Here, we designed a ring-like unit cell with multiple deformation modes. In Figure 1A, the ring-like unit cell comprises four “z”-shaped structures (also in the gray frame in Figure 1C), which are folded with mountain (solid line) and valley (dashed line) creases and formed into 3D configurations from the 2D patterns by taping the edges (see purple edges in Figure 1A and see also Supplementary Figure S1), where the opposite “z”-shaped structures are centrosymmetric. γ 1 and γ 2 are important planar design angles on the facets of the opposite “z”-shaped structures, as shown in Figure 1A. For a regular unit, the “z”-shaped structure with γ 1 is perpendicular to that with γ 2 , that is, β = 90 ° . The length parameters m , n , and q are identical for the four “z”-shaped structures (in this study, m = n = 2 q ), and only the angle parameters γ 1 and γ 2 are different. Deformation angles θ 1 and   θ 2 are defined as the dihedral angles between two facets, corresponding to the “z”-shaped structures with γ 1 and γ 2 , respectively (Figure 1B top, pink, and blue dihedrals). Based on the requirement of geometrical compatibility (the same height in z-direction), there is a relation between θ 1 and   θ 2 (Figure 1B bottom; see Eq. 11). The panels of the kirigami structure are assumed to be rigid. The Miura origami is a specific case of our kirigami design, which is shown in Section 8 of SI.

The ring-like unit samples were fabricated using the Strathmore 500 Series 3-ply Bristol card stock that was laser cut based on a design pattern generated using Mathematica 11.2. The edges of the given panels were glued to be connected (Figure 1A) to build 3D units for flexible foldability. See SI Section 6 for details.

The calculation methods for the sizes of a ring-like unit and four-unit combination comprising four ring-like units and the design method for targeted Poisson’s ratios are given in SI Sections 1–4.

The top box, camera, and bottom box were connected and moved together with the test head under the control of a universal testing machine. The sample was put on a motionless substrate. When the bottom box touched the sample, it was deformed and captured using a camera. Then, the binary images of the sample at different times were obtained, and the sample in each image was enclosed within a minimum enclosed rectangle, and then, the x- and y-sizes were obtained. Finally, the real size of the sample was obtained by using a conversion factor of 0.4167 mm/pixel. The experimental setup is shown in Figure 3F.

Once θ 1 is fixed, the height in the z-direction of the whole ring-like structure is determined, and then, θ 2 is determined. With different initial settings of θ 1 = θ 2 = 0 and θ 1 = 360 ° − θ 2 = 0 , there are two deformation paths, that is, ① → ② → ③ and ④ → ⑤ → ⑥ , in the θ 1 −   θ 2 space (see Figure 1B for the plot and Figure 1C for the configurations of ① ∼ ⑥ ). For γ 1 < γ 2 , the two paths are disconnected. However, for γ 1 = γ 2 , the two paths are connected at point O, and points ② and ⑤ overlap, resulting in the connected “X”-shaped paths (Figure 1B plot). Figure 1C shows the discrete configuration examples ① ∼ ⑥ of a unit with γ 1 < γ 2 and γ 1 = γ 2 and the convex ( θ 2 < 180 ° ) and concave ( θ 1 > 180 ° ) parts in a unit cell. Figures 1B,C show that these discrete configurations are realized by continuously changing θ 1 . In fact, for γ 1 < γ 2 , configuration   ② can be switched to ⑤ with panel bending, and vice versa. Here, we study only the case of γ 1 ≤ γ 2 , as the case of γ 1 ≥ γ 2 can be known by swapping the current γ 1 and γ 2 , which means that the unit is rotated by 90 ° . See SI Sections 1 and 2 for the detailed geometric model.

Here, the ring-like units possess various deformation behaviors in the x- and y-directions. Figure 1D shows the isotropic deformations for γ 1 = γ 2 = 60 ° along paths ① → ② and ⑤ → ⑥ , where the nominal x- and y-strains are equal ( Δ x x 0 = Δ y y 0 , see the overlapped blue and red nonsolid lines, where the nonsolid lines are pertaining to γ 1 = γ 2 = 60 ° , and the solid lines are pertaining to γ 1 = 45 ° and γ 2 = 60 ° ), and the anisotropic deformations ( Δ x x 0 ≠ Δ y y 0 ) for γ 1 = γ 2 = 60 ° along paths ② → ③ and ④ → ⑤ and for γ 1 = 45 ° and γ 2 = 60 ° along all paths. The nominal z-strain is always symmetrical about θ 1 = 180 ° (see Figure 1D inset). Here, the nominal strain is defined as Δ s s 0 = s s 0 − 1 , where s = x , y , z denotes the dimension and s 0 = s | θ 1 = 180 ° denotes the initial dimension. Figure 1D implies that we may obtain arbitrary combinations of different Poisson’s ratios along the x- and y-directions. The relations between the nominal z-strain and the nominal x- and y-strains are shown in Supplementary Figure S3, which is a variant of Figure 1D.

There are two degrees of freedom (DOFs) of the ring-like unit, that is, θ 1 and β . To explain the two DOFs, Figures 2a and c show that the ring-like unit can be skewed by β ≠ 90 ° with fixed θ 1 compared to that with β = 90 ° (Figure 2B), with the opposite “z”-shaped structures remain parallel. We defined the dimensions of the units by aligning the AB side in the y-direction, as shown in Figure 2A, and found that with the continuous change of β , the ring-like unit shows a continuous shearing deformation mode. The size changes in the x- and y-directions under different values of β are shown in Figures 2E, F, respectively. This shows that θ 1 controls the opening and closing mode of the structure, while β controls the shearing mode.

To demonstrate the analytical geometrical model with experimental data, we compressed a paper-made ring-like unit (Figure 3) in the z-direction using a universal testing machine. Here, the unit with γ 1 = 45 ° and γ 2 = 60 ° is programmed into four discrete patterns ( ① , ③ , ④ , and ⑥ corresponding to Figure 1C; the sample cannot stay at θ 1 = 180 ° , i.e., transitional patterns ② and ⑤ ) and compressed in the z-direction. From the top, a camera is used to obtain a video of the unit deformation process to calculate the x- and y-dimensions (see Figure 3F for the experimental setup and Supplementary Videos S1–4). The measured x- and y-dimensions are shown in Figures 3A–D and compared to the analytical curves. When the unit is shrunk in a given direction under compression, the scraping and friction between the unit and substrate become apparent; thus, the measured dimensions oscillate around the analytical values (see Figures 3B–D). Figure 3E is used to show the sample layout direction compared with the model. The results of the model and experiment agree with each other well. The experimental setup is shown in Figure 3F.

After validating the analytical model, we focused on the inverse design of a unit with β = 90 ° for targeted Poisson’s ratios as the x- and y-dimensions can be clearly defined with β = 90 ° (see Figure 2B). Poisson’s ratios in the x- and y-directions under compression in the z-direction are calculated by ν z x = − d x / x d z / z and ν z y = − d y / y d z / z [31] based on the definition in Figure 1B. Even with γ 1 = γ 2 , the ring-like unit shows various Poisson’s ratios in different paths. In Figure 4A, with γ 1 = γ 2 = 60 ° for deformation path ① → ② , we have ν z x > 0 and ν z y > 0 ; for ② → ③ , ν z x < 0 and ν z y > 0 ; for ④ → ⑤ , ν z x > 0 and ν z y < 0 ; and for ⑤ → ⑥ , ν z x < 0 and ν z y < 0 . Therefore, we can expect more various Poisson’s ratios of a unit cell with γ 1 ≠ γ 2 . Taking γ 2 = 60 ° as example, we plotted Poisson’s ratio curves ( ν z x − ν z y relations) with γ 1 = 60 ° , 59 ° , and 55 ° for paths ① → ② , ② → ③ , ④ → ⑤ , and ⑤ → ⑥ as shown in Figure 4B. If we plot the ν z x − ν z y curves within γ 1 ≤ γ 2 = 60 ° , the curves would fill the regions A (pink), B (blue), C (yellow), and D (green), which are the half of ν z x − ν z y space (i.e., regions A, B, C, and D and regions A′, B′, C′, and D′ are symmetric about ν z x = ν z y , see Figure 4B). Also, this does not limit to γ 2 = 60 ° , which implies that the curves would fill the other half of ν z x − ν z y space (i.e., regions A’+B’+C’+D′) by swapping the x-size and y-size (or swapping γ 1 and γ 2 ), which means that a unit cell is rotated by 90 ° . For example, if the design parameters γ 1 = τ and γ 2 = ω obtain the point ( ν z x , ν z y ) = ( ν τ , ν ω ) in the Poisson’s ratio space, then the point ( ν z x , ν z y ) = ( ν ω , ν τ ) can be obtained by γ 1 = ω and γ 2 = τ .

The optimization of the structures for given Poisson’s ratios in the x- and y-directions can be achieved by minimizing the error, δ = ( ν z x − ν x ) 2 + ( ν z y − ν y ) 2 , subjected to 0 < θ 1 ,   θ 2 < 360 ° , and 0 < γ 1 ≤ γ 2 < 90 ° (see SI for details), where ( ν x , ν y ) are the targeted Poisson’s ratios. To test this method, we chose six points, I, I′, II, III, III′, and IV, in the Poisson’s ratio space to represent the six targeted pairs of Poisson’s ratios, as shown in Figure 4C, where points I, II, III, and IV are four vertices of a rectangle, and I′ and III′ are the symmetrical points of I and III about ν z x = ν z y . As a result, we obtained θ 1 = 106.8 ° ,   θ 2 = 84 ° , γ 1 = 46.2 ° , and γ 2 = 60 ° for point I with ( ν x , ν y ) = ( 0.2 ,   0.05 ) . At point I′ with ( ν x , ν y ) = ( 0.05 ,   0.2 ) , we have the structure with θ 2 = 106.8 ° ,   θ 1 = 84 ° , γ 2 = 46.2 ° , and γ 1 = 60 ° , which is shown as the structure I′ in Figure 4D. Other optimal design results for points II, III, III′, and IV are also shown in Figures 4C,D (with the resulting parameters listed in the caption of Figure 4), where Figure 4C shows the ν z x − ν z y curves passing through the six targeted points (left) and the corresponding parameters in the 3D design space ( θ 1 , γ 1 , γ 2 ) (right), and Figure 4D presents the contour plots and minima of δ and the corresponding structural configurations. Taking configurations II and IV as examples, the convex and concave parts result in positive and negative Poisson’s ratios, respectively, but the specific values are determined by the design angles γ 1 and γ 2 . This optimization method provides an effective tool to design a ring-like unit with targeted Poisson’s ratios, and the minimum value of δ can be less than 10 − 12 for each case.

After understanding the deformation mechanism of one ring-like unit, here we focus on that with multiple units. Four identical ring-like units can be combined to form a four-unit combination, which can be periodically replicated to construct a 2D cellular structure, as shown in Figure 5A (in the gray shade, the four-unit combination is displayed with a yellow–pink–yellow–pink pattern because the diagonal ring-like units are with the same configuration). Although the four ring-like units are identically designed with γ 1 = 45 ° , γ 2 = 60 ° , and β = 90 ° , four deformation patterns of each ring-like unit (see Figure 1C, ① , ③ , ④ , and ⑥ ) make 4 × 4 = 16 discrete patterns for the four-unit combination. Figure 5B introduces one pattern of four-unit combinations in detail, and the other 15 patterns are shown in Figure 5C. Figure 5B shows the relation between the dimensionless sizes x c / n and y c / n (where x c and y c are defined as the sizes of the four-unit combination in the x- and y-directions, see also Supplementary Figure S2 and Eq. (14)). The four-unit combination is with two ring-like units in pattern ① (yellow) and the other two in pattern ③ (pink) (see Figure 5B). Here, the four ring-like units have the same value of θ 1 within 0 ≤ θ 1 ≤ 180 ° to maintain the same height in the z-direction, which is necessary given the geometrical compatibility requirements for building 3D cellular structures. In the plot of Figure 5B, there is a sharp corner in the x c / n − y c / n curve since, before the corner (stage 1), the yellow unit dominates the x c value, and then, the pink unit dominates the x c value (stage 2). Similar sharp corners, as well as smooth x c / n − y c / n curves, can be found in the other 15 patterns in Figure 5C. In fact, the 16 patterns can be divided into four groups based on the pattern of the diagonal ring-like units in yellow (groups #1, #2, #3, and #4 in Figure 5C, where the symbol “#a, b” means No. b in group #a), and within each group, the pattern of the yellow unit remains unchanged. Some size ranges related to the x- and y-dimensions of the four-unit combination are identical. For example, in group #2, the first two plots and last two plots individually have the same x c range, while the first and fourth plots and the second and third plots individually have the same y c range. This implies that different four-unit combinations may be again connected through the side with identical sizes to create a new unit (comprising 4n (n = 2, 3, 4, … ) basic ring-like units), then the number of deformation patterns of the resulting structure is greatly increased. This operation can be recursively implemented within a group (see the combination of #3.1 and #3.4 in Figure 5D) or across groups (see the combination of #3.1 and #2.1 in Figure 5D).

The shearing deformation of the 2D cellular structures with β ≠ 90 ° compared to the non-shearing case with β = 90 ° is shown in Figure 6. Three representatives of the 16 discrete patterns are chosen, as shown in Figures 6A–C, and the top edges of the upper-left-most yellow structures are aligned with the y-direction (Figure 6A). This is to show a rotation-like effect of the 2D cellular structures with shearing deformations, although the directions of key sides are fixed. This figure shows the potential ability to control mechanical waves in the shearing directions (see the possible shearing forces in Figure 6B), which is beyond the topic of this work but will be an interesting further work. The structures in Figure 6 (column 1–3) are shown on the x–y plane, and they can be stacked in the z-direction layer by layer to build 3D cellular structures (see Figure 6, column 4).

Finally, we explore another design of cellular structures consisting of multiple ring-like units with β = 90 ° (see Figure 7). Different from the manner in Figure 6, the 2D cellular structure is constructed by symmetrically connecting the ring-like units with the same pattern in the x- and y-direction, and the 3D cellular structure is also built by stacking 2D structures. Similar to the units discussed earlier, both the 2D and 3D structures are flat-foldable. By taking advantage of our analysis of a unit cell, we can design new 2D and 3D cellular structures exhibiting anisotropic and isotropic deformation in the x–y plane under uniaxial compression/tension in the z-direction. Similar to a ring-like unit with γ 1 = 45 ° and γ 2 = 60 ° (Figure 1D, solid lines), a 3D cellular structure comprising the same units shows two anisotropic deformation behaviors: for ② → ① , we have ν z x > 0 and ν z y > 0 but ν z x ≠ ν z y ; and for ② → ③ , ν z x < 0 and ν z y > 0 (Figure 7A); likewise, the 3D cellular structure shows two other anisotropic deformation behaviors: for ⑤ → ④ , ν z x > 0 , and ν z y < 0 ; and for ⑤ → ⑥ , ν z x < 0 and ν z y < 0 but ν z x ≠ ν z y (Figure 7B). Although ν z x and ν z y have the same sign, their values are different because of γ 1 ≠ γ 2 (see also a ring-like unit in Figure 1D with γ 1 = 45 ° and γ 2 = 60 ° , solid lines). Moreover, isotropic behaviors can be realized with the same value of γ 1 and γ 2 . Figure 7C shows two isotropic deformation behaviors with γ 1 = γ 2 = 60 ° : for ② → ① , ν z x > 0 and ν z y > 0 ; for ② → ⑥ , ν z x < 0 , and ν z y < 0 (see also a ring-like unit in paths ② → ① and ⑤ → ⑥ in Figure 1D with γ 1 = γ 2 = 60 ° , nonsolid lines, notice that configurations ② and ⑤ are identical, as shown in Figures 1B,C). These cellular structures also have only one DOF with β = 90 ° (folding and unfolding by manipulating one parameter, i.e., the deformation angle θ 1 ; see Figure 1B), which can be easily controlled as an origami robot. Using this construction method, we can realize the deformation behavior of a 3D cellular structure only based on that of a single unit.

The cellular structures in Figure 7 with 6 × 6 × 6 units look like 2.5D because for each unit, the height is less than the length or the width. But in essence, the structures are 3D since the layers can be continuously stacked along the z-direction, such as with 6 × 6 × 12 units. Here, we used 6 × 6 × 6 units for clear visualization.