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Origami Lattices and Folding-induced Lattice Transformations

Lattices and their underlying symmetries play a central role in determining the physical properties and applications of many natural and engineered materials. By bridging the lattice geometry and rigid-folding kinematics, this study elucidates that origami offers a comprehensive solution to a long-standing challenge regarding the lattice-based materials: how to systematically construct a lattice and transform it among different symmetric configurations in a predictable, scalable, and reversible way? Based on a group of origamis consisting of generic degree-4 vertices, we successfully construct all types of 2D and 3D Bravais lattices, and demonstrate that they can undergo all diffusionless phase transformations via rigid-folding (i.e., dilation, extension, contraction, shear, and shuffle). Such folding-induced lattice transformations can trigger fundamental lattice-symmetry switches, which can either maintain or reconstruct the nearest neighborhood relationships according to a continuous symmetry measure. This study can foster the next generation of transformable lattice structures and materials with on-demand property tuning capabilities.


I. INTRODUCTION
Lattices and their underlying symmetries play a central role in determining physical properties-such as band structure, compressibility, and elastic modulus-of many natural [1,2] and engineered materials [3,4]. Such a lattice-property relationship is particularly evident in metamaterials [5][6][7]. Therefore, purposefully designing and adjusting lattice topology can significantly expand the achievable material property range [8][9][10]. However, due to the lack of a flexible and versatile platform, constructing lattice structures from the ground up is challenging, and once the material is synthesized, its constituent lattice typically cannot be modified. Several studies have been proposed for tuning the underlying lattice structures of metamaterials; however, they mainly stay at a case study level, and the range and freedom of lattice adjustment are limited [11][12][13]. Therefore, there is a lack of an integrated and scalable approach for reconfiguring lattice structures on demand, let alone transforming their symmetry properties. Here we offer a solution to fill this gap by exploring and exploiting origami folding. The research will answer the long-standing question of how to systematically construct a lattice and transform it among different symmetric configurations in a predictable, scalable, and reversible way. Due to the broad importance of lattice structure and its transformations, answers to this question will be extremely valuable in many fields including lattice and symmetry theory, metamaterials, adaptive materials and structures, and origami science itself.
Origami has become a popular subject among mathematicians, educators, physicists, and engineers owing to the seemingly infinite possibilities of transforming two-dimensional (2D) sheets into three-dimensional (3D) shapes via folding [14][15][16][17][18]. Historically, such folding-induced shape transformations have been examined based on the spatial positions and orientations of its facet surfaces and crease lines [19,20]. For example, many origami-based mechanical metamaterials are analyzed by considering the folding as coordinated facet rotations with respect to the hingelike creases-essentially a linkage mechanism [21,22]. In this study, we examine the origami folding through a different lens by asking how folding can spatially arrange and rearrange the characteristic entities in the origami. These characteristic entities can be the vertices where crease lines intersect, or the center points of crease lines and facets. By treating these entities as the elements of a lattice (aka lattice points), we uncover that origami offers a remarkably comprehensive framework to construct all 2D and 3D Bravais lattices, induce all types of diffusionless lattice transformations (aka dilation, extension, contraction, shear, and shuffle), switch the lattice structure among different symmetric configurations, and even reconstruct the nearest-neighbor relationships based on a continuous symmetry measure. Therefore, this result can fundamentally advance our capability to engineer high-performance material systems by incorporating the rich design and kinematics of origami folding. . The inclusions can be selected according to targeted applications, such as conducting element [23] and sonic barrier rods [24,25]. Assembly of multiple origami sheets can further enrich the 3D design space [Figs. 1(c)-1(e)]. By establishing correlations between the origami geometry and the lattice configuration (see details in Appendix A for 2D Bravais lattices, and Appendix B for 3D Bravais lattices), we discover that all five types of 2D and 14 types of 3D Bravais lattices-a well-established description of the lattice configuration according to symmetry-can be constructed on a group of simple and rigid-foldable origamis consisting of generic degree-4 vertices [26,27]. The mapping between origami geometry and lattice configuration is not one-to-one in that multiple origami designs and lattice-point assignments can satisfy the reflection and rotational symmetry requirements of one specific Bravais lattice. Such a property is beneficial to lattice construction in that it provides abundant candidate origami patterns as well as enough flexibility to arrange lattice points. For a specific Bravais lattice, the symmetry requirements, as well as the constraints on axial distances and axial angles (Appendixes A and B), could be used as guidelines for origami design and lattice-point assignment. Once they are prescribed, the corresponding lattice at any intermediate state during folding could be uniquely determined.

B. Folding-induced diffusionless phase transformation and discrete symmetry switches
Besides being a versatile platform for lattice construction, origami also provides effective mechanisms to transform these lattices via folding. By translating the folding-induced facet and vertex rearrangement into the corresponding lattice-point movements (Fig. 2), we discover that folding can impart all kinds of diffusionless phase transformations via attaining both lattice-distortive strains and shuffles. The former further includes dilation, contraction and extension, and shear [28]. Appendix C thoroughly examines the mathematical correspondence between origami folding and 2D and 3D lattice transformations. For each type of lattice transforamtion, a transformation matrix is formulated to quantitatively describe the changes of lattice vectors during folding. Particularly, correlations between origami's kinematic properties and lattice transformations are also established. For 2D lattices of vertex inclusions, we show that origami sheets with a negative in-plane Poisson's ratio (e.g., Miura-ori [22]) can induce 2D dilations [ Fig. 2(a)]; those with a positive inplane Poisson's ratio (e.g., eggbox pattern [29]) can bring about 2D contractions and extensions [ Fig. 2(b)]; and those with in-plane shear deformations (e.g., single-collinear (SC) origami [27]) can trigger 2D shear components [ Fig. 2(c)]. Two-dimensional shuffle, manifested as a small movement of a lattice point (red) within the unit cell, can also be achieved in a Miura-ori-based lattice [ Fig. 2(d)]. For 3D lattices of vertex inclusions, 3D dilation is achievable by exploiting the tridirectional auxeticity of a stacked general flat-foldable (GFF) structure [27]  shuffle is evident when the lattice point locating at the center vertex of the Miura-ori cell translates within the 3D unit lattice cell during folding [ Fig. 2(h)]. It is worth noting that the shear or shuffle components do not occur alone, and they are always accompanied by dilation or contraction and extension. These rich connections between origami kinematics and lattice-point movements can enable us to program a broad range of lattice transformations-according to the Cohen's classification, for example [30]-by purposefully designing the crease patterns and their folding motions.
More importantly, these folding-induced diffusionless transformations can fundamentally switch the lattices' underlying symmetry groups. For instance, the 2D contraction and extension could add an additional reflection symmetry to the rectangular lattice based on Miura-ori and switch it into a square lattice [i.e., the symmetry group switches from D 2 to D 4 , Fig. 2(b)]. The 2D shuffle, on the other hand, could break down a reflection symmetry of a centered rectangular lattice and evolve it into a non-Bravais lattice [D 2 to D 1 , Fig. 2 Remarkably, with a carefully designed crease pattern, origami folding can switch the corresponding lattice among more than two Bravais types, which could appear either alternatively or repeatedly. Taking the Miura-ori pattern as an example, by prescribing the crease design parameters (a, c, and γ ), the corresponding 2D lattices of rod inclusions could reach four types of 2D Bravais lattice: rectangular (R), centered rectangular (CR), square (S), and hexagonal (H) lattices [24]. For instance, if c/a = 1 and γ = 60 • , the Miura-ori-based lattice of rod inclusions will undergo a series of discrete switches: from being a hexagonal lattice (θ = 0 • ) to a square lattice (θ = 54.7 • ), to a hexagonal lattice again (θ = 70.5 • ), and finally to a rectangular lattice (θ → 90 • ) [ Fig. 3(a)]. Note that between these switches, the origami lattices are non-Bravais, which are not denoted in Fig. 3(a). Detailed parametric analysis and another example of discrete symmetry switches can be found in the Supplemental Material, S1 [31].

C. Folding-induced continuous symmetry evolutions in origami lattices
So far, we have been focusing on discrete folding configurations where the corresponding lattices are strictly symmetric. However, folding is a continuous process; when the origami is folded slightly away from these strictly symmetric configurations, the corresponding lattice no longer possesses certain symmetry in a strict sense but is still very close. Therefore, it is necessary to derive a quantitative method to analyze the folding-induced and continuous changes in the lattice symmetry. Here we adopt the concept of continuous symmetry measure (CSM), which was first proposed in the field of structural chemistry [32,33]. Given a shape and a symmetry group G (such as D 2 and D 4 ), the continuous symmetry measure, S(G), quantifies the minimal displacement that the points of the object have to undergo in order to be transformed into a shape with G symmetry. Hence, an object features a zero S(G) when it is strictly G symmetric; if this object has to undergo larger displacements to acquire G symmetry, its S(G) value increases accordingly. In other words, rather than a "black and white" discrete approach to describe symmetry (that is, either symmetric or asymmetric), the CSM offers a  FIG. 3. Discrete and continuous symmetry measures as well as the reconstruction of the nearest-neighbor relationship of the origami lattices. Based on a Miura-ori sheet with c/a = 1, γ = 60 • , (a) demonstrates the discrete switches of the lattice from a hexagonal, to a square, to another hexagonal, and finally to a rectangular type. Correspondingly, the symmetry switches from D 6 , to D 4 , to D 6 , and finally to D 2 . Based on the same design, (b) displays the CSM values of the two unit cells (i) and (ii) with respect to symmetry groups D 6 , D 4 , and D 2 , where the arrows denote the folding angles when reconstructions of the nearest-neighbor relationship happen. For clear illustration, lattices at specific folding configurations are provided in (c). The unit cell (i) deforms via weak transformations when θ < 54.7 • ; as the folding reaches and passes θ = 54.7 • , a reconstructive transformation occurs and a new unit cell (i ) with reconstructed nearest lattice-point neighbors is generated. Similar reconstructive transformation happens on unit cell (ii) when the folding reaches and passes θ = 70.5 • , with a new unit cell (ii ) being generated. In examining the overall lattice, we observe another type of reconstructive transformation. When reaching and passing θ = 44.4 • , the nearest neighbors change from those lattice points constituting the unit (i) to those constituting the unit cell (ii); the lattice mainly represents D 6 and D 4 symmetry before and after θ = 44.4 • , respectively. Similar reconstructions also occur at θ = 62.4 • and 78.4 • . At each folding configuration, the lattice with the nearest neighbors is denoted by a shaded polygon with solid lattice points. more continuous "gray" scale to characterize the strength of a particular type of symmetry group. Detailed mathematical description of this concept and its application in quantifying origami lattice can be found in the Supplemental Material, S2 [31], and in [32,34]. By adopting this measure, we are able to describe how much the origami lattice possesses certain symmetry and understand its nearest symmetric correspondence at a certain folding configuration.
As an example, we revisit the case in Fig. 3(a) by evaluating the CSM of the origami lattice with respect to symmetry groups D 6 , D 4 , and D 2 , which correspond to the hexagonal, square, and rectangular (or centered rectangular) lattices, respectively. We select a hexagon unit cell (i) and a parallelogram unit cell (ii) of the lattice and calculate their CSM values with respect to different symmetry groups throughout the folding range from θ = 0 • (fully flat) to θ = 90 • (fully folded) ( Fig. 3(b) (see detailed calculation procedures in the Supplemental Material, S2 [31]). At θ = 0 • , S i (D 6 ) = 0 for cell (i), meaning that the lattice is of hexagonal type and strictly exhibits D 6 symmetry. As θ increases, S i (D 6 ) gradually increases, indicating a loss of D 4 symmetry; however, S ii (D 4 ) decreases for cell (ii), which is a signal of strengthening D 6 symmetry. At θ = 54.7 • , S i (D 6 ) reaches a local maximum whereas S ii (D 4 ) = 0; this suggests that the lattice is far away from D 6 symmetry but fully acquires the D 4 symmetry (aka square lattice). As θ further increases, S i (D 6 ) starts to decrease but S ii (D 4 ) increases. At θ = 70.5 • , S i (D 6 ) returns to zero while S ii (D 4 ) reaches a local maximum, implying that the lattice fully regains D 6 and loses D 4 symmetry. At the final stage of folding, S ii (D 2 ) converges to zero while S ii (D 4 ) climbs quickly, indicating the emergence of a rectangular lattice with D 2 symmetry when θ approaches 90 • .

D. Breaking and reconstructing of the nearest-neighbor relationship in origami lattices
To understand the physics underpinning the foldinginduced discrete symmetry switches and CSM evolutions, we carefully examine the lattice-point movements during these transformations. Around those configurations with strict symmetry, even with a relatively small range of folding, the latticepoint movements can be sufficient to switch the symmetry group but not break the nearest-neighbor relationships. For example, the contraction and extension shown in Fig. 2(b) can switch the symmetry group from D 2 to D 4 by slightly moving the four lattice points in the unit cell such that the corresponding rectangle changes to a square, but the unit cell always consists of the same four lattice points during this process. Such a transformation mechanism that maintains the nearest-neighbor relationship is defined as weak, which is manifested as smooth variations of the CSM value in Fig. 3(b). However, near some critical folding configurations, the lattice points within the unit cell move in a way that they are no longer the nearest neighbors, so that a new unit cell of the same type is generated by incorporating a different set of lattice points. Such a mechanism involving breaking and reconstructing of nearest-neighbor relationships is defined as reconstructive. 1 For example, in Fig. 3(c), the hexagonal shape formed by the lattice points of unit cell (i) first undergoes a weak transformation, from θ = 0 • to near 54.7°, as it becomes a houselike shape. When θ reaches and passes 54.7°, the original nearest-neighbor relationship cannot be maintained.
Specifically, one of the lattice points in unit cell (i) is replaced by a new lattice point such that a new unit cell (i ) with the nearest lattice-point neighbors is reconstructed. In the CSM plot, such reconstructive transformation is manifested as the nonsmooth maxima corresponding to this particular unit cell (e.g., the maxima at θ = 54.7 • for unit cell (i) and the maxima at θ = 70.5 • for unit cell (ii)).
The reconstructive mechanisms discussed so far are defined on a particular type of unit cell, and we can extend its definition by considering different types of unit cells in the origami lattice simultaneously. At any folding configurations, the unit cell with a relatively lower value of the CSM better represents the prominent symmetry of the overall lattice. Therefore, among different types of unit cell, when there is a change in terms of the smallest CSM, the nearestneighbor relationship of the lattice can be also considered to experience a reconstruction. The difference here is that such a reconstruction occurs between unit cells of different types. In the example shown in Fig. 3, S i (D 6 ) is lower than S ii (D 4 ) before θ reaches 44.4°, indicating that the lattice points in the unit cell (i) constitutes the nearest-neighbor relationship of the overall lattice, and the overall lattice mainly represents D 6 symmetry; however, S i (D 6 ) outstrips S ii (D 4 ) as θ passes 44.4°, suggesting that the lattice points in the unit cell (ii) become the nearest neighbors, and the overall lattice mainly exhibits D 4 symmetry. Based on this generalized definition of the reconstructive mechanism, it is evident from Fig. 3 that folding can trigger multiple and successive reconstructions of the nearest-neighbor relationships (e.g., at θ = 44. Three-dimensional Bravais lattices constructed on the generic four-vertex origami can also exhibit symmetry switches and evolutions by folding. We find that lattices of the same centering type (primitive, body-centered, basecentered, and face-centered lattices) possess the potential to evolve among different crystal families, which need further exploration. It is also worth highlighting that unlike certain martensitic transformations that are irreversible [1,2], the above origami folding-induced transformations are always reversible even after multiple reconstructions of the nearestneighbor relationship.

III. SUMMARY AND DISCUSSIONS
This study uncovers and elucidates the comprehensiveness of the capability to construct and reconfigure lattice structures by uniquely exploring origami folding. All five types of 2D and 14 types of 3D Bravais lattices can be constructed by exploiting the origami design space and by carefully selecting its characteristic points for lattice-point assignment. All 2D and 3D diffusionless lattice transformations-including dilation, extension and contraction, shear, and shuffle-can be achieved in such origami lattices via rigid folding. More importantly, we discover that folding can induce continuous and reversible evolutions of the lattice symmetry, and these symmetry evolutions can either maintain or reconstruct the nearest-neighbor relationship in a predictable manner.
Since the lattice structure and its symmetry properties directly govern many physical properties, the origami lattice construction and transformation offers us great freedom to architect programmable and adaptive metamaterials. Here the lattice points can be occupied by any components used for metamaterials, including geometric entities [9,35], mechanical units [36], acoustic modules [37], electromagnetic devices [7,23], and photonic elements [7,38], etc. Attaching these components to the origami scaffold gives us metamaterials and metastructures (from nanometer-scale DNA origami [39] to meter-scale origami space structure [16]) whose mechanical, thermal, acoustic, optical, or electromagnetic properties can be effectively tailored by folding on demand. For example, our previous research demonstrated how embedding cylindrical rods to a Miura-ori sheet can result in a transformable lattice [24,25,40]. Folding, in this case, can effectively tune the lattice symmetry and the lattice axial lengths and significantly change the sound wave propagation behavior in terms of band gaps (see a detailed example in the Supplemental Material, S3 [31]). Such an origami lattice of rod inclusions could be used as sonic barriers for noise mitigation. Moreover, in addition to the global lattice transformation via folding, perturbing the positions of specific lattice points can introduce local defects (e.g., pop-through defects [41]) into the origami lattice, which may impart additional property programmability to the lattice-based metamaterials or metastructures.
Finally, it is worth emphasizing that, unlike the application case studies in [24,25], this research is unique and significant in that it uncovers a fundamental and comprehensive theoretical framework on the correlations between origami geometries and lattice configurations. By establishing origami as a versatile scaffold to construct and transform lattices with continuous symmetry evolutions, this theoretical framework can open the door to engineering metamaterials and metastructures that can alter their underlying lattice structures on demand. The results of this research can serve as design guidelines and analytical tools for future studies on the engineering of lattice control via origami scaffolds of different size scales, for many other applications.

APPENDIX A: CONSTRUCTIONS OF 2D BRAVAIS LATTICES WITH DEGREE-4 VERTEX ORIGAMIS
There are five types of 2D Bravais lattices: oblique, rectangular, centered rectangular, square, and hexagonal lattices. For simplicity, they are hereafter referred to as O, R, CR, S, and H, respectively. In this Appendix section, we show how they can be constructed based on degree-4 vertex (4-vertex) origamis by providing the detailed correlations between origami geometries and lattice configurations. Note that there are multiple origami designs and multiple characteristic entity assignments that can be used to create one specific type of 2D Bravais lattice. Thus, the constructions given here serve as examples to elucidate the underlying geometric correlations.
To obtain an oblique lattice, a single-collinear (SC) origami pattern is used. Figure 4

SC unit (d) Square
Miura-ori unit

(c) Centered rectangular (e) Hexagonal
Miura-ori unit where H, W , and L are the height, width, and length of a folded SC unit, respectively [middle, Fig. 4(a)]. By projecting the folded SC unit onto the reference x-o-y plane, we obtain a polygon, whose shape is described by angles ξ , η, and ϕ S [middle and right, Fig. 4(a)].
The other four types of 2D Bravais lattices can be constructed by using Miura-ori patterns. Figure 4(b) shows the geometry of a Miura-ori unit. A Miura-ori unit can be characterized by two crease lengths (a and c) and one sector angle (γ ) [left, Fig. 4(b)]. Folding of the Miura-ori unit can also be described by the folding angle θ (θ ∈ [0, 90 • ]), defined as the dihedral angle between one of its facets and the reference x-o-y plane. Vertices 1, 2, 3, 5, 6, and 7 are coplanar in the x-o-y plane. The outer dimensions of the Miura-ori unit are given by [22] where H, W , L are the height, width, and length of a folded Miura-ori unit, respectively [middle, Fig. 4(b)]. Projecting the folded unit onto the x-o-y plane, a polygon is also obtained. By treating the coplanar vertices 1, 2, 3, 5, 6, and 7 as lattice points and placing inclusions on them, we can construct 2D origami lattices of vertex inclusions. Figure 5 shows the polygons that are projected from a partially folded origami unit and the corresponding 2D Bravais lattices. For each 2D Bravais lattice, the detailed correlations between the origami geometries and lattice geometries are given in Table I.   TABLE I

APPENDIX B: CONSTRUCTIONS OF 3D BRAVAIS LATTICES WITH DEGREE-4 VERTEX ORIGAMIS
Three-dimensional Bravais lattices can be categorized into seven systems. In each system, the lattice points in a unit cell can follow four different centering types. Specifically, the seven systems are triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic [42]. The four centering types are primitive, base-centered, body-centered, and face-centered. However, some combinations of lattice systems and centering types create the same lattice. After considering such redundancy, there are 14 unique types of 3D Bravais lattices.
We use three types of origami structures to construct 3D Bravais lattices. They are a stacked SC structure in the bulged-out configuration, a stacked Miura-ori structure in the bulged-out configuration, and a stacked double-collinear (DC) structure. In what follows, we formulate the geometries of the three stacked origami structures, and then detail the cor-relations between the origami geometries and the 3D lattice configurations.
The stacked SC structure consists of two SC units [ Fig. 6(a)]. Each SC unit is characterized by crease lengths a x , b x , and c x , and sector angles γ x1 and γ x2 . Here the subscript x takes the values of A or B and represents the bottom unit A or the top unit B, respectively. Without loss of generality, we assume γ x1 < γ x2 . The following geometry constraints have to be satisfied to ensure the kinematic compatibility between the two SC units during the whole rigid-folding process [27]: Folding of the stacked SC structure is still a single-degreeof-freedom mechanism. It can be described by a folding angle θ A (or θ B ), which is defined as the dihedral angle between a facet of the bottom (or top) unit and the reference x-o-y plane [ Fig. 6(a)]. θ A and θ B are not independent of each other but satisfy the following relationship: In the bulged-out configuration −90 • θ A 0, and in the nested-in configuration 0 < θ A 90 • . The outer dimensions of the SC stacked structure in the bulged-out configuration are The stacked SC structure is used to construct the triclinic and the monoclinic lattices. To characterize the triclinic lattice, two additional angles ϕ 1 and ϕ 2 [ Fig. 7(a)] are defined: The stacked Miura-ori structure consists of two Miura-ori units [ Fig. 6(b)]. Each Miura-ori unit is characterized by two crease lengths a x and b x , and one sector angle γ x , where the subscript x also takes the values of A or B, denoting the bottom unit A or the top unit B, respectively. The following geometry constraints have to be satisfied to ensure the kinematic compatibility between the two units during rigid folding [22]: Similarly, folding of the stacked Miura-ori structure is a single-degree-of-freedom mechanism. It can be described by a folding angle θ A (or θ B ), which is defined as the dihedral angle between a facet of the bottom (or top) unit and the reference x-o-y plane [ Fig. 6(b)]. θ A and θ B are not independent of each other but satisfy the following relationship: In the bulged-out configuration −90 • θ A 0, and in the nested-in configuration 0 < θ A 90 • . The outer dimensions of the Miura-ori stacked structure in the bulged-out configuration are The stacked Miura-ori structure is used to construct the orthorhombic, the tetragonal, the hexagonal, and the cubic lattices.
The stacked DC structure is obtained by stacking two identical DC units [ Fig. 6(c)]. Each DC unit has two pairs of creases that are collinear, and it can be characterized by two crease lengths a, c and one sector angle γ . Folding of a DC unit can be described by the dihedral angle ρ between its facets. Hence, the outer dimensions of a folded DC unit are H = c cos(ρ/2), W = 2c sin(ρ/2). (B8) The stacked DC structure is used to construct a specific rhombohedral lattice with axial angles equaling 60 • . To obtain this lattice, two collinear creases have to remain unfolded. Based on these three types of stacked origami structures, the 14 types of 3D Bravais lattices can be constructed according to Fig. 7. For each 3D Bravais lattice, a unit cell is shown on the corresponding origami structure. The detailed correlations between the origami geometries and the lattice geometries are listed in Table II.

APPENDIX C: ACHIEVING DIFFUSIONLESS PHASE TRANSFORMATIONS THROUGH ORIGAMI FOLDING
Diffusionless phase transformations can be distinguished between transformations dominated by lattice-distortive strains and those where shuffles play a significant role. Lattice-distortive strains can transform the lattice from one Bravais type to another, and the shuffle refers to small movements of a lattice point within the lattice unit cell. In this section, we thoroughly examine the examples given in Fig. 2 (detailed parameters of these examples are provided in Table III). We show that based on different origami patterns, all types of 2D and 3D diffusionless transformations can be achieved by rigid folding. For each type of transformation, a lattice transformation matrix is formulated, which is further correlated to the kinematic properties of 2D and 3D origami structues. Such correlations are summarized in Table S4 in the Supplemental Material [31].

2D Dilation
Two-dimensional dilation is observed in a lattice of vertex inclusions based on a Miura-ori, which is an example of 2D Bravais lattice shown in Figs. 2(a) and 8(a). Folding of a Miura-ori unit can be described by the dihedral folding angle θ (θ ∈ [0, 90 • ]) between its facet and the reference x-o-y plane [ Fig. 4(b)]. Hence, the lattice vectors corresponding to two different configurations (with folding angles θ and θ ) can be formulated as Here, the length L and the width W of the Miura-ori unit are defined in Eq. (A2). Since there is no diffusion involved, we can formulate a transformation matrix U 2D-D to quantitatively describe the lattice transformation due to folding from angle U 2D-D is a diagonal matrix. When θ < θ, the two diagonal elements of U 2D-D are always larger than 1, suggesting that the lattice dilates along both lattice vectors. Note that the Miura-ori unit exhibits a negative Poisson's ratio in the L and W directions during folding [22]: When the length L increases (or decreases) due to folding, the width W increases (or decrease) accordingly. Therefore, the negative Poisson's ratio is the kinematic origin of the 2D dilation.

2D Contraction and extension
Two-dimensional contraction and extension is observed when transforming an eggbox-pattern-based lattice of vertex inclusions, which is an example of 2D Bravais lattice shown in Figs. 2(b) and 8(b). The eggbox pattern [29] consists of four identical parallelogram facets characterized by crease lengths a, b and a sector angle γ ; its length L and width W are given by where α and β are two angles between the boundary creases and a perpendicular line, used for describing the folding motion (0 α, β γ ) [ Fig. 8(b)]. They are not independent of each other but follow the relationship cos α cos β = cos γ .
Hence, the lattice vectors corresponding to two different configurations (with angles α and α ) can be expressed as Since there is no diffusion, the folding-induced lattice transformation from angle α to α can be described by a transformation matrix U 2D-C/E such that Here U 2D-C/E describes the lattice-distortive strains that transform the lattice; the subscript "2D-C/E" stands for  U 2D-C/E is also a diagonal matrix. When α > α, sin α / sin α > 1 but 0 < sin β / sin β < 1, suggesting that the lattice contracts along one lattice vector but expands along the other vector, and vice versa. Note that the eggbox unit exhibits a positive Poisson's ratio in the L and W directions during folding [29]: When the length L increases due to folding, the width W decreases instead, and vice versa. Therefore, the positive Poisson's ratio is the kinematic origin of the 2D contraction and extension.

2D Shear
A two-dimensional shear component is observed when we transform a lattice of vertex inclusions based on a single collinear (SC) origami, which is an example of 2D Bravais lattice shown in Figs. 2(c) and 8(c). Folding of the SC unit can be described by the dihedral angle θ (θ ∈ [0, 90 • ]) defined between the facet and the reference x-o-y plane [ Fig. 4(a)] [27], so that the lattice vectors corresponding to two different configurations (with folding angles θ and θ ) yield The length L, width W , and the angle ϕ S of the SC unit are defined in Eq. (A1). Since there is no diffusion, the folding-induced lattice transformation from angle θ to θ can be described by a transformation matrix U 2D-S such that Here U 2D-S describes the lattice-distortive strains that transform the lattice; the subscript "2D-S" means two-dimensional shear. U 2D-S can be formulated as U 2D-S is not a diagonal matrix. In addition to the nontrivial diagonal elements that describes the contraction and (or) extension along the lattice vectors, there is also a nonzero off-diagonal element describing the distortion induced by shearing. By substituting the expressions of L, W , and ϕ S [i.e., Eq. (A1)] into Eq. (C12), correlations between the transformation matrix U 2D-S and the rigid-folding kinematics can be established.
Note that the SC origami is characterized by its ability in achieving in-plane shearing deformation during folding, which manifests as the changes of the angle ϕ S . For example, by folding the SC unit (a = b = c, γ 1 = 36 • , γ 2 = 72 • ) from θ = 15 • to θ = 65 • , the angle ϕ S changes from 108.4 • to 115.8 • . As a result, the in-plane shearing deformation mechanism is the root cause of the 2D shear component in lattice transformations.

2D Shuffle
Two-dimensional shuffle plays a significant role in transforming a particular type of Miura-ori-based lattice of vertex inclusions shown in Figs. 2(d) and 8(d). Generally, this lattice as a whole is not a Bravais type due to the additional lattice point within the unit cell. As a result, the two lattice vectors used in the previous three cases are no longer sufficient to describe this lattice, because the lattice point inside the unit cell cannot be obtained by translating the two vectors. This kind of lattice is essentially a collection of two congruent Bravais lattices that are shifted from one another. Mathematically, such multilattice [43] can be characterized by a group of lattice vectors l i and a shift vector p. That is, l i describes the constituent Bravais lattice, and p describes the offset between the two congruent lattices. Here, folding of the Miura-ori unit is still described by the dihedral angle θ (θ ∈ [0, 90 • ]) between its facet and the reference x-o-y plane [ Fig. 4(a)] [27], so that the vectors corresponding to two different configurations (with folding angles θ and θ ) can be expressed as The dimensions L, W , and J of the Miura-ori unit are defined in Eq. (A2). Since there is no diffusion, we can then describe the folding-induced lattice transformation from angle θ to θ as a combination of deformation and shift. Therefore, we can formulate the transformation matrix T 2D-SH such that Here the subscript "2D-SH" means two-dimensional shuffle, and T 2D-SH can be written as T 2D-SH is not a diagonal matrix, and it reflects how the latticedistortive strain and the shuffle component coexist in the transformation. U 2D-D is the submatrix corresponding to 2D dilation, which describes the lattice-distortive strains of the constituent 2D Bravias lattice. It can be replaced by other transformation matrices (i.e., U 2D-D , U 2D-C/E , or U 2D-S ) if different origami structures are employed. Other elements in the third row of T 2D-SH quantify the 2D shuffle component, which satisfy the following relationship: Substituting the expressions of L, W , and J [i.e., Eq. (A2)] into Eqs. (C13)-(C15), correlations between the matrix T 2D-SH and the rigid-folding kinematics can be established. Note that changes in the relative positions among vertices (or other characteristic entities) are ubiquitous in origami folding, and it is the underlying mechanism that induces the shuffle in lattice transformation. The shuffle component can be significant, and it can even move a lattice point from one unit cell to another, triggering a break and reconstruction of the nearest-neighbor relationship [e.g., Fig. 3(c) in the main text].

3D Dilation
Three-dimensional dilation is observed when transforming a lattice of vertex inclusions based on a stacked generic-flatfoldable (GFF) structure in the bulged-out configurations [27] [a kind of 3D Bravias lattice shown in Figs. 2(e) and 9(c)]. The stacked GFF structure consists of two GFF units A and B [ Fig. 9(a), left]. Each unit is characterized by crease lengths a x , c x and sector angles γ x1 , γ x2 . The subscript x takes the values of A or B, denoting the bottom unit A or the top unit B, respectively. Without loss of generality, we assume γ x1 < γ x2 . The following geometry constraints have to be satisfied to ensure the kinematic compatibility between the two units: Folding of the stacked GFF structure is still a single-degreeof-freedom mechanism, and it can be described by the dihedral angle ρ A1 between two facets of the bottom unit A [ Fig. 9(a), right]. Alternatively, we can describe its folding by using the dihedral angles θ Ai (or θ Bi ) (i = 1, 2, 3, 4) between the facets of the bottom unit A (or top unit B) and the reference x-o-y plane. θ Ai and θ Bi are not independent of each other but satisfy the following constraints: When ρ C1 A1 ρ A1 < 180 • , the structure is in nested-in configurations and 0 • < θ Ai 90 • . In particular, when ρ A1 = ρ C1 A1 , the structure self-locks so that θ A1 = θ A3 = 90 • and 0 • < θ A2 = θ A4 < 90 • . On the other hand, when 180 • ρ A1 360 • , the structure is in bulged-out configurations and −180 • θ Ai 0 • .

3D Contraction and extension
Three-dimensional contraction and extension is observed when transforming a 3D Bravais lattice of vertex inclusions based on a stacked Miura-ori structure in the bulged-out configuration (−90 • < θ A < 0) [27] [a kind of 3D Bravias lattice shown in Figs. 2(f) and 9(d)]. Construction of the stacked Miura-ori structure is introduced in Fig. 6(b), and its outer dimensions are given in Eq. (B7).
Based on the assignment of lattice points on the stacked Miura-ori structure given in Fig. 9(d), the lattice vectors corresponding to two different configurations (with folding angle θ A and θ A ) can be expressed as Since there is no diffusion, the folding-induced lattice transformation from angle θ A to θ A can be described by a transformation matrix U 3D-C/E such that U 3D-C/E (a diagonal matrix) describes the lattice-distortive strains that transform the lattice; the subscript "3D-C/E" means three-dimensional contraction and extension. U 3D-C/E can be formulated as By substituting the expressions of L, W , and H [i.e., Eq. (B7)] into Eq. (C28), one can correlate U 3D-C/E to the rigid-folding kinematics.
The observed 3D contraction and extension has a close relation to the Poisson's ratios of the stacked Miura-ori structure (bulged-out configurations, with −90 • < θ A < 0). Based on the outer dimensions given in Eq. (B7), the Poisson's ratios v W L and v HL have opposite signs in that Hence, in the folding range −90 • < θ A < 0, if L| θ A < L| θ A , the negative Poisson's ratios v W L indicates that W θ A < W | θ A ; however, the positive Poisson's ratio v HL suggests that H| θ A > H| θ A . As a result, the lattice contracts in the L and W directions but extends in the H direction. In other words, the opposite Poisson's ratios are the origin of the 3D contraction and extension.

3D Shear
Three-dimensional shear is observed when transforming a 3D Bravais lattice of vertex inclusions based on a GFF sheet [27] [a kind of 3D Bravias lattice shown in Figs. 2(g) and 9(e)]. A GFF unit is characterized by crease lengths a, c and sector angles γ 1 , γ 2 . Without the loss of generality, we assume γ 1 < γ 2 . Its folding can be described by the dihedral angle ρ 1 or the folding angles θ i (i = 1, 2, 3, 4) (defined as the dihedral angle between the facets and the reference x-o-y plane). Folding of a GFF unit can also be divided into two stages. In the first stage, ρ 1 decreases from 180 • to a critical value ρ C 1 , while θ i (i = 1, 2, 3, 4) increase from 0 • but remains smaller than 90 • . At the end of the first stage, ρ 1 reaches a critical value ρ C 1 , and θ 1 and θ 3 reach 90 • , while θ 2 and θ 4 remain smaller than 90 • . In the second stage, ρ 1 continues decreasing towards 0, and θ 1 and θ 3 keep increasing (θ 1 = θ 3 > 90 • ), but θ 2 and θ 4 start to decrease and remain θ 2 = θ 4 < 90 • . Details of the folding kinematics are discussed in [27]. Specifically, Unlike the three aforementioned cases, here, three lattice vectors are not sufficient to describe the 3D non-Bravais lattice here due to the additional lattice point inside the unit cell. Therefore, a shift vector p has to be incorporated in addition to the lattice vectors l i [43], similar to the 2D shuffle scenario. The lattice vectors l i describe the constituent Bravais lattice, and the shift vector p describes the offset between the two congruent lattices. Specifically, these vectors corresponding to two different configurations (with folding angle θ A and θ A ) can be formulated as