Theorem on the Compatibility of Spherical Kirigami Tessellations

Introduction. Prescribed cuts on kirigami structures can induce desired deformations across various scales [1, 2, 3, 4, 5, 6, 7, 8]. This concept has been utilized to design metamaterials [9, 10, 11, 12, 13], morphable structures [14, 15, 16, 17, 18, 19], nanocomposites [20], soft robotics [21], and mechanical actuators [22]. In these applications, deployability and multistability play significant roles in determining the energy landscapes and morphing routes of the deploying process. The deformation of deployable (rigidly deployable, to be exact) kirigami structures can be idealized as continuous rotations of rigid panels connected by flexible hinges at the corner [18]. Thus, from the viewpoint of geometry, rigid deployability means having a series of piecewise isometric transformations between the undeployed configuration and any deployed configurations along the path of deployment. By contrast, multistability emerges if piecewise isometric transformations only exist at a finite number of states on the deployment path [9, 17], which is physically equivalent to discontinuous connections between stress-free configurations ¹¹1 Here we only consider stress-free stable configurations, which are determined solely by geometry.. Existence of such isometric transformations is also referred to as compatibility, a term originally proposed to design rigidly deployable origami tessellations [24, 25, 26].

In order to achieve desired energy landscapes with deployability or multistability upon deployment of kirigami structures, pioneering studies on geometrical and topological design principles have been carried out [27, 28, 29, 30, 31, 32, 17, 18, 33]. However, most existing works focus on classical patterns with planar symmetry [34], e.g., the well-known rotating squares [35] and kagome patterns [36], while freeform slit distributions can greatly expand the configuration space of kirigami structures [17, 18, 33]. Besides, nearly all the current works on morphable kirigami consider cutting flat sheets to engineer the deployed shapes in two or three dimensions, while very few are focused on cutting curved surfaces of non-Euclidean geometry. As an example of kirigami on curved developable surfaces, cylindrical shells with prescribed slits have been found to have unusual energy barriers with pop-up deformations compared to flat sheets [7]. Generically, kirigami perforated on non-developable surfaces (e.g., spherical surface) can benefit the design of shape-adaptive devices such as wearable sensors [37] and curvy imagers [38, 39]. But relevant research is still absent. Therefore, it is of great significance to develop general theories on the deployments and energy landscapes of kirigami structures covering freeform cutting patterns and Euclidean and non-Euclidean design spaces.

In this Letter, we focus on spherical quadrilateral kirigami (SQK) tessellations—the geodesic cuts divide curved sheets into $M$ columns and $N$ rows of arrayed quadrilateral panels [Fig. 1 (see Supplemental Material: Sec. IX [40] for fabrication details)] that are (ideally) connected by free rotational hinges at the corner ²²2We focus on patterns with a compact configuration. There are also kirigami patterns with no compact states, for which the cuts are voids of actual area, such as the lattice kirigami [27] and rhombi-slit kirigami [46].—and prove the following compatibility theorem:

First, we demonstrate the validity of this theorem for basic $3\times 3$ SQK tessellations [Fig. 2(a)] by investigating the corresponding compatibility condition ³³3In our discussion, the term tessellation stands for $M\times N$ kirigami patterns with $M\geq 3$ and $N\geq 3$; we refer to patterns with $M=2$ or $N=2$ as kirigami strip, which are always rigidly deployable (see Fig. S1 and Supplemental Material: Sec. I [40]).. Each solution of the compatibility condition stands for a unique compatible configuration. We will verify that the number of such solutions is either one or two, depending on the geometry of the given kirigami pattern. Then, the proof is accomplished by the fact that the compatibility of an $M\times N$ tessellation requires any of its $3\times 3$ parts to be compatible. Further, we give a corollary for implementing a spherical tessellation which guarantees two compatible configurations, and another one for a floppy planar tessellation with infinite compatible states. Finally, we design a reconfigurable kirigami structure with a dome-like deployed compatible configuration.

Compatibility theorem. We start by investigating $3\times 3$ SQK tessellations built on a sphere with Gaussian curvature $K$. Based on spherical trigonometry [43], we restrict the slits to be minor arcs of great circles, and employ notations in Fig. 2(a) to formulate the compatibility condition. One can observe that, first, there are four connected quadrilateral slits $C_{i}$ surrounding the inner panel, with side lengths $a_{i}$, $b_{i}$, $c_{i}$, $d_{i}\in(0,\pi/\sqrt{K})$, and opening angles $\alpha_{i}$, $\beta_{i}$, $\gamma_{i}$, $\delta_{i}\in(0,\pi)$ for $i=1,2,3,4$. Second, since the slits are geodesic lines at the undeployed state, the side lengths satisfy $a_{i}+b_{i}=c_{i}+d_{i}$. Given the side lengths of a slit $C_{i}$, the opening angles $\beta_{i}$, $\gamma_{i}$, and $\delta_{i}$ are uniquely determined as functions of $\alpha_{i}$ (see Supplemental Material: Sec. II [40]). As a result, the shape of a slit relies on its neighbor by the conserved relations $\alpha_{i}=\pi-\beta_{i+1}$, for $i=1,2,3,4$ ( $\beta_{5}$ is defined as $\beta_{1}$, and the same cyclic relationship follows below for other quantities). That is, the opening angles in adjacent slits (say, $\beta_{i}$ and $\beta_{i+1}$) are in a one-to-one correspondence, which we denote by $\cos\beta_{i}=g_{i}(\cos\beta_{i+1})$. The explicit expressions of $g_{i}$ are provided in Supplemental Material: Sec. II [40]. Then we define the loop function $g\triangleq g_{1}\circ g_{2}\circ g_{3}\circ g_{4}$ for $(g_{i}\circ g_{i+1})(\cdot)=g_{i}[g_{i+1}(\cdot)]$. From the expressions of $g_{i}$, we observe that $g$ is smooth on the feasible domain $\cos\beta_{1}\in[-1,c_{r}]$ for $c_{r}\in(-1,1]$. The upper bound $c_{r}$ represents the stage where at least one of opening angles $\alpha_{i}$ and $\gamma_{i}$ reaches $\pi$ upon deployment, and the lower bound $-1$ corresponds to the undeployed state. A valid compatible configuration requires that the value of $\cos\beta_{1}$ is preserved around a loop of operations by $g_{i}$, so that the compatibility condition reads

Each root of Eq. (1) represents one compatible state of a $3\times 3$ SQK tessellation. A trivial solution is $\cos\beta_{i}=-1$ at the undeployed state. To explore other compatible configurations, we will next show that $g$ is a strict convex function on $[-1,c_{r}]$. The first derivative of $g$ with respect to $\cos\beta_{1}$ is given by $g^{\prime}(\cos\beta_{1})=\prod_{i=1}^{4}g^{\prime}_{i}(\cos\beta_{i+1})$, in which $g^{\prime}_{i}$ can be explicitly expressed as (see Supplemental Material: Sec. II [40])

where we have $\alpha_{i}=\pi-\beta_{i+1}$, and $\beta_{i}$, $\gamma_{i}$, $\delta_{i}$ are functions of $\alpha_{i}$ under given side lengths $a_{i}$, $b_{i}$, $c_{i}$, and $d_{i}$. Checking the right-hand side of Eq. (2), we find $g^{\prime}_{i}>0$ on $(-1,c_{r})$. Also, we can prove $g^{\prime\prime}_{i}>0$ under the condition $a_{i}+b_{i}=c_{i}+d_{i}$ (see Supplemental Material: Sec. III [40]). It then follows that $g^{\prime\prime}>0$ on $(-1,c_{r})$. Adding the smoothness of $g$, we conclude that $g$ is a strict convex function on $[-1,c_{r}]$. As a result, Eq. (1) has at most two roots, and equivalently, a $3\times 3$ SQK tessellation has at most two compatible configurations. It further follows that an $M\times N$ tessellation has at most two compatible states as well, because the number of its compatible configurations cannot exceed that of any of its $3\times 3$ parts. Therefore, Theorem 1 is proved.

Examples and verification. Theorem 1 asserts that an SQK tessellation can only have up to one compatible configuration away from its undeployed state. Here we demonstrate a special class of SQK tessellations that are assured to have the deployed compatible configuration. Assuming $a_{i}=c_{i}$ and $b_{i}=d_{i}$ for $i=1,2,3,4$, the explicit expressions of the loop function—denoted by $g^{e}$ in this case—can be derived as (see Supplemental Material: Sec. III [40])

in which $P,Q=\prod_{i=1}^{4}\cos^{2}[(a_{i}\pm b_{i})\sqrt{K}/2]$. Then, we solve Eq. (1) and obtain $\cos\beta_{1}=\pm 1$, indicating two compact compatible configurations [as shown in Fig. 2(b)]. Conversely, if a $3\times 3$ SQK tessellation is compatible at two compact states, we can conclude $a_{i}=c_{i}$ and $b_{i}=d_{i}$, following the two conditions $a_{i}+d_{i}=b_{i}+c_{i}$ at $\cos\beta_{i}=1$ and $a_{i}+b_{i}=c_{i}+d_{i}$ at $\cos\beta_{i}=-1$. In general, for an $M\times N$ SQK tessellation, we can investigate all of its $3\times 3$ parts and obtain:

This corollary is instructive to design SQK patterns with specified compatible states. For a $3\times 3$ SQK tessellation, we define $k^{b}_{i}$ and $k^{d}_{i}$ as the ratios by which the slit $C_{i}$ is divided by the intersecting slits $C_{i-1}$ and $C_{i+1}$, i.e., $k^{b}_{i}=b_{i}/(a_{i}+b_{i})$ and $k^{d}_{i}=d_{i}/(c_{i}+d_{i})$. Fixing the boundary vertices, the undeloyed kirigami is uniquely determined by $k^{b}_{i}$ and $k^{d}_{i}$ via solving a non-linear equation system, and the deployed state is then given by applying the deformation induced from the reference opening angle $\beta_{1}$. If we assign the cutting ratios with $k^{b}_{i}=k^{d}_{i}$, the tessellation will be compatible at two compact configurations. We can further optimize $k^{b}_{i}$ and $k^{d}_{i}$ to shift the second compatible configuration from $\cos\beta_{1}^{\star}=1$ to $\cos\beta_{1}^{\star}\in(-1,1)$. These formulations are provided in Supplemental Material: Sec. IV [40].

A $3\times 3$ SQK tessellation with $k^{b}_{i}=k^{d}_{i}=0.4$ is illustrated in Fig. 2(b). The slits are prescribed on a spherical square of side length $\pi/3$ and Gaussian curvature $K=1$. This tessellation has two compact states at $\cos\beta_{1}^{\star}=\pm 1$ [Fig. 2(b), left and right], and is incompatible at $\cos\beta_{1}\in(-1,1)$ [Fig. 2(b), middle]. An optimized tessellation compatible at $\cos\beta_{1}^{\star}=0$ is illustrated in Fig. 2(c). One can observe that the kirigami pattern only changes slightly after optimization [i.e., $k^{b}_{i}\approx k^{d}_{i}\approx 0.4$, as shown in Fig. 2 (c), left], while the compatible state is converted from $\cos\beta_{1}^{\star}=1$ to $0$ [Fig. 2(c), right]. This high sensitivity to the small changes of the reference pattern arises from the high non-linearity of the loop function $g$ with respect to $a_{i}$, $b_{i}$, $c_{i}$, and $d_{i}$. The plots of loop functions for these two tessellations are shown in Fig. 2(d). We can see that both curves are convex and intersect with the $45^{\circ}$ rising line twice, indicating that there exist two compatible configurations. For comparison, the $3\times 3$ SQK tessellations with a single compatible state are shown in Fig. S3.

Rigid-to-floppy transition. If the Gaussian curvature $K$ is sufficiently small, we can expect that the SQK tessellations are approximately located on a plane, degenerating into planar quadrilateral kirigami (PQK) tessellations. When $K=0$, Eq. (3) becomes $g^{e}(x)=x$, so that the compatibility condition always holds for $3\times 3$ PQK tessellations with $a_{i}=c_{i}$ and $b_{i}=d_{i}$ (i.e., the slits form parallelograms). Otherwise, if $a_{i}\neq c_{i}$ or $b_{i}\neq d_{i}$, we can verify that the degenerate $g_{i}^{\prime}$ and $g_{i}^{\prime\prime}$ are always positive (see Supplemental Material: Sec. V. [40]). Hence, Theorem 1 still holds under this circumstance. Generally, we can investigate all the $3\times 3$ parts of $M\times N$ PQK tessellations and obtain:

In the inspiring work [18], Choi et al. proved that a planar kirigami tessellation with kite-shape slits is rigidly deployable if and only if all the slits are rhombuses. Corollary 2 further extends the design space of quadrilateral kirigami, as a floppy mechanism, to a much broader domain, i.e., from rhombus to parallelogram slits.

Corollaries 1 and 2 are obtained from the geometric compatibility. Actually, they reflect the physical insights on the rigid-to-floppy transition from curved to flat kirigami. We now examine the connection between Gaussian curvature and the rigidity of SQK tessellations. To this end, we develop a single-spring model—the kirigami is represented by a system of hinge-connected rigid panels, except that one hinge is replaced by a linear spring with stiffness $k_{S}$, as illustrated in Fig. 2(b). In this way, the compatible configuration corresponds to a zero elongation of the spring, whereas an incompatible configuration corresponds to a non-zero elongation. Then, the incompatibility or rigidity of the system can be characterized by the elastic energy $E_{S}=(k_{S}/2)\Delta_{S}^{2}$, where $\Delta_{S}$ is the elongation of the spring. We use Taylor’s series to expand the scaled energy $E_{S}/(k_{S}b_{1}^{2})$ at $K=0$ (see Supplemental Material: Sec. VI.A [40]):

where $L=\max\{a_{1},b_{1},...,a_{4},b_{4}\}$, and $L^{2}K\ll 1$. The leading term in Eq. (4) reflects the competing roles of the spherical surface area ($\sim 1/K$) and the slit size ($\sim\sum_{i=1}^{4}a_{i}b_{i}\sim L^{2}$). It clearly and explicitly shows that, when $K=0$, the energy is zero for any value of $\beta_{1}$, which means that there is an entire path of zero-energy deployments on a plane; however, when $K\neq 0$, there are only two isolated zero-energy states: $\beta_{1}=\pi$ (undeployed) and $\beta_{1}=0$ (deployed and compact). Moreover, Eq. (4) can also be related to the loop function $g^{e}$ as follows (see Supplemental Material: Sec. VI.A [40]):

which indicates that the degree of incompatibility $|g^{e}(\cos\beta_{1})-\cos\beta_{1}|$ characterizes the magnitude of the elastic energy. Since $g^{e}(\cos\beta_{1})-\cos\beta_{1}=0$ only has two roots [Fig. 2(d)], Eq. (5) also reveals two zero-energy configurations. Note that the apparent rigidity of the two-configuration result will vanish if we introduce additional freedom to move in the radial direction (see Supplemental Material: Sec. VIII. [40]).

The evolution of the energy upon deployment can be simulated by a multispring model, in which the kirigami is represented by hinge-connected springs along the edges and diagonals of panels. In this model, the elastic energy of the deployed tessellations can be written as $E\left(\bf{Y}\right)=\sum\nolimits_{n}k_{n}[l_{n}\left(\bf{Y}\right)-l^{0}_{n}]^{2}/2$, where $\bf{Y}$ is the array of panel-vertex positions, $l_{n}$ the spring length numbered by the index $n$, $l^{0}_{n}$ the rest length at the undeformed state, and $k_{n}$ the spring stiffness (set to be $1/l^{0}_{n}$). The deployed configurations are determined by incrementally increasing the kinematic parameter $\cos\omega$—defined by $\cos\beta_{1}$ of the lower-left $3\times 3$ tessellation [Fig. 1(b)]—from the undeployed state. At each step, we minimize the energy $E\left(\bf{Y}\right)$ taking positions of all the vertices as variables, which are constrained to form an SQK pattern, and enforce the opening angle $\omega$ (see Supplemental Material: Sec. VI.B [40]). Figure 3(a) demonstrates the energy curves of SQK shells perforated on a spherical square of fixed side length $s=\pi/3$ for different Gaussian curvatures. While decreasing $K$ from 1 to 0.4 [Fig. 3(b)], the energy barrier drops significantly. If $K=0$, the SQK tessellation degenerates to a PQK tessellation, which is rigidly deployable (floppy) with zero energy of deformations. We illustrate the deployed configurations of these kirigami tessellations in Fig. S6. We additionally show the energy curves with deployed compatible states at $\cos\omega\in(-1,1)$ in Fig. S7.

Shape-morphing structures. As a final demonstration for potential applications of our theorem, we address how to design the cutting patterns to achieve compatible deployed configurations with desired shapes. We start from a $6\times 6$ square SQK shell structure of side length $s=0.465\pi$ and Gaussian curvature $K=1$, and then optimize the locations of vertices via minimizing the distance between the deployed outer vertices and the boundary of a spherical dome of radius $r=1$ and height $h=1.2$. Details of the optimization framework are given in Supplemental Material: Sec. VII [40]. Figure 4 demonstrates the shell structure (see Supplemental Material: Sec. IX [40] for fabrication details). The covering area of the undeployed pattern and the deployed dome can be calculated by $\tan(S_{\rm squa.}/4)=\sin^{2}(s/2)\sqrt{\sec s}$ and $S_{\rm dome}=2\pi rh$, respectively. Thus, the expansion ratio of the area is $S_{\rm dome}/S_{\rm squa.}\approx 2.0$. According to Theorem 1, the deployed configuration is at an isolated compatible state, which is ideally rigid, so that it can form a stable structure for potential applications such as tents and roofs (see Movie 1 in Supplemental Material [40]). Moreover, shape-morphing mechanisms can be realized by PQK tessellations with different topologies [44].

Conclusion and discussion. In summary, we show that spherical quadrilateral kirigami tessellations can only be compatible at isolated configurations, whereas planar kirigami quadrilateral tessellations with parallelogram slits have infinite and continuous compatible states. We develop single-spring and multispring models to explicitly analyze and simulate the evolution of energy along the deployment paths for different values of Gaussian curvature, which characterizes the rigid-to-floppy transition. Since the deformation energy drops appreciably near the flat surface, the slightly curved kirigami shells are expected to be a promising candidate of pseudo-mechanisms, as those presented in Refs. [45, 46].

The compatibility theorem and its corollaries reveal the role of curvature in determining the deployment behaviors of kirigami systems. The effect of curvature on constrained morphology of stable structures can also be observed in various physical phenomena in nature such as the growth of nanoshells [47] and rigid colloidal crystals [48] on spherical substrates. More curvature-induced scenarios, such as the buckling of non-Euclidean kirigami shells—the counterpart of buckling-induced planar kirigami [10]—can be further investigated. Finally, the compatibility in our theorem relies purely on the size-independent geometry of the prescribed slits, so that the applicability is rooted in diverse materials and various scales.