The metamaterial unit, as shown in Fig. 1A, is composed of 4 isosceles right-angled triangular blocks with waist length a and thickness t, connected by 4 hinges (black arcs), which form a planar 4-bar linkage. A kinematic analysis of the unit indicates that it is a single degree of freedom (DOF) and bifurcates at α = 180° and β = 0°, where α and β are dihedral angles, thus leading to 2 distinct motion paths. On path 1 (the black line), the unit behaves as a planar 4-bar linkage with all 4 hinges active during motion following the relationship β = 180° − α. At the bifurcation point marked by a star, the upper and lower pairs of adjacent triangular blocks contact each other to form new, larger blocks that are geometrically similar to the initial ones. The unit then transitions from a 4-bar linkage to a 2-bar linkage, also with a single DOF, and moves along path 2 (the blue line) where α varies from 90° to 270°, while β remains at 0°. During this motion, the 2 hinges enclosed in the larger blocks do not rotate and are thus considered inactive (gray arcs), leaving only 2 active hinges.

This kinematic bifurcation behavior of the unit, when tessellated in a plane, allows the metamaterial to reconfigure into a variety of configurations and form hierarchies. To illustrate this, first consider the simplest case when 2 units are connected by 2 additional hinges (red arcs) to form a 1 × 2 metamaterial, as shown in Fig. 1B. Initially, the metamaterial can be considered as an assembly of two 4-bar linkages and still moves with a single DOF. When it reaches the bifurcation point where each unit becomes a 2-bar linkage, it transitions into a planar 4-bar linkage like the unit, but now each large triangular block consists of 2 basic ones. After reconfiguration, the number of active hinges is reduced from 10 to 4. Further extending the metamaterial to 2 × 2 by connecting two 1 × 2 metamaterials with 2 hinges (red arcs) as shown in Fig. 1C, it can still preserve the single DOF motion and kinematic bifurcation (detailed derivation in Note S1). Therefore, the 2 × 2 metamaterial first reconfigures into an assembly of two 4-bar linkages in which each triangular block comprises 2¹ basic ones, and then into a single 4-bar linkage where each triangular block comprises 2² basic ones. Correspondingly, we can define the hierarchy of the metamaterial based on the size of triangular blocks in the 4-bar linkages. Still taking the 2 × 2 metamaterial as an instance, at configuration I, only basic triangular blocks exist, and thus, the metamaterial has a single-level configuration with 4 level 0 units. At configuration II, each triangular block is formed by 2¹ basic ones, and the metamaterial still has a single-level configuration with 2 level 1 units. Similarly, at configuration III, the metamaterial has a single-level configuration with one level 2 unit, as each block is composed of 2² basic ones. The number of active hinges at the 3 levels is respectively 22, 14, and 6. It is worth noting that the metamaterials can actually generate more motion paths and bifurcation points than we have presented in Fig. 1, which are not discussed here as they are not closely related to the hierarchical levels mentioned above. See more details in Movie S1 and Note S1.

More generally, a 2^{⌊i/2⌋}\times 2^{⌊\left(i+1\right)/2⌋} metamaterial, constructed by 2ⁱ level 0 units, can be continuously reconfigured from level 0 to level i (i > 0), thus forming i + 1 single-level configurations, where ⌊i/2⌋ and ⌊(i + 1)/2⌋ denote the floor function (representing the largest integer less than or equal to the given value). We have recursively proven that these hierarchical metamaterials maintain a single DOF throughout their motion paths except at bifurcation points (Note S1).

Beyond single-level configurations, the metamaterial can also be reconfigured to contain units of different levels, thus forming various multi-level configurations. To explore this, we recursively analyze the number of multi-level configurations and the correlation with the number of units. It is obvious that the 1 × 2 metamaterial in Fig. 1B can only form single-level configurations, and thus, we start from the 2 × 2 metamaterial in Fig. 1C. If the 2 level 0 units at the bottom are reconfigured to one level 1 unit, while the other 2 at the top remain unchanged, a multi-level configuration with 18 active hinges featuring 2 level 0 units and one level 1 unit can be obtained (the blue one in Fig. 2A). Alternatively, reconfiguring the top 2 level 0 units into one level 1 unit while keeping the bottom 2 units unchanged yields another multi-level configuration with the same number of active hinges (the gray configuration in Fig. 2A). Since the gray configuration can be obtained by vertically mirroring the blue configuration, they are symmetry-equivalent [53] and expected to yield identical stiffness. Therefore, in this study, we exclude configurations that are symmetry-equivalent by symmetry operations (second-order rotational symmetry, vertical mirroring, or horizontal mirroring), using a developed screening procedure. See Note S2 and Supplementary Code for details. We retain only one representative configuration from each set of mutually symmetry-equivalent configurations, thereby identifying the symmetry-unique configurations. Thus, the 2 × 2 metamaterial has only one symmetry-unique configuration. For the 2 × 4 metamaterial, its multi-level configurations can be derived from the 2 × 2 metamaterial. Specifically, the 3 single-level and 2 multi-level configurations of the 2 × 2 metamaterial can be combined to form C_5^1{C}_5^1=5^2=25 configurations for the 2 × 4 metamaterial. Excluding the 3 single-level configurations (at levels 0, 1, and 2, respectively) constructed from the self-combinations of the single-level configurations, there are 5² − 3 = 22 multi-level configurations. Among them, 8 symmetry-unique configurations are shown in Fig. 2B: 5 level 0&1 configurations with varying numbers of level 0 and level 1 units, 1 level 0&2, 1 level 1&2, and 1 level 0&1&2.

Further, for the 4 × 4 metamaterial, the number of multi-level configurations is deduced from the 2 × 4 metamaterial. Through a similar analysis, it is found to have (22 + 4)² − 4 = 672 multi-level configurations, among which 81 are symmetry-unique ones. Figure 2C illustrates the paper prototypes of 4 representative multi-level configurations: 0&1, 0&1&2, 0&2&3, and 0&1&2&3.

To generalize, we denote the total number of multi-level configurations for a 2^{⌊i/2⌋}\times 2^{⌊\left(i+1\right)/2⌋} metamaterial with the highest level i as P_ⁱ, given by:

Adding the number of single-level configurations, the total number of configurations for metamaterials can be expressed as P_i + i + 1, where the number of symmetry-unique configurations can be determined by an automated screening procedure (Note S2). Figure 2D illustrates the exponential increase in the number of total and symmetry-unique configurations from 1 × 1 to 4 × 8 metamaterials. For instance, a 4 × 8 metamaterial would have 458,330 configurations, including 3,828 symmetry-unique configurations, fully demonstrating the reconfigurability of these hierarchical metamaterials.

Finally, all the configurations, including both the single-level and multi-level ones, can be mutually transferred through kinematic bifurcation, and the resultant metamaterial maintains a single DOF motion except for the bifurcation points (detailed derivation in Note S1). As an instance, the reconfiguration process of a 4 × 4 metamaterial among 10 configurations (5 single-level and 5 multi-level) is demonstrated in Movie S2.

The hierarchical reconfiguration, which leads to the variation of active hinges and single DOF motion behavior of the metamaterial, enables tunable stiffness over a wide range. Without losing generality, first consider the 2 × 2 metamaterial at the multi-level configuration as shown in Fig. 3A to illustrate this. Defining the angle between the vertical side of the metamaterial outer contour and the triangular block as θ/2, the active hinges of the metamaterial can be categorized into 6 groups (#1 to #6) with dihedral angles of 90° − θ, θ, 90° − θ, 90° + θ, 180° − θ, and θ, respectively. Among these, hinges #1 to #4 are the 4 basic hinges in the initial printed configuration of the metamaterial (Fig. S7). Hinge #5 is reconfigured from hinge #2, and hinge #6 is formed by merging 2 hinges (#4). The other 3 hinge types (#7 to #9) discussed in this paper can also be derived from the 4 basic hinges through reconfiguration or merging processes (see Notes S4 and S5 and Fig. S8 for details). When compressed along the x axis, governed by the single DOF motion kinematics, all the active hinges rotate by the same magnitude ∆θ. Since the triangular blocks are much stiffer than the hinges, the strain energy of the metamaterial, U, can be calculated as the summation of the strain energy of the hinges. Subsequently, the stiffness can be obtained by taking the second derivative of the strain energy with respect to the width W=2acos\left(\mathrm{\pi }/4-\theta /2\right) of the metamaterial as followsin which N is the number of active hinges and k is the rotational stiffness of each hinge. This rotational stiffness was measured after 14 rounds of thermal reconfiguration to ensure consistent performance, since the stiffness of the hinges stabilizes after this initial period (see Fig. S9 and Note S5 for details). For a given metamaterial, if θ is maintained when it is reconfigured to different levels so as to keep its outer contour unchanged, Eq. (2) is always applicable. Therefore, the stiffness in a particular configuration is only linearly correlated to the number of active hinges, provided that the hinge rotational stiffness is not affected by reconfiguration. Figure 3B presents all 4 configurations of the 2 × 2 metamaterial at different levels, which have 6, 14, 18, and 22 active hinges, respectively. Correspondingly, the stiffness can be tuned by a ratio of 3.67, the ratio of the maximum stiffness to the minimum stiffness.

More generally, the above conclusion can be extended to a 2^{⌊i/2⌋}\times 2^{⌊\left(i+1\right)/2⌋} metamaterial, which is constructed from 2ⁱ level 0 units and has a total of N_T = 3 × 2^{i + 1} − 2 hinges (Note S1). This metamaterial can be reconfigured into configurations composed of various units at different levels, where the number of level j units is denoted as u_j​ (j varies from 0 to i and u_j​ ≤ 2^{i – j}; details in Note S3). Given that inactive hinges only exist in level j units (j ≥ 1), the number of active hinges in this metamaterial is determined by subtracting the inactive hinges contributed by each level j unit (j ≥ 1). To distinguish from the overall metamaterial, we define the total number of hinges in a level j (j ≥ 1) unit as n_j,T, and the number of inactive hinges in that unit as n_j,inact. Specifically, a level j unit has a total of n_j,T = 3 × 2^{j + 1} − 2 hinges, but only 6 of them are active (Note S1), resulting in n_j,inact = 3 × 2^{j + 1} − 8 inactive hinges. Therefore, the number of active hinges of the metamaterial can be calculated as

Using the same approach, the compressive stiffness of the 2^{⌊i/2⌋}\times 2^{⌊\left(i+1\right)/2⌋} metamaterial in the x direction can also be derived as follows

The stiffness tunability ratio of the metamaterial can be calculated aswhere N_max = N_T = 3 × 2^{i + 1} − 2 is the total number of hinges in the metamaterial (all units at level 0) and N_min = 6 corresponds to the metamaterial at the single-level configuration with one level i unit. The stiffness tunability ratio is proportional to 2ⁱ, the number of level 0 units.

Moreover, the resolution of the tunable stiffness, R, which represents the difference between 2 neighboring stiffness values, is determined by the increase in inactive hinges when 2 level j units are reconfigured into one level j + 1 unit.

Finally, the compressive stiffness of the metamaterial in the y direction can also be obtained following the same procedure, which also shows a linear correlation with the number of active hinges (Note S3).

When the size of the metamaterial reaches 4 × 4 (i = 4), it includes 86 configurations with 20 different active hinge numbers of 6, 14, 22, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, and 94, leading to a wider tunability ratio of 15.67 (Fig. 3C). Further extending the metamaterial to 16 × 16 (i = 8) will result in a stiffness tunability ratio of 255.67 [Eq. (5)].

The 4 × 4 metamaterial was designed, fabricated, and tested to validate the theoretical model. Three-dimensional (3D) printing was adopted to manufacture the specimens with thermoplastic polyurethane (TPU) (Note S4). Quasi-static uniaxial compression tests were conducted on 6 typical configurations—levels 0, 1, 0&1&2, 2, 3, and 4—with 94, 62, 54, 30, 14, and 6 active hinges, respectively (Fig. 4D). The experimental setup is shown in Fig. 3E (see details in Movie S3 and Note S5). The experimental and theoretical stiffness values are compared in Fig. 3E, demonstrating a clear difference in stiffness between these configurations and a good agreement between the theoretical and experimental results. Furthermore, the stiffness of other 4 multi-level configurations (level 1&2, 2 different level 0&1&2, and level 0&3), each with 54 active hinges (Fig. S10C), was experimentally measured and found to be consistent with theoretical predictions (Fig. S10D and E). These results confirm that the theoretical model accurately predicts the stiffness of metamaterials.

To visualize the stiffness of the hierarchical metamaterial in real time, we integrate reconfigurable circuits in the metamaterial to map the transitions among units at different levels to the multichannel electrical outputs. This establishes a linear relationship between structure, properties, and information, enabling the metamaterial to display the number of active hinges and stiffness in various configurations without the need for external sensors. To illustrate this, we first consider the reconfiguration between single-level configurations from 2 level j units to 1 level j + 1 unit, during which the reduction in the number of active hinges N can be obtained from Eq. (6). These 2 configurations can be abstracted as the binary mechanical inputs A_j, where A_j = 0 for level j and A_j = 1 for level j + 1, to obtain the corresponding electrical outputs. When j = 0 (Fig. 4A), a logic operator is designed for this reconfiguration, including a buffer gate and a NOT gate, where the power input terminal V_cc and the electrical outputs Q₁ and Q₂ are represented by the blue dot and nodes ① and ②, respectively. Each pink LED (marked by rectangles) indicates that 4 hinges are active when it is lit, and the yellow LED is always lit as the minimum number of active hinges for a metamaterial is N_min = 6. By defining the number of lit LEDs as q, the number of active hinges N can be indicated by q. The circuits to implement the buffer gate are marked by navy lines on the surface of the metamaterial. At level 0 (A₀ = 0), the basic triangular blocks in the metamaterial are discrete. Therefore, the output node ① is not connected, and Q₁ = 0. When the 2 level 0 units are reconfigured into 1 level 1 unit (A₀ = 1), these basic triangular blocks are merged into larger blocks to connect the circuits across different blocks. This connection activates the output node ①, and thus, Q₁ = 1. The output Q₁ is then used as the input to the NOT gate. When Q₁ = 0, both the pink and yellow LEDs are lit, and q = 2, which indicates 10 active hinges. When Q₁ = 1, the pink LED is shorted and turned off (white), and only the yellow LED is still on, with q = 1 and 6 active hinges. When j = 1 (Fig. 4B), reconfiguring 2 level 1 units into 1 level 2 unit reduces the number of active hinges by 8, thus requiring 2 pink LEDs. Figure 4B shows a similar logic operator consisting of a buffer gate and a NOT gate, yielding 3 outputs Q₁ to Q₃ (represented by nodes ① to ③), where Q₁ and Q₂ are identical. At level 2 (A₁ = 1), 2 output nodes ① and ② are activated (Q₁ = Q₂ = 1), which shorts the 2 pink LEDs. This changes the number of lit LEDs from 3 to 1, indicating that the number of active hinges N in this metamaterial varies from 14 to 6. Note that these are just 2 examples of circuit designs among many possible solutions (see Fig. S11 for details on other feasible circuit arrangements). Further extending to the reconfiguration between the single-level configurations from 2 level j units to 1 level j + 1 unit (j > 1), the logic operators for j = 1 with outputs Q₁ to Q₃ can still be applied.

In addition to single-level configurations, this method is also applicable to multi-level configurations of metamaterials. For ease of description, if the metamaterial has multiple inputs A_j for the same level j, they are notated sequentially as A_j,s. Taking the 2 × 2 metamaterial as an example, its logic diagram (Fig. 4C) contains 3 pairs of buffer and NOT gates. The first 2 pairs serve as logic operators for j = 0 (Fig. 4A) with inputs A_0,1 and A_0,2, respectively, while the third pair serves as the logic operator for j = 1 (Fig. 4B) with input A₁. The 3 pairs use a common yellow LED. This design thus results in 5 outputs Q₁ to Q₅ (represented by nodes ① to ⑤), where outputs Q₃ and Q₄ are identical. According to the logic diagram, the global circuits of this metamaterial can be constructed, where outputs Q₁ to Q₄ (nodes ① to ④) control the 4 pink LEDs, and output Q₅ (node ⑤) controls the yellow LED. The number of active hinges N can be encoded through a linear sparse mapping with the number of lit LEDs q, as follows:where the number of lit LEDs q can be calculated by q=\overline{A_{0,1}}+\overline{A_{0,2}}+2\overline{A_{1,2}}+1. As also seen in Fig. 4C, the number of active hinges at different configurations, from a single-level configuration at level 0 (q = 5, 22 active hinges) to a multi-level one at level 0&1 (q = 4, 18 active hinges) and then to a single-level configuration at level 1 (q = 3, 14 active hinges), can be shown by the states of LEDs.

More generally, this stiffness visualization method can be extended to a 2^{⌊i/2⌋}\times 2^{⌊\left(i+1\right)/2⌋} metamaterial (i ≥ 1). The global circuits for the metamaterial are constructed hierarchically by first combining 2^{i − 1} logic operators for j = 0 with inputs A_0,s (s = 1, 2, …, 2^{i − 1}) to form the initial circuit, and successively superimposing 2^{i − j − 1} logic operators for each level j with inputs A_j,s​ (s = 1, 2, …, 2^i−j-1) until j = i − 1. The global circuits, denoted as C_global, are superimposed as follows

Its number of lit LEDs can be expressed as

Then, the number of active hinges can be determined using Eq. (7).

The 4 × 4 metamaterial (i = 4) was designed, fabricated, and tested to validate this visualization method. As shown in Fig. 4D, the global circuits for this metamaterial are constructed based on Eq. (8) and the logic diagram in Fig. S12A. It generates a maximum of q_max = 23 LEDs [all units at level 0, see Eq. (7), where N = N_max​ = 94 and N_min​ = 6], including 1 highlighted in yellow and the other 22 in pink. Six typical configurations—levels 0, 1, 0&1&2, 2, 3, and 4—with 94, 62, 54, 30, 14, and 6 active hinges, respectively, demonstrate the theoretical results across all hierarchical levels from 0 to 4 (Fig. 4D). The specimen (Fig. S12C to F) integrates 3D-printed TPU structures with conductive elastomer circuits (cast in grooves on the bottom surface) and is reconfigured using snap-fit structures (Note S6). The 7-segment display is used to show the number of active hinges N, which is calculated from the number of lit LEDs q [see Eq. (7)], with specific implementation details given in Note S6. The experimental results (Fig. 4E and Movie S4) show that the number of active hinges as the 4 × 4 metamaterial reconfigures among 10 configurations (5 single-level and 5 multi-level) can be accurately visualized by the lit LEDs and the 7-segment display, which demonstrate the reliability of the proposed design for real-time stiffness visualization in hierarchical metamaterials without external sensors.

For the 4 × 4 metamaterial in Fig. S7A, all 94 hinges are active, with triangular blocks of side length a = 18 mm and thickness t = 20 mm, and the initial configuration at angle θ = 45°. It was manufactured using a Raise3D Pro2 3D printer via fused filament fabrication (FFF) with TPU. Printing parameters included a nozzle temperature of 240 °C, printing-bed temperature of 70 °C, and a speed of 18 mm/s. For the visual model, graphite conductive adhesive was injected into the grooves on the bottom surface to form the circuit. Additional details on hinges, circuits, and reconfiguration are provided in the Supplementary Materials.

The experimental setup for the quasi-static uniaxial compression experiments in Fig. 3E and Movie S3 to measure the stiffness of the metamaterial is as follows. The horizontal test machine used has a stroke of 800 mm and a load cell of 50 N, as shown in Fig. S10A. The specimen was placed on the fixed plate and compressed by the load plate connected to the load cell. Polytetrafluoroethylene (PTFE) films were applied on the plates, and an even layer of lubricating oil was applied to minimize friction between the specimen and the plates. Displacement control was applied in the compression experiments, and the loading rate was chosen to be 2 mm/min to eliminate dynamic effects. The final compression displacement was selected to be 3.99 mm, corresponding to a 3% compression strain. Additional details about the hinge stiffness measurements, the quasi-static uniaxial compression experiments in the y direction, and the real-time stiffness visualization experiments are provided in Notes S4 to S6.

The experimental results (compressive stiffness) in Fig. 3E were derived from the force–displacement curves (Fig. S10B) measured in quasi-static experiments, using a linear fitting method. Statistical analysis was performed using Microsoft Excel 2021 and MATLAB R2020b.