Wallpaper Groups and Auxetic Metamaterials
Brendan Burns Healy, Aparna Deshmukh, Elliott Fairchild, Caroline J. Merighi, Konstantin Sobolev
WWallpaper Groups and Auxetic Metamaterials
Brendan Burns Healy ∗ , Aparna Deshmukh † , Elliott Fairchild † , Caroline J. Merighi ‡ , and Konstantin Sobolev ‡ Abstract.
We examine a fundamental material property called Poisson’s ratio, which establishes the relationshipfor the relative deformation of a physical system in orthogonal directions. Architects and engineershave designed advanced systems using repeating patterns that can potentially exhibit auxetic be-havior, which is the property of having a negative Poisson’s ratio. Because two-dimensional crosssections of each of these patterns has an associated wallpaper group, we can look for useful correla-tions between this geometric information and the two-dimensional response as defined by Poisson’sratio. By analyzing the data, we find two properties of the wallpaper group that correlate with moreeffective Poisson’s ratio required for applications. This paper also contains an introduction to wall-paper groups and orbifold notation and an appendix contains some literature references recreated touse our preferred notation.
Key words. metamaterials, auxetic systems, wallpaper groups
AMS subject classifications.
1. Introduction.
It has long been an area of interest in materials science and engineeringto study how particular small-scale arrangements of a material can contribute to aggregateproperties seemingly in contradiction with the large-scale properties of the bulk material itself.Such arrangements of material are called metamaterials , and many classes of these metama-terials have proven useful in countless applications.While much of the scientific literature focuses on metamaterials whose target properties arephotonic and electromagnetic in nature, this paper will concentrate on mechanical propertiesof these systems. Metamaterials designed for mechanical optimization are useful for theirresistance to permanent deformation and durability under strain. Mechanical metamaterialsare of interest for advanced design and construction of concrete/reinforced concrete structuressuch as buildings and roads as well as medical applications like durable and flexible prosthetics.For more examples, see [20, Section 1.3].One class of particular interest is those systems which undergo a particular type of defor-mation, or shape change, when force is applied. In two dimensions, a system under stress, orforce in a particular direction, will usually see points shift (undergo strain ) in that direction,as well as the orthogonal ( transverse ) direction.
Definition 1.1 (2-dimensional Poisson’s ratio).
Let x be the direction in which the force isapplied to a metamaterial system, and z be an orthogonal direction. Then we let (cid:15) x , (cid:15) z denotehow much deformation occurs in each of those directions. The
Poisson’s ratio of the materialin these directions is given by ν xz := − (cid:15) z (cid:15) x . A material is auxetic in these directions if ν xz < . ∗ Department of Mathematical Sciences, University of Wisconsin–Milwaukee, Milwaukee, WI ([email protected]). † Department of Civil Engineering, University of Wisconsin–Milwaukee, Milwaukee, WI. ‡ Department of Mathematical and Statistical Sciences, Marquette University, Milwaukee, WI. More precisely, given a displacement field u , we have (cid:15) x = ∂ u x ∂x and (cid:15) z = ∂ u z ∂z . a r X i v : . [ m a t h . M G ] F e b B. B. HEALY, A. DESHMUKH, E. FAIRCHILD, C. J. MERIGHI, AND K. SOBOLEVFigure 1.
An example of an auxetic system in the relaxed state (left) and under strain in the horizontaldirection (right).
We note here that it is possible for a material to be auxetic in some directions, but notauxetic in other directions. In other words, both isotropic auxetic metamaterials, those forwhom the Poisson’s ratio in 2-dimensional space is independent of the directions x and z , and anisotropic auxetic metamaterials, those for whom the ratio depends on the directions x, z ,exist. For more background on auxetic metamaterials, the reader is referred to [15], [19], [18].As most auxetic systems are periodic in nature, we can gain insight into them by studyingtheir inherent periodic geometries. This idea has been pursued by Borcea and Streinu in theirdevelopment of an expansive mathematical theory of auxetic behavior in arbitrary dimensionsby studying one-parameter deformations of periodic lattices (see [4], as well as [6], [5]). Here wewill zoom in on a particular property of 2-dimensional auxetic systems; we study the associatedsymmetry groups, called wallpaper groups , which will be introduced and discussed in the nextsection. Some consideration of the wallpaper group for an auxetic system has already beenstudied in [23]. In this paper, we extend this relationship by analyzing how the wallpapergroup corresponding to an auxetic system contributes to its deformation properties.Section 2 is a self-contained introduction to wallpaper groups, including orbifold notation.We discuss the geometry of the topological space obtained by taking a quotient of the Euclideanplane by a given wallpaper group considered as a subgroup of the isometries of the plane.In Section 3 we outline one possible metric for wallpaper groups as a way of correlatingthe symmetry of a metamaterial system and its auxetic properties, which themselves will beprimarily be measured using Poisson’s ratio. In order to do this, we discuss how one wallpapergroup may be considered to represent more symmetry than another. We also discuss howmotivation for the guiding questions of the present article are highly influenced by findings ofGrima, Mizzi, Azzopardi, and Gatt, who in [14] demonstrate that a purposeful loosening ofthe symmetries of a system negatively impact its auxetic performance.In Section 4 we list the data we are collecting from the literature to examine for correla-tions. For each paper we draw from, we give a quick summary of the relevant portions of thepaper, in order to contextualize the systems under analysis. Finally, Section 5 summarizes theconclusions we draw from this data, and Appendix 6 lists a few further questions of interest tothe authors. In particular we find that reflective symmetry has a beneficial effect on Poisson’sratio, and that rotational symmetry of some orders are better than others.
2. Wallpaper Groups.
Any two-dimensional pattern that can be extended infinitely to fillthe Euclidean plane can be classified based on the types of symmetry present in that pattern.We can describe which type(s) of symmetry are present in the pattern by identifying the symmetry group , or group of isometries, associated to that pattern. Recall that an isometry is
ALLPAPER GROUPS AND AUXETIC METAMATERIALS 3 a distance preserving homeomorphism of the plane E .All isometries of the Euclidean plane are of one of the following four types: translation,rotation, reflection, or glide reflection. A wallpaper group is a symmetry group of the planethat contains two (linearly) independent translations. Wallpaper groups are classified by howmany (if any) of each type of the other three symmetries are present in the patternThere are exactly 17 wallpaper groups up to abstract isomorphism, a fact proved by EvgrafFedorov in 1891 [11] and independently by George Pólya in 1924 [21]. One notable part of thisclassification is that a wallpaper group can only contain rotations by ◦ , ◦ , ◦ , or ◦ .This comes from the crystallographic restriction theorem (see, for instance, [8]), which in twodimensions can be formulated as: Every discrete isometry group of the Euclidean plane withtranslations spanning the whole plane, any isometry of finite order must have order 1, 2, 3,4 or 6. Note that all reflections have order . Rotations of order 2 are ◦ rotations, order are ◦ rotations, order are ◦ rotations, and order are ◦ rotations. The fundamental domain of a wallpaper pattern is the small-est piece of the pattern that will create the entire pattern when moved around by the isometriesof the wallpaper group. You can think of it as the smallest “tile” that you would need to coverthe whole plane without overlaps of gaps. The ways you would need to move the tile around bytranslations, rotations, reflections, and/or glide reflections are the elements of the wallpapergroup associated with the pattern.
Figure 2.
The region outlined in purple is a fundamental domain for this pattern.
The wallpaper group acts on the fundamental domain by identifying its vertices and edges.If we quotient E by these identifications, we get a particular orbifold , which is a topologicalobject encoding the wallpaper group data. Essentially, one can take the fundamental domain Recall that the order of a group element g is the smallest possible integer such that g n is the identity(trivial) element. Note that any isometries of order 1 are trivial so we do not need to keep track of them. B. B. HEALY, A. DESHMUKH, E. FAIRCHILD, C. J. MERIGHI, AND K. SOBOLEV and “fold” and “glue” it based on the symmetries of the group. The shape that results is ourorbifold. For wallpaper groups, there are 17 possible (2-)orbifolds that can result from thisprocess , and they correspond exactly to the 17 wallpaper groups. We can characterize whichquotient space (orbifold) we get using orbifold notation , which in turn gives notation for thewallpaper groups.For example, the symmetries of the pattern shown in Figure 2 are an order 4 rotationand two orthogonal lines of reflection (mirror symmetry). The quotient of this space by thewallpaper group that consists of these symmetries (aka the orbifold) has a cone point of order (this comes from the order rotation) and a corner point of order (which comes from theintersection of the two lines of reflection. Visual representations of this orbifold are shown inFigures 3 and 4. Figure 3.
Orbifold identifications in the fundamental domain of the pattern shown in Figure 2.
There are several different conventions for naming the 17 wallpa-per groups. In this paper, we use orbifold notation [7]. It is a natural choice because it makesit relatively easy to “see” the topology. In this notation system, each wallpaper group has aunique orbifold symbol that consists of a string of characters. Each character (and its locationin the string) indicates a characteristic of the orbifold.While orbifold notation can be used to describe more than just wallpaper groups, here we When starting with the plane, that is. We caution the reader that there are many other two-dimensionalorbifolds, but their universal covers are not E . Other notation systems for wallpaper groups are also in use, including Coxeter and crystallographic nota-tion.
ALLPAPER GROUPS AND AUXETIC METAMATERIALS 5Figure 4.
The quotient orbifold ∗ (green region left uncolored). will only discuss the characters used in the orbifold symbols of the 17 wallpaper groups. Thesecharacters are: • o : denotes the group with only translational symmetry • × : denotes a glide reflection (a cross-cap on the orbifold) • ∗ : indicates whether numerals denote cone points or corner points • , , , : denote the order of each cone/corner pointEach numeral after an asterisk corresponds to a corner point , which is a point of rotationthat lies on the boundary of the quotient orbifold (see the blue point in Figure 4. Each numeralbefore an asterisk (or in an orbifold symbol that does not contain an asterisk) corresponds toa cone point , which is a point of rotation that does not lie on a line of mirror symmetry, i.e.is in the interior of the orbifold (see the black point in Figure 4. The numeral indicates theorder of the rotation. For example, the orbifold symbol for the orbifold in Figure 3. is ∗ .The before the ∗ tells us there is a cone point of order 4 (a center of rotation by ◦ thatdoes not lie on a line of reflection). The after the ∗ tells us there is one corner point of order (a center of rotation by ◦ that does lie on a line of reflection).
3. Symmetry and Auxetic Properties.
The goal of this paper is to explore whether usefulcorrelations exist between the geometric data of a metamaterial system and its auxetic proper-ties. This investigation is conducted primarily via re-analyzing data on auxetic systems foundin the mechanical engineering literature. We summarize auxetic systems and Poisson’s ratiosthat have been investigated in prior papers and then identify the wallpaper group associated toeach system. We then look for relationships between the geometric data of the wallpaper groupand the Poisson’s ratio of the system, and discuss our conclusions in Section 5. Through thisanalysis we provide some preliminary insights into the geometric properties correlated withbetter performing auxetic systems, in order to aid in the design of effective metamaterials inthe future.To motivate this examination, we turn to one existing result on how the Poisson’s ratioof an auxetic system varies with its regularity. In [14], Grima, Mizzi, Azzopardi, and Rubenfind that the auxetic properties of metamaterial system degrade with a controlled destructionof the symmetry. In particular, they begin with an ordered, highly symmetric repeating two-dimensional pattern of cuts, then allow for those cuts to rotate randomly. As this random
B. B. HEALY, A. DESHMUKH, E. FAIRCHILD, C. J. MERIGHI, AND K. SOBOLEV variation is allowed to increase, they find that the Poisson’s ratio monotonically increases(which is a less desirable behavior). While the author’s conclusion is that minor variationshave little effect on the system’s performance, we may take the monotonic behavior as evidencefor our hypothesis that more symmetry yields better systems.Before we can explicitly state our hypothesis, we must carefully state what we mean by“more symmetric”. Intuitively, a wallpaper group which is the same as another except that itcontains reflections, for example, should be considered “more symmetric”. Similarly, a patternis “less symmetric” than another if it only has 3-fold, rather than 6-fold, rotation. To dothis comparison formally, we refer to work of Coxeter [9]. In this paper, Coxeter investigates(among many other things) the relationships that the wallpaper groups have to each other inan abstract algebraic sense. He answers the question as to which groups abstractly embed assubgroups of the other. This relation almost forms a poset (Partially Ordered SET) structureon the wallpaper groups, except that unfortunately some pairs of groups mutually embed ineach other. However, if we treat those groups as equivalent, we can form a partial order onthe equivalence classes of these groups.
Definition 3.1.
Let
G, H be wallpaper groups. We say that G is more symmetric than H ,written H ≤ G , if there is a path (downward) from [ G ] to [ H ] in Figure 5. The use of the notation [ G ] is to recall that we want to consider equivalent any two groupswhich share a box in Figure 5. The following is a routine exercise in set theory (indeed, theproof is exactly the construction of the above figure from the information of [9]). Proposition 3.2.
Let W be the set of equivalence classes of wallpaper groups. Then ( W , ≤ ) is a poset. We can now formally state the primary question of interest for the present article.
Hypothesis
Definition 3.4.
A metamaterial system is called anisotropic if there exist directions x, z suchthat ν xz (cid:54) = ν zx . A system which is not anisotropic is said to be isotropic . Definition 3.5.
A metamaterial system is called chiral if its associated wallpaper group doesnot contain a reflection. A system is called achiral otherwise.
While this latter definition may not be worded the same as the description of the propertyfound in the literature, it captures the same phenomenon (see Section 2.2 in [15]).
In order to faithfully compare auxetic behaviors across systems, we com-pare systems created by the same authors in the same project. This guarantees that we arecomparing systems which used the same or similar parameters within their metamaterial sys-tems, and all simulation was done with the same program and settings. Just as the Poisson’sratio of a system can be dependent on the directions of strain, the magnitude of strain canaffect a system’s auxeticity. While the study of deformation mechanisms of structures over
ALLPAPER GROUPS AND AUXETIC METAMATERIALS 7 *632632 3*3/*333333 2*22 / *2222 4*2*442 44222*22 × ××× o Figure 5.
A partial order of (equivalence classes of) the seventeen wallpaper groups, color-coded by thehighest order rotation in the group. We say a group G is “more symmetric” than another group H if there isan directed path from G to H . For example, ∗ is more symmetric than × because there is a path from ∗ to × (via ∗ ). large (finite) strains is important to applications of these systems, many of the experimentsand applications involving auxetic systems focus on behavior over infinitesimal strains. Convention (Infinitesimal Strain and Finite Strain).
We restrict our analysis to the initial(infinitesimal strain) Poisson’s ratio of systems and denote it as ν xz or ν zx .The symmetries of an auxetic system can potentially change with the state of the system.Consider, for example, the rotating squares model proposed by Grima and Evans [13] whichis presented in Figure 6. In the completely closed state, the geometry is a regular tessellationof squares with wallpaper group ∗ . Upon expansion, the system loses symmetry along itsdiagonals, changing its wallpaper group to . B. B. HEALY, A. DESHMUKH, E. FAIRCHILD, C. J. MERIGHI, AND K. SOBOLEVFigure 6.
The rotating squares system has different symmetries in its closed and partially open states
Convention (Wallpaper group associated to partially open state).
We define the wallpapergroup of an auxetic system to be the one associated to the system’s partially open state.For a practical guide to determining the wallpaper group associated to a repeating pattern,refer to the flowchart on page 128 of [24]. For ease of reference, this flowchart is recreated inthe appendix of this paper using orbifold notation instead of crystallographic notation.Some systems under consideration will be anisotropic and have two different Poisson’s ratiolisted by the source literature.
Convention (Anistropy vs. Isotropy).
For anisotropic systems, the values ν xz , ν zx willbe averaged , and the resulting value will be considered the Poisson’s ratio of the system.
4. Data.
Our data come from two sources, [16] and [2]. While it would clearly be prefer-able to have more examples of auxetic systems, the limitations of comparing like systems leaverelatively few parts of the literature suitable for analysis. As mentioned in the previous section,in order to control for choices made by the authors, we will only compare systems coming fromthe same paper. In order to get meaningful analysis, we used sources that had examples withmany different wallpaper groups. The papers listed here were the only ones we found that metthese criteria, however this data is sufficient to suggest certain conclusions that we describe inSection 5.We briefly summarize the aims of each paper and the parameter choices that the authorsmade, before listing the data we extract from each.
A systematic approach to identify cellular auxeticmetamaterials . Körner and Liebold-Ribeiro [16] introduce a set of auxetic systems derivedfrom the triangular, quadratic, and hexagonal lattices. In particular, they consider all the‘vibrations’, or sinusoidal perturbations, of these lattices and look at the special cases whereall parts of the lattices move together at the same frequency (these special vibrations arecalled the eigenmodes of the lattices). After specifying the parameters L and t below to be L = 2 . mm for the hexagon, L = 5 mm for the square and triangle and t = 0 . mm, the authorsmodel the lattices in Abaqus 6.13 with the material properties of titanium and calculate thefirst eigenmodes of each of the lattices.The authors proceed to create periodic lattices out of the eigenmodes and calculate theinitial (infinitesmal) Poisson’s ratio through simulations in Abaqus 6.13 for the resulting sys-tems. Eight of these systems show auxetic behavior; these auxetic systems their wallpapergroups are given in the table below. ALLPAPER GROUPS AND AUXETIC METAMATERIALS 9
Picture of System Wallpaper Group Poisson’s Ratio [16] ∗ ν xz = ν zx = − . ∗ ν xz = − . , ν zx = − . ν xz = ν zx = − . ν xz = ν zx = − . ∗ ν xz = ν zx = − . ∗ ν xz = − . , ν zx = − . ν xz = ν zx = − . ν xz = ν zx = − . Comparative study of auxetic geometries by means of computer-aided design and engineering . Elipe and Lantada [2] had the goal of creating a library ofvarious auxetic systems. In order to do this, they consider a variety of different metamaterialsystems and simulate their mechanical properties using the programs Solid Edge and NX-8.0(Siemens PLM Solutions). To control for extraneous parameters, the authors attempt to makethe dimensions of the systems as uniform as possible (see [2, Section 2.1]).In the following table, we remark that the first system was considered in [2] in two differentconfigurations. While both configurations have the same wallpaper group, the version omittedfrom the table had a less optimal Poisson’s ratio. We consider better version of this systemonly. Additionally, the authors also investigate a handful of systems with positive Poisson’sratio - we do not consider these systems in our data. In the table below we reproduce picturesof the systems considered by Elipe and Lantada.Picture of System Wallpaper Group Poisson’s Ratio [2] ∗ ν xz = − . ∗ ν xz = − . ∗ ν xz = − . ν xz = − . ν xz = − . ∗ ν xz = − . ∗ ν xz = − . ∗ ν xz = − . ALLPAPER GROUPS AND AUXETIC METAMATERIALS 11
Picture of System Wallpaper Group Poisson’s Ratio [2] ν xz = − . ∗ ν xz = − . ∗ ν xz = − . ν xz = − . ∗ ν xz = − . ν xz = − . ∗ ν xz = − .
5. Conclusions.
We begin by listing the data from the previous section in a few suggestiveways. This first table compares the Poisson’s ratio of systems which have reflections to thosewhich do not.Source With reflection (achiral) Without reflection (chiral)[16] -0.2, -0.2, -0.3, -0.9 -0.2, -0.1, -0.5, -0.4[16] averaged -0.4 -0.3 [2] -1.68, -0.789 -0.504 -0.177, -0.326, -0.628-0.901, -0.81, -0.071, -0.926 -0.273, -0.248-0.966, -1.18, -0.315[2] averaged -0.8142 -0.330
Based on this information, we posit that achiral auxetic systems exhibit lower Poisson’sratio than their chiral counterparts. This finding supports Hypothesis 3.3.Next we consider the largest degree of rotation associated to each pattern. In examiningthe poset structure present in Figure 5, we see that there are is a natural way to categorize the wallpaper groups by rotation. We will consider 6 and 3 fold rotations together, 4 fold rotationstogether, and everything that lies below both of these together. Because we did not observeany systems with no rotational symmetry whatsoever, this last class will include exactly thosepatterns with ◦ rotations.Source 6 & 3 fold 4 fold 2 fold[16] -0.2, -0.2, -0.2, -0.3, -0.5 -0.9-0.1, -0.4[16] averaged -0.22 -0.4 -0.9 [2] -0.628,-0.273 -0.504, -0.315 -1.68, -0.177-0.248 -0.326, -0.901 -0.789, -0.071-0.81, -0.966 -0.926,-1.18[2] averaged -0.383 -0.637 -0.804 While the data in the first table supported Hypothesis 3.3, this table tells a different story.In fact, the data suggests that fewer rotations give better auxetic properties! This may bebecause there is more freedom to choose better parameters if one isn’t restricted to a many-fold rotational structure. However, the authors conjecture that the ‘failure’ of the higher foldpatterns has more to do with the correlation with chirality. Indeed, in our data set, none ofthe 6-fold patterns were achiral (we found no examples of the wallpaper group *632), meaningthat the collection of systems with 6-fold rotational symmetry were handicapped, in a certainsense.In [2], we can take a closer look at what is happening. Sorting these systems by theirwallpaper group, we obtain the following:Wallpaper Group Poisson’s Ratio Averaged Poisson’s Ratio ∗ -0.504 -0.504 ∗ -0.81, -0.901, -0.966, -0.315 -0.748 -0.326 -0.326 -0.177 -0.177 ∗ -1.68, -0.071 -0.876 ∗ -0.789, -0.926, -1.18 -0.965 -0.273, -0.248, -0.628 -0.383 Looking at only the achiral systems and appealing to Figure 5, we in fact see that lesssymmetric systems have lower (better) Poisson’s ratios. But we can be even more specific. Sys-tems with rotation centers not on reflection lines perform better than systems whose rotationsare all on reflection lines. In fact, the wallpaper group with the lowest average Poisson’s ratiohad all rotations not on reflection lines. We obtain a similar result from the systems in [16];the achiral systems with some rotation centers not on reflection lines had average Poisson’sratio -0.467, while those having rotation centers all on reflection lines had an average Poisson’sratio of -0.2. This question is motivated by the discussion of rotational centers in [16] (see the
ALLPAPER GROUPS AND AUXETIC METAMATERIALS 13Figure 7.
Frequency of wallpaper groups found in [16] and [2]. observation regarding non-auxetic modes in Section 4.1).We also note briefly that no systems in these papers had wallpaper groups appearing inthe lower half (the bottom 5 nodes) of Figure 5. While this lack of data does not allow us toconclude anything in particular about metamaterial systems with these wallpaper groups (ifthey exist), we believe that they are not found because they would not have more desirableproperties than the systems currently found in the literature. The fact that the current liter-ature on auxetic systems does not include any systems with these “less symmetric” wallpapergroups could be because they are not as desirable from a practical points of view. This sug-gests that more symmetric wallpaper groups are of more interest when it comes to producingauxetic metamaterials (up to the restriction described in the previous paragraph).Additionally, we found no wallpaper groups that had glide reflections (that were not com-positions of translations and reflections that were themselves elements of the wallpaper group).Put another way, we never had to answer “yes” to the “Glide reflection?” question in the flow-chart in Appendix A. Consequently, we may wonder if the groups × , ∗ × , and/or ×× canoccur as wallpaper groups of auxetic systems.Figure 7 shows that while the most common individual group in the two source papers has − fold rotation, in aggregate − fold patterns were more common.In summary, it appears that there are meaningful correlations between the mathematicalstructure (wallpaper group) of a two-dimensional auxetic system and the physical properties(Poisson’s ratio) of that system. Our analysis suggests that an auxetic system whose wallpapergroup contains at least one reflection is more likely to have a better (lower) Posisson’s ratio thanan auxetic system whose wallpaper group contains no reflections. It also suggests that auxeticsystems whose wallpaper groups have - or -fold highest order rotations have worse (higher)Poisson’s ratios than systems whose wallpaper groups have - fold highest order rotations,which in turn have worse Poisson’s ratios than systems whose wallpaper groups have -foldhighest order rotations. We acknowledge that our data are somewhat limited. However, theseearly results suggest that when designing auxetic metamaterial systems where a low Poisson’sratio is desirable, that it may be advantageous to focus on systems whose underlying geometric structure is a wallpaper group with reflections and -fold rotations.
6. Further Questions.
There are many further questions that could be asked about therole that geometry and topology play in the study of metamaterials. We list a few questionshere that we believe would be particularly enlightening to investigate.
Question
Question
Question - and - fold maximum rotations. Does this pattern hold more broadly? For example, are systemswith -, - or - fold rotation more likely to be anisotropic than systems with -fold rotation? Question fractal , a geometric object which is self-similar atvarious scales. Does the Hausdorff or Renyi dimension of the iteration of this scale-invariancemake useful predictions about the behavior of an auxetic system?
ALLPAPER GROUPS AND AUXETIC METAMATERIALS 15
Appendix A. Wallpaper Group Resources.
Highest order rotation?Reflection?Allrotationcenters onreflectionlines? ∗
333 3 ∗ Reflection?Allrotationcenters onreflectionlines? ∗
442 4 ∗ Reflection? ∗
632 632
Reflection?Reflectionlinesin twodirections? Glidereflection?Allrotationcenters onreflectionlines? ∗ ∗ ∗ × Reflection?Glidereflection? Glidereflection? ∗ × ∗∗ ×× ◦ This chart is a guide for identifying the wallpaper group associated with any two-dimensionalrepeating pattern that fills the plane. It is based on a similar chart in [24], but adapted toemphasize the orbifold structure and notation discussed in Section 2. The first question toask is always “What is the highest order rotation in the pattern?” The order of a rotation isthe minimum number of times the rotation must be applied before the pattern returns to thestarting position. For example, if the smallest rotation is ◦ , it is an order rotation because it takes four ◦ rotations to create a full ◦ rotation. The highest order rotation correspondsto the largest digit in the orbifold signature of the wallpaper group. The next question to askis “Are there any reflections?” If there are one or more reflections, the orbifold signature ofthe wallpaper group will contain the ∗ symbol. The question(s) to ask after determining thehighest order rotation and whether or not there are any reflections depend on the previousanswers. It should be noted that there are alternative questions that one could ask after thequestion about whether or not there are reflections. REFERENCES [1]
International tables for crystallography. Vol. A: Space-group symmetry , Springer, 5. ed., reprinted withcorrections ed., 2005.[2]
J. C. Alvarez Elipe and A. Diaz Lantada , Comparative study of auxetic geometries by means ofcomputer-aided design and engineering , Smart Materials and Structures, 21 (2012), p. 105004, https://doi.org/10.1088/0964-1726/21/10/105004.[3]
S. Babaee, J. Shim, J. C. Weaver, E. R. Chen, N. Patel, and K. Bertoldi ,
3d soft metamaterialswith negative poisson’s ratio , Advanced Materials, 25 (2013), p. 5044–5049, https://doi.org/10.1002/adma.201301986.[4]
C. Borcea and I. Streinu , Geometric auxetics , Proceedings of the Royal Society A: Mathematical,Physical and Engineering Sciences, 471 (2015), p. 20150033.[5]
C. S. Borcea and I. Streinu , Auxetic deformations and elliptic curves , Computer Aided GeometricDesign, 61 (2018), p. 9–19, https://doi.org/10.1016/j.cagd.2018.02.003.[6]
C. S. Borcea and I. Streinu , Auxetic Regions in Large Deformations of Periodic Frameworks , vol. 71,Springer International Publishing, 2019, p. 197–204, https://doi.org/10.1007/978-3-030-16423-2_18,http://link.springer.com/10.1007/978-3-030-16423-2_18.[7]
J. H. Conway and D. H. Huson , The orbifold notation for two-dimensional groups , Structural Chem-istry, 13 (2002), p. 247–257, https://doi.org/10.1023/A:1015851621002.[8]
H. S. M. Coxeter , Introduction to geometry , Wiley Classics Library, John Wiley & Sons, Inc., NewYork, 1989. Reprint of the 1969 edition.[9]
H. S. M. Coxeter and W. O. J. Moser , Generators and Relations for Discrete Groups , SpringerBerlin Heidelberg, 1972, https://doi.org/10.1007/978-3-662-21946-1, http://link.springer.com/10.1007/978-3-662-21946-1.[10]
L. D’Alessandro, V. Zega, R. Ardito, and A. Corigliano ,
3d auxetic single material periodicstructure with ultra-wide tunable bandgap , Scientific Reports, 8 (2018), p. 2262, https://doi.org/10.1038/s41598-018-19963-1.[11]
E. Fedorov , Symmetry in the plane , Proceedings of the Imperial St. Petersburg Mineralogical Society,2 (1891), p. 345–390. In Russian.[12]
R. Gatt, L. Mizzi, J. I. Azzopardi, K. M. Azzopardi, D. Attard, A. Casha, J. Briffa, andJ. N. Grima , Hierarchical auxetic mechanical metamaterials , Scientific Reports, 5 (2015), p. 8395,https://doi.org/10.1038/srep08395.[13]
J. N. Grima and K. E. Evans , Auxetic behavior from rotating squares , (2000).[14]
J. N. Grima, L. Mizzi, K. M. Azzopardi, and R. Gatt , Auxetic perforated mechanical metamaterialswith randomly oriented cuts , Advanced Materials, 28 (2016), p. 385–389, https://doi.org/10.1002/adma.201503653.[15]
P. U. Kelkar, H. S. Kim, K.-H. Cho, J. Y. Kwak, C.-Y. Kang, and H.-C. Song , Cellular auxeticstructures for mechanical metamaterials: A review , Sensors, 20 (2020), p. 3132, https://doi.org/10.3390/s20113132.[16]
C. Körner and Y. Liebold-Ribeiro , A systematic approach to identify cellular auxetic materials ,Smart Materials and Structures, 24 (2015), p. 025013, https://doi.org/10.1088/0964-1726/24/2/025013.[17]
R. Lakes , Foam structures with a negative poisson’s ratio , Science, 235 (1987), pp. 1038–1041.
ALLPAPER GROUPS AND AUXETIC METAMATERIALS 17 [18]
T.-C. Lim , Auxetic materials and structures , Springer, 2015.[19]
P. Ma , A review on auxetic textile structures, their mechanism and proper-ties , Journal of Textile Science & Fashion Technology, 2 (2019), https://doi.org/10.33552/JTSFT.2019.02.000526, https://irispublishers.com/jtsft/fulltext/a-review-on-auxetic-textile-structures-their-mechanism-and-properties.ID.000526.php.[20]
M. Mir, M. N. Ali, J. Sami, and U. Ansari , Review of mechanics and applications of auxetic struc-tures , Advances in Materials Science and Engineering, 2014 (2014), p. 1–17, https://doi.org/10.1155/2014/753496.[21]
G. Pólya , Xii. Über die analogie der kristallsymmetrie in der ebene
S. M. Sajadi, C. F. Woellner, P. Ramesh, S. L. Eichmann, Q. Sun, P. J. Boul, C. J. Thaem-litz, M. M. Rahman, R. H. Baughman, D. S. Galvão, and et al. ,
3d printing: 3d printedtubulanes as lightweight hypervelocity impact resistant structures (small 52/2019) , Small, 15 (2019),p. 1970284, https://doi.org/10.1002/smll.201970284.[23]
M. Stavric and A. Wiltsche , Geometrical elaboration of auxetic structures , Nexus Network Journal,21 (2019), p. 79–90, https://doi.org/10.1007/s00004-019-00428-5.[24]