The Maekawa-Justin theorem is a locally necessary but not sufficient condition; for a crease pattern to be globally flat-foldable, all its vertices must be locally flat-foldable, which is a necessary but not sufficient condition. Any flat-foldable vertex must have even degrees (precondition of the Kawasaki-Justin theorem), which leads to a necessary criterion for global flat foldability: the two-colourability of a crease pattern (see Fig. 1e). A graph with all even degree vertices can be called a Eulerian graph, and it has been proven that Eulerian graphs satisfy two-face colouring 6 . However, flat foldability is hard 7 . The readers are referred to refs. 7,8 for more details.
During the folding-induced transformation, if the deformation of the sheet is only concentrated along the creases, without bending or stretching the panels, it is referred to as rigid origami. Such origami structure can be folded while keeping all regions of the paper flat and all crease lines straight 6 . Analysis of rigid foldability is of interest because of the increasing use of new materials other than paper for origami applications. Whereas paper is quite forgiving if the panels must deform, other materials such as metal, wood and stiff plastics are not. The well-known Miura-ori pattern, eggbox pattern, waterbomb pattern and Yoshimura pattern are all rigid origami, but the square twist 9,10 and hypar 11 patterns are not rigid origami.
A condition for rigid foldability given a single origami vertex 12,13 specifies that the product of rotation matrices about all the creases (say N in number) of a vertex should be the identity matrix (I) 14,15 , for any kinematically admissible configurations, using the Belcastro-Hull theorem:
where R ρ and α Q are the transformation matrices in terms of the turn- ing angles ρ i and panel angles α i , respectively (see Fig. 1i). These matrices are given by:
1 0 0 0 cos -sin 0 sin cos , ( ) = cos -sin 0 sin cos 0 0 0 1 (5)
Origami, the ancient art of paper folding, has proven to be a powerful concept, inspiring innovations in science, engineering and beyond. Folding can transform one geometric form to another, usually from two dimensions to three dimensions. This transformation creates new possibilities for scientists and engineers to design multifunctional machines, lightweight structures and architected materials.
The ideas of folding-induced origami structures can be found in nature 1 (Fig. 1a), and in thin walled structures undergoing sudden large deformation 2,3 (Fig. 1b,c). A recent noteworthy engineering application of origami is the starshade structure, which is illustrated in Fig. 1d.
Mathematically, an origami structure is locally a 2D, discrete manifold, which is characterized by a set of creases, lines on the manifold where (sharp) folding occurs and folding angles of the creases that determine the amount of folding. The creases divide the manifold into 2D pieces, called panels. The creases can either be straight or curved lines. When all the creases are straight lines, the panels are polygonal in shape 4 . The points of intersection of the creases are referred to as the vertices. The local structure of an origami design refers to any portion of the origami that is away from boundaries and intersections of multiple panels 5 . Some origami do not have any boundaries, forming enclosed polyhedra. In the remainder of this Primer, unless otherwise stated, origami refers to patterns with straight creases and thin panels.
Depending on the direction of the folding of the crease, when folded up creases are categorized as mountain folds and when folded down they are valley folds. The mountain/valley (M/V) assignments are relative because, depending on the viewing angle, mountain folds can be viewed as valley folds, and vice versa, but they are always pointing to opposite directions (locally). Such a convention leads to the crease pattern, a blueprint for origami structures, as shown in Fig. 1e.
Following the instruction of the crease pattern, typical origami folds up from a flat sheet into a 3D shape through isometric, or nearly isometric, transformation, without subjecting the sheet to stretch or tearing. Such an origami structure, with a flat initial state, is called developable (for example, Hypar origami). Theoretically, when the thickness of the sheet is assumed to be zero, the volume encompassed by the origami structure in its developed state is zero. To determine whether an origami structure is locally developable from its crease pattern, the N panel angles (for example, α i N { , =1 … } i ), or sector angles, meeting at a vertex can be added up. If their sum is 360° (or π 2 ), this vertex can be flattened onto a plane (see Fig. 1f). Mathematically, this condition is expressed as:
(1)
If all vertices of an origami structure are developable, this origami is globally developable. This rule only applies to vertices with neighbourhoods that are locally 2D manifold.
When the folding-induced transformation includes a state (other than the developed state) in which the entire origami structure can be flattened onto a plane, typically with overlapping of panels, the corresponding origami is called flat-foldable (for example, the Kresling tube). Flat foldability is a new, independent property from developability, for example, the eggbox is not developable but flat-foldable. The folded Primer and ρ ≈ 53.13°3
. The sign of the turning angle is obtained using the righthand thumb rule with the thumb pointing along each of the creases in a consistent direction (either inward or outward from the vertex) as the fingers curl along the arrows marked for ρ i , as shown in Fig. 1i. If the direction of the thumb must change to follow the arrow marked for a turning angle, then this suggests a change of sign. From these choices of angles, it can be shown that the product of the rotation matrices about all four creases would be an identity matrix.
The matrix approach is quite practical and useful when analysing origami structures with several vertices and creases, as the implementation can be naturally carried out using a computer program. This is typically done by modelling origami using Denavit-Hartenberg-based analysis 16 that is widely used in the analysis of linkages. The connection between linkages and origami is very useful, especially for thick origami 17,18 .
To ensure both sufficiency and necessity for rigid foldability, other detailed conditions (in addition to the Belcastro-Hull theorem 12,13 ) must be carefully checked, such as the bird's feet condition on M/V assignment 19 , which is both necessary and sufficient for a single vertex. Based on the rank deficiency of the kinematic Jacobian matrix (a linear expansion of the Belcastro-Hull theorem), a counting rule to calculate the generic degrees of freedom (DOFs) of a finite origami pattern has been developed 14 . When the generic DOF is greater than zero, the corresponding pattern is rigidly foldable; otherwise, no decisive conclusion can be drawn. The counting rule is given as follows 14 :
where N GDOF denotes the generic number of DOFs, N EO denotes the number of boundary edges of a pattern, k is the number of sides of a polygonal panel and N k P denotes the number of k-sided polygons in an origami pattern. Based on the kinematic Jacobian matrix, a pattern with only triangular panels is guaranteed to be rigidly foldable if it has more than three boundaries 7 . However, this condition is too strict practically. For example, many degree-four origami patterns (patterns with only degree-four vertices) are rigidly foldable, but they cannot be determined by this formula and require case by case discussion 20,21 . The panel angles and M/V assignment are both critical for rigid foldability (Fig. 1h). Some ways to predict the folding behaviour of rigid origami are to use mathematical tools such as spherical trigonometry 4 or computational origami simulations 22 .
It is important to realize that flat foldability, developability and rigid foldability are independent features of origami; having one feature does not imply the other (Table 1 and Fig. 1j). Different features are suited for particular engineering applications. For example, robotic actuation requires a quick deployment of motion and release of energy, and hence non-rigid origami with multistability is often exploited for those purposes. On the other hand, sandwich composite cores require mass production and high stiffness, and hence developable rigid origami patterns are often used.
Origami patterns are commonly crafted from thin sheets (approaching zero thickness). To apply them to real engineering applications, thickness accommodation imposes additional constraints. In general, thick origami is treated on a case by case basis, and thus many methods exist to accommodate panel thickness 23 . By modelling thick origami using spherical linkages 18 , the folding creases remain unchanged and the panels are either tapered 24 or offset 25 for compact folding with least physical interference of panels. A kinematic approach has been proposed for rigid origami of thick panels, involving the replacement of spherical linkages with spatial linkages at origami vertices consisting of four, five and six creases 17 . This is a comprehensive approach, which is capable of reproducing motions kinematically equivalent to those of zero-thickness origami (Fig. 1k). Meanwhile, to achieve the thick-panel folding of the non-developable vertex, auxiliary panels as intermediate links can be introduced to construct a plane-symmetric spatial linkage, which delivers compact folding 26 . Alternatively, a parallel-crease method can be used to create space for the panel thickness 27 ; however, this method introduces extra DOFs as a fourcrease vertex is transferred into an eight-bar linkage with at least two DOFs. A recent contribution to the thick-panel origami consists of applying kirigami to the thick panels, whose advantage is to obtain the most compact folding of the Miura-ori patterns with uniform thickness 28 . Additional work has been done on the physical forms of crease lines, such as rolling-contact joints 29 or compliant joints, which results in the variable kinematic models for the folding process. As there are many methods to treat thick-panel origami, it is difficult to determine which is the most efficient method without considering the practical case of interest. For example, for large-scale deployable structures, the folding ratio and the stiffness take the highest priority, whereas for micro-scale structures, the fabrication and flexibility take the main role. Hence, other methods are expected in future developments.
From an engineering perspective, a few representative origami patterns are considered in this Primer, including the Miura-ori pattern, the eggbox pattern, the waterbomb pattern and the Kresling tube.
Table 1 | A sample of origami patterns with featured properties, and applications Pattern Featured properties a Engineering applications Miura-ori 1 DOF, developability, rigid foldability, flat foldability, auxeticity Space structures 72 , metamaterials 30 , frequency-selective surfaces 170 , robotics 171 Blockfold 1 DOF, developability, rigid foldability, flat foldability, auxeticity Foldcore 83 Eggbox 1 DOF, rigid foldability, flat foldability Sandwich structures 172 , metamaterials 30 Waterbomb Developability, rigid foldability, flat foldability Smart materials 173 , robotics (origami wheel) 117,174 , origami stents 175 Yoshimura Developability, rigid foldability, flat foldability Folded concrete structures 168 Kresling tube Flat foldability, multistability Robotics 34,46 , impact mitigation 176 Morph 1 DOF, rigid foldability, flat foldability, reversible auxeticity Metamaterials 31,121 Barreto-Mars 1 DOF, developability, rigid foldability, flat foldability, auxeticity Solar cells 177 Flasher 1 DOF, developability, auxeticity Solar sails 178 Ron-Resch Developability, rigid foldability Energy absorption 179 Miura-oribased tubes 1 DOF, rigid foldability, flat foldability Robotics 41 Square twist Developability, flat foldability, auxeticity, multistability Metamaterials 9 , programmable antennas 10 Hypar origami Developability, multistability Metamaterials 4 Origami snapology Rigid foldability Metamaterials 180 , wave guide 181 Trimorph 1 DOF, rigid foldability, flat foldability, reversible auxeticity, multistability Metamaterials 32 This is not an exhaustive list. DOF, degree of freedom. a Incomplete list. 0123456789();:
These representative patterns are used as examples for origami experiments and manufacturing with desired rigid or non-rigid behaviour (see Experimentation). In the Results section, geometric descriptions of origami structures are provided using these patterns, and their geometric mechanics features are discussed. The unusual properties of origami structures have enabled engineering applications across different fields and scales, including sustainable and resilient buildings, mechanical metamaterials, robotics and medical devices. Furthermore, basic information for testing origami and standard formats to share origami designs are recommended. The limitations and opportunities in the context of design and manufacturing of origami, and an outlook for the future of origami in engineering are envisioned.
This section presents experimental equipment and manufacturing techniques to create and test origami patterns. These patterns display a range of properties, including a tunable Poisson's ratio 30,31 , multistability [32][33][34][35][36] , tunable stiffness [37][38][39][40] and others, such as shape morphing [41][42][43] and acoustics 44,45 . The experimental evidence of such properties requires careful sample manufacturing and the design of ad-hoc experimental set-ups. These set-ups are particularly important as the samples undergo large deformations simultaneously in longitudinal and transverse directions, owing to their intrinsic non-linear folding mechanisms. Different laboratory equipment is required depending on the type of experiment to be performed, for example, qualitative or quantitative. In the latter case, loading frames, prototyping machines, cameras, transducers and acquisition systems are needed to simultaneously test origami patterns and to record experimental data.
Origami tessellations have a long history 7 . They can be realized by different manufacturing techniques and base materials. Some of the most common materials and techniques are addressed below.
Paper and polymer-based models. The simplest way to create origami patterns is by folding craft paper (for example, Mi-Teintes, Canson), cardboard, polyester film (for example, Grafix Drafting Film) or composite film (for example, Durilla Durable Premium Ice Card Stock) 38,[46][47][48] . The folding lines can be marked or perforated with evenly spaced cuts. Laser cutters are used to perforate thin flat sheets along the folding lines, as shown in Fig. 2a. Other electronic cutting machines (for example, Silhouette CAMEO, Silhouette America) can also be used to perforate thin sheets that can be folded using origami principles. The main advantage of paper-based origami is its ease of realization. Furthermore, craft paper sheets have a very small thickness, thus preventing panel thickness accommodation issues when extreme folding is achieved. Although very effective, this method can induce some difficulties in quantitative testing as the paper-based panels are very flexible. In fact, out-of-plane deformation may not only be localized on the creases but also in the panels through flexural deformation. This could lead to difficulty in the experimental validation of the underlying theory as the hypothesis of rigid foldability may no longer hold.
Computer numerical control milled models. A more effective method to create origami samples suitable for mechanical testing consists of assembling base components such as unit cell strips, panels and hinges, which are cut by a computer numerical control milling machine (such as the Roland MDX 540) 49 , as shown in Fig. 2b. The working principle of the milling machine concerns material removal through a rotating cutting tool driven by computer-aided design/computer-aided manufacturing (CAD/CAM) software. The base components can be made from different materials, such as polymeric (polycarbonate, polypropylene) or metallic (thin aluminium or steel sheets) materials. The main advantage of this manufacturing method is its versatility and precision, which permits the realization of complicated shapes and multistable origami tessellations. In particular, this method allows fine-tuning of the energy landscape together with the mechanical properties of the tessellation, by varying the crease thickness, panel geometry and base material 32 . An attractive material to create zero-energy creases (freerotating hinges) is polypropylene as it guarantees excellent folding performance and fatigue resistance. On the other hand, if creases with specific rotational stiffness are required, these can be achieved by means of solid rubber (for example, silicon rubber) 32 .
technologies have enabled the construction of intricate and complex topologies at different scale levels 50 . In this context, 3D printing represents an alternative method to creating origami samples, as shown in Fig. 2c. The origami models are built layer by layer from a previously prepared 3D CAD file. Various materials and 3D printing technologies can be used depending on specific needs. For instance, origami tessellations have been realized by fused filament fabrication 51,52 , material jetting 53 , selective laser sintering 54 , stereolithography 5,55 , digital light processing 56,57 and, at the micro-scale, by two-photon polymerization laser lithography 58 . In addition, 4D printing adds the transformation over time (fourth dimension) to 3D printing, which has been used to create multifunctional shape-morphing and self-foldable origami-based structures and materials 42,[59][60][61][62] . The main benefit of additive manufacturing technique is its ability to make multi-material parts during a single printing step, thus avoiding complicated assemblage processes.
Once the manufacturing is completed, the next step is to fold along the creases and assemble the folded components to achieve the final origami structure. Developable patterns, such as the Miura-ori, can be realized by folding a single flat sheet of material along the crease lines. By contrast, non-developable patterns, such as the Eggbox, can be obtained through the folding and assembling of several sub-parts or pieces 49 . In the case of paper-based origami, the union among different parts is usually achieved using flaps and either double-sided tape or paper glue. On the contrary, assembled plastic models are realized by glueing several modular base components. Conveniently, each unit cell or component should have several seats and/or extensions to allow an easy assembly.
The quality and integrity of the sample is fundamental for the success of the experiments. Some discrepancies between theoretical prediction and experimental results are commonly related to sample defects such as ageing, manufacturing and fatigue issues. For instance, if a crease is broken or damaged, an imperfection is introduced into the tessellation, deeply influencing the experimental results. For such reasons, before executing any mechanical test on origami tessellations (for example, uniaxial testing), a careful visual sample check should be done to verify the integrity of the pattern and its actual dimensions in the rest configuration should be verified with measuring devices (tape measure and caliper). In fact, the dimension of the sample in the rest configuration represents a common reference point for experiments,
0123456789();: 0123456789();:
Primer theory and simulations 49 . Thus, the dimensions of the sample are an essential parameter when load versus displacement experimental data are compared with theoretical or numerical simulations (for example, in MERLIN 63 ).
Origami tessellations are reconfigurable structures that exhibit simultaneous deformations in both the longitudinal and transverse directions. Further, origami systems can be highly stiff in certain directions and flexible in other directions 37 . Hence, different types of experimental apparatus and set-ups may be required to investigate the behaviour of origami structures depending on the nature of deformations and loading. Recently, experimental set-ups to perform the uniaxial tests have been proposed to demonstrate the unique Poisson's ratio behaviour of origami metamaterials 49 . In this work, it has been shown that the gripping mechanism, connecting the ends of the sample to the loading frame, plays a key role in the quality of the results obtained from the testing machine. The standard way to connect the sample to the loading frame is by clamping its ends to the machine, as shown in Fig. 2d. The drawback of this approach is that the gripping system prevents the free deployment of the pattern in the transverse direction, leading to non-uniform transverse deformation. This is evident by observing the shape assumed by the sample during tension and compression testing. In a tension test, the sample deforms into a dog-bone shape; in compression, it deforms into a barrel shape. The proposed gripping mechanism 49 consists of a system of rail and sliders that allows proper attachment of the sample while permitting free transverse motion during the folding/unfolding process, as shown in Fig. 2e. Such a system, called a Saint-Venant fixture, eliminates Saint-Venant end effects during uniaxial testing experiments. This advanced set-up permits a free deployment of the constrained sample, thus ensuring that the origami tessellation remains truly periodic even when deformed in a tension or compression test. To obtain high-resolution quantitative information from experiments, a loading frame machine is required for folding/unfolding of the origami tessellation. The loading frame should be equipped with a load cell and a displacement transducer for recording the applied load as a function of the sample length. The whole experimental apparatus should be arranged horizontally to reduce gravitational effects and avoid out-of-plane instabilities that could arise during the experiments. Moreover, a Teflon plate should be placed underneath the sample to reduce friction and stick-slip phenomena. For the monitoring of the longitudinal and transverse deformation of the sample, a high-resolution camera must be placed orthogonally to the testing platform to record the experiments. This allows recording of the motion of a selected array of points, identified by coloured markers, via a digital imaging correlation and tracking method. To facilitate the post-processing analysis via digital imaging correlation, the colour of the markers should be chosen to enhance the contrast with the tessellation.
Different origami patterns may require different experimental set-ups. For instance, the Kresling pattern couples axial displacement (contraction/expansion) with twist, leading to non-rigid origami behaviour. To experimentally investigate this behaviour, fixtures that decouple the deformation modes are needed 64 . Figure 3 illustrates the experimental set-ups for either compression or torsion tests on individual Kresling cells as well as generic Kresling arrays. The arrays can be composed of an odd or an even number of cells, without any constraint on the chiral arrangement of any cell or group of cells. Both set-ups comprise two fixtures that connect the top and bottom surfaces of the Kresling origami to the loading frame machine, as illustrated in Fig. 3a. Both fixtures are equipped with multiple miniaturized magnets, ensuring connections between the Kresling and the loading frame, as shown in Fig. 3a,d. In both set-ups, the bottom fixture is the same and effectively restrains both rotational and axial movements, preventing any undesired rigid motion of the Kresling samples during the tests. Figure 3c,f shows the snapshots recorded during the experiments. During the tests, axial force and twisting moment are recorded using a force/torque sensor. Additionally, the axial displacement can be measured with a displacement transducer. For further details, the reader is referred to ref. 64.
The main data acquired during origami uniaxial testing are the load applied at one end of the sample, the length evolution of the sample and the transversal and longitudinal deformation during the folding/ unfolding process. A data acquisition system usually consists of a load cell and a displacement transducer to collect loads and displacements, respectively. The acquisition system could be part of the loading frame used to perform the test or external to the testing machine. In the former case, the data acquisition is more straightforward but not as versatile as in an external system, where there are no limits on the number and type of transducers. In the case of an external acquisition system, it is common to use a cDaq (by National Instruments) or Arduino systems interfaced with a PC through software written either in Lab-VIEW or MATLAB/Simulink. Analysis of the experimental data (load/ displacement) can be performed in a spreadsheet (such as Office Excel, LibreOffice Calc, IWork Numbers) or in programming languages (such as Mathematica, MATLAB or Python). A tailored experimental method is needed for monitoring the transversal and longitudinal deformation of the pattern together with the execution of the test. For this purpose, a digital imaging correlation analysis and tracking method is essential for the frame by frame analysis of the recordings of the experiments and for estimating the motion of the markers previously located on the vertices of the pattern unit cell. Several strategies can be adopted to perform this analysis. The simplest way is manually analysing a limited number of frames extracted from the recorded movie through opensource software (for example, VLC, Quick Time combined with GIMP, or nanoCad). If a frame by frame analysis of the movie is required, this can be performed by ImageJ filament open-source software or through in-house software written in Mathematica, MATLAB or Python.
The process of shape morphing in origami tessellations necessitates the utilization of external sources for actuation. The most employed methods for actuating origami structures include pneumatic actuation and magnetic actuation. However, the emergence of 4D printing has introduced the possibility of using stimuli-responsive materials that can respond, for instance, to light, heat or humidity (hygroscopic control).
Fluidic actuation. Fluidic actuation of origami involves using pressurized air or gas to control and manipulate origami structures 51 . In this context, as the origami samples need to be inflated they must be made impermeable to prevent air leakage. Typically, this is accomplished by coating origami structures with a thin layer of polydimethylsiloxane or by manufacturing them using plastic sheets with creases engraved onto them. To operate this method, the structure must be linked to an air supply system, such as a compressor or a simple air pump. This system allows for the regulation of both air pressure and flow rate. As this actuation approach is tethered, the structure being activated
0123456789();: Primer Ball bearing Free rotating plate Kresling Magnet Bottom plate (non-rotating) a c d f e b Torsional actuator and rotational transducer Linear actuator and displacement transducer Force/torque sensor 0123456789();: Primer needs to be equipped with an inlet for connecting the air source to the origami structure through flexible tubing.
Magnetic actuation. Magnetic actuation involves the utilization of magnetic fields to govern and manipulate origami structures that are crafted from materials that respond to magnetic forces 65,66 . This is accomplished by integrating magnetic components into the origami structures, which are magnetized beforehand. Usually, these magnetic components are created by blending a silicone rubber precursor with specific proportioning of magnetic microparticles. The magnetic characteristics of these components are determined using a magnetometer. Subsequently, actuation is typically achieved using, for example, a 3D Helmholtz coil system capable of generating a uniform magnetic field, the direction and intensity of which can be altered by adjusting the current flowing through the coils. The primary advantage of this method is that it enables untethered control with fast actuation.
Typical properties exhibited by origami tessellations are characterized by tunability and programmability. Tunable properties arise by virtue of the various folded states of the system and can be changed through the application of an active stimulus (for example, force). The compliant folding of origami structures enables in situ control of the property they exhibit, making them excellent candidates for tunability. It has been demonstrated that several important mechanical properties, such as Poisson's ratio 30,31 , elastic band gaps 44,67 , thermal expansion coefficients 68 and anisotropic stiffness 37,69 , can be tunable in origami metamaterials. As the folding of origami is generally a smooth continuous process, the variation in the properties that are being tuned is also gradual. Programmability, on the other hand, refers to obtaining a property of interest based on variations in the design (for example, geometry). This enables an abrupt change in the properties of the origami tessellation with respect to the change in the programmability parameter 70 . Programmability parameters are typically associated with the geometric features of the panels or local defects that can be induced in the origami tessellations. In this section, the theoretical geometric results pertaining to kinematics of origami patterns, modelling frameworks of non-rigid origami and some relevant properties (Poisson's ratio, wave dynamics and multistability) are addressed.
The configuration of an origami structure is characterized by its crease pattern as well as the dihedral angle between panels that define the folded state. The edges of the panels correspond to the folding creases or boundaries of the structure, and the points of intersection of the creases are the vertices. The configurational analysis of origami can be carried out using concepts of spherical trigonometry to determine direct relations between the dihedral angles of the creases meeting at a vertex 71 , and further determine the DOFs of the vertex. The DOFs of the entire origami structure, which may comprise several vertices, can be found using compatibility constraints imposed on the creases connecting adjacent vertices.
The Miura-ori pattern. The Miura-ori pattern was originally designed for use in space applications such as deployable solar arrays 72 . The pattern can also be found in certain plant leaves in nature 73 . The Miura-ori is a periodic 2D tessellation (or lattice) with each unit cell having four parallelogram-shaped panels as shown in Fig. 4. The Miura-ori pattern is composed of degree-four vertices where each vertex has either three mountain folds and one valley fold or vice versa. The angle α denotes the smaller panel angle (Fig. 4). When all the panels are rigid, the Miura-ori pattern possesses a single DOF. All the dihedral angles between adjacent panels depend on one single arbitrary folding angle, such as the crease angle ϕ as shown in Fig. 4. The crease angle (ϕ or ψ) uniquely describes any partially folded state of the structure. In the fully developed state, all the dihedral angles are equal to π. In the com- pletely flat-folded state, the dihedral angles are either 0 or 2π and the structure is rigid. The relation between the two crease angles (ϕ and ψ) of the Miura-ori is given by:
The eggbox pattern. The eggbox pattern takes its name from its appearance in the partially folded state. The eggbox is a 2D tessellation formed with two types of degree-four vertices that are non-developable. One of the vertices has four mountain (or valley) creases. The other vertex has two mountain creases and two valley creases. The pattern has parallelogram-shaped panels arranged in a fully symmetric way as shown in Fig. 4. When all the panels are rigid, the eggbox pattern deforms as a deployable single-DOF structure. The eggbox pattern exhibits smooth folding from one flat-folded state, where two dihedral angles are zero and the other two are π, to the other flat-folded state where the dihedral angles swap the values. Similar to the Miura-ori, the relation between the crease angles (ϕ and ψ) of the eggbox is given by:
The waterbomb pattern. The most widely studied waterbomb origami 74 comprises degree-six vertices, as shown in Fig. 4. Depending on the tessellation, the degree-six waterbomb unit cell can fold in different curved configurations. Figure 4 shows the tessellation of a waterbomb unit cell with four mountain and two valley creases intersecting at a common vertex 75 . In general, the degree-six waterbomb unit cell has three DOFs. However, if the folding mode is restricted to be locally symmetrical (four mountain creases with the same dihedral angle), then the configuration of the structure at any partially folded state can be obtained from a single folding angle chosen as an independent variable. The relation between the folding angles ξ and γ is given by 76 :
The Kresling tube. The triangulated cylinder/tube of the Kresling pattern undergoes non-rigid folding. The tessellation is formed by connecting unit cells along one direction as shown in Fig. 4. The crease pattern of a single unit cell is shown in Fig. 4 and is characterized independently by the parameters a, b, α and n, where n represents the number of edges (or cells) in each unit cell. The parameter c is also defined as shown in the figure. The unit cell can exhibit bistability depending on its geometric parameters and materials. The two stable states of the unit cell are denoted zero and one, and the corresponding folded configurations are characterized by the twist angles ψ 0 or ψ 1 , respectively, and the unit cell heights h 0 or h 1 , respectively. The two stable states are characterized by a geometric equivalence of the crease pattern parameters a, c and α. Therefore, enforcing these parameters
0123456789();: Primer Miura-ori Eggbox Waterbomb Kresling γ γ γ γ 2π-ξ ψ α α a a α a α a α α α α a a ф b b b b b b b c b b b Tessellation Tessellation Crease pattern before assembly ψ a α α ф b Unit cell Unit cell Unit cell Unit cell Folded state Top view Folded states Cylinder Tube Developed state Developed state 2a 2a M V V h i ψ i ψ α a α α α ф b Folded states b b 2a α a b α α Nature Reviews Methods Primers | (2024) 4:40 11 0123456789();:
Primer to be the same in the two stable states, they can be calculated 71 as follows:
( )
Some of the pioneering studies on origami structures focused on their rigid origami kinematics, which are of key interest for deployable structures and robotic applications. Some well-known software in this category includes the Rigid Origami Simulator that runs a projectioncorrection algorithm for solving linearized compatibility equations 22 , the GPU-accelerated Origami Simulator 77 and the Rhino plug-in Crane 78 .
However, in the past decade, interest in the use of origami for mechanical and civil engineering applications such as energy absorption, vibration control and load-bearing has gained prominence. Investigation of origami for such applications requires modelling the non-rigid behaviour of origami and carrying out structural analysis simulations. Besides shell-based finite element analysis, structural analysis of origami is typically carried out using an efficient reduced order model called the bar and hinge model 79 , which captures the folding deformations, bending of panels and in-plane stretching of panels. In this model, the panels are replaced by bars that are placed along all the creases of the origami pattern as well as the panel diagonals. The stiffness associated with folding and panel bending is modelled through rotational springs or hinges between the triangulated trusstype panels. Some of the first bar and hinge models 79 were formulated to capture small deformations of origami tessellations during structural analysis. Later, improvements were made to bar and hinge models to capture various features such as isotropic panel stretching 37 , complex panel-bending deformations and large non-linear deformations 63 . For example, Fig. 5a shows bending of an eggbox tessellation using MERLIN software that implements a non-linear mechanics formulation associated with the bar and hinge model 63 . Figure 5b shows representative simulation results by MERLIN, obtained from a uniaxial tension test of origami tessellations, that show good agreement with experimental data.
Poisson's ratio is an important property that dictates the deformation of materials. The magnitude of Poisson's ratio of linear elastic isotropic materials is restricted within a small range from -1 to +0.5. Interestingly, origami metamaterials can exhibit Poisson effects that vary with the folded geometry of the structure 30 and are beyond the conventional range as they are typically not isotropic. For example, the Miura-ori pattern can exhibit in-plane Poisson's ratio values all the way from negative infinity to zero, achieving the extreme values when it is in a developed state or the flat-folded state. The negative Poisson effect in Miura-ori makes it an auxetic metamaterial. On the other hand, the eggbox pattern displays tunable in-plane Poisson's ratio values from zero to positive infinity as it folds from one flat-folded state to the other. The Poisson's ratio results for Miura-ori and eggbox origami metamaterials are shown in Fig. 5b, where good agreement between theory and experiments is obtained. Recently, novel origami patterns were discovered, namely Morph 31 (see Fig. 5b) and Trimorph 32 , which can exhibit any real value of Poisson's ratio (negative infinity to positive infinity) depending on their folded configuration. Origami metamaterials exhibit a wide variation in the value of Poisson's ratio as they are being folded or deformed. However, recent research has shown that it is also possible to design origami patterns in such a way that the Poisson's ratio value is held nearly constant under large deformations 80 . All these results indicate the capability of origami metamaterials to be programmed or tuned to display desired Poisson effects.
Origami metamaterials have two major features that are useful for applications related to wave propagation and vibrations. First, the high contrasting levels of stiffness in the system in terms of crease folding, panel bending and panel stretching can lead to elastic band gaps, which denote ranges of frequencies where waves do not propagate. Second, the frequency range of the band gaps and other related characteristics can be easily tuned by compliant folding of origami. A technique called Bloch wave reduction can be employed to carry out wave propagation studies in periodic media such as origami tessellations 44 . Although typical calculations to study dynamics of tessellated structures could involve the consideration of several unit cells, the Bloch wave approach reduces the calculation effort to a single cell within the periodic system by virtue of its translational symmetry.
The unit cell, which forms the basis of the four tessellations considered, with geometrical parameters dictating their mechanics and configurations (left column). Miura-ori (from left to right): unit cell, unit module, tessellation in a deployed state and two images showcasing different folded states of a physical Miura-ori tessellation. Eggbox: non-developable nature of the pattern illustrated by its unit cell (left) requires that several strips cut from a flat sheet (middle-left) be assembled and glued together to obtain the desired tessellation (middle-right). Different folded states of a physical eggbox tessellation (right). Waterbomb (from left to right): unit cell, unit module, crease pattern and folded state. Kresling (from left to right): perspective and folded state top views of an eight-sided polygon unit cell, and one-piece crease pattern with reverse crease directions used to create the four-storey Kresling tube (right). a, b, side lengths; α, panel angle; ξ, γ, folding angles; φ, ψ, crease angles; c, height of the pattern; h, height of the Kresling unit cell; i (h i , ψ i ), 0 or 1 for the two stable states; M, mountain assignment; V, valley assignment.
Unlike several lattice structures, where the structural interactions are restricted to the nearest neighbouring nodes of a unit cell, origami metamaterials may exhibit non-local structural behaviour 44 . Hence, the application of Bloch boundary conditions should be careful enough to ensure that such (beyond nearest neighbour) nodal interactions are considered for accurate prediction of dynamic properties. Using such an approach, researchers have studied wave dynamics and bandgap structures in origami metamaterials. For example, modal characteristics of standard Miura-ori and eggbox patterns revealed the presence of elastic band gaps that are tunable by virtue of their folded configuration and programmable by virtue of their panel geometry 44 . That is, the frequency ranges of the band gaps obtained could be controlled (in theory) by subjecting the origami tessellation to folding, or by redesign of panel sizes or angles. In another study on wave dynamics, 1D origami-based lattice structures with triangulated cylinders were designed and experimentally found to exhibit rarefaction waves for applications in impact mitigation 45 , as shown in Fig. 5c. There are two aspects that influence the tunable dynamic properties of origami lattice structures: the tunable geometry of the system by virtue of folding; and the mass and stiffness of panels and hinges. The former aspect is typically independent of the base material with which the origami structure is made. The latter aspect depends on the material used, and therefore influences the natural frequency ranges and, to some extent, the modal characteristics. Hence, the material should be chosen based on the target frequencies relevant to the engineering application. However, the tunability of the dynamic properties can be explored as a function of the choice of base material.
Some origami structures experience multiple stable states such that each of them locks a particular configuration. These structures with multiple stable states can serve as the basic unit cell for architected materials, creating properties that differ from those of traditional materials. For example, the snap-through action (for example, the deformation process between different stable states) of the Kresling origami can be activated by non-contact forces, such as those generated by magnetic fields 46 . In addition, the multistable origami concept can be used to create a tessellation with switchable mechanical properties:
the Trimorph pattern, made by tessellation of tristable origami unit cells, can switch between different metastable phases to produce distinct stiffness, anisotropy and Poisson's ratio 32 , as shown in Fig. 5d.
Conceptually, the multistability in origami structures usually comes from two types of incompatibility. The first is the incompatibility between the stress-free states of the folding hinges between the origami panels. This happens when the zero-energy, stress-free states of the folding hinges (or rest angles) of an origami structure are not compatible with its rigid folding kinematics, such that the folding hinges can never be all stress-free at the same time 81 . The second type is the incompatible panel geometries that forbid rigid folding, such as the square twist, Kresling pattern and many others 11,36,45,46,70,82 . Overcoming rigid foldability requires bending and stretching of the panels, which creates energy barriers between stable states where panels are usually flat and unstretched. Both types of incompatibility could be present in one pattern, at the same time, such as in the Trimorph pattern, in which the line defect is caused by the first source whereas the point defect is caused by the second source 32 .
Owing to the inherent cross-disciplinary nature of its design principles, origami is a rich source of inspiration for creating cutting-edge materials and structures with a broad range of applications. A particular example of successful engineering application is foldcore sandwich panels 83 . A non-exhaustive list includes applications in engineering 15 , physics 31 , material science 30 , microrobotics 84 , waveguiding 85 , impact mitigation 45 , space structures (starshade origami), solar technologies 86 and artificial muscles 87 . Some distinguishing characteristics of origami are scalability and high deployability 84,[88][89][90] . Scalability broadens the spectrum of applications across multiple length scales as the behaviour of origami structures is primarily governed by geometry in several cases. High deployability enables extremely reconfigurable shapes and tunable mechanical properties in static and dynamic regimes to be achieved. Thus, origami principles can inspire design of materials and structures with myriad applications 87,[91][92][93] . Origami has generated ideas for futuristic infrastructure development, has inspired the creation of several artificial materials concepts, including metamaterials, has been used to create soft robots and medical devices and has enabled efficient transport of large structures in space applications. It is to be noted that the application areas discussed below are not mutually exclusive. Applications towards one topic (for example, mechanical metamaterials or soft robotics) can have relevance to applications related to another topic (for example, infrastructure or medical devices).
Novel origami-based mechanisms have been implemented in architecture to create responsive building skins and adaptive diagrid façades capable of maximizing solar shading, acoustic performance, energy efficiency and structural performance 86,94 . The energy efficiency interventions on buildings, such as wall insulation, usually remain fixed over time regardless of the climatic conditions. On the contrary, the use of kinetic origami-based building skins can lead to maximizing energy efficiency at all hours of the day, optimizing the relation between internal comfort and exterior climate conditions. For example, some studies showed that kinetic origami-based facades optimize daylight by approximately 50% from March to September and about 30% from October to February, compared with the case of static façades, thus improving indoor visual and thermal comfort 95,96 . Another example is the futuristic shape-adaptive shading origami-based system installed on the facades of Al Bahar Towers, located in the financial centre of Abu Dhabi 97,98 . The motile façade installed on these towers reduces the internal temperature by 50% with a substantial decrease in energy consumption for air conditioning, diminishing CO 2 emissions by 1,750 tons per year 99 . Furthermore, origami-based systems are starting to be used to create kinetic solar arrays capable of tracking sun motion and, thus, maximizing solar energy intake 100,101 .
Origami principles of reconfigurability and deployability can offer a valuable contribution to creating robust and resilient buildings, potentially minimizing material usage, thus leading to a dramatic reduction of embodied CO 2 emission 96,102,103 . In this context, there have been several efforts by researchers to obtain large-scale deployable structures inspired by origami behaviour. For example, pneumatic and multistable origami structures have been shown to allow the design of large-scale structures that can be deployed from a very compact configuration 33 , as shown in Fig. 6a (top). Another example involves lightweight canopy structures that can be realized by coupling and stacking of origami tubes in different directions 37 , as shown Fig. 6a (middle). This leads to deployable roofs with high out-of-plane stiffness. Other researchers used modified geometries to create novel accordion-type shelters with improved structural stability and stiffness 104 , as shown in Fig. 6a (bottom).
Metamaterials are artificial materials that can exhibit exotic properties superior to constituent materials 105 . Mechanical metamaterials are a subclass of metamaterials whose properties of interest are mechanical in nature, such as acoustic, thermal or elastic. Typically, metamaterials are obtained by repeating a unit cell whose geometry is designed once and cannot vary over time. Contrastingly, origami can transform its geometry continuously from a folded to an unfolded configuration, thus widening the design space. As metamaterial properties are strictly related to the unit cell geometry, origami permits achieving extremely tunable and reprogrammable mechanical properties [106][107][108] . Moreover, the relationship between stiffness of the panels and the creases provides opportunities to design the energy landscape of the resultant metamaterials [109][110][111] . This is useful to attain multistability, self-deployment or mechanical computing systems 112 . Origami design principles have also been successfully applied to conceive origami-based metamaterials exhibiting auxetic behaviour 30,113,114 , tunable Poisson's ratio 36,38 , self-locking 115,116 , high strength-to-weight ratio 117,118 and tunable stiffness 39,111,119,120 . Some researchers have shown that, by properly designing kinematic paths, it is possible to achieve lockable and flatfoldable modes 38,121 (Fig. 6b). In addition to extreme static mechanical properties, origami-based metamaterials permit the achievement of tunable dynamic properties. For instance, the morphing of the origami tessellation leads to controlling electromagnetic or elastic waves 122,123 , deflecting light 124 , and opening and widening band gaps 44,85,[125][126][127] . Origami metamaterials can also exhibit interesting coupling behaviours that are not typically observed in conventional materials, such as shear-normal coupling 32 and compression-twist coupling effects 128 .
In recent years, origami has been a source of inspiration for several applications in robotics. Soft robotics applications require gradual changes in stiffness and reconfigurability. Origami structures are well known to exhibit multistability, tunable stiffness and shape morphing, and hence can meet the design requirements for soft robotics. Exploiting origami-based design principles, robotic arms have been realized 0123456789();: Primer through the uniaxial repetition of Kresling unit cells capable of multidirectional morphing, grasping objects and exploring hard-to-reach areas via magnetic actuation 46,65,129,130 , as shown in Fig. 6c. Within this framework, origami morphing capabilities allow the realization of grippers 131 , artificial hands that gently grasp fragile objects without breaking them [132][133][134] . Origami design principles can also inspire the creation of programmable artificial muscles with multidimensional actuation 87 . Origami robots capable of multimodal deformations 51,129 and navigation (for example, crawling, jumping and swimming) indicate the potential for advanced applications in this field.
The progress in rapid prototyping technology and the rising demand for increasingly miniaturized tools fostered the implementation of origami design in the biomedical industry for in vivo or ex vivo purposes. The scalability, high deployability and extreme packaging capabilities of origami make them particularly suitable for minimally invasive medical devices 135,136 . The auxetic nature of several origami patterns permits the conception of novel surgical tools that can access the human body through a small incision, travel in a very compact state to the intervention destination and deploy in their functional shape to execute the surgery. Origami designs have been explored for facilitating the execution of biopsy 137 , MRI-guided radiofrequency ablation and catheter insertion 138 or inspection of hard-to-reach sites 139 . Origami-based deployable surgical reactors have been designed for potential use in face-lift operations 140 and adopted to create novel orthopaedic implants 141,142 and tissue scaffolds 143,144 . Origami-based self-folding microrobots have been realized to enable encapsulation, gastrointestinal microsurgeries 145 and drug delivery 146 . Origami-based structures have been conceived to assist retinal microsurgery 89,147 and as a support system for optimizing the insertion of flexible instruments in robot-assisted procedures 148 . Within this framework, origami-based microgrippers Primer have been designed for the capture and retrieval of objects and biopsies 149,150 , minimizing invasiveness and potential human errors.
Space structures, such as solar arrays or satellites, are large-scale structures that need to be compactly transported and deployed in target orbits. Efficient transport of such structures requires them to be lightweight and able to be folded into a small volume. Origami patterns can be implemented in the design of extremely lightweight, densely packed membranes and devices to be deployed once placed at target locations in space. Deployable membranes can serve as solar sails, propelled by solar wind, for exploration missions; as solar power arrays for satellites, for solar energy collection, conversion and transmission; as reflect array antennas; as space telescopes; and as protective shields 151,152 . The Miura-ori pattern was a key invention that enabled compaction and deployment of space structures 72,[153][154][155] . Design of origami-based solar sails has evolved over years, for example, with a different deployment technique that uses radial segments, wrapped in spirals around a central hub, that unfold tangentially 156 . The pattern resembles a moonflower just after opening, and is nicknamed 'origami flasher' 101,157,158 (starshade origami). Another variant, a four-quadrant square solar sail is extended in the diagonals by telescopic booms. The recent Kresling pattern 1,2,159,160 allows cylinders to be actively folded and unfolded, similar to bellows, and is applied to the Sunshield project of the IXO-Space Telescope 161 , the Mars Rover drill protection and for origami antennas of space-ground communication 81 .
Physical samples of origami are made with various methods. In general, the kinematics of origami structures is primarily driven by its geometry and is less sensitive to the materials being used, which is one of the many virtues of origami designs. However, deviations in the geometric structure, such as misalignment in the crease pattern, could lead to discrepancy from the expected behaviour. In a previous study, experiments and numerical simulations showed that geometric imperfections could hinder the foldability of origami structures and increase its compressive stiffness 47 . This effect is less apparent when the origami
A discrete Fourier transformationbased technique that enables efficient analysis of infinitely periodic lattice systems.
The crease pattern of a flat-foldable origami structure (for example, crane) can be coloured such that no two neighbouring regions are assigned the same colour while using as few colours as possible.
The pattern of creases left on the surface after an origami structure has been unfolded.
Marks left on the material surface after a fold has been unfolded.
Vertices delimited by four creases or four panels. For example, parallelogram-based origami is composed of degree-four vertices.
Origami that can be unfolded into a flat sheet without overlapping or deformation of the panels.
The angle between adjacent panels, which describes the current configuration.
The frequency range within which elastic wave propagation is prevented through the medium.
Appendages on the panels that can be placed and glued onto the matching seats.
Origami that can be folded into a flattened state of zero volume.
A sandwich structure consisting of thin stiff facesheets and thick, low-density core made of an architected material (for example, origami).
The angles that are required for one panel of an adjacent pair to rotate until it meets the other panel in a consistent direction, which could be either the dihedral angle or its supplementary angle.
The Japanese art of cutting and folding thin sheets of materials (for example, paper) from flat into three-dimensional objects.
Mechanisms built from stiff bars connected by freely rotating joints (rigid links).
The art of folding paper (or other surfaces) into three-dimensional shapes, usually from uncut squares or other continuous shapes.
The basic elements of an origami structure that occupy the area bounded by creases or borders of the surface.
The negative of the ratio of transverse strain to longitudinal strain in a material subject to uniaxial loading.
Origami that can be folded while keeping all regions of the surface (for example, paper) flat and all crease lines straight.
A small region near the point of load with non-uniform distribution of stress, where the effect of the exact form of the loading cannot be ignored.
Grooves made by material removal for precise placement of joints.
Coverings on a surface, using one or more patterns (tiles) with no overlaps and no gaps.
Primer is made with compliant (or soft) materials but becomes severe for stiff materials. For Miura-ori, the square residual of the Kawasaki-Justin condition (for example, Kawasaki excess) correlates approximately linearly with the increase in stiffness of imperfect Miura-ori structures 47 . Small geometric imperfections stiffen the originally programmed mode but generally do not alter the mode shape.
Origami structures are usually shared in data formats that store meshes, which must include at least two pieces of information: the coordinates of the vertices (or nodes); and the groups of vertices that belong to each face (or panel). A typical example of such a format is the OBJ format, which is supported by the Rigid Origami Simulator 77 , the Origamizer 162 and MERLIN 63 . Although the STL format is the most common for 3D printing and animation editing, it is not suitable for origami because all faces in STL must be triangular, whereas polygonal faces are common in origami designs. Recently, researchers formulated the flexible origami list data structure (FOLD) format that is dedicated to origami design 163 . Compared with the standard industrial formats, the FOLD format not only stores the static information of an origami design but also allows frames to be collected that depict the folding process of an origami structure. For flat-folded origami, the FOLD format includes the topological stacking order of faces that overlap geometrically to distinguish different folded states. The FOLD format is gradually becoming popular among the origami community. Origami Simulator 21 and MERLIN 63 support the FOLD format. Note that all formats mentioned above apply to both partially folded origami and its flat crease pattern.
Most current developments in origami engineering are based on an adaptation of existing origami patterns or are those that gained attention by serendipitous discovery. Principles-based systematic design of new origami patterns that can exhibit exotic engineering properties is a challenging problem. Although there have been developments in the recent years towards inverse design of origami for engineering applications, they are still in the rudimentary stages. The theoretical or numerical design of origami patterns for target requirements is an active area of research [164][165][166] and more developments are needed in this direction, to make origami engineering versatile across applications. The challenges arise from two contexts -one related to the choice of design variable space and the other related to the choice of engineering properties that are explored. Further, in most cases, the designs must satisfy requirements related to developability, flat foldability or rigid folding, which constraints the solution space. In terms of applications, the design of crease hinges, especially in the context of thick origami, is not systematically investigated and hinders largescale and reliable use of origami structures made from thick sheets of material or thick-panel prisms.
Behaviour of origami patterns is particularly influenced by the presence of imperfections in their geometry. Most studies on origami patterns demonstrate the results through prototypes made from regular symmetric structures with no imperfections. However, the presence of geometric imperfections in the patterns can lead to unexpected behaviour. Hence, extensive studies by incorporating imperfections owing to manufacturing defects should be performed while investigating the suitability of origami for applications.
Limitations of efficient manufacturability of origami structures also exist, especially in the context of non-developable patterns.
Most current methods involve manual assembling of individual modules to form the final tessellated structure. Further, in the context of developable patterns, the tessellated structures need manual folding through creases marked on the entire sheet. Advanced manufacturing techniques such as 3D printing can be investigated to seek potential alternatives to these limitations.
Exploring micro-scale sample experimentation remains a relatively unexplored territory demanding further investigation. From a mathematical point of view, origami structures exhibit scale-independent properties. However, from an engineering perspective, larger-scale experiments behave differently from microscale origami, as the process of miniaturizing origami introduces significant complexities, owing to the changing underlying physics governing the mechanical response at different scales. These complexities encompass various factors, including the substantial impact of Van der Waals forces on the mechanical response of micro-scale origami, the intricate challenges associated with manufacturing owing to limited materials and manufacturing technologies capable of achieving nanometre precision, as well as the development of miniature testing platforms.
Origami structures can be highly kinematic in nature with multiple DOFs. Thorough analysis of the structure's mobility should be carried out before adapting origami for applications. If unwanted modes of mobility are found to exist, appropriate constraints should be applied on the structure to lock or guide it into desired configurations.
Although origami science is a more mature field, origami engineering has the potential to revolutionize the fields of engineering and design by enabling the creation of complex, lightweight and adaptable structures across scales, which can be manipulated and deployed in various applications. Beyond modelling and manufacturing of origami, automated folding and actuation can lead to major advances especially with multi-physics considerations (for example, magnetic actuation 46,167 ). Thus, origami engineering has potential for continued advances in several fields ranging from robotics to space structures, architecture and medical technology. The latter field holds great potential, especially as medical doctors and engineers collaborate in a true interdisciplinary fashion including actual experiments in animals with potential for further findings in humans. Modelling of an origami structure is problem-specific, and therefore a single framework may not be readily applicable. Therefore, researchers have adopted context-specific modelling approaches for origami. As origami ideas permeate different fields, several origami modelling frameworks and packages are expected to be developed in the future. Alongside, one could expect various design approaches to also be developed. The material used to make the origami may play a key role and challenge the development of manufacturing processes for origami-based structures. Although many recent studies on origami use paper-based models to demonstrate ideas, real applications -depending on the length scale of interest -could use various materials ranging from polymeric materials at the micron scale 58 to construction materials such as steel or concrete 168 that could help reshape our infrastructure towards more sustainable solutions. Other fascinating materials to be explored are biological materials and living tissues, which could revolutionize the medical field 169 .
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