransferases, GlucUronic acid substitution of Xylan (GUX)-1 and GUX2 that are required for the addition of both glucuronic acid and 4-O-methylglucuronic acid branches to xylan in Arabidopsis stem cell walls. The gux1 gux2 double mutants show loss of xylan glucuronyltransferase activity and lack almost all detectable xylan substitution. Unexpectedly, they show no change in xylan backbone quantity, indicating that backbone synthesis and substitution can be uncoupled. Although the stems are weakened, the xylem vessels are not collapsed, and the plants grow to normal size. The xylan in these plants shows improved extractability from the cell wall, is composed of a single monosaccharide, and requires fewer enzymes for complete hydrolysis. These findings have implications for our understanding of the synthesis and function of xylan in plants. The results also demonstrate the potential for manipulating and simplifying the structure of xylan to improve the properties of lignocellulose for bioenergy and other uses.