Smooth doubly curved origami shells with reprogrammable rigidity

Domain: neuroscience
Published: 2026-02-13
Editorial outcome: minor_revision

Abstract
Origami tessellations can transform flat sheets into curved yet inherently compliant surfaces that only approximate curvature and are unable to reconcile a fundamental trade-off among load-bearing capacity, curvature precision, and stiffness reprogrammability. We resolve this conflict by introducing a tileable crease pattern that folds into smooth, doubly curved shapes, enabling structural locking with minimal sagging under load. Solving an inverse problem, we compute fold patterns that match prescribed smooth surfaces with double, variable, and constant curvature. By strategically embedding tendons with varying pre-tension, we demonstrate reversible transformations from ultrasoft, formless states into rigid, load-bearing structures with in-situ tunable stiffness spanning orders of magnitude. This work unlocks a paradigm for folding doubly curved origami metamaterials, enabling flat-pack transport and scalable deployment of smooth, load-bearing shells.

Introduction
Rigid, smoothly curved surfaces are essential across science and engineering, impacting fields from aerospace and product design to optics1,2 and biomechanics3–5. A material that needs to be dynamically shaped into a smoothly curved surface often requires reconciling antagonistic functions, such as morphing its shape and/or adapting its properties without compromising rigidity and smoothness in the deployed state. Typically, the trade-off between rigidity, curvature smoothness, and property tunability cannot be attained in current technology, including wearable supports and exoskeletons6–9, deployable orthopedic implants10, soft robotics11,12, and deployable solar sails13 or antennas14. Soft robotic metamaterials, for instance, often rely on either inflatable systems with soft skins15 or origami structures featuring jagged, corrugated textures16,17. The former lacks surface rigidity, whereas the latter lacks surface smoothness. Soft skins are also vulnerable to abrasion in rough environments, such as those encountered in planetary exploration. Additionally, jagged surfaces are unsuitable for applications requiring low friction for drag reduction, for example, in swimming robots. Moreover, conformal curved contours are often necessary in orthopedic implants to match the complex geometries of curved bones. While recent origami-based deployable implants have enabled compact insertion and in situ expansion, their designs typically deploy into faceted forms that cannot form smooth, doubly curved geometries, a critical requirement for conformal, load-bearing integration of implants in complex anatomical sites10.

Similar challenges arise in wearable assistive technology and exoskeletons, which must conform to the curved shape of the human body while providing structural support. Existing concepts, on the other hand, often rely on discrete folds7 or rigid, chainmail-like elements6,9, resulting in non-smooth surfaces that can cause discomfort or even pain during prolonged wear or when compressed against the body. Currently, a technology that can deploy into a shell tessellation with a smoothly curved surface and tunable load-bearing capacity remains an unresolved challenge across various domains.

Origami principles offer a promising solution for creating deployable shells with desired curvature18–28 and metamaterials with programmable stiffness29–33 at multiple length scales. However, deployable origami shells composed of thin sheets are often floppy under mechanical loads due to the existence of zero-energy modes and the additional compliance of their non-rigid sheets; thin sheets can bend and twist easily, especially in origami incorporating curved creases or deformable panels. On the other hand, existing tessellable concepts typically rely on straight-crease patterns to deploy into a piecewise-linear approximation of curvature, with precision dependent on the pattern size18–28. Although refining the crease pattern size improves the smoothness of the approximated curves, it inevitably decreases the global thickness of the origami shell. Since flexural rigidity scales cubically with thickness, this reduction significantly compromises the load-bearing capacity of the shell, thereby creating an inherent trade-off between curvature smoothness and load-bearing capacity.

Curved creases, by contrast, can naturally generate folded structures with smoothly curved surfaces. Primarily explored in art34–38—beginning with the Bauhaus experiments of Josef Albers and the systematic explorations of David Huffman, and later expanded through the swirling pleated sculptures of Erik and Martin Demaine—curved-crease forms have more recently found practical applications in engineering39–42. In a previous work42, the authors introduced a hybrid curved-straight crease pattern, geometrically constructed to enable tiling into cylindrical surfaces with a prescribed global curvature of constant value. The intrinsic nature of this pattern and its kinematic constraints, however, make it impossible to generate smooth doubly-curved surfaces, fold into shapes with varying curvature, and deploy structural shells with reprogrammable flexural rigidity. These properties have significant implications that are crucial in a broad range of applications, such as doubly curved antenna reflectors for space, tunable and deployable curved structures, ergonomically shaped exoskeletons and wearables, aerodynamically shaped bodies for automotive and aerospace, and reconfigurable doubly curved soft robots with adaptive variable curvature.

One strategy for generating stiffness in deployable structures is to systematically integrate a network of pre-stretched tendons that partially or fully constrain the folding degrees of freedom, akin to cables in a structural tensegrity system43–45. Cables have previously been employed to control folded geometry46 or induce mechanical bistability47 in origami structures. More recently, tensegrity principles have been demonstrated effective for enhancing the axial stiffness during the deployment of single-degree-of-freedom, collapsible, straight-crease origami structures, such as Miura, Kresling, and modular Miura tubes48,49. Most of them feature unidirectionally tiled, rigid-foldable origami patterns, which primarily resist compression and tension only. These concepts, however, cannot be easily extended to in-plane origami tessellations made of bendable panels, which can fold into shell structures capable of resisting bending actions and multi-axial forces, a common scenario encountered in everyday technology.

In this work, we introduce the doubly curved lens-box, a lockable and rigid-ruling foldable50 hybrid crease pattern that combines curved and straight creases. Once deployed in its locked state, its tessellated pattern conforms to a doubly curved surface which is smooth in one direction and piecewise linear in the other. Unlike existing concepts that are floppy, our doubly curved lens-box is supplemented with an array of tendons, here denoting cable-like elements that resist tension only. These tendons can stiffen the folded pattern by contraction, reminiscent of cables in a tensegrity structure43–49. Since our origami building block tessellates into a smooth, continuous surface, the resulting origami shell forms a periodic metamaterial whose effective properties can be tailored via geometric design and actively reprogrammed through tendon pre-tension post fabrication. The tendons guide folding and enable the underlying folded pattern to shift reversibly and continuously from a relaxed configuration to a rigid, multiaxially load-bearing state. This evokes phase transformation in solids, where the amount of tendon stretching, as opposed to the transition temperature, governs folding and property tunability, a process that squeezes adjacent building blocks in a manner analogous to the jamming transition51. The concept introduced in this work, on the other hand, does not rely on gravity51, fluid pressure9, and environmental stimuli.

Results
Geometry of the doubly curved lens-box pattern
Our doubly curved lens-box origami pattern comprises two types of distinct yet complementary building blocks (Fig. 1a): curved-crease lens units (greenish yellow shades), which resemble a lens shape, and flat-foldable straight-crease waterbomb connector units (blue shades), which join the straight creases of adjacent lens units. Each lens unit consists of two reflected, non-symmetric curved arcs enclosing a middle lens panel (M), separating an upper leg panel (U) from a lower leg panel (L). Hybrid-crease unit cells, formed by merging lens-units and two dissimilar-size waterbomb connectors, can tile the entire plane. Upon folding, their shared linear edges form straight creases, whereas the two-dimensional arc creases (Fig. 1a-top) deploy into their three-dimensional curved counterparts (Fig. 1a-bottom), generating an array of out-of-plane arched surfaces joined by flat panels. At a specific folding configuration, when the facets of each connector unit become coplanar (i.e., flat-folded), the origami reaches a locked state. The geometric interplay between connectors and lens units of varying sizes and shapes yields a doubly curved tessellated surface that can be smooth along the lens direction (blue curvature, \kappa I) and a piecewise-linear approximation of uniform curvature along the orthogonal direction (orange curvature, \kappa II). It also enables the formation of variable curvature geometries.

To understand the folding kinematics of our doubly curved pattern, we idealize its deformation. We recall first that a straight-crease pattern folds through the mere rotation of its hinges (creases) without flexing its faces, a property known as rigid-foldability. In contrast, folding a curved-crease pattern requires the additional deflection of its panels. For developable curved-crease lens units, we assume a sheet of vanishing thickness deforms only through bending. This is underpinned by the energy scaling law: the stretching energy scales with the panel thickness t, while the bending energy scales with t3, making bending the dominant form of deformation. This assumption enables us to describe the kinematic of the lens unit by segmenting each curved panel into a non-crossing ruled surface (Fig. 1b), forming the rulings, a family of rule segments (see Supplementary Note 1). If the rulings remain unaltered during the entire folding process, the curved-crease is rigid-foldable50. To distinguish this from the rigid-foldability definition used in straight-crease origami, we hereafter refer to it as rigid-ruling foldability. Proving the existence of an invariant ruling pattern is essential for describing folding and achieving rigid-ruling foldability. This property has significant implications as it enables the use of even non-deformable materials to create the discretized, ruled segmentation version of the rigid-ruling foldable origami structures that transition from a flat state to a locked, approximately curved shape.

We demonstrate the existence of the ruling pattern and the folded geometry shown in Fig. 1b using theories of general curved-crease folding50,52–54, insights into the ruling segmentation developed in our earlier work on the hybrid Miura lens-box crease pattern42, and differential geometry under specific assumptions (see Supplementary Notes 1 and 2). We assume folding mapping to be isometric, i.e., a length-preserving transformation, with no twisting deformation in the lens panel M. The rulings within the lens panel M are thus parallel segments, and the geometry of the folded lens unit remains symmetric with respect to the plane bisecting M. Additionally, we assume that curved creases fold smoothly, generating surfaces with C2 continuity (second derivative) and kink-free folded creases. This assumption is justified by the energetic cost of non-smooth folding, which would require the formation of kinks or additional creases. These configurations are less energetically favorable because the bending energy required for smooth folding is much lower than the stretching energy needed to create kinks or new creases. Our experimental observations qualitatively support this assumption.

The folded geometry of the generalized lens unit is constructed by establishing the relations between pairs of points on creases connected through rule segments (see Supplementary Note 2.1). The unfolded (flat) C2 curved-crease arc is denoted by the parametric function l(s), where s corresponds to the x-coordinate of the crease point (blue dot in Fig. 1b). We introduce the folding parameter vR, which measures the distance between the apices of the right cone-ruled segments of the U and L panels in the folded state (Fig. 1b-bottom), and denote the x-coordinate of the corresponding cone apices in the unfolded state as u. Let \mu(vR) denote the angle between the base edge of the lower leg panel, treated as a directed vector from its left endpoint to its right endpoint, and the positive x-axis in the folded state, measured counterclockwise as positive. Denoting the orthogonal projection of the folded panel M onto the x-z plane by the parametric function \g(s)—where s now represents the arc length—we derive the fundamental geometric constraint that governs the smooth folding of the curved crease (see Supplementary Note 2.1.1) as

l'(0) + vR \cdot l(0) \cdot u \sin \mu \cdot g'(0) = g(0) \cos 2\mu \cdot \sqrt{1 - g'(0)2} + \tan \mu = 0 (1)

Furthermore, \g(s) can be explicitly expressed as a function of the unfolded crease l(s) (see Supplementary Note 2.1.1), allowing the entire folded geometry to be described through a single independent folding parameter (vR). Hence, the lens unit exhibits a single degree of freedom (DOF).

Rigid-ruling foldability and locking
Having described the geometry of our 1-DOF rigid-ruling foldable lens unit, we now discuss the folding of the connector and the entire pattern (see Supplementary Note 2.2). We show that the connector is also rigid-foldable with 3-DOF, similar to the original waterbomb pattern55,56 (see Supplementary Note 3). This can reduce to either 2-DOF with symmetric folding or 1-DOF when its tail panels come into contact at a specific folding state (Fig. 1a-bottom and Supplementary Note 3.3, Supplementary Fig. 9). Once the connector flat-folds fully, the lens unit can no longer fold as it has reached a locked state (Fig. 1a-bottom). While local flat-foldability can be verified using Kawasaki’s57 and Maekawa’s Theorems58,59, these are insufficient for assessing global flat-foldability in our 4-vertex connector unit. To design a flat-foldable connector, we geometrically construct its flat-folded state and derive constraints that prevent edge and vertex penetration (see Supplementary Note 2.2.1). Satisfying these flat-foldability conditions also ensures its full-range rigid folding motion.

Assessing the rigid-ruling foldability of a multi-vertex pattern such as our doubly curved lens-box tessellation is NP-hard60. Here, we can only prove the rigid-ruling foldability of a single doubly curved lens-box unit and its unidirectional tessellations (see Supplementary Note 3.4). In particular, we use Freeform61 rigid origami simulations to demonstrate that our pattern deforms isometrically and remains rigid-ruling foldable once tessellated in plane (see Supplementary Note 3.5). To achieve this, we first approximate the curved-crease fold pattern with a piecewise linear version consisting of a finite number of ruling segments, forming a discrete ruled surface. While the mobility of the resulting pattern remains independent of the ruling discretization, its folded geometry converges to the smoothly folded curved-crease shape as the number of ruling segments increases. Although we cannot fully analyze the folding mechanics of the tessellated pattern through rigid-folding simulations alone, we observe that in practice, folding becomes more challenging when the sector angle \beta < \pi/2 (Fig. 1a-top), and the folding difficulty increases as \beta decreases. We attribute this phenomenon to the kinematics of the origami pattern, where some creases exhibit a non-monotonic evolution of their dihedral angles and sharply reverse their folding direction. While this behavior is manageable for a single unit, it becomes strongly amplified when several connector units are tessellated and required to fold compatibly (see Supplementary Note 3.3.2 and Supplementary Fig. 9).

Locking into smoothly curved surfaces
With the geometric construction described above, we now address a more interesting question: Given a smooth 3D surface \Omega, can we find a 2D doubly curved lens-box tessellation that, upon locking, can be isometrically embedded onto \Omega while maintaining C1 or C2 smoothness along one principal direction? This is equivalent to finding a locked pattern where the panel M conforms to a prescribed meridional curvature \kappa I = 1/R GI (the curvature of \g, shown as the blue curve in Fig. 1a-bottom), while its tessellation along the azimuthal direction approximates the given curvature \kappa II = 1/R GII (orange curve in Fig. 1a-bottom) in a piecewise linear manner (see Supplementary Notes 4 and 5). This problem is explicitly formulated as an inverse problem that can be solved through constrained optimization (see Supplementary Note 6). Constraints for the lens units involve smooth folding of the crease and C1 smoothness along the meridional direction for two adjacent units (see Supplementary Note 4.1). For the connectors, constraints include achieving flat-foldability without overlap or collision, and preventing panel protrusion beyond the target surface (see Supplementary Note 2.2.1). Additionally, the effect of panel thickness is considered when matching the meridional curvature, as flat-folded connectors typically have a non-negligible thickness of \sim 8t.

While our formulation can determine planar tessellations that conform to double-curvature surfaces with uniform curvature (see Fig. 1a), folding these patterns remains challenging, and convergence is not always guaranteed. Moreover, for variable-curvature surfaces, obtaining such tessellations is generally impossible. We develop here an alternative strategy to tackle these problems: an explicit, additive tessellation approach that resembles a layer-by-layer 3D printing fabrication (see Supplementary Note 5.2). We begin by partitioning the generator curve of the given surface into sections of constant curvature. Initially, we construct a layer of the locked units that matches the specified surface curvatures, \kappa I and \kappa II. Subsequent layers are then independently constructed using the geometry established in the preceding layer. As a result, the unfolded crease patterns for each layer remain separate (disconnected) but can be connected to neighboring layers along the meridional direction once locked. In contrast to double-curvature surfaces, cylindrical surfaces can be fabricated by folding a single, continuous sheet of paper with a tessellated pattern, whose unit cells remain seamlessly connected throughout the entire folding process, from the unfolded to the locked state (see Supplementary Fig. 13a, b, and Supplementary Note 5.1). We showcase the versatility of our approach with closed-form relations enabling constant- and varying-curvature cylindrical surfaces, including a closing ring, logarithmic spiral, and a chair-like form (Fig. 2a, b, c), as well as other non-C1 periodic tessellations (see Supplementary Note 5.1.4). Figure 2a-g shows the results of our analysis applied to a geometrically diverse set of cylindrical and doubly-curved surfaces, with paperboard prototypes fabricated through laser perforation and manual folding (see Methods). Cylindrical surfaces are folded from a single sheet (no adhesive), whereas doubly curved surfaces are assembled by folding strips into layers in the x–y plane and bonding along the z-direction.

Scaling the origami pattern enables to determine its curvature design space; however, in practice, the attainable curvature is bounded by thin-panel criteria on the base sheet (e.g., t/l crease \lesssim 0.01–0.05, t/R c \lesssim 0.01, where l crease is a characteristic crease length). Achieving larger curvature (smaller R c) generally requires finer tessellations and thinner base material, which in turn reduces the global thickness of the folded shell.

Embedding tendons into origami units
Our doubly curved lens-box unit exhibits multi-DOFs before locking. Once locked, it becomes stiff under uniform in-plane compression (Fig. 3a top-right). However, upon tessellation into an aperiodic shell, the overall structure’s DOFs increase, creating internal mechanisms. In practice, additional non-rigid deformation modes may also emerge due to the panel flexibility (Fig. 3b). While some origami tessellations30,33, such as our lockable lens-box pattern without tendons (see Fig. 3c), can achieve limited stiffness or maintain partial rigidity under specific compressive loads and boundary conditions, achieving global rigidity in existing origami tessellations made of compliant panels remains a significant challenge.

Here, we postulate that the addition of strategically oriented tensile members, namely tendons (Fig. 3c), to the doubly curved lens-box units can suppress all DOFs of the system, ensuring multiaxial rigidity regardless of the direction of the applied loads and boundary conditions. Since rigidity under compression is achieved through locking, the added members must only warrant rigidity under tension: a harmony between tensile and compressive members reminiscent of tensegrity structures62–64. As illustrated in Fig. 3c, these tendons pass through the top-middle and bottom corners of the flat-folded connector units. By stretching the tendons, the partially folded units flex towards their locked state (Fig. 3c-right). When the connectors flat-fold, the tendons reach their shortest length, and the locked units squeeze their neighboring units. Pre-tension in the tendons acts analogously to applying in-plane pressure on individual building blocks; it enables preserving the integrity of the entire shell structure regardless of any external compressive force.

We show that the addition of tendons not only prevents deformations arising from multiple rigid-ruling folding motions, but can also restrain the non-rigid-ruling deformation modes (e.g., Fig. 3a, b) upon strategic tendon arrangement. For example, the paperboard curved structure in Fig. 3d shows nearly zero deflection against a twisting action when rigidified using the tendons, unlike its glued counterpart, which severely deforms under identical conditions. To determine the tendon position and layout illustrated in Fig. 3c, d, we resort to structural rigidity analysis and convert our locked unit into an equivalent triangulated network of bars and joints (see Supplementary Note 7). The bottom tendons are self-stressed, a feature advantageous for imparting first-order stiffness to the frame63,64.

Our model aims to capture the structural rigidity of the doubly curved lens-box solely in its locked state, where mobility reduces to zero under compression. Here, we assume that our bent panels, defined by the ruling segmentation, have zero flexural stiffness, and the rulings are modeled as creases with zero torsional stiffness. This analysis examines the inextensional deformation modes—finite and infinitesimal DOFs—that our locked origami may possess. To this end, we made the following assumptions: (i) the creases do not bend, following the asymptotic analysis of the Föppl–von Kármán equations which show that, at the thin panel limit, the energy required to bend a strip of straight crease is five times larger than the energy required to stretch it65,66; (ii) the connector unit panels in the flat-folded state resist bending due to their four- or eight-layer thickness, making bending energetically less favorable compared to that of single-layer panels in the lens unit.

While our simplified truss-based model qualitatively demonstrates the rigidity of our tendon-origami structure, it is not suitable for analyzing its mechanical response. A more advanced computational model—accounting for panel bending stiffness and interfacial contacts between connector panels, as well as geometric, material, and contact nonlinearities—would be required to fully unveil the underlying mechanics and failure mechanisms. This is beyond the scope of this paper, the focus of ongoing research.

In situ tunable load-bearing capacity
To precisely adjust the flexural rigidity of our tendon-origami shells (Fig. 4a), we incorporate a simple one-way rotational gear mechanism designed to stretch individual tendons incrementally (see Methods and Supplementary Note 8). By pulling the tendons, the partially folded soft shell gradually reconfigures into its stiff (locked) state with zero-DOF (Supplementary Movies 1 and 2). In this configuration, the tendon-origami shell internally jams, pressing the adjacent building blocks with a force proportional to the tendon stretch (Fig. 4a). This process, which to some extent recalls jamming in granular materials, inhibits rigid-ruling and non-rigid-ruling deformation modes. While tendons experience stretching, our lens units tolerate compressive loads in harmony, gradually transitioning from a soft to a stiff state. This mode-shift can reversibly, continuously, and quickly enhance the flexural modulus of our tendon-origami shells in operando.

We quantify changes in mechanical properties as a function of the tendon pre-tension T by performing a series of three-point bending experiments (see Methods and Supplementary Note 8) and calculating the apparent elastic flexural modulus B* of the shell (Fig. 4b). Stretching a tendon with initial length l 0 and Young’s modulus E t by l G creates the pre-tension T = E t l G / l 0 (see Supplementary Note 8.2). If the average stiffness K of the initial elastic regime (Fig. 4b), and the initial planar dimensions L, W, and the thickness h 0 of the shell are known, then B* = K L 3 / 4 W h 0 3. Our results in Fig. 4c show that the apparent flexural modulus increases convexly with the applied tendon pre-tension due to the (i) in-plane-bending coupling (geometric/stress stiffening) in the pre-stressed shell and (ii) staggered engagement of the top tendons at steps 4, 7, 9, and 10 to re-flatten the unloaded shape. This nonuniform engagement causes the effective pre-stress (top and bottom) to rise nonlinearly with the nominal increment in the bottom tendon pre-tension, yielding an exponential-like trend in the bending stiffness. We terminated the loading at the final reported point, where additional pre-tension triggered cell distortion and local buckling. Beyond this onset of instability, further pre-tension did not translate into a reliable stiffness increase for this configuration. (The lowest modulus value remained unregistered due to the challenge of measuring the initial tendon length in a formless relaxed state).

Figure 4d illustrates our paper-based arched deployable shell, exhibiting a remarkable load-to-weight ratio (approximately 162), surpassing that of the glued (rigid) counterpart (see Fig. 4e) by a factor of 4. During actuation, the tendon-origami shell can attain various configurations depending on its initial state and external loads. For instance, Fig. 4f–h shows the shell actuating under its own weight (14g), can transition in multiple stable forms before locking into its prescribed load-bearing arc. In Fig. 4f–g, the tendon-origami shell is stabilized by panel bending and unit self-contact under a moderate boundary constraint at the edges (manually held during the demonstration). In Fig. 4h, the shell attains free-standing equilibrium states (① - ⑦) where gravity is balanced by partial, non-uniform tendon pre-tension, panel bending rigidity, and unit contact; a full pre-tension of the tendons produces the final locked configuration (⑦).

Tendons enable precise control over the flexible shell kinematics, facilitating a wide range of motions and configurations, particularly suitable for soft robotics applications in constrained environments, e.g., endoscopic surgery. Unlike existing metamaterials with continuously tunable modulus9,67,68 that rely on high temperatures67, electrical/magnetic fields68, or pneumatic pressure9—posing risks during use or failure—our design remains safe and functional even in the event of partial failure. This is primarily because our tendon-origami system relies on lockable building blocks with multiple independent tendons, enabling load redistribution upon failure of a few tendons. As a result, partial failure does not compromise the overall integrity or functionality of the system. In contrast, systems controlled by pressurized fluids can fail catastrophically upon puncture, whereas tendon-origami mechanisms remain robust, typically unaffected by the failure of a single tendon, offering a reliable approach for programming stiffness and motion in a robotic system.

Methods
Material and fabrication
We fabricated all origami prototypes from 0.21-mm-thick cellulose paperboard sheet (200 gm−2 Fabriano Craft paper) by laser-perforating flat sheets along the prescribed crease pattern (CM1290 laser cutter, SignCut Inc.) and then manually folding them. Fold lines were produced using 1.2-mm-long cuts, uniformly spaced at 1-mm intervals. To hold the tendon-free origami shells in their locked configurations, adjacent facets of the waterbomb connector units were glued in their flat-folded state using a commercial polyvinyl acetate (PVA) paper adhesive. Sample manufacturing, including laser-cutting, folding, and tendon implementation, was conducted at ambient temperature (22-23°C) and a relative humidity of ~30-40%. The tensile Young’s modulus and strength of our paperboard sheets were measured to be approximately 7.9GPa and 57MPa, respectively, in the machine direction (MD), which are roughly twice the values measured in the cross direction (CD)30.

The gear-based tuning mechanism were made of high-density wood fiberboard (Masonite Panel) with a thickness of 3mm for all planar parts, such as gears, stoppers, and panels, along with hardwood dowel rods with a diameter of 5mm and a length of 20mm for the shafts (see Supplementary Notes 8.2 and Supplementary Fig. 23). Tendons are made of monofilament nylon fishing line (Red Wolf, Canadian Tire Corporation, Canada) with a nominal load capacity of 3.6kg and a diameter of about 0.1mm. While we did not perform experiments for measuring the properties of the fishing line, its Young’s modulus varies between 1.5 and 2.4GPa, and its ultimate strength between 0.25 and 0.9GPa69.

Mechanical testing
Experiments were conducted using an electrodynamic mechanical testing equipment (StepLab, STEP Engineering S.r.L., Resana, Treviso, Italy) equipped with a 1 kN load cell (AEP Transducers, Cognento, Italy; Type TSC3) and operated under displacement-control conditions. To maintain quasi-static loading condition, we applied a displacement ramp corresponding to a nominal strain rate of 10−3s−1. Consistent with the fabrication procedure, all tests were performed at ambient temperature and a relative humidity of around 30–40%. For three-point bending experiment, specimens were supported on two cylindrical rollers (30mm diameter), and loading was applied by lowering an indenter with a smooth cylindrical steel head of the same diameter. Displacement was obtained from the crosshead readings, which provides sufficient accuracy for our purpose, as discussed previously30.

Tendon pre-stress tuning mechanism
To tune the tendon prestress, we fabricated a gear-based tuning mechanism (Supplementary Note 8.2). The mechanism enables controlled pretensioning and can be locked after applying a prescribed tendon stretch to maintain the applied prestress and the locked shell configuration. The tuning mechanism was attached to the origami shell in the locked state using three screws (Supplementary Fig. 23c). The gears permit rotation in one direction only, with reverse motion prevented by stoppers. A rubber band was used to pull the stopper towards the gear shaft, thereby conditioning the stopper’s motion. Gears and stoppers were bonded to their shafts, allowing the shafts to rotate freely within the mounting holes. Tendons were secured by passing their ends through a small transverse hole drilled in each shaft.

Construction of the ‘Doubly Curved Lens-Box’ Pattern
The doubly curved lens-box unit combines two folding motifs: a lens-unit and a variation of the waterbomb. Its geometry consists of one lens unit and two non-uniform waterbomb connector units. To analyze the overall folding kinematics, we study these components separately, starting with the lens unit. During folding, we assume that all free boundaries (edges) of the lens unit remain undeformed (rigid) and that the straight creases of the waterbomb connectors do not bend. The detailed characterization of our doubly curved lens-box pattern, including its ruling segmentation and geometric construction, is presented in Supplementary Notes 1 and 2.

Rigid-ruling foldability
Rigid-foldability describes the property of an origami pattern to fold through the sole rotation of its creases without facet deformation. The proof of rigid-ruling foldability of our doubly curved lens-box pattern and related discussion are provided in Supplementary Note 3.

Tessellation generation for target surface mapping
The necessary conditions and constraints for smooth folding of the lens unit, flat foldability of the waterbomb connector units, smooth connection of the lens-box units upon tessellation, and the general rigid-ruling foldability of the entire crease pattern are provided in Supplementary Notes 1–4. Based on these relations, we identify tessellations that match the first (global) principal curvature \kappa I of the target surface along the lens panel, i.e., arc-lengths, and approximate the second principal curvature \kappa II. Supplementary Note 5 formulates the construction of generalized cylindrical surfaces with \kappa II = 0, and Supplementary Note 6 presents a numerical optimization procedure to search for locked lens-box embeddings of intrinsically curved (non-zero Gaussian curvature) target surfaces. We denote the radii associated with the first and second global principal curvatures by R GI and R GII, respectively, and use R c for the local radius of the curvature of the lens panel M.

Rigidity analysis
We analyze the rigidity of the origami shell structure under the following assumptions. First, the curved-crease unit is converted into an equivalent rigid-origami model defined by its ruling pattern (Supplementary Fig. 21). A diagonal crease is added (introduced only in the rigidity model) to the quadrilateral panels of the lens panel to allow twisting deformation modes. Second, all panels are assumed to be infinitely rigid, and the rulings and crease lines are modeled as frictionless rotational hinges. Third, to simplify the formulation, we replace the rigid origami structure with a triangulated network of inextensible struts connected through pin joints. Additional details are provided in Supplementary Note 7.

Data availability
All data needed to evaluate the conclusions in the paper are present in the main text, the Supplementary Information, and the other supplementary files. Source data are provided with this paper.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This manuscript presents a new origami structure called the doubly curved lens-box, which combines curved and straight folds. By embedding pre-tensioned tendons, the system enables reversible transformations from soft, compliant states to rigid, load-bearing structures with tunable stiffness. The authors present an inverse design framework for generating tessellations that conform to prescribed curved surfaces and validate the concept with paperboard prototypes, showing potential for applications in fields such as exoskeletons, deployable aerospace structures, robotics, and adaptive architectural systems.\n\nThe idea is interesting and the presentation is generally clear. I have some suggestions for the authors to improve the manuscript:\n1. It is strongly suggested that the authors stick to the terminology of “rigid-ruling foldability,” rather than using rigid foldability and rigid-ruling foldability interchangeably, because by the most well-accepted definition, curved origami cannot be rigid foldable because it necessarily involves panel bending.\n2. Line 151-152: this claim is wrong: “This property has significant implications as it enables the use of even non-deformable materials to create rigid-foldable origami structures…” Again, curved origami necessarily involves panel bending – no matter the amount, which is impossible for strictly non-deformable materials to achieve. Therefore, a softer tone is needed, such as saying “even stiff materials.” This is a problem throughout the manuscript. The authors have to be very careful of making strong claims as the proposed designs are not strictly “rigid origami.”\n3. The authors mentioned about difficulties during folding, and indicated that some geometric parameters lead to deviation from rigid-ruling foldability. It is then important to investigate such phenomenon by conducting FEA, or bar-and-hinge model analysis, at least on a few particular patterns as preliminary reults. It is not ok just saying such critical study is out-of-scope, when this matters to the claims in this manuscript.\n4. On a related note, in the constrained optimization, the authors should report how much the constraints are violated in the end, which is very likely, as in Ref. 20.\n5. Line 246-247: It should be clearly indicated in Fig. 2 about whether a pattern is folded from an entire piece or glued from many separate parts. The cut-and-glue strategy is fine, but it is important to clearly show this to the readers.\n6. All demonstrations in this work are made of paperboards. However, paperboards are quite forgiving for stretching deformation, which, by theory, should not present in this work. Therefore, it is important to show material versatility of the proposed designs by trying a different material, especially materials with large in-plane stiffness.\n7. There are very little explanations for Fig. 4F-H. Please elaborate on how the different configurations are stabilized.", "reviewer_2": "This paper presents a new origami folding pattern -- called \"doubly curved lens-box\" that can develop a flat sheet precursor into a doubly curved surface (with a smooth curvature and piecewise curvature). This is a significant step forward from the author's previous paper, which showed only a single curved surface. In addition, the authors proposed a novel method of locking the folded curvature with embedded tendons, demonstrating the practical use of this new origami pattern for different engineering functions.\n\nOverall, this paper presents an impressively thorough framework for the design and kinematic analysis of the doubly curved lens-box. The tendon locking is a nice addition. Therefore, I recommend publication with the following suggestions for further improvement.\n\nReachable design space: Although the authors have demonstrated several fascinating examples in Fig. 2, it is never entirely clear what range of curvatures the doubly curved lens-box can achieve. Is there a theoretical limit on the obtainable kappa_I and kappa_II?\n\nRigid foldability: Is local rigid foldability only a necessary condition for global rigid foldability? That is, have you encountered a situation where the doubly curved lens box is rigid-foldable individually, but the assembly of different lens boxes is no longer rigid-foldable?\n\nTendon Routing: Are the number and orientation of tendons optimized in the experiments, or are they distributed based on engineering intuition?\n\nFig. 4C shows that as the pre-tension linearly increases, the flexural modulus grows exponentially. This observation is quite surprising. Can the author elaborate on why? What is the physical underpinning of this behavior? Is there a threshold on tendon pre-tension, beyond which the flexural modulus can no longer increase?\n\nI am not sure if the big-picture comparison in Fig. 5a is entirely fair, as other published studies did not utilize tendon-locking to increase their modulus. Perhaps the authors could display two curves based on their new folding pattern, one with the tendon and one without."}, "author_response": "Response to Reviewers’ Comments\nManuscript Number: NCOMMS-25-57487\nManuscript Title: Smooth, doubly curved origami shells with reprogrammable\nrigidity\nArticle Type: Research Article (Physical Sciences/Engineering)\nAuthors: Morad Mirzajanzadeh and Damiano Pasini\nWe thank the reviewers for their valuable feedback and constructive comments. Our detailed\nresponse (light blue) to each of their remarks is provided below along with our revised portions of\nthe manuscript (dark blue and indented).\nReviewer #1:\nThis manuscript presents a new origami structure called the doubly curved lens-box, which\ncombines curved and straight folds. By embedding pre-tensioned tendons, the system enables\nreversible transformations from soft, compliant states to rigid, load-bearing structures with tunable sti(cid:431)ness. The authors present an inverse design framework for generating tessellations that\nconform to prescribed curved surfaces and validate the concept with paperboard prototypes,\nshowing potential for applications in fields such as exoskeletons, deployable aerospace structures,\nrobotics, and adaptive architectural systems.\nThe idea is interesting and the presentation is generally clear. I have some suggestions for the authors\nto improve the manuscript:\nComment # 1-1:\nIt is strongly suggested that the authors stick to the terminology of “rigid-ruling foldability,” rather\nthan using rigid foldability and rigid-ruling foldability interchangeably, because by the most well-\naccepted definition, curved origami cannot be rigid foldable because it necessarily involves panel\nbending.\nResponse: We thank the reviewer for this suggestion. We have revised the manuscript to use ‘rigid-\nruling foldability’ consistently, replacing ‘rigid foldability’ where appropriate. We retain ‘rigid\nfoldability’ only when referring to straight-crease cases, such as the connector units, or quoting\nprior literature.\nPage 1 of 16\nComment # 1-2:\nLine 151-152: this claim is wrong: “This property has significant implications as it enables the use of\neven non-deformable materials to create rigid-foldable origami structures…” Again, curved origami\nnecessarily involves panel bending – no matter the amount, which is impossible for strictly non-\ndeformable materials to achieve. Therefore, a softer tone is needed, such as saying “even sti(cid:431)\nmaterials.” This is a problem throughout the manuscript. The authors have to be very careful of\nmaking strong claims as the proposed designs are not strictly “rigid origami.\nResponse: Thank you for pointing out this seemingly contradictory statement.\nWhile a curved-crease origami with non-deformable materials/panels is not expected to bend and\nfold, our approach circumvents this global rigidity by discretizing the curved creases into ruling-line\nsegments, e(cid:431)ectively converting the system into its equivalent, straight-crease origami composed\nof rigid panels. This approximation preserves the overall folded geometry and kinematics. By\nintroducing torsional sti(cid:431)ness at the hinge locations corresponding to the original ruling lines, we\ncan also emulate the bending response of a curved-crease origami that is made of thin, deformable\npanels. As a result, the shape and mechanical behavior of the original deformable-panel origami\ncan be reproduced (approximately) in its thick-panel/non-deformable-panel, rigid-foldable\ncounterpart. To clarify this, we have revised the statement as follows:\n“This property has significant implications as it enables the use of even non-\ndeformable materials to create the discretized, ruled segmentation version of the\nrigid-ruling foldable origami transitioning from a flat state to a locked, approximately\ncurved shape.”\nPlease note that we also used similar clarifications in our original discussion section, as it reads:\n“Demonstrated as rigid-ruling foldable, the discretized, ruled segmentation version\nof our patterns can be adapted to non-deformable materials or thick-panel origami.”\nAdditionally, we have checked the entire manuscript to make sure the claims/statements that have\nbeen made are clear and compatible with curved-crease origami restrictions, as recommended.\nComment # 1-3:\nThe authors mentioned about di(cid:431)iculties during folding, and indicated that some geometric\nparameters lead to deviation from rigid-ruling foldability. It is then important to investigate such\nphenomenon by conducting FEA, or bar-and-hinge model analysis, at least on a few particular\npatterns as preliminary reults. It is not ok just saying such critical study is out-of-scope, when this\nmatters to the claims in this manuscript.\nResponse: We thank the reviewer for highlighting this important point. We agree that this particular\naspect deserves additional analysis and explanation. Although we briefly discussed this in the\noriginal SI, we now clarify the underlying mechanism yielding complex folding at certain angles in\nPage 2 of 16\nthe revised version of the manuscript and SI. We first explain the challenges associated with FE\nsimulations attempting to model the folding of our double-curvature lens-box tessellation, and why\ncurrent FE/bar-and-hinge modelling approaches are not yet well suited to systematically investigate\nrigid-ruling foldability and the folding complexity we observe for certain angles.\nRigid-ruling foldability and limitations of FE/bar-and-hinge models.\nRigid-ruling foldability is a geometric property: the ruling pattern must remain invariant throughout the\nmotion. Standard FEA or compliant bar-and-hinge models, by design, allow panel stretching and non-\nguided bending, i.e., bending in directions other than ruling; these deformations enable motions that violate\nruling invariance and isometric deformation. Hence their sole use cannot serve to test rigid-ruling\nfoldability. For that reason, our endorsement of rigid-ruling foldability relies on geometric analysis and\nkinematic simulation tools that enforce isometric deformations and prescribed rulings (Freeform Origami).\nThe challenge in assessing the rigid-foldability of multi-vertex origami patterns, including ours, has also\nbeen reported in another work (60) as mentioned in the original manuscript “Assessing the rigid-ruling\nfoldability of a multi-vertex pattern such as our doubly curved lens-box tessellation is NP-hard (60).”.\nFolding analysis struggles for FE analysis.\nOur building blocks comprise lens units and waterbomb-family connector units. The kinematics of\na single lens unit is relatively straightforward only if it is idealized as a 1-DOF mechanism and if we\nimpose our assumptions including folding symmetry and rigid-ruling foldability, as discussed in the\nmanuscript and SI. In folding a physical prototype, however, it is often possible to accidentally bend\npanels along alternative ruling patterns, thus it is challenging to maintain rigid-ruling foldability and\nfolding symmetry. The existence of several admissible ruling patterns implies that a lens unit\ne(cid:431)ectively has a manifold of DOFs and admissible kinematic paths.\nThe connector units have an even more complex (3-DOF) rigid-folding motion, which we illustrate\nwith the configurational space plotted in Supplementary Fig. 9b (shown again below) for a\nsimplified, symmetric-folding case (reduced to 2 DOFs). The surface plot reveals the unbounded\nnature of the kinematic paths: any line (slice) on this surface corresponds to a possible (symmetric)\nfolding path. For non-symmetric folding, which often occurs during physical paper folding, this\ndesign space becomes even richer. The blue line shown in Supplementary Fig. 9b represents only\none of these possible paths, where a linear relation between certain folding angles exists (up to the\nturning point marked by the open circle in Supplementary Fig. 9d, left panel). In tessellated origami\npatterns, the number of DOFs and the range of admissible kinematic paths increase substantially,\nmaking the analysis and characterization of the folding process considerably more complex.\nPage 3 of 16\na b Eqs. (29-31)\nEqs. (33-36)\nc\nd\nFig. S9. Full-range configuration space of 2-DOF waterbomb connector unit.\nGiven the complex kinematic landscape described above, it is extremely challenging to prescribe\nboundary conditions in the FE model of our initially flat multi-DOF origami pattern so that all crease\nrotations evolve along one specific theoretical path of folding . The situation is even more complex\nin FE models with deformable thin materials, where rigid-ruling foldability can be easily violated,\nand several additional folding paths become accessible during folding. Relaxing any of the idealized\nassumptions leads to the emergence of kinematic paths that are unknown and therefore not\ndirectly comparable with our analytical model. In contrast, for single-DOF origami patterns with a\nPage 4 of 16\nunique folding path, it is relatively straightforward to simulate the folding process, as is often\ndemonstrated in the literature.\nIn our view, the main challenge for FE or bar-and-hinge simulations is to address an inverse\nproblem: determining the boundary conditions and actuation (location, magnitude, and sequence)\nthat will drive the system along a targeted folding path in a high-dimensional configuration space.\nTo our knowledge, this inverse problem has not yet been systematically addressed in the literature\nand constitutes a challenging research direction in its own, with potential impact on revealing paths\nfor automated folding of complex origami patterns. Existing rigid-folding simulation tools, such as\nFreeform Origami (also used in this work), rely on a geometric/kinematic constraint solver (but not a\nphysical force-based folding simulator). They cannot provide direct information for analyzing the\nspecific folding mechanics or complexities we observe.\nWhile we could, in principle, force FE simulations to follow a theoretically possible kinematic path\nby prescribing specific rotational boundary conditions for individual connectors (hinges) with non-\nmonotonic amplitude and thereby bring the origami into its folded state, such an analysis would\nartificially bypass the practical folding problem and thus obscure the folding di(cid:431)iculty raised by the\nreviewer. Conversely, if we prescribe the rotation of only a few creases and leave all other creases\nto fold freely, it becomes practically impossible for the FE model to find the experimentally\nobserved (or desired) folding path in this high-dimensional configuration space.\nTo address the reviewer’s concern within these limitations, rather than forcing an FE model along an\nimposed path, we now provide a detailed kinematic explanation of the observed folding di(cid:431)iculty,\nsupported by additional parametric exploration of the configuration space (Supplementary Fig. 9d\nand related discussion).\nUnderlying mechanics of the folding di(cid:431)iculty.\nFigure S9D illustrates the evolution of the folding angles −𝜌 , −𝜌 , and 𝜌 as functions of 𝜌 for four\n(cid:2869) (cid:2870) (cid:2873) (cid:2871)\ndi(cid:431)erent sector angles 𝛽. For 𝛽 ≥ 𝜋/2, the folding angles −𝜌 and 𝜌 increase monotonically with\n(cid:2869) (cid:2873)\n𝜌 . In contrast, for 𝛽 < 𝜋/2, the angles −𝜌 and 𝜌 initially increase monotonically up to a turning\n(cid:2871) (cid:2871) (cid:2869) (cid:2873)\npoint (marked by the open circle, corresponding to the contact point between the tail panels) and\nthen decrease immediately, before rising again. In physical terms, the angles first rotate in one\ndirection, reach a maximum or minimum at the turning point, and then continue rotating in the\nopposite direction. In other words, −𝜌 cannot be treated as a smooth, globally monotone function\n(cid:2869)\nof 𝜌 at that configuration. This indicates a kinematic singularity in the configuration space when\n(cid:2871)\nthe motion is parameterized by 𝜌 , even though the underlying rigid-folding motion remains\n(cid:2871)\ncontinuous in the configuration-parameter space.\nThe above qualitatively explains why folding becomes more di(cid:431)icult than in the monotonic cases\n(𝛽 = 90∘ or 110∘). More specifically, the following are the reasons:\n• If we try to “drive” the mechanism mainly by actuating a panel that primarily changes 𝜌 ,\n(cid:2871)\nthen near the turning point, crease 1 and crease 5 tend to halt and then reverse their\nrotation direction. If the manual actuation does not accommodate this reversal cleanly, the\npanels tend to bend, or the origami appears to “lock”.\n• For monotonic curves, each value of 𝜌 corresponds to a unique configuration. For curves\n(cid:2871)\nwith a turning point, some configurations with the same −𝜌 and 𝜌 correspond to distinct\n(cid:2869) (cid:2873)\nvalues of 𝜌 , reflecting the more complex, multi-valued relation between the folding angles.\n(cid:2871)\nPage 5 of 16\n• The turning point is associated with a singular posture when the motion is parameterized by\n𝜌 . The mechanism remains kinematically admissible, but small imperfections in motion or\n(cid:2871)\nloading can deviate it from the ideal rigid-folding path, inducing additional panel bending,\nand render the folding somewhat “fussy”.\n• In paperboard origami with finite thickness and crease sti(cid:431)ness, this turning point can\ncorrespond to an energy peak or plateau. When manually folded, the origami may be\nperceived as undergoing a “stall” or a subtle snap-through event as the mechanism\ntraverses the singular configuration\nWe agree that a dedicated FE/bar-and-hinge study, aimed specifically at solving the inverse\nactuation problem for such multi-DOF patterns, would be a valuable future work, but it lies beyond\nthe scope of the present study.\nTo address the reviewer’s concern, we have expanded the SI (added the new Section 3.3.2.\n‘Kinematic Origin of Folding Di(cid:431)iculty’) and now provide a detailed analytical explanation of the\nfolding di(cid:431)iculty we observed.\n“3.3.2 Kinematic Origin of Folding Difficulty\nThe two sets of Eqs. (33) and (36) trace a feasible 1-DOF rigid-folding motion of our\nconnector waterbomb unit after the initial panel contact (red dot), corresponding to the\nsecond branch in Fig. S9b and to the curve beyond the red dot (and unfilled circles) in Fig.\nS9d. Our results show that the sector angles 𝛽 > 𝜋/2 yield a strictly monotonic (increasing\nor decreasing) evolution of all dihedral angles along this path. However, for 𝛽 < 𝜋/2, the\ndihedral angles 𝜌 and 𝜌 are no longer monotonic: they first change in one direction, then\n(cid:2869) (cid:2873)\npass through an extremum and reverse.\nDuring the manual folding process, we observed that while a single unit can be folded\nrelatively easily for almost any 𝛽, folding a tessellated pattern becomes significantly more\ndifficult for smaller sector angles 𝛽 < 𝜋/2. We associate this increased difficulty with the\nkinematics of folding an origami pattern, where some creases exhibit a non-monotonic\nevolution of their dihedral angles, i.e., the folding rate 𝑑𝜌 /𝑑𝜌 (with 𝑖 = 1,2,5) changes its\n(cid:3036) (cid:2871)\nsign along the motion (see Fig. S9d). At these turning points, 𝜌 can no longer be treated as a\n(cid:3036)\nsmooth, globally monotonic function of 𝜌 ; the configuration corresponding to the\n(cid:2871)\nextremum is a kinematic singularity when the motion is parameterized by 𝜌 . In practice,\n(cid:2871)\nthis means that as 𝜌 continues to change, certain creases must momentarily “stall” and\n(cid:2871)\nthen reverse rotation, which is difficult to realize through simple manual actuation and\ntends to promote panel bending or apparent locking.\nIn a tessellated lens-box pattern, many connector units share panels and must fold\ncompatibly. For 𝛽 < 𝜋/2, every connector must pass through its non-monotonic turning\npoint, meaning that several creases across the tessellation must stall and reverse rotation\nin a coordinated way. Small mismatches between neighboring units then accumulate,\nforcing additional panel bending and local deviations from the ideal rigid-folding path, which\nmanifest as apparent locking or “fussiness” during manual folding. We now turn our\nattention to assessing the rigid-foldability of a full lens-box pattern.”\nPage 6 of 16\nWe have also revised the related part of the manuscript as\n“In particular, we use Freeform(61) rigid origami simulations to demonstrate that our\npattern deforms isometrically and remains rigid-ruling foldable once tessellated in plane\n(see Supplementary Note 3.5). To achieve this, we first approximate the curved-crease fold\npattern with a piecewise linear version consisting of a finite number of ruling segments,\nforming a discrete ruled surface. While the mobility of the resulting pattern remains\nindependent of the ruling discretization, its folded geometry converges to the smoothly\nfolded curved-crease shape as the number of ruling segments increases. Although we\ncannot fully analyze the folding mechanics of the tessellated pattern through rigid-folding\nsimulations alone, we observe that in practice, folding becomes more challenging when the\nsector angle 𝛽 < 𝜋/2 (Fig. 1a-top), and the folding difficulty increases as 𝛽 decreases. We\nattribute this phenomenon to the kinematics of the origami pattern, where some creases\nexhibit a non-monotonic evolution of their dihedral angles and sharply reverse their folding\ndirection. While this behavior is manageable for a single unit, it becomes strongly amplified\nwhen several connector units are tessellated and required to fold compatibly (see\nSupplementary Note 3.3.2 and Supplementary Fig. 9).”\nComment # 1-4:\nOn a related note, in the constrained optimization, the authors should report how much the\nconstraints are violated in the end, which is very likely, as in Ref. 20.\nResponse: As suggested by the reviewer, we have now quantified the residual constraint violation\nand stationarity quality for all optimized lens-box units.\nFor each of the optimized lens-box geometries, we evaluated every constraint at the final solution\n𝑝(cid:3036)∗. For nonlinear equality constraints Φ(cid:3037)(cid:3435)𝑝(cid:3036)(cid:3439) = 0 (Eq. (3), Eq. (20), and Eq. (43)), the maximum\nabsolute residual (cid:3627)Φ(cid:3037)(cid:3435)𝑝(cid:3036)∗(cid:3439)(cid:3627) is always below 10(cid:2879)(cid:2869)(cid:2870). All nonlinear inequality constraints Ψ(cid:3038)(cid:3435)𝑝(cid:3036)(cid:3439) ≤ 0\n(e.g., geometric bounds, angle limits, non-penetration, no-overlap, and rigid-ruling foldability) and\nall simple variable box bounds (𝑝 (cid:3044) ≤ 𝑝(cid:3036) ≤ 𝑝 (cid:3044) ) are satisfied with zero violation within numerical\n(cid:3028) (cid:3044) (cid:3029)\nprecision.\nWe also assessed first-order optimality. For each optimized design, we restarted the interior-point\nsolver from that solution and recorded the solver diagnostics. The solver-reported feasibility\nresiduals are on the order of 10(cid:2879)(cid:2869)(cid:2873)–10(cid:2879)(cid:2869)(cid:2874), and the KKT (Karush–Kuhn–Tucker) (stationarity)\nresiduals are on the order of 10(cid:2879)(cid:2876) . The complementarity residuals are on the order of 10(cid:2879)(cid:2870)(cid:2868) or\nsmaller.\nThese results indicate that our final solutions satisfy all constraints up to numerical precision and\nmeet the KKT optimality conditions within 𝒪(10(cid:2879)(cid:2876)).\nTo address the reviewer’s comment, we have added this quantitative assessment to the SI:\n“We implement the numerical optimization algorithm in Mathematica using the\ninterior-point method (FindMinimum). The initial guesses for the design variables are\nchosen based on our understanding of the problem and the geometric attributes of\nthe lens box unit cell; they are then refined iteratively to approach a feasible solution.\nPage 7 of 16\nThe optimization is considered successful if the algorithmreturns a solution that\nsatisfies all constraints within a given numerical tolerance (i.e., the maximum\nresidual among all constraints is smaller than 10(cid:2879)(cid:2876)) over several hundred iterations.\nAfter convergence, we evaluate the numerical quality of each solution in two ways:\n(i) Feasibility. We substitute the optimized design variables 𝑝(cid:3036)∗ back into all\nconstraints and measure any violation. For each equality constraint Φ(cid:3037)(cid:3435)𝑝(cid:3036)(cid:3439)= 0, we\ncompute the residual (cid:3627)Φ(cid:3037)(cid:3435)𝑝(cid:3036)∗(cid:3439)(cid:3627) across all unit-cell designs. The maximum equality\nresidual is below 10(cid:2879)(cid:2869)(cid:2870). All inequality constraints Ψ(cid:3038)(cid:3435)𝑝(cid:3036)(cid:3439)≤ 0 (e.g., geometric\nbounds, angle limits, non-penetration, no-overlap, and rigid-ruling foldability) and all\nsimple variable box bounds (𝑝 (cid:3044) ≤ 𝑝(cid:3036) ≤ 𝑝 (cid:3044) ) are satisfied with zero violation within\n(cid:3028) (cid:3044) (cid:3029)\nnumerical precision.\n(ii) First-order optimality. We restart the interior-point solver from the optimized\nsolution and record the solver diagnostics. The solver-reported feasibility residuals\nare on the order of 10(cid:2879)(cid:2869)(cid:2873)−10(cid:2879)(cid:2869)(cid:2874), the KKT (Karush-Kuhn-Tucker ) (stationarity)\nresiduals are on the order of 10(cid:2879)(cid:2876), and the complementarity residuals are on the\norder of 10(cid:2879)(cid:2870)(cid:2868) or smaller.\nThese values indicate that our designs are numerically feasible to machine precision\nand satisfy the KKT optimality conditions at the 𝒪(10(cid:2879)(cid:2876)) level.”\nNotation update. We now write inequality constraints in the standard ≤ 0 form (equivalent to the\nprevious > 0 convention via negation). This change is purely notational and does not a(cid:431)ect the\nfeasible set of results.\nComment # 1-5:\nLine 246-247: It should be clearly indicated in Fig. 2 about whether a pattern is folded from an entire\npiece or glued from many separate parts. The cut-and-glue strategy is fine, but it is important to\nclearly show this to the readers.\nResponse: We thank the reviewer for the helpful suggestion. We have revised the manuscript and\nFig. 2 caption to explicitly state the fabrication method for each prototype. The revised manuscript\nnow includes:\n“Figure 2a-g shows the results of our analysis applied to a geometrically diverse set\nof cylindrical and doubly-curved surfaces, with paperboard prototypes fabricated\nthrough laser perforation and manual folding. Cylindrical surfaces are folded from a\nsingle sheet (no adhesive), whereas doubly curved surfaces are assembled by\nfolding strips into layers in the x–y plane and bonding along the z-direction.”\nPage 8 of 16\nThe caption of Fig. 2 now includes the following statement:\n“In panels (a–c), the origami patterns are folded from a single sheet. In panels (d–g),\nthe patterns are assembled by folding strips into layers in the x–y plane, which are\nthen glued along the z-direction to form the complete surface.”\nComment # 1-6:\nAll demonstrations in this work are made of paperboards. However, paperboards are quite forgiving\nfor stretching deformation, which, by theory, should not present in this work. Therefore, it is\nimportant to show material versatility of the proposed designs by trying a di(cid:431)erent material,\nespecially materials with large in-plane sti(cid:431)ness.\nResponse: We appreciate this suggestion. To further demonstrate the potential of our origami\nfabricated with other materials, we would like to share unpublished prototypes that successfully\ndemonstrate doubly curved lens-box folding using materials with high in-plane sti(cid:431)ness (to\nminimize the possibility of in-plane stretching) in the pictures below. These prototypes are\nfabricated from an aramid–epoxy laminate ([0/90] ) which has a typical in-plane tensile modulus of\ns\nabout 40 – 70 GPa (around an order of magnitude larger than paperboard), ensuring that the\nkinematics are not reliant on material stretch. The results show that it is feasible to fold our doubly\ncurved lens-box patterns from these composite sheets by only adjusting the laser-cutting\nparameters to create creases that fold without failure. While these e(cid:431)orts are ongoing, we share\nthese early results with the reviewer to demonstrate the feasibility of folding our patterns using\nmaterials with very high in-plane sti(cid:431)ness, thus better matching the theoretical assumptions used\nin our paper.\nPage 9 of 16\n(180 gsm sheet)\n60 mm\n(79 gsm sheet)\n[Figure Redacted]\nComment # 1-7:\nThere are very little explanations for Fig. 4f-h. Please elaborate on how the di(cid:431)erent configurations\nare stabilized.\nResponse: Thank you for the helpful comment. In these demonstrations, the tendon–origami\nstructure remains compliant because tendons are not fully pre-tensioned.\n•\nFig. 4f and g: The origami shapes are stabilized by panel bending and unit self-contact under\na light boundary constraint at the edges (manual holding during the demo).\n•\nFig. 4h: The configuration is free-standing, arising from a balance of gravity, partial (non-\nuniform) tendon pre-tension, panel bending rigidity, and unit contact. Fully pre-tensioning\nthe tendons leads to the final locked state shown as configuration ⑦. Each intermediate\nincrement (not measured quantitatively) corresponds to a specific distribution of pre-\ntension in the top and bottom tendons.\nPage 10 of 16\nTo make this clear, we have added the following statement in the revised manuscript:\n“In Fig. 4f and g, the tendon-origami shell is stabilized by panel bending and unit\nself-contact under a moderate boundary constraint at the edges (manually held\nduring the demonstration). In Fig. 4h, the shell attains free-standing equilibrium\nstates (① - ⑦) where gravity is balanced by partial, non-uniform tendon pre-tension,\npanel bending rigidity, and unit contact; a full pre-tension of the tendons produces\nthe final locked configuration (⑦).”\nPage 11 of 16\nReviewer #2:\nThis paper presents a new origami folding pattern -- called \"doubly curved lens-box\" that can\ndevelop a flat sheet precursor into a doubly curved surface (with a smooth curvature and piecewise\ncurvature). This is a significant step forward from the author's previous paper, which showed only a\nsingle curved surface. In addition, the authors proposed a novel method of locking the folded\ncurvature with embedded tendons, demonstrating the practical use of this new origami pattern for\ndi(cid:431)erent engineering functions.\nOverall, this paper presents an impressively thorough framework for the design and kinematic\nanalysis of the doubly curved lens-box. The tendon locking is a nice addition. Therefore, I\nrecommend publication with the following suggestions for further improvement.\nComment # 2-1:\nReachable design space: Although the authors have demonstrated several fascinating examples in\nFig. 2, it is never entirely clear what range of curvatures the doubly curved lens-box can achieve. Is\nthere a theoretical limit on the obtainable kappa_I and kappa_II?\nResponse: Thank you for the thoughtful question about the attainable values of curvature. Our\norigami approach is scalable: in the zero-thickness, inextensible-panel idealization, arbitrarily large\ncurvatures (small radii) are achievable by refining the crease pattern. In practice, though, the\nrealizable curvature is limited by the thin-panel assumption we made for the base sheet and by the\ne(cid:431)ective global thickness of the folded origami shell.\nTo remain in the bending-dominated regime and avoid in-plane stretch, we require (i) a small sheet-\nthickness-to-crease-length ratio, e.g., 𝑡/ℓ ≲ 0.01−0.05 (thin-plate range), and (ii) a small\n(cid:2913)(cid:2928)(cid:2915)(cid:2911)(cid:2929)(cid:2915)\nthickness-to-radius (of the bent panel) ratio, e.g., 𝑡/𝑅 ≲ 0.01 (equivalently, surface strain 𝜀 ≈\n(cid:3030)\nt/(2𝑅 ) ≪ 1) [R1]. As curvature increases (i.e., 𝑅 decreases), these constraints typically\n(cid:3030) (cid:3030)\nnecessitate either finer patterns and/or thinner base sheets. Therefore, both the base-material\nthickness 𝑡 and the global thickness of the origami shell determine the practically attainable\nenvelope of curvature values.\nIf the reviewers’ question seeks a numerical curvature range for a given sheet thickness, that is\nmaterial- and pattern-dependent, this lies outside the scope of this paper; a general closed-form\nbound is non-trivial and remains an open question of research.\nTo address the reviewer’s comment, we have now included the following statement in the\nmanuscript:\n“Scaling the origami patternenablesto determine its curvature design space;\nhowever, in practice, the attainable curvature is bounded by thin-panel criteria on\nthe base sheet (e.g., 𝑡/ℓ ≲ 0.01−0.05, 𝑡/𝑅 ≲ 0.01, where ℓ is a\n(cid:2913)(cid:2928)(cid:2915)(cid:2911)(cid:2929)(cid:2915) (cid:3030) (cid:2913)(cid:2928)(cid:2915)(cid:2911)(cid:2929)(cid:2915)\ncharacteristic crease length). Achieving larger curvature (smaller 𝑅 ) generally\n(cid:3030)\nrequires finer tessellations and thinner base material, which in turn reduces the\nglobal thickness of the folded shell.”\nPage 12 of 16\n[R1] Calladine, C.R., 1983. Theory of shell structures. Cambridge university press.\nComment # 2-2:\nRigid foldability: Is local rigid foldability only a necessary condition for global rigid foldability? That\nis, have you encountered a situation where the doubly curved lens box is rigid-foldable individually,\nbut the assembly of di(cid:431)erent lens boxes is no longer rigid-foldable?\nResponse: Yes. Unit-level (local) rigid-(ruling) foldability is a necessary but not su(cid:431)icient\ncondition for the global rigid foldability of a tessellation. We have encountered configurations\nwhere an individual doubly curved lens-box is rigid-(ruling) foldable, yet specific 2D assemblies are\nnot, due to kinematic incompatibilities at shared creases (e.g., closure constraints imposed by\nperiodicity).\nThat said, the outcome depends on the assembly. When lens-box units are tessellated only along\ntheir longitudinal (y) direction (Fig. 1a), the inter-unit kinematics matches, and the unit-level rigid-\n(ruling) foldability is su(cid:431)icient to obtain global rigid-(ruling) foldability. More complex 2D\ntessellations can introduce additional compatibility constraints that may break global rigid\nfoldability.\nAs noted in the main manuscript, in general, “Assessing the rigid-ruling foldability of a multi-vertex\npattern such as our doubly curved lens-box tessellation is NP-hard (60).”, which explains why\nsu(cid:431)iciency can fail in some assemblies.\nComment # 2-3:\nTendon Routing: Are the number and orientation of tendons optimized in the experiments, or are\nthey distributed based on engineering intuition?\nResponse: We selected the tendon count and routing using a simple, rigidity-driven design\nprocedure rather than ad-hoc intuition. Specifically, we (i) ensured structural rigidity of the locked\nlens-box unit under compression, and (ii) minimized the number of tendons subject to\nmanufacturability constraints (placing holes as close as the practical limit to the origami vertices\nand along fold-compatible directions). This yields a sparse, symmetric routing that stabilizes the\nunit while limiting non-rigid deformations in the lock state.\nHere, we did not perform a global combinatorial optimization over all possible tendon counts and\nlocations. This means other routings could also work and may yield di(cid:431)erent rigidity/sti(cid:431)ening\ntrade-o(cid:431)s; systematically exploring these is an interesting direction for future work.\nPage 13 of 16\nComment # 2-4:\nFig. 4c shows that as the pre-tension linearly increases, the flexural modulus grows exponentially.\nThis observation is quite surprising. Can the author elaborate on why? What is the physical\nunderpinning of this behavior? Is there a threshold on tendon pre-tension, beyond which the\nflexural modulus can no longer increase?\nResponse: Thank you for this helpful question.\nMechanism. The tested panel (pre-stressed shell) contains two tendon sets (top and bottom). In\nour experiment, the bottom tendons were incrementally stretched at every step (nominally linear),\nwhile the top tendons were additionally stretched only at selected steps, i.e., 4,7,9 and 10, to re-\nflatten the unloaded shell (as noted in the original Fig. 4b-c caption). This staggered engagement\nraises the overall pre-stress, and increases the apparent bending sti(cid:431)ness via in-plane–bending\ncoupling (stress sti(cid:431)ening). Because the top-tendon stretches occur in discrete events (and with\nlarger magnitudes at later steps), the e(cid:431)ective pre-stress (top + bottom) grows convexly with the\nstep count rather than strictly linearly. The apparent flexural modulus, therefore, follows a convex\ntrend that is well fit by an exponential curve.\nThreshold. We halted pre-tensioning at the final reported point because further stretching\nproduced visible cell distortion, local panel buckling, and large out-of-plane bending; beyond this\nonset of instability, additional pre-tension did not yield any further reliable increase in the apparent\nflexural modulus for this configuration. The precise threshold depends on pattern geometry, tendon\nrouting, and material properties, and a rigorous value would require a dedicated stability analysis\n(e.g., tracking the first occurrence of local wrinkling/buckling or a slope change in the sti(cid:431)ness–pre-\ntension curve).\nTo clarify these points, we have revised the related part in the manuscript into:\n“Our results in Fig. 4c show that the apparent flexural modulus increases convexly\nwith the applied tendon pre-tension due to the (i) in-plane-bending coupling\n(geometric/stress sti(cid:431)ening) in the pre-stressed shell and (ii) staggered engagement\nof the top tendons at steps 4, 7, 9, and 10 to re-flatten the unloaded shape. This\nnonuniform engagement causes the e(cid:431)ective pre-stress (top and bottom) to rise\nnonlinearly with the nominal increment in the bottom tendon pre-tension, yielding\nan exponential-like trend in the bending sti(cid:431)ness. We terminated the loading at the\nfinal reported point, where additional pre-tension triggered cell distortion and local\nbuckling. Beyond this onset of instability, further pre-tension did not translate into a\nreliable sti(cid:431)ness increase for this configuration. (The lowest modulus value\nremained unregistered due to the challenge of measuring the initial tendon length in\na formless relaxed state.)”\nPage 14 of 16\nComment # 2-5:\nI am not sure if the big-picture comparison in Fig. 5a is entirely fair, as other published studies did\nnot utilize tendon-locking to increase their modulus. Perhaps the authors could display two curves\nbased on their new folding pattern, one with the tendon and one without.\nResponse: Thank you for noting this. Our original plot demonstrates both trends, i.e., the origami\nalone without tendon stretch (dashed curve) and the tendon-sti(cid:431)ened case (solid curve). This is\nwhy our pattern’s sti(cid:431)ness curve starts from a small bending rigidity; we also show that it can be\ncontinuously increased by applying tendon stretching, though this progression is not explicitly\ndepicted on the original plot. In the initial stages, our sti(cid:431)ness remains at its lowest value, even\namong the literature, due to our floppy tendon-origami structure with multiple DOFs.\nWe also emphasize that the literature shown here is not readily compatible with integrating a\nsimilar tendon mechanism: A necessary condition for a comparable approach is the presence of\ncontact-based locking mechanisms within the origami structure.\nTo address the reviewer’s comment, we have clarified the original data point representing our work\nwithout tendons (T=0; large light blue circle) and highlighted the application of tendons and its\nincreasing pre-tension (T > 0) e(cid:431)ect on sti(cid:431)ness in our revised Fig. 5a, as shown below:\n“\n”\nWe also revised the first sentence of the Discussion to highlight that prior literature does not use\ntendons:\nPage 15 of 16\n“This work has introduced a hybrid origami tessellation combining straight and\ncurved creases, capable of isometric folding and locking into structural shells with\nsmooth doubly-curved or cylindrical surfaces with varying curvature, thereby\naddressing the trade-off between flexural rigidity and curvature smoothness in\ntraditional tendon-free origami tessellations (see Fig. 5a).”\nThe caption of Fig. 5 has been revised to:\n“Fig. 5. Trade-off between rigidity and curvature smoothness, and potential\napplications. a Qualitative rigidity-smoothness plot for surfaces obtained from\nexisting tendon-free origami tessellations, illustrating… ”\nPage 16 of 16"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have addressed my comments, and I can recommend the manuscript for publication now.", "reviewer_2": "I believe the authors have effectively addressed my comments and clarified the ambiguities in the original manuscript.\nTherefore, I support its publication."}, "author_response": ""}]

Decision letter