Wallpaper group kirigami
WWallpaper group kirigami
Lucy Liu † , Gary P. T. Choi † , L. Mahadevan , ∗ Harvard College, Cambridge, MA, USA Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA Departments of Physics, and Organismic and Evolutionary Biology, Harvard University, Cambridge, MA, USA † These authors contributed equally to this work. ∗ To whom correspondence should be addressed; E-mail: [email protected]
Abstract
Kirigami, the art of paper cutting, has become the subject of study in mechanicalmetamaterials in recent years. The basic building blocks of any kirigami structures arerepetitive deployable patterns, the design of which has to date largely relied on inspirationsfrom art, nature, and intuition, embedded in a choice of the underlying pattern symmetry.Here we complement these approaches by clarifying some of the connections betweenkirigami patterns and symmetry in terms of the wallpaper groups, the set of seventeen planesymmetry groups that fully characterize the space of periodic tilings of the plane. We startby showing that deployable kirigami patterns in any of the seventeen wallpaper groups canbe constructed, and then design symmetry-preserving cut patterns that can be effectivelyapplied for achieving arbitrary size change throughout deployment. We further prove thatdifferent symmetry changes can be achieved by controlling the shape and connectivity of thetiles. Finally, we connect these results using the underlying lattice structures of the kirigamipatterns. All together, our work provides a systematic approach for creating deployablestructures with any prescribed size and symmetry properties, thereby paving the way for thedesign of a wide range of metamaterials by harnessing kirigami.
Kirigami, the creative art of paper cutting, has recently transformed from a beautiful art forminto a promising approach for the science and engineering of shape and thence function. Byintroducing architected cuts into a thin sheet of material, one can achieve deployable structures1 a r X i v : . [ c ond - m a t . s o f t ] F e b ith auxetic properties while morphing into pre-specified shapes. This has led to a number ofstudies on the geometry, topology and mechanics of kirigami structures (1–5). Most of thesestudies start with a relatively simple set of basic building blocks of kirigami patterns that take theform of triangles (6) or quads (7), although on occasion they take inspiration from art in the formof ancient Islamic tiling patterns (8), which are periodic. The periodicity of the pattern allowsus to easily scale up the design of a deployable structure without changing its overall shape.Recently, there have been attempts to explore generalizations of the cut geometry (9, 10) and cuttopology (11) moving away from purely periodic deployable kirigami base patterns. However, itis still unclear how one might explore such base patterns systematically. Since the deploymentof a kirigami structure is largely driven by the local rotation of the tiles, it is natural to ask whatclass of symmetries and size changes of the deployed structure can be achieved by controllingthe tile geometry and connectivity.A natural place to begin in our quest to address this question is to turn to the class oftwo-dimensional repetitive patterns that tile the plane, which are characterized by the planecrystallographic groups (the wallpaper groups ) (12). A remarkable result by Fedorov (13) andP´olya (14) is that there are exactly seventeen distinct wallpaper groups with different propertiesin terms of the rotational, reflectional, and glide reflectional (i.e. the combination of a reflectionover a line and a translation along the line) symmetries. Furthermore, the crystallographicrestriction theorem tells us that the order of rotational symmetry in any wallpaper group patterncan only be n = 1 , , , , . Table 1 lists the seventeen wallpaper groups (represented usingthe crystallographic notations) with their symmetry properties (15). While wallpaper groupshave started to form the basis for planar electromagnetic metamaterials (16, 17) and topologyoptimization (18), they do not seem to have been explored in the context of kirigami-basedmechanical metamaterials, with only a few patterns identified (19). Here, we remedy this andconsider all 17 of the wallpaper groups for the design of deployable kirigami patterns.2 otational Reflectional symmetrysymmetry Yes No Any mirrors at ◦ ? p4Yes: p4m No: p4g3-fold Any rotation center off mirrors? p3Yes: p31m No: p3m12-fold
Any perpendicular reflections? Any glide reflection?Yes; Any rotation center off mirrors?
No: Yes: pgg No: p2Yes: cmm No: pmm pmg1-fold
Any glide reflection axis off mirrors? Any glide reflection? (none) Yes: cm No: pm Yes: pg No: p1Table 1: Characterization of the seventeen wallpaper groups (15).
The first question that naturally arises is whether all seventeen wallpaper groups can be used fordesigning deployable kirigami patterns. We answer this question by establishing the followingresult:
Theorem 1.
For any group G among the seventeen wallpaper groups, there exists a deployablekirigami pattern in G . To see this, we construct explicit examples of periodic deployable structures in all seventeenwallpaper groups (Fig. 1). Key reflection axes, glide reflection axes and rotation centers arehighlighted and can be used together with Table 1 for determining the wallpaper group type foreach of them. Note that all patterns in Fig. 1 are rigid-deployable, i.e. there is no geometricalfrustration in the deployment of them (see also SI Video S1).More examples of periodic deployable patterns are given in Fig. 2. Fig. 2 a - b shows tworigid-deployable patterns derived from the p6 example in Fig. 1. Fig. 2 c - e show three rigid-deployable patterns derived from the standard kagome pattern. Fig. 2 f - l shows seven patternsderived from the standard quad pattern. Fig. 2 m - n show two rigid-deployable p4g patterns, with3 p31m p6m p6 p6 p6m p4 p4 p4m p3 p3 p6 pgg pgg cmm p2 p2 p4 pg pg cm p1 p1 p1 pm cm cm cm p1 p1 pmg pgg p4g pmm pmg pmg cmm p2 pmm p3m1 p3 p3 p31m p3 p31m p4g pgg cmm p4m p4g p4m Figure 1:
Examples of periodic deployable kirigami patterns in the seventeen wallpapergroups . For each example (with the crystallographic notation boldfaced), we show a portion ofthe initial contracted state, an intermediate deployed state and the fully deployed state. Tiles withdifferent shapes are in different colors. Key reflection axes (red dotted lines), glide reflectionaxes (blue dotted lines) and rotation centers (red dots) that can be used for determining theirwallpaper group type are highlighted.different underlying topologies that lead to different wallpaper group changes under deployment.Fig. 2 o shows a rigid-deployable p4 pattern with the same topology as the pattern in Fig. 2 n . Itis noteworthy that not all deployable kirigami patterns are rigid-deployable. Fig. 2 p shows two4 jd i pg pmgpg h p3 p3p3p6 p6p6 a g pg pgpg onml p31m p31mp3 pm pgpg ce k cm pmcm p4g p31mp4g p1p1p4g p4p4p4g p31mp4 p4p4 p p4m p4mp4 f p3 p3m1p3 b Figure 2:
More examples of periodic deployable patterns. a - b , Two rigid-deployable patternsderived from the p6 example in Fig. 1. c - e , Three rigid-deployable patterns derived from thestandard kagome pattern. f - l , Seven rigid-deployable patterns derived from the standard quadpattern. m , A rigid-deployable p4g pattern consisting of squares and rhombi. Note that thepattern becomes p1 once deployed. n , Another rigid-deployable p4g pattern created by breakingthe rhombi in m into triangles. This time, the pattern becomes p4 throughout the deployment. o , A rigid-deployable p4 pattern. Note that it has the same underlying topology as n . p , Twobistable Islamic tiling patterns (8) which are not rigid-deployable. Geometrical frustration existsat the intermediate deployments, while the initial and final states shown are frustration-free.Key examples of the reflection axes (red dotted lines), glide reflection axes (blue dotted lines)and rotation centers (red dots) that can be used for determining their wallpaper group type arehighlighted.bistable p4g and p3m1 Islamic tiling patterns (8), which exhibit geometrical frustration at theintermediate states of the deployment while being frustration-free at the contracted and finaldeployed states. Note that Theorem 1 focuses on the initial (contracted) state of deployable5irigami patterns. In fact, from Fig. 1 and Fig. 2, we can also see examples of periodic deployablepatterns with final deployed shape in any of the seventeen wallpaper groups:• p6m: See the p6m → p31m → p6m example in Fig. 1.• p6: See the p6 → p6 → p6 example in Fig. 2 a .• p4m: See the p4m → p4g → p4m example in Fig. 1.• p4g: See the pmg → pgg → p4g example in Fig. 1.• p4: See the p2 → p2 → p4 example in Fig. 1.• p31m: See the p31m → p3 → p31m example in Fig. 2 c .• p3m1: See the p3m1 → p3 → p3m1 example in Fig. 2 d .• p3: See the p3m1 → p3 → p3 example in Fig. 1.• cmm: See the pgg → pgg → cmm example in Fig. 1.• pmm: See the cmm → p2 → pmm example in Fig. 1.• pmg: See the pg → pg → pmg example in Fig. 2 g .• pgg: See the pg → pg → pgg example in Fig. 2 h .• p2: See the p2 → p2 → p2 example in Fig. 2 i .• cm: See the pg → pg → cm example in Fig. 1.• pm: See the cm → cm → pm example in Fig. 2 k .• pg: See the pg → pg → pg example in Fig. 2 l .6 p1: See the cm → p1 → p1 example in Fig. 1.Therefore, we have the following theorem: Theorem 2.
For any wallpaper group G among the seventeen wallpaper groups, there exists adeployable kirigami pattern with its final deployed shape in G . After showing the existence of deployable kirigami patterns in all seventeen wallpaper groups forboth the contracted and deployed states, it is natural to ask whether some of the wallpaper groupsare more advantageous over the others in terms of deployable kirigami design. In particular, onemay wonder whether the size change of a deployable pattern is limited by its symmetry. It isclear that the size change is not limited by the reflectional symmetry or the glide reflectionalsymmetry. Here we show that the size change can in fact be arbitrary for any given rotationalsymmetry:
Theorem 3.
For any deployable wallpaper group pattern with n -fold rotational symmetry, wecan design an associated pattern with n -fold rotational symmetry and arbitrary size change. The result is achieved by designing certain expansion methods for augmenting a given patternwith n -fold symmetry without breaking its symmetry. Two expansion methods are introducedbelow (see Fig. 3 and SI Video S2). To achieve significant size change while preserving rotational symmetry, expansion cuts can beintroduced to select rotating units in the pattern. Using a 4-fold expansion cut on a square in a1-fold, 2-fold or 4-fold pattern, we can achieve an expansion of the pattern without changingits rotational symmetry (Fig. 3 a ). Using a 3-fold expansion cut on a triangle in a pattern with7 bc d e S S s S s s s a a Figure 3:
Symmetry-preserving expansion . a , An expansion cut pattern on a square with 4-foldrotational symmetry (top left). The pattern can be refined hierarchically to achieve a larger sizechange (bottom left). These expansion cut patterns can be utilized for augmenting deployablepatterns with -, -, or -fold rotational symmetry, such as the p4 pattern in Fig. 1, to achievean arbitrary size change while preserving the rotational symmetry (right). b , An expansion cutpattern on a regular triangle with 3-fold rotational symmetry (top left). The pattern can be refinedhierarchically to achieve a larger size change (bottom left). These expansion cut patterns canbe utilized for augmenting deployable patterns with -, -, or -fold rotational symmetry, suchas the p6 pattern in Fig. 1 to achieve an arbitrary size change while preserving the rotationalsymmetry (right). c , Another type of expansion cuts on a p4 pattern produced by placingadditional rectangular units between tiles. The first row shows the contracted, intermediate andfully deployed state of an augmented p4 pattern with 1 expansion layer. The second row showsthe contracted and deployed state of an augmented p4 pattern with 2 expansion layers. d , Anaugmented p4m pattern constructed in a similar manner. e , The top row shows the contractedand deployed state of a deployable p4 pattern, with the shaded blue regions representing a unitcell and its deployed shape. The bottom row shows an augmented version of it with 1 level of“ideal” expansion cuts of infinitesimal width.1-fold, 3-fold or 6-fold rotational symmetry, we can achieve an expansion of the pattern withoutchanging its rotational symmetry (Fig. 3 b ).While the above expansion cuts are introduced on a square and a triangle only, it is easy tosee that similar expansion cuts can be introduced on any tiles with 4-fold and 3-fold rotational8 b c Figure 4:
More symmetry-preserving expansion cut patterns. a , An expansion cut patternthat can be introduced on any tile with 6-fold rotational symmetry. The cut pattern achievesan expansion throughout deployment while preserving the 6-fold symmetry of the tile. b , Anexpansion cut pattern that can be introduced on any tile with 2-fold rotational symmetry. Thecut pattern achieves an expansion throughout deployment while preserving the 2-fold symmetryof the tile. c , An expansion cut pattern with 2-fold rotational symmetry that preserves both therotational symmetry and reflectional symmetry. The expansion cut pattern is derived from thepattern in Fig. 3 a , with four copies of it placed appropriately to form a pattern that preserves thereflectional symmetry throughout the deployment. The top row shows the deployment of thepattern with 1 level of cuts. The bottom row shows the deployment of the pattern with 2 levels ofcuts.symmetry respectively. Fig. 4 a shows a symmetry-preserving expansion cut pattern that canbe introduced on any 6-fold tile (e.g. a regular hexagon). The cut pattern preserves the 1-fold,2-fold, 3-fold or 6-fold rotational symmetry of the entire kirigami pattern. Fig. 4 b shows asymmetry-preserving expansion cut pattern that can be introduced on any 2-fold tile (e.g. arectangle). The cut pattern preserves the 1-fold or 2-fold rotational symmetry of the entirekirigami pattern. It can be observed that by increasing the level of cuts, we can achieve a largersize change.Note that the pattern in Fig. 3 a can be utilized for augmenting a given periodic deployable9attern to achieve an arbitrary size change, while the reflectional symmetry of the given patternmay be lost. Fig. 4 c shows an expansion cut pattern with 2-fold rotational symmetry derivedfrom it. We suitably reflect the pattern to form an expansion cut pattern on a square consistingof 16 triangles and 4 squares. Note that the new cut pattern is not only with 2-fold rotationalsymmetry but also reflectional symmetry. Therefore, it can be utilized for augmenting a givenperiodic deployable pattern with - or -fold rotational symmetry while preserving both itsrotational symmetry and reflectional symmetry. Another way to design an associated pattern with increased size change is to add rotating unitsbetween adjacent tiles of the original pattern (Fig. 3 c - d ). More specifically, we augment a givendeployable pattern by adding thin rectangles between adjacent tiles, which allow for greaterexpansion when the pattern is deployed. Analogous to the above-mentioned method, it is possibleto preserve the rotational symmetry of the given pattern by appropriately placing the additionalunits. Again, it is possible to preserve the reflectional symmetry of the contracted state or eventhe deployed state of certain patterns using this method (for example, the pattern in Fig. 3 d with an even number of expansion layers). We remark that this method introduces gaps to thecontracted state of the new pattern. To quantify the size change achieved by our proposed symmetry-preserving expansions, weconsider the p4 pattern in Fig. 3 e and denote the side length of the larger and smaller squaresin the original pattern as S and s , with s ≤ S . We measure the size change of the pattern upondeployment by selecting a unit cell in the contracted state and comparing its area to that of acorresponding unit cell in the deployed state (Fig. 3 e , the shaded regions in the top row). It is10asy to see that the contracted unit cell has area S + s and the deployed unit cell has area ( S + s ) . Therefore, the base size change ratio is r = ( S + s ) S + s . (1)This ratio simplifies to when S = s and when S = 2 s .Each expansion cut creates a new unit that, upon deployment, rotates to further separatethe original tiles of the base pattern. In the fully deployed state, we let a i be the additionalvertical and horizontal separation introduced by each cut in the i -th round of expansion cuts. Theexpansion cuts also shave area off of the original tiles in order to form the new rotating units. Let b i be the width that the squares of side length s lose from each cut in the i -th round of expansioncuts.With n rounds of expansion cuts, the unit cell’s area after deployment will be ( S + s +2 (cid:80) ni =1 ( a i − b i )) and the size change ratio will be r n = ( S + s + 2 (cid:80) ni =1 ( a i − b i )) S + s . (2)The values of elements in a i and b i depend on the shape and width of expansion cuts. If weconsider “ideal” expansion cuts of length s and infinitesimal width, then a i = s √ and b i = 0 forall i (Fig. 3 e , bottom row). For these ideal cuts, the size change ratio after n rounds of expansionwould be r n = ( S + s + 2 n s √ ) S + s = ( S + s + √ ns ) ( S + s ) . (3)This suggests that the size change ratio scales approximately with n , and we can achieve anarbitrary size change by choosing a sufficiently large n .Similarly, one can perform an analysis on the size change of the triangle expansion cut patternin Fig. 3 b . We select a unit cell in the contracted state and compare it to the corresponding unitsin the deployed and expanded states. Unit cells are represented as shaded areas in Fig. 5. Let S b s n c n S Figure 5:
Size change in the triangle-hexagon deployment pattern. a , The contracted anddeployed states of the pattern with no expansion cuts used. The shaded areas represent the unitcell used to calculate the pattern’s size change ratio. b , On the left, an image of the triangle-hexagon pattern with one round of expansion cuts deployed. On the right, a close-up of anirregular octagon formed by deployment of the expansion cuts. The octagon is broken into thesmaller triangles and rectangles we use to determine its area.be the side length of the hexagons and s be the side length of the triangles; a regular hexagonwill have area √ S and a regular triangle area √ s .The contracted state unit cell consists of one hexagon and two triangles, which together havearea √ S + √ s . The deployed state unit cell has three additional rectangles, each with area Ss . Then the deployed unit cell area is √ S + √ s + 3 Ss , and the base size change ratio is r = 3 √ S + √ s + 6 Ss √ S + √ s . (4)For this pattern, expansion cuts as done in Fig. 3 b introduce gaps with area equivalent tothat of the irregular octagons shown in blue in Fig. 5 b . We assume expansion cuts are done in amanner that preserves the equilateral triangle shape of the green tiles.Let a i be the additional separation the i -th round of expansion cuts adds between each pairof adjacent triangles and hexagons, so each pair’s closest vertices are now c n = (cid:80) ni =1 a i apart.Let b i be the side length each equilateral triangle loses in the i th round of expansion cuts, so s n = s − (cid:80) ni =1 b i is the triangle’s remaining side length after n rounds of expansion cuts.Each octagon can be broken into smaller rectangles and triangles as seen in the figure: four12 − − triangles of area a n √ , two rectangles of area c n s n , two rectangles of area Sc n √ ,and a center rectangle of area Ss n . The triangles are − − because maximal deploymentoccurs when the triangles and hexagons of the original pattern are as separated as possible. Thisoccurs when each edge between a triangle vertex and a hexagon vertex bisects both vertex angles.The octagon’s total area will then be c n √ + c n s n + Sc n √ Ss n .After n rounds of expansion cuts, the expanded unit cell consists of a regular hexagon withside length S , two equilateral triangles with side length s n , and three irregular octagons asdescribed above. This unit cell has area √ S + √ s n + 3( c n √ + c n s n + Sc n √ Ss n ) . Thenthe size change ratio is r n = 3 √ S + √ s n + 6( c n √ + c n s n + Sc n √ Ss n )3 √ S + √ s , (5)where s n = s − (cid:80) ni =1 b i and c n = (cid:80) ni =1 a i .Now, if we consider “ideal” expansion cuts of length s and infinitesimal width, we have a i = s and b i = 0 for all i . It follows that c n = ns and s n = s . Therefore, with these idealexpansion cuts we have r n = 3 √ S + √ s + 6( n s √ + ns + Sns √ Ss )3 √ S + √ s , (6)which scales approximately with n and is unbounded. This shows that we can achieve anarbitrary size change using the triangle expansion cut pattern with suitable refinements.We remark that by the above symmetry-preserving expansion methods, one can also easilyachieve an arbitrary perimeter change. Now, we study how the kirigami patterns change in terms of the wallpaper groups throughoutthe deployment. More specifically, what are the possible symmetry changes throughout thedeployment? We have the following result: 13 heorem 4.
Gain, loss, and preservation of symmetry are all possible throughout the deploymentof a kirigami pattern.
To see this, note that from Fig. 1 and Fig. 6 we can observe different types of symmetrychange as a pattern expands from its contracted state to its deployed state:• Rotational symmetry gained: pmg → pgg → p4g (permanent)• Rotational symmetry lost: p4g → pgg → cmm (permanent), p6m → p31m → p6m(temporal)• Rotational symmetry preserved: p6 → p6 → p6m• Reflectional symmetry gained: pgg → pgg → cmm (permanent)• Reflectional symmetry lost: p3m1 → p3 → p3 (permanent), p4g → pgg → cmm (temporal)• Reflectional symmetry preserved: p4m → p4g → p4m• Glide reflectional symmetry gained: p1 → p1 → p4m (permanent)• Glide reflectional symmetry lost: cm → p1 → p1 (permanent), cmm → p2 → pmm(temporal)• Glide reflectional symmetry preserved: pg → pg → cmWe remark that although some patterns preserve rotational, reflectional or glide reflectionalsymmetry, the rotation centers and reflection axes do not necessarily remain fixed. For instance,the pattern cmm → p2 → pmm has rotation centers off mirrors at the initial state, while allrotation centers lie on mirrors at the final deployed state. For the pattern pm → cm → cm, thenumber of reflection axes decreases throughout deployment, while the number of glide reflectionaxes remains unchanged. 14 cbd p1 p2p1 PBCD A O
PBCD OA TQ S Q p31m p1p1 f p1 p4mp1 e p2 p6mp2 Figure 6:
Exploring possible symmetry changes . a , A general deployed quad pattern with2-fold rotational symmetry. The parallelograms can be changed in pairs (red and blue, or yellowand green) to other shapes whose vertices form parallelograms. b , For the case where the centerof rotation (COR) O is at the center of a rift, consider two corresponding points P , Q within tilesand denote the two opposite vertices of the rift by A and C . We can show that ∆ AP O ∼ = ∆ CQO throughout deployment or contraction, which implies that the contracted pattern is also with2-fold rotational symmetry. c , For the case where the COR O is at the center of a tile. d , Forthe case where the COR O at the intersection point of two tiles, we construct a deployablepattern that achieves a 1-fold to 2-fold symmetry change. e , Using a variation of the standardquad kirigami topology, we can achieve a 1-fold to 4-fold symmetry change as well as a gain inreflection. f , Introducing floppiness can lead to a large variety of symmetry changes, such as a2-fold to 6-fold symmetry gain (p2 → p2 → p6m) or a loss in all symmetries (p31m → p1 → p1). For each pattern, tiles with different shapes are in different colors. Note that several pattern examples in Fig. 1 are with the standard quad kirigami topology, whereunit cells containing four tiles arranged as seen in the p4m example in Fig. 1 are connected in alarger grid. It can be observed that these patterns exhibit 1-fold, 2-fold, and 4-fold rotationalsymmetry, and some of them can even achieve a 2-fold to 4-fold rotational symmetry change(the p2 and pmg examples). Are other rotational symmetry gains possible for patterns with thistopology, like changes from 1-fold to 2-fold or 4-fold rotational symmetry?15ere we consider a general deployed pattern with the standard quad kirigami topology and2-fold rotational symmetry (Fig. 6 a ). Note that for simplicity we use parallelograms to representthe tiles. These parallelograms can be changed to other shapes so long as the four vertices whereeach shape connects to other tiles form parallelograms, and congruent parallelograms are formedby vertices of the red and blue shape pair, and the yellow and green shape pair. We examine thethree possible cases for where O , a center of rotation (COR) of the deployed state, may lie: atthe center of a rift (a gap that forms when tiles separate during deployment), the center of a tile,or a point where two tiles connect. (i) O is at the center of a rift In the unit cell containing the rift and its four adjacent tiles, consider any two points P and Q which lie within tiles and map to each other after a 180-degree rotation around O . Then O isthe midpoint of P Q , and we can construct the congruent triangles shown in Fig. 6 b involving P, Q, O , and two opposite vertices of the rift A and C . AO = OC because the center of aparallelogram bisects its diagonals. Since opposite parallelograms across the rift are congruent,the rift will be a parallelogram.The rift changes shape during deployment, but the tiles themselves are rigid, so P and Q area fixed translation from A and C respectively. Then P A = CQ and ∠ P AB = ∠ QCD at anypoint in deployment. Throughout deployment the rift remains a parallelogram, so AO = OC and ∠ P AO = ∠ QCO . Therefore, ∆ AP O ∼ = ∆ CQO at all stages of deployment.It follows that O remains a midpoint of P Q , so in the contracted state, a 180-degree rotationaround O still maps P and Q to each other. Then 2-fold rotational symmetry is preserved and O remains a COR for the contracted unit cell. The full pattern is a grid of unit cells, and whenwe rotate it around O , each unit cell is rotated right onto another unit cell. As the unit cell has2-fold rotational symmetry, the rotational symmetry of the entire pattern is preserved. Thus,16he contracted pattern must have 2-fold rotational symmetry, so a 1-fold contracted state cannotdeploy to a 2-fold or 4-fold state in this case. (ii) O is at the center of a tile As shown in Fig. 6 c , we now consider the outlined unit cell centered at O . Within this cell, bychoosing two points P and Q which map to each other through 180-degree rotation, constructingcongruent triangles, and applying the argument from case (i), we can again see that as the angleschange through deployment, P and Q remain a 180-degree rotation around O apart. Therefore,the unit cell retains 2-fold rotational symmetry around O . Once again, O also remains a COR forthe full pattern due to its grid structure. This shows that a 1-fold contracted state cannot deployto a 2-fold or 4-fold state in this case. (iii) O is at the intersection point of two tiles As shown in Fig. 6 d , we construct an explicit example of a deployable pattern with a 1-foldcontracted state and a 2-fold deployed state (p1 → p1 → p2). Note that for this case 4-foldsymmetry is not possible in the deployed state as a 90-degree rotation maps a tile to a rift.We conclude that for patterns with the standard quad kirigami topology, 1-fold to 4-foldrotational symmetry gain is not possible, and that 1-fold to 2-fold gain is possible only in case(iii). This type of analysis offers a systematic way to understand how deployment affects patternsymmetry. Note that the above analysis has only focused on the standard quad kirigami topology. If weconsider other cut topologies, we can achieve a larger variety of symmetry changes. For instance,using a variation of the standard quad kirigami topology, one can achieve a pattern with a 1-foldto 4-fold symmetry change and a gain in reflectional symmetry (Fig. 6 e ).17or the patterns we have considered so far, each tile has at least two vertices connectedto vertices of neighboring tiles, and so the motions of all tiles are interrelated. However, onecan also consider changing the underlying topology of certain kirigami patterns such that someof the tiles have only one vertex connected to another tile, thereby increasing the floppinessof the patterns. Fig. 6 f shows two patterns with floppy rhombus or triangle tiles. During andafter deployment, these floppy tiles have only one vertex’s position determined and are freeto rotate around that fixed vertex. The p2 → p2 → p6m pattern exhibits a 2-fold to 6-foldsymmetry change and a gain in reflectional symmetry, while the p31m → p1 → p1 pattern losesall symmetries throughout deployment. Now, we present a more detailed analysis of the possible symmetry changes in terms of the gain,loss and preservation of reflectional, glide reflectional and rotational symmetries. : In Section 4.1, we have explored the possible rotational symmetry gainfor quad patterns. In fact, by considering more general periodic deployable patterns (possiblywith floppy tiles), we can show that an n -fold to m -fold rotational symmetry gain is possible forany n, m ∈ { , , , , } with n | m and m > n :• → : See the p1 → p1 → p2 example in Fig. 6 d , and the patterns in Fig. 2 g - h .• → : See the pmg → pgg → p4g example and the p2 → p2 → p4 example in Fig. 1.• → : See the p3 → p3 → p6 example in Fig. 1.• → : See the p1 → p1 → p31m example in Fig. 7 a .• → : See the p2 → p2 → p6m example in Fig. 6 f .18 a p1 p6mp1 b Figure 7:
More examples of deployable kirigami patterns with rotational symmetry gainthroughout deployment. a , A p1 → p1 → p31m example derived from the example in Fig. 6 f ,with the shape of the triangles modified. Deployment of a unit cell is shown on the left, anddeployment of a larger pattern section is shown on the right. b , A p1 → p1 → p6m examplederived from the example in Fig. 6 f , with the geometry of the pattern and the connectivity of thetriangles modified. Deployment of a unit cell is shown on the left, and deployment of a largerpattern section is shown on the right.• → : See the p1 → p1 → p4m example in Fig. 6 e .• → : See the p1 → p1 → p6m example in Fig. 7 b . Reflectional symmetry : Gain of reflectional symmetry can be observed for all n = 1 , , , , :• n = 6 : See the p6 → p6 → p6m example in Fig. 1.• n = 4 : See the p4 → p4 → p4m example in Fig. 1.• n = 3 : See the p3 → p3 → p3m1 example in Fig. 2 b .• n = 2 : See the pgg → pgg → cmm example in Fig. 1.• n = 1 : See the pg → pg → pmg example in Fig. 2 g .19 lide reflectional symmetry : Gain of glide reflectional symmetry can be observed for all n = 1 , , , , :• n = 6 : See the p6 → p6 → p6m example in Fig. 1.• n = 4 : See the p4 → p4 → p4m example in Fig. 1.• n = 3 : See the p3 → p3 → p3m1 example in Fig. 2 b .• n = 2 : See the p2 → p2 → p6m example in Fig. 6 f .• n = 1 : See the p1 → p1 → p31m example in Fig. 7 a . We can see that an m -fold to n -fold rotational symmetry loss is possible for any n, m ∈{ , , , , } with n | m and m > n :• → : See the cmm → p1 → p1 example in Fig. 8 a .• → : See the p4m → pmm → pmm example in Fig. 8 b and the p4m → pmg → pmgexample in Fig. 8 c .• → : See the p6m → p3 → p3 example in Fig. 8 d .• → : See the p31m → p1 → p1 example in Fig. 6 f .• → : See the p6m → pmg → pmm example in Fig. 8 e .• → : See the p4g → p1 → p1 example in Fig. 9 f .• → : See the p6m → p1 → p1 example in Fig. 8 f .Loss of reflectional and glide reflectional symmetries can be easily achieved by breaking theconnectivity of the tiles (see Section 5 for a more detailed discussion).20
4m pmgpmg c e cmm p1p1 a d p4m pmmpmm b p6m p3p3 f p6m pmmpmgp6m p1p1 Figure 8:
More examples of deployable kirigami patterns with rotational symmetry lostthroughout deployment. a , A cmm → p1 → p1 example derived from the standard quadpattern. b , A p4m → pmm → pmm example derived from the standard quad pattern. c , A p4m → pmg → pmg example derived from the standard quad pattern. d , A p6m → p3 → p3 examplederived from the standard kagome pattern. e , A p6m → pmg → pmm example derived fromthe standard kagome pattern. f , A p6m → p1 → p1 example derived from the standard kagomepattern. It can be observed that preservation of n -fold rotational symmetry throughout deployment ispossible for all n = 1 , , , , :• n = 6 : See the p6 → p6 → p6m example in Fig. 1.• n = 4 : See the p4m → p4g → p4m example in Fig. 1, and the p4m → p4 → p4m examplein Fig. 2 f .• n = 3 : See the p31m → p3 → p31m example in Fig. 1, and the p3m1 → p3 → p3m1example in Fig. 2 d . 21 n = 2 : See the cmm → p2 → pmm example and the pgg → pgg → cmm example inFig. 1.• n = 1 : See the cm → p1 → p1 example and the pm → cm → cm example in Fig. 1.Preservation of reflectional and glide reflectional symmetries can also be observed (see thep6m → p31m → p6m example and the p4m → p4g → p4m example in Fig. 1).Furthermore, it is possible to design periodic deployable patterns with n -fold rotationalsymmetry that stay in the same wallpaper group throughout deployment:• n = 6 : See the p6 → p6 → p6 example in Fig. 2 a .• n = 4 : See the p4 → p4 → p4 example in Fig. 2 o .• n = 3 : See the p3 → p3 → p3 example in Fig. 2 e .• n = 2 : See the p2 → p2 → p2 example in Fig. 2 i .• n = 1 : See the p1 → p1 → p1 example in Fig. 1. From the above results on the gain, loss and preservation of rotational symmetry, we have thefollowing theorem:
Theorem 5.
For any n, m ∈ { , , , , } with m ≥ n and n | m , it is possible to design adeployable kirigami pattern that achieves an n -fold to m -fold rotational symmetry changethroughout deployment, and a pattern that achieves an m -fold to n -fold rotational symmetrychange throughout deployment. Similarly, from the above results on the gain, loss and preservation of reflectional and glidereflectional symmetry, we have the following theorems:22 heorem 6.
For any n = 1 , , , , , it is possible to design a deployable kirigami pat-tern with n -fold rotational symmetry that achieves any target reflectional symmetry change(gain/loss/preservation) throughout deployment. Theorem 7.
For any n = 1 , , , , , it is possible to design a deployable kirigami patternwith n -fold rotational symmetry that achieves any target glide reflectional symmetry change(gain/loss/preservation) throughout deployment. Observing the close relationship between the wallpaper group of a periodic deployable kirigamipattern and its underlying topology, we analyze different patterns in terms of their latticerepresentations (Fig. 9). In the lattice representation, each tile is represented by a node. Anedge between two nodes exists if their corresponding tiles are connected to each other. Herewe introduce a cyclic notation for the lattice representation of a periodic kirigami pattern.Starting from a tile with the lowest connectivity, we denote a as its number of neighbors. Wethen consider all neighbors of the tile and choose the one with the lowest connectivity, anddenote its number of neighbors as a . We continue the process until the sequence repeats (i.e. a , a , . . . , a k , a , a , . . . ), and use ( a , a , . . . , a k , a ) to represent the lattice. We remark thatwhile each sequence does not necessarily correspond to a unique lattice structure, it helps usunderstand the connectivity of any given periodic deployable kirigami pattern. Below, we analyzethree lattice types we observed in periodic deployable kirigami patterns (see Fig. 10 for moreexamples). Note that the only regular polygons that can tile the plane are the triangle, square, and hexagon.Therefore, the only regular lattices are (4 , (Fig. 9 a ), (3 , (Fig. 9 b ) and (6 , (Fig. 9 c ). Note23 d fe b c Figure 9:
Lattice representations of kirigami patterns . a , The regular (4 , lattice repre-senting the standard quad topology (e.g. the p4m, p4g, pmg patterns in Fig. 1), with each tileconnected to exactly four adjacent tiles. b , The regular (3 , lattice representing the standardkagome topology (e.g. the p6m, p3m1 and p3 patterns in Fig. 1). c , The regular (6 , lattice.Note that the triangles in it make the structure rigid. d , Using the expansion cuts introduced inFig. 3, we can turn a deployable structure with the regular (4 , lattice into another deployablestructure with a (2 , , lattice. e , Using the expansion tiles introduced in Fig. 3, we can turnthe triangles in a rigid lattice into another polygon, thereby producing novel deployable patternswith a (2 , , lattice (left) or a (2 , , lattice (right). Both examples shown here are p6m → p6 → p6. f , Breaking certain connections in a given lattice yields a 1-fold deployable structurewith another lattice representation. The example shown here is a p4g → p1 → p1 pattern with a (1 , , , , lattice (see also the p31m pattern in Fig. 6 f with a (2 , , , lattice).that the regular (4 , lattice and the regular (3 , lattice correspond to the standard quad kirigamitopology and the standard kagome kirigami topology, both of which are deployable. On thecontrary, the rigidity of the triangles (3-cycles) in the regular (6 , lattice prevents it from beingdeployable. Another type of lattice we observed can be viewed as an augmented version of the regular lattice,with certain tiles inserted in a rotationally symmetric way. One example is the topology of thep6 pattern in Fig. 1, which is a deployable (3 , , lattice. Interestingly, the two symmetry-24 b Figure 10:
More deployable lattice structures. a , The topology of the p6 pattern in Fig. 1is a deployable (3 , , lattice. b , The topology of the p4g and p4 patterns in Fig. 2 n - o is adeployable (2 , , lattice.preserving expansion methods we introduced in Fig. 3 provide us with a systematic way ofcreating new symmetric augmented lattice of deployable structures from any given pattern. Forinstance, the expansion cuts allow us to turn the regular (4 , lattice into another deployablestructure with a (2 , , lattice (Fig. 9 d ) while preserving the 4-fold rotational symmetry of thelattice. The expansion tiles also effectively add vertices along the edges of the rigid triangles ina given lattice, thereby turning them into other polygons and making the structure deployable,with the rotational symmetry preserved. Fig. 9 e shows an example of turning a rigid (6 , latticeinto a deployable (2 , , lattice (top), and an example of turning a rigid (4 , lattice into adeployable (2 , , lattice (bottom). This shows that the expansion methods are not only usefulgeometrically but also mechanically for the design of deployable kirigami patterns. By removing certain connections in the lattice of a given kirigami pattern, we can break thesymmetry of the lattice and hence achieve a large variety of changes in the rotational, reflectional25r glide reflectional symmetries throughout deployment. For instance, one can obtain a floppy (1 , , , , lattice as shown in Fig. 9 f and a floppy (2 , , , lattice as shown in the p31mpattern in Fig. 6 f , which lose all symmetries throughout deployment. This shows that it ispossible to create a kirigami pattern in any wallpaper group G with the deployment path G → p1 → p1. Using trimmed lattice with carefully designed tile geometries, it is also possible toachieve a symmetry gain such as the 2-fold to 6-fold rotational symmetry gain in the floppy p2pattern in Fig. 6 f . One can consider the space of the seventeen wallpaper groups as a directed graph G = ( V , E ) ,where the vertex set V consists of the seventeen nodes representing the seventeen wallpapergroups, and the directed edge set E consists of directed arrows indicating all possible groupchanges throughout deployment of the kirigami patterns. Based on the patterns we have identifiedin this paper, we construct a subgraph (cid:101) G = ( V , (cid:101) E ) of G where (cid:101) E are obtained from the patternswe have identified (see Fig. 11). It is easy to see that (cid:101) G is connected, which suggests that G isalso connected.By Theorem 1, the out-degree of any vertex in G is at least 1, which is evident from the graph (cid:101) G . Similarly, by Theorem 2, the in-degree of any vertex in G is at least 1. We can easily see thateach wallpaper group in the graph (cid:101) G is the endpoint of some paths.By the floppy lattice construction introduced above, the in-degree of p1 in G should be exactly17. Note that in the graph (cid:101) G in Fig. 11, we have omitted all G → p1 → p1 changes except forthose explicitly described in the figures in this paper.26 fl ection Without re fl ection Figure 11:
The graph (cid:101) G of possible group changes observed in the patterns we have identi-fied. The group change of each example pattern considered in this paper is represented using onedistinct color. For instance, the red arrows from p6m to p31m and from p31m to p6m correspondto the p6m → p31m → p6m standard kagome pattern in Fig. 1, and the orange arrow from p6 top6m corresponds to the p6 → p6 → p6m pattern in Fig. 1. We remark that (cid:101) G is only a subgraphof G , the graph of all possible group changes. 27 Discussion
In this work, we have explored the connection between kirigami and the symmetry associatedwith the planar wallpaper groups. We have shown that it is possible to create deployable patternsusing all of the seventeen wallpaper groups, and further studied the size change, symmetrychange and lattice structure of these patterns, yielding key insights into the design of deployablekirigami structures. The ability to control the size, perimeter and symmetry changes in theseexamples makes kirigami a very viable paradigm for art, architecture and technology in suchinstances as the design of energy-storing devices (20), electromagnetic antennae (21, 22) etc.A natural limitation of planar periodic deployable patterns is in their rotational symmetry,with the possible orders of rotation being n = 1 , , , , only. A class of planar tessellationsclosely related to the wallpaper groups are the aperiodic quasicrystal patterns (23) that can alsotile the plane. While lacking translational symmetry, quasicrystal patterns can exhibit rotationalsymmetry not found in any wallpaper group patterns. Natural next steps include the possibilityof creating deployable structures based on quasicrystal patterns, and extending our study ofsymmetries of deployable patterns to 3D for the design of structural assemblies (24). References
1. B. G.-g. Chen, B. Liu, A. A. Evans, J. Paulose, I. Cohen, V. Vitelli, and C. D. Santangelo,“Topological mechanics of origami and kirigami,”
Phys. Rev. Lett. , vol. 116, no. 13, p. 135501,2016.2. Y. Tang and J. Yin, “Design of cut unit geometry in hierarchical kirigami-based auxeticmetamaterials for high stretchability and compressibility,”
Extreme Mech. Lett. , vol. 12,pp. 77–85, 2017. 28. L. A. Lubbers and M. van Hecke, “Excess floppy modes and multibranched mechanisms inmetamaterials with symmetries,”
Phys. Rev. E , vol. 100, no. 2, p. 021001, 2019.4. M. K. Blees, A. W. Barnard, P. A. Rose, S. P. Roberts, K. L. McGill, P. Y. Huang, A. R.Ruyack, J. W. Kevek, B. Kobrin, D. A. Muller, et al. , “Graphene kirigami,”
Nature , vol. 524,no. 7564, p. 204, 2015.5. T. C. Shyu, P. F. Damasceno, P. M. Dodd, A. Lamoureux, L. Xu, M. Shlian, M. Shtein, S. C.Glotzer, and N. A. Kotov, “A kirigami approach to engineering elasticity in nanocompositesthrough patterned defects,”
Nat. Mater. , vol. 14, no. 8, p. 785, 2015.6. J. N. Grima and K. E. Evans, “Auxetic behavior from rotating triangles,”
J. Mater. Sci. ,vol. 41, no. 10, pp. 3193–3196, 2006.7. J. N. Grima and K. E. Evans, “Auxetic behavior from rotating squares,”
J. Mater. Sci. Lett. ,vol. 19, no. 17, pp. 1563–1565, 2000.8. A. Rafsanjani and D. Pasini, “Bistable auxetic mechanical metamaterials inspired by ancientgeometric motifs,”
Extreme Mech. Lett. , vol. 9, pp. 291–296, 2016.9. G. P. T. Choi, L. H. Dudte, and L. Mahadevan, “Programming shape using kirigami tessella-tions,”
Nat. Mater. , vol. 18, no. 9, pp. 999–1004, 2019.10. G. P. T. Choi, L. H. Dudte, and L. Mahadevan, “Compact reconfigurable kirigami,”
Preprint,arXiv:2012.09241 , 2020.11. S. Chen, G. P. T. Choi, and L. Mahadevan, “Deterministic and stochastic control of kirigamitopology,”
Proc. Natl. Acad. Sci. , vol. 117, no. 9, pp. 4511–4517, 2020.12. B. Gr¨unbaum and G. C. Shephard,
Tilings and patterns . WH Freeman & Co., 1986.293. E. Fedorov, “Simmetrija na ploskosti,”
Zapiski Imperatorskogo Sant-Petersburgskogo Min-eralogicheskogo Obshchestva , vol. 28, pp. 345–390, 1891.14. G. P´olya, “ ¨Uber die analogie der kristallsymmetrie in der ebene,”
Zeitschrift f¨ur Kristallo-graphie , vol. 60, pp. 278–282, 1924.15. P. G. Radaelli,
Symmetry in crystallography: Understanding the international tables . OxfordUniversity Press, 2011.16. W. J. Padilla, “Group theoretical description of artificial electromagnetic metamaterials,”
Opt. Express , vol. 15, no. 4, pp. 1639–1646, 2007.17. C. M. Bingham, H. Tao, X. Liu, R. D. Averitt, X. Zhang, and W. J. Padilla, “Planarwallpaper group metamaterials for novel terahertz applications,”
Opt. Express , vol. 16,no. 23, pp. 18565–18575, 2008.18. Y. Mao, Q. He, and X. Zhao, “Designing complex architectured materials with generativeadversarial networks,”
Sci. Adv. , vol. 6, no. 17, p. eaaz4169, 2020.19. M. Stavric and A. Wiltsche, “Geometrical elaboration of auxetic structures,”
Nexus Netw. J. ,vol. 21, no. 1, pp. 79–90, 2019.20. W. Liu, M.-S. Song, B. Kong, and Y. Cui, “Flexible and stretchable energy storage: recentadvances and future perspectives,”
Adv. Mater. , vol. 29, no. 1, p. 1603436, 2017.21. X. Liu, S. Yao, B. S. Cook, M. M. Tentzeris, and S. V. Georgakopoulos, “An origamireconfigurable axial-mode bifilar helical antenna,”
IEEE Trans. Antennas Propag. , vol. 63,no. 12, pp. 5897–5903, 2015. 302. S. A. Nauroze, L. S. Novelino, M. M. Tentzeris, and G. H. Paulino, “Continuous-rangetunable multilayer frequency-selective surfaces using origami and inkjet printing,”
Proc.Natl. Acad. Sci. , vol. 115, no. 52, pp. 13210–13215, 2018.23. D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-rangeorientational order and no translational symmetry,”
Phys. Rev. Lett. , vol. 53, no. 20, p. 1951,1984.24. G. P. T. Choi, S. Chen, and L. Mahadevan, “Control of connectivity and rigidity in prismaticassemblies,”