Volume 70 Received 2 April 2013  Symmetry groups of singlewall nanotubes Ma. Louise Antonette N. De Las Peñas,^{a}^{*} Mark L. Loyola,^{a} Antonio M. Basilio^{b} and Eko Budi Santoso^{a} ^{a}Mathematics Department, Ateneo de Manila University, Loyola Heights, Quezon City, 1108, Philippines, and ^{b}Chemistry Department, Ateneo de Manila University, Loyola Heights, Quezon City, 1108, Philippines This work investigates the symmetry properties of singlewall carbon nanotubes and their structural analogs, which are nanotubes consisting of different kinds of atoms. The symmetry group of a nanotube is studied by looking at symmetries and color fixing symmetries associated with a coloring of the tiling by hexagons in the Euclidean plane which, when rolled, gives rise to a geometric model of the nanotube. The approach is also applied to nanotubes with nonhexagonal symmetry arising from other isogonal tilings of the plane. Keywords: singlewall nanotubes; structural analogs; symmetry groups; color symmetry. 
In the past few years there has been a vast amount of fundamental and applied research on singlewall carbon nanotubes (e.g. Iijima & Ichihashi, 1993; Endo et al., 1996; Saito et al., 1998; Damnjanovic et al., 1999; Reich et al., 2004). Structural analogs, such as boron nitride (BN), boron carbide (BC_{3}) and boron carbon nitride (e.g. BCN, BC_{2}N), have also been studied (see, for instance, Chopra et al., 1995; Miyamoto et al., 1994; Liu et al., 1989; Yap, 2009).
In materials research, structural analogs are important in the design of new materials. Analogs of existing molecules and compounds are considered because alteration of some components of the previous material can lead to improvements in some properties. Carbon nanotubes, for instance, are structurally strong, with their electronic properties dependent on their chirality (Dresselhaus et al., 1995). The BCN analogs, on the other hand, are expected to maintain the mechanical properties but with variations in other properties, e.g. electrical properties. Carbon nanotubes can be conducting or semiconducting while BCN nanotubes are insulating (Yap, 2009).
The study of symmetry properties of nanotubes has been relevant for gaining deep insight into their physical (quantum numbers, optical activity, conductivity, among others), electronic, mechanical and magnetic properties which determine their potential applications. Studies on symmetry groups of carbon nanotubes and their structural analogs have been carried out (see Damnjanovic et al., 2001; Cotfas, 2006; Barros et al., 2006; Loyola et al., 2012, and references therein). There are also works pertaining to symmetry groups of nanotubes with nonhexagonal symmetry (e.g. Milosevic & Damnjanovic, 2006; Damnjanovic et al., 2007; Arezoomand & Taeri, 2009). These tubes, such as those rolled up from twodimensional rectangular and rhombic lattices, include the incommensurate ones  structures without translational symmetries. Commensurability properties contribute to the physical and electronic characteristics of nanotubes (Milosevic & Damnjanovic, 2006; Damnjanovic et al., 2007; Damnjanovic & Milosevic, 2010).
In our paper, the objective is to give an alternative approach to determine the symmetry groups of singlewall nanotubes. We use the theory of colorings of tilings in a cylindrical orbit space arising from colored isogonal tilings in the plane as a tool to describe and model singlewall nanotubes and characterize their symmetry properties. Colorings of isogonal tilings are employed as a means to determine different atomic configurations that exist and are theoretically possible for a nanotube.
For carbon nanotubes and their structural analogs, the method involves studying color symmetries associated with a vertex coloring of the tiling by hexagons in the plane which, when folded to form a cylinder, serves as a geometric model for a nanotube. We derive the symmetry group of the nanotube from the color fixing group of the associated coloring of the hexagonal tiling. In addressing nanotubes with nonhexagonal symmetry, we look at color fixing groups associated with their corresponding isogonal tilings on the plane. The color fixing symmetries also shed light on the commensurate properties of the nanotubes.
In our previous study (Loyola et al., 2012), the symmetries of a structural analog of a carbon nanotube are analyzed directly from the threedimensional symmetries of a given coloring on a cylinder which models the nanotube. On the other hand, in this work, our starting point is to work on colorings of isogonal tilings and to examine the colorings in cylindrical orbit space evolving from these colorings. From this technique, we are able to provide formulas for the color groups and color fixing groups for the colorings in the orbit space using the symmetries of the corresponding planar colorings as basis. The approach offers a convenient way to study symmetries of nanotubes by looking at twodimensional isometries. It also facilitates the characterization of the line group structure of the symmetry group of the nanotube based on the plane crystallographic group nature of the symmetry group of the colored tiling in the plane.
Color symmetry theory has been used in the past to describe crystal structures (Harker, 1978; Schwarzenberger, 1984; Senechal, 1988a) and quasicrystal/nonperiodic structures (Baake, 1997; Lifshitz, 1997; Scheffer & Lück, 1999; Bugarin et al., 2008, 2013). Crystallographers have used colored symmetrical patterns and tilings in several ways, such as in describing arrangements of atoms in a crystal and in deriving magnetic symmetries of crystals and quasiperiodic crystals. This paper highlights the contribution of color symmetry to describe monoperiodic structures, determine atomic arrangements pertaining to nanotube structures and understand their symmetry properties.
The results in this work cover a general class of colorings including nonperfect colorings  colorings with color symmetries that form a proper subgroup of the symmetry group of the corresponding uncolored tiling. Modeling nanotubes with atoms that are not equally distributed and studying their symmetry properties require methods in arriving at nonperfect colorings of isogonal tilings and an understanding of how these planar colorings are realized in the cylinder. This study extends and encompasses the ideas on perfect colorings provided in related literature (see, for example, Senechal, 1979, 1988b).
The outline of the paper is as follows. In §2, we discuss a geometric model of a singlewall carbon nanotube and the derivation of its symmetry group. §3 presents the setting pertaining to structural analogs of carbon nanotubes. In §4, we outline the concepts in color symmetry theory relevant to the study. The main results on symmetry groups of the structural analogs are given in §5. These results are applied in §6 to derive the symmetry groups of BN, BC_{3}, BCN and BC_{2}N nanotubes. In §7, we extend the approach and results of the previous sections to include nanotubes with nonhexagonal symmetry. A short discussion on commensurate and incommensurate nanotubes is included. Finally, in §8, a summary of the paper is presented together with the future outlook of the work.
A geometric model of a singlewall carbon nanotube is constructed by wrapping up a graphene sheet into a seamless cylindrical tube. The graphene sheet is modeled by a tiling of the Euclidean plane by hexagons whose vertices represent carbon atoms and whose edges represent carbon bonds. The sheet (or the hexagonal tiling ) is rolled along a vector , called the chiral vector of the tube, emanating from the center of a fixed hexagonal tile, which we assign as the origin O(0, 0), and terminating at the center of another tile so that these centers coincide in the tube (Fig. 1). The vector becomes the circumference of the tube.
 Figure 1 (a) The hexagonal tiling together with the translation vectors , and the chiral vector . (b) The [6, 2] carbon nanotube obtained by rolling up the hexagonal tiling in (a) along the chiral vector . 
The hexagonal tiling has symmetry group G = (in IUCr notation) generated by the sixfold (counterclockwise) rotation a about O, the reflection b about the line through O in the direction of , and the translations x, y (Fig. 1a). The chiral vector can be expressed as , where . In which case, we obtain an [m,n] nanotube. If n = 0, m = 0 or m = n, we arrive at a zigzag nanotube; if m = 2n, 2m = n or m = n we have an armchair nanotube. In any other cases, we obtain a chiral nanotube. The chiral vectors for the achiral (zigzag and armchair) nanotubes are displayed in Fig. 2(a).
 Figure 2 (a) Chiral vectors associated with achiral nanotubes. A (b) chiral and (c) zigzag carbon nanotube, together with the z axis, axis of a twofold rotation and mirror plane of the reflection passing through the z axis. 
Let X consist of points in corresponding to the vertices of the hexagonal tiling and let L be a subgroup of the symmetry group G of generated by the translation l = x^{m}y^{n}, where . We consider the orbit space of all Lorbits of points in X. Each belongs to a set if and only if v may be obtained from u via translation by an integer power of l.
Let us denote by the normalizer of L in G. The set of right cosets of L in N_{G}(L) acts on the orbit space by left multiplication. That is, if , and , then . It is a well known result from the theory of manifolds (Senechal, 1988b; Ratcliffe, 2006) that given the subgroup L (or any torsionfree subgroup) of G, the symmetry group G^{*} of is isomorphic to .
In our geometric model of a singlewall carbon nanotube with chiral vector , a point of corresponds to an atom in the nanotube and is the symmetry group of the nanotube.
For the chiral [m, n] carbon nanotube with chiral vector , we let . Among the elements of that normalize L_{C} are the twofold rotation a^{3} and the translations x and y. We have N_{G}(L_{C}) = . This implies that the symmetry group of the chiral nanotube is G_{C}^{*} , where d = gcd (m,n). The elements in G_{C}^{*} can be described as follows: G_{C}^{*} contains the dfold rotation about the z axis brought about by the translation x^{m/d}y^{n/d}. Other symmetries in G_{C}^{*} include a screw rotation about the z axis brought about by a translation which is neither parallel nor perpendicular to l_{C} and the twofold rotation , which results from the twofold rotation a^{3}, about an axis passing through the origin and perpendicular to the z axis. We have .
The symmetry groups of nanotubes and other monoperiodic structures are also described in terms of line groups (Damnjanovic et al., 1999, 2001; Damnjanovic & Milosevic, 2010). These are symmetry groups of structures in Euclidean space that are periodic only in a single direction (periodicity is assumed to be in the direction of the z axis). There are 13 infinite families of line groups formed by taking products of an infinite cyclic group generated by either a translation, a screw rotation, or a glide reflection, and an axial point group. A complete list of these families and their corresponding group generators is found in Damnjanovic et al. (2001) and Damnjanovic & Milosevic (2010). Line groups form an important family of groups which include rod groups, also known as crystallographic line groups (Kopský & Litvin, 2002; Evarestov & Panin, 2012). In the case of the chiral nanotube, its symmetry group G_{C}^{*} belongs to line group family 5.
Let us now consider the [m,0] zigzag carbon nanotube with chiral vector and . Observe that aside from the twofold rotation a^{3}, the nontranslation elements of G that normalize L_{Z} include the reflections that are either parallel or perpendicular to l_{Z}. We thus have and . The symmetry group G_{Z}^{*} consists of the mfold rotation , a screw rotation and the twofold rotation . As in the chiral nanotube, arises from the twofold rotation a^{3}. The reflection a^{3}b about the vertical axis passing through the origin O gives rise to a reflection with a vertical mirror plane passing through the z axis. The symmetry group of the [m,0] nanotube is .
For the [2n,n] armchair carbon nanotube with chiral vector , we let . We obtain similar computations as in the zigzag case, we have N_{G}(L_{A}) = . The symmetry group G_{A}^{*} is generated by the nfold rotation , a screw rotation , the twofold rotation and the reflection with a vertical mirror plane passing through the z axis and brought about by the reflection a^{4}b. That is, , . The symmetry groups of the zigzag and armchair nanotubes belong to line group family 13.
We present in Table 1 a complete list of the symmetry groups of singlewall carbon nanotubes according to chirality. An illustration of a chiral and a zigzag nanotube is given, respectively, in Figs. 2(b) and 2(c).

Aside from carbon nanotubes, nanotubes consisting of different types of atoms and having an underlying hexagonal symmetry are of interest in materials research. These are called structural analogs of carbon nanotubes. These nanotubes are structurally similar to singlewall carbon nanotubes and can also be described by rolling up the tiling by hexagons of the Euclidean plane into a cylindrical tube. In this case, the vertices of represent atoms corresponding to different types of chemical elements. To distinguish an atom from another, we assign different colors to the vertices and we obtain a vertex coloring of . Rolling up the colored tiling along a vector will give rise to a colored tiling on the cylinder which will represent or model a structural analog of a carbon nanotube.
As an example, let us consider the vertex 3coloring of the hexagonal tiling given in Fig. 3(a) with an equal distribution of colors. We roll the colored tiling along the vector to obtain the colored tiling on the cylinder in Fig. 3(b). We arrive at a model of a [6, 0] nanotube made up of an equal distribution of three kinds of atoms, such as a BCN nanotube.
 Figure 3 (a) A vertex coloring of the hexagonal tiling with an equal distribution of three colors. (b) A 3coloring of the [6, 0] nanotube representing a BCN nanotube. 
There are other vertex 3colorings of the hexagonal tiling that give rise to models of BCN nanotubes. See, for example, the colorings presented in Fig. 7. The three colors in the colorings are also equally distributed and suggest other arrangements of atoms on a BCN nanotube, distinct from that given in Fig. 3(a).
In arriving at vertex colorings of the hexagonal tiling , our basis is a framework for coloring tilings appearing in De Las Peñas et al. (2006, 2011). The framework facilitates a systematic enumeration of vertex kcolorings of a given tiling using the subgroup structure of the symmetry group of the uncolored tiling. The reader can refer to these works for more details on the methodology. In our work, we apply the method hand in hand with GAP (The GAP Group, 2008) and Mathematica (Wolfram Research, 2011) to arrive at the colorings. In the next section, we discuss the formalism pertaining to vertex colorings of the hexagonal tiling and the corresponding colorings on the cylinder as well as notions from color symmetry theory that are important points of consideration in determining the symmetry groups of structural analogs of carbon nanotubes.
Let G denote the symmetry group of the uncolored hexagonal tiling and X the set of vertices of . If is a set of k colors, we define a vertex kcoloring of to be an onto function . Each is assigned a color in C. The coloring determines a partition where f^{1}(c_{i}) is the set of elements of X assigned color c_{i}.
Let H be the subgroup of G which consists of symmetries in G that effect a permutation of the colors in C. Then if, for every , there is a such that h(f^{1}(c)) = f^{1}(d). This defines an action of H on C where we write hc: = d if and only if h(f^{1}(c)) = f^{1}(d). The group H is called the color group and the elements of H are referred to as the color symmetries of the given coloring of the hexagonal tiling . The group K of symmetries in H that fix the colors is called the color fixing group associated with the coloring. Note that, if H = G, then the coloring of is called a perfect coloring.
Note that, in obtaining a coloring on the cylinder from the colored tiling on the plane, the subgroup of G defining the cylinder or nanotube must be a subgroup of K. This is to ensure that two different colors do not overlap in the cylinder. Given a function that describes a vertex coloring of , we obtain the onto function that gives the corresponding coloring on the cylinder.
Let be the color group of the coloring on the cylinder and let , . Then for every , there is a such that (Lg)(f^{*1}(c_{i})) = f^{*1}(c_{j}) or (Lg)(c_{i}) = c_{j}. It follows that gc_{i} = c_{j}. Thus and . Conversely, if , then we have . Moreover, if denotes the color fixing group of the coloring on the cylinder, then if and only if . Thus, we have the following result.
Let H and K denote, respectively, the color group and color fixing group of a vertex coloring of the hexagonal tiling . Then is the color group and is the color fixing group for the corresponding coloring on the cylinder.
In our study, the main objective is to determine the symmetry properties of a structural analog of a carbon nanotube. In as much as this nanotube is represented by a coloring on the cylinder, our focus is to study the symmetry group of this colored pattern or its color fixing group. From the above theorem, this is given by . To understand the structure of K^{*}, a starting point would be to look at the color group of the coloring on the cylinder. These ideas are discussed in more detail in the next section.
Consider a colored cylinder of k colors that models a structural analog of a carbon nanotube consisting of k different atoms. The color group acts on the set C of k colors and, consequently, there exists a homomorphism , where Perm(C) is the group of permutations of C. The kernel of is . Thus, or .
To characterize the structure of , we use as our basis the twodimensional symmetries present in N_{K}(L), . The planar symmetries in N_{K}(L) give rise to cylindrical symmetries which define the line group structure of . The correspondence between the symmetries is presented in Table 2. In Table 3, we give the line group family structures of K^{*} with their corresponding generators. The element denotes a screw rotation about the z axis. We have the translation with vector parallel to the z axis and a glide reflection about a plane passing through the z axis. On the other hand, is a twofold rotation about an axis perpendicular to the z axis, while is a dfold rotation about the z axis. Moreover, is a reflection about a plane perpendicular to the z axis while is a reflection about a plane passing through the z axis. Finally, is a 2dfold rotoinversion about the z axis.


In the chiral case, for instance, N_{H}(L) is either of plane crystallographic group type p1 or p2. This gives rise to two possibilities for N_{K}(L), or p2. If , certain translational symmetries in N_{K}(L) will result in a screw rotation and a rotation about the z axis in K^{*}. This implies that K^{*} belongs to the first family of line groups. If , aside from the translational symmetries, we have the twofold rotation in N_{K}(L) that will give rise to a corresponding twofold rotation in K^{*} about an axis perpendicular to the z axis. Hence, K^{*} belongs to line group family 5.
For the achiral cases, we consider the following plane crystallographic group types for N_{K}(L), namely, p1, p2, cm, pm, c2mm, p2mm, pg, p2mg and p2gg. These are all the possibilities for N_{K}(L) satisfying (Senechal, 1985; Rapanut, 1988).
(i) The first case is when or p2. Consequently, K^{*} belongs to line group family 1 or 5, respectively.
(ii) Suppose or pm. If , a translation t, which is neither parallel nor perpendicular to l (Fig. 4a), will yield a nontrivial screw rotation in K^{*}. On the other hand, if , such a translation (called in Fig. 4b) will yield a trivial screw rotation in K^{*}. (A screw rotation in K^{*} is referred to as trivial if its rotation and translation components are also symmetries in K^{*}.)
 Figure 4 Respective translations t and on the lattice corresponding to (a) and (b) . 
To characterize K^{*} further, we consider the reflections in N_{K}(L). If or pm, then it contains reflections in one direction. If the axes of reflections are parallel to l, then we obtain a reflection about a plane perpendicular to the z axis. In this case, K^{*} belongs to line group family 4 if , or to line group family 3 if . On the other hand, if the axes of reflections are perpendicular to l, then there is a reflection about a plane passing through the z axis. We obtain K^{*} to be of line group family 8 if , or of line group family 6 if .
(iii) If or p2mm, then N_{K}(L) contains reflections in two directions. Consequently, we obtain a reflection about a plane perpendicular to the z axis as well as a reflection about a plane passing through the z axis. If , we have a similar case as , where there is a translation that will yield a nontrivial screw rotation, implying that K^{*} belongs to line group family 13. On the other hand, if , then K^{*} belongs to line group family 11.
(iv) If , then N_{K}(L) contains glide reflections in one direction and no reflections. If a glide reflection axis is parallel to l, we get a rotoinversion about the z axis and thus K^{*} belongs to line group family 2. On the other hand, if a glide reflection axis is perpendicular to l, we get a glide reflection about a plane passing through the z axis. In this case, K^{*} belongs to line group family 7.
(v) If , then N_{K}(L) contains glide reflections and reflections. As in (iv), if the glide reflection axis is parallel to l, we get a rotoinversion about the z axis. In this case, the reflection has an axis perpendicular to l, so we also obtain a reflection about a plane passing through the z axis. Thus, K^{*} belongs to line group family 9. On the other hand, if the glide reflection axis is perpendicular to l, we get a glide reflection about a plane passing through the z axis and a reflection about a plane perpendicular to the z axis. Thus, K^{*} belongs to line group family 12.
(vi) Lastly, if , then N_{K}(L) contains glide reflections in two directions and no reflections. In this case, since the glide reflections in N_{K}(L) have axes which are both parallel and perpendicular to l, then K^{*} contains a rotoinversion about the z axis together with a glide reflection about a plane passing through the z axis. Hence K^{*} possesses line group family 10 symmetries.
We give a summary of the discussion above in Table 3.
If we consider the coloring in Fig. 3(a), which gives rise to a BCN nanotube, we find that its color fixing group is . Now, depending on the chiral vector of the nanotube, we get different plane crystallographic group structures for N_{K}(L). For instance, in the chiral case, when , we obtain N_{K}(L) = . From the result given in Table 3, K^{*} belongs to line group family 1. For the zigzag nanotube with , we find that the axis of reflection a^{3}b is perpendicular to the chiral vector . Hence, the symmetries of K^{*} belong to line group family 8. On the other hand, for the armchair nanotube with . But this time, the axis of reflection a^{3}b is parallel to the chiral vector . Thus, K^{*} belongs to line group family 4. For the other achiral nanotubes, , so K^{*} belongs to line group family 1. Hence, for a BCN nanotube obtained from the 3coloring in Fig. 3(a), the symmetry group belongs to line group family 1, 4 or 8 depending on the chirality.
In this part of the paper, we apply the results given in the previous section to derive the symmetry groups of particular structural analogs of a carbon nanotube, such as BN, BC_{3}, BCN and BC_{2}N nanotubes.
Note that the restrictions on the values of m, n for the chiral vector in order to arrive at BC_{3}, BCN and BC_{2}N nanotubes are presented in Table 4.

The vertex 2coloring of the hexagonal tiling given in Fig. 5, when folded along a chiral vector, will result in a model of a BN nanotube. The vertices of the hexagonal tiling are assigned two colors, which are equally distributed, to represent the boron and nitrogen atoms. The color fixing group of the coloring is given by . Depending on the chirality of the nanotube, we calculate N_{K}(L) and determine its plane crystallographic group type. We arrive at the line group structure of presented in Table 5. For a chiral nanotube, and K^{*} belongs to line group family 1. For the zigzag nanotubes, . The axes of the reflections in N_{K}(L) are perpendicular to l. Thus, K^{*} belongs to line group family 8. For the armchair nanotubes, N_{K}(L) is also of type cm. However, the axes of reflections in N_{K}(L) are parallel to l. In this case, K^{*} belongs to line group family 4.

 Figure 5 A vertex 2coloring of the hexagonal tiling that gives rise to a BN nanotube. 
Note that the coloring on the cylinder obtained from the 2coloring presented in Fig. 5 can also serve as a model for other structural analogs of carbon nanotubes, such as gallium nitride (GaN) and aluminium nitride (AlN) nanotubes. These analogs also possess an equal distribution of two atoms. Our calculations suggest that these nanotubes also have the same symmetry groups as the BN nanotubes. The same symmetry group structure of BN and GaN nanotubes has been reported in Damnjanovic et al. (2001), Alon (2001) and Evarestov et al. (2010).
In obtaining geometric models for BC_{3} nanotubes, we construct vertex 2colorings of the hexagonal tiling using two colors that appear in the ratio of 3:1, as shown in Fig. 6. The red and yellow colors represent carbon and boron, respectively. The symmetry groups of the BC_{3} nanotubes arising from the colorings given in Fig. 6 are summarized in Table 6. The symmetry groups of the BC_{3} nanotube arising from the coloring given in Fig. 6(a) are also reported in Damnjanovic et al. (2001). The symmetric arrangements of the boron and carbon atoms appearing in Figs. 6(b) and 6(c), which suggest other atomic configurations of a BC_{3} nanotube, appear in Wang et al. (1996) and Azevedo & de Paiva (2006).

 Figure 6 Vertex 2colorings of the hexagonal tiling that give rise to BC_{3} nanotubes. (a) K = , (b) K = , (c) K = . 
As discussed previously, a model for a BCN nanotube will arise by considering vertex 3colorings of the hexagonal tiling with an equal distribution of three colors. We present in Fig. 7 two other 3colorings that will give rise to BCN nanotubes. The atomic configurations suggested by these colorings are among the most stable carbonboronnitride ternary graphitelike monolayers as reported in Azevedo & de Paiva (2006). The symmetry groups of the corresponding BCN nanotubes are given in Table 7.

 Figure 7 Vertex 3colorings of the hexagonal tiling that give rise to BCN nanotubes. (a) , (b) . 
In this part of the paper, we enumerate various 3colorings of the hexagonal tiling associated with BC_{2}N nanotubes and present their corresponding symmetry groups. The colors we use in the colorings are yellow, red and blue that appear in the ratio of 1:2:1. They are used to represent boron, carbon and nitrogen, respectively. We adopt the methodology given in De Las Peñas et al. (2006, 2011) and obtained each coloring by specifying a color group H which is a subgroup of the symmetry group of the hexagonal tiling. Eliminating the colorings with adjacent boron and nitrogen atoms, we arrive at the colorings presented in Fig. 8. In constructing these colorings, we consider H such that . In these examples, the set of vertices of the tiling form two equal orbits under H. The symmetry groups for the resulting BC_{2}N nanotubes are presented in Table 8. Results indicate that we have line group families 5 and 13 for the BC_{2}N (type I) nanotubes and line group families 1, 3 and 6 for the BC_{2}N (type II) nanotubes (refer to Damnjanovic et al., 2001 and Pan et al., 2009 for comparison). For the BC_{2}N (type III) nanotubes (Liu et al., 1989), we also obtain line group families 1, 3 and 6. Our calculations suggest other types of nanotubes other than those of types I, II and III.

 Figure 8 Vertex 3colorings of the hexagonal tiling that give rise to BC_{2}N nanotubes. Colorings corresponding to type I, II and III BC_{2}N nanotubes are labeled. (a) , (b) , (c) , (d) , (e) , (f) , (g) , (h) , (i) , (j) , (k) , (l) . 
The approach discussed in the previous sections to determine symmetry groups of carbon nanotubes and their structural analogs can be adapted to a more general setting, when the tubes may not necessarily have hexagonal symmetry.
Synonymous to a graphene sheet, a monolayer consisting of a single type of atom may be modeled geometrically using an isogonal tiling of . Isogonal tilings are vertextransitive and include, among others, the 11 Archimedean tilings and the tilings whose faces are the unit cells of the five twodimensional Bravais lattices (Grünbaum & Shephard, 1978). In the model, vertices of represent atoms and edges represent bonds. Just like a carbon nanotube, a cylindrical tube is obtained by wrapping up the monolayer of atoms (or the tiling ) along a chiral vector . If x, y with vectors , are the generating translational symmetries of the symmetry group G of , then the chiral vector of the nanotube can be expressed as , where , and we obtain an [m, n] singlewall nanotube. Employing the orbit space approach discussed in §2, the resulting nanotube has symmetry group , where L is the subgroup of G generated by the translation with vector .
To illustrate these ideas, we consider a nanotube rolled up from a monolayer made up of carbon atoms located at the vertices of a 4 ·8^{2} tiling shown in Fig. 9(a). This isogonal tiling has faces consisting of regular octagons and squares. Its symmetry group is given by generated by the fourfold (counterclockwise) rotation a about O, the reflection b about the line through O in the direction of , and the translations x,y. If we fold this 4 ·8^{2} tiling along the chiral vector , which determines the subgroup , we obtain an [m,n] TUC_{4}C_{8} nanotube (Arezoomand & Taeri, 2009; Heydari & Taeri, 2009). For example, a [4, 0] TUC_{4}C_{8} nanotube is presented in Fig. 9(b).
 Figure 9 (a) The 4 ·8^{2} tiling together with the translation vectors and the chiral vector . (b) The [4, 0] TUC_{4}C_{8} nanotube obtained by rolling up the 4 ·8^{2} tiling in (a) along the chiral vector . 
We compute the symmetry group of the TUC_{4}C_{8} nanotube and determine its line group structure. The computations based on chirality are given in Table 9. The results indicate that in addition to line group families 5 and 13, to which the symmetry group of a singlewall carbon nanotube belongs, the symmetry group of a TUC_{4}C_{8} nanotube may also belong to line group family 11. The results we obtain are consistent with the symmetry groups of the TUC_{4}C_{8} nanotubes computed by Arezoomand & Taeri (2009).

Now, in obtaining nanotubes consisting of k different atoms, we consider a kvertex coloring of . To ensure that no two types of atoms or colors coincide in the tube, the subgroup L determined by the tube's chiral vector must be a subgroup of the color fixing group K of the coloring. The symmetry group of the nanotube is given by .
In analyzing the twodimensional symmetries present in N_{K}(L), we remark that if a symmetry normalizes the subgroup L (whose nonidentity elements consist of translations), then g must send an element of L to itself or to its inverse. It follows that g cannot be a rotation of order greater than 2. Hence, the possible plane crystallographic group types for N_{K}(L) are p1, p2, cm, pm, c2mm, p2mm, pg, p2mg and p2gg. As mentioned in §5, this list includes all the possible crystallographic types for N_{K}(L) satisfying , where H is the color group of the colored tiling. Thus, Table 3 gives a complete characterization of all the possible line group structures of nanotubes rolled up from colored isogonal tilings with a crystallographic color fixing group.
We show in Fig. 10(a) a vertex 3coloring of the 4 ·8^{2} tiling. Its color fixing group is given by . The colored tiling is rolled along the vector to obtain the nanotube given in Fig. 10(b). This nanotube points to a possible structural analog of the [4, 0] TUC_{4}C_{8} nanotube (Fig. 9b) consisting of three kinds of atoms that appear in the ratio of 1:2:1. We obtain N_{K}(L) = K and the symmetry group K^{*} of the nanotube belongs to line group family 6.
 Figure 10 (a) A vertex coloring of the 4 ·8^{2} tiling using three colors. (b) A 3coloring of the [4,0] TUC_{4}C_{8} nanotube suggesting a structural analog made up of three kinds of atoms that appear in the ratio 1:2:1. 
Nanotubes that arise by rolling up an isogonal tiling may result in commensurate and incommensurate structures. A nanotube is said to be commensurate if the infinite cyclic group of its corresponding line group symmetry structure contains translations along the z axis. Otherwise, the nanotube is said to be incommensurate.
In our work, the commensurability of a given nanotube is determined by finding a translation in N_{G}(L) or N_{K}(L) (when is colored) that is perpendicular to the translation l that generates L. Recall from Table 2 that such a translation in N_{G}(L) or N_{K}(L) gives rise to a respective translation in G^{*} or K^{*} along the z axis.
An incommensurate nanotube has a symmetry group belonging to either line group family 1 or 5 (Damnjanovic et al., 2007; Damnjanovic & Milosevic, 2010). This is due to the fact that for each of the remaining 11 line group families, a suitable power p of its infinite cyclic group generator results in a translation (not necessarily the minimal one) in the line group. For line group families 2, 3, 6, 9, 11 with infinite cyclic group generator (a translation), we have p = 1; for families 7, 10, 12 with infinite cyclic group generator (a glide reflection), p = 2; and for families 4, 8, 13 with infinite cyclic group generator (a screw rotation with rotation component of order 2d, where d is the order of the line group's generating rotation), we have p = 2d. See Table 3 for details of each line group family's generators. Moreover, Table 3 clarifies that if N_{G}(L) or N_{K}(L) is one of the plane crystallographic groups cm, pm, c2mm, p2mm, pg, p2mg and p2gg, then we obtain a commensurate nanotube. An incommensurate nanotube may arise if N_{G}(L) [respectively, N_{K}(L)] is of type p1 or p2. We remark further that incommensurate nanotubes only occur when N_{G}(L) or N_{K}(L) has an underlying parallelogramic (oblique), rectangular or rhombic (centered rectangular) lattice structure. This is because a translation perpendicular to l is always present if the lattice of N_{G}(L) or N_{K}(L) is square or hexagonal. Thus, a singlewall carbon nanotube or a TUC_{4}C_{8} nanotube is always commensurate regardless of its chiral vector.
We present in Fig. 11(a) a rectangular tiling with symmetry group having basis vectors
The group G is generated by the twofold (counterclockwise) rotation a about O, the reflection b about the line through O in the direction of , and the translations x,y. Depending on the chiral vector along which it is rolled, this rectangular tiling gives rise to both commensurate and incommensurate nanotubes.
 Figure 11 (a) The rectangular tiling with translation vectors x and y as given in equation (1). (b) A commensurate nanotube obtained by rolling up the rectangular tiling in (a) along the chiral vector . (c) An incommensurate nanotube obtained by rolling up the rectangular tiling in (a) along the chiral vector . 
The nanotube in Fig. 11(b) is obtained by rolling the rectangular tiling along the vector . We have . This tells us the nanotube is commensurate with its symmetry group G^{*} belonging to line group family 11. On the other hand, the nanotube in Fig. 11(c) is obtained by rolling the same tiling along the vector . Observe that and does not contain a translation perpendicular to x^{4}y^{2}. This nanotube is incommensurate and its symmetry group G^{*} belongs to line group family 5.
As a last example, we present a 2coloring of the rectangular tiling (Fig. 12a) that gives rise to a nanotube (Fig. 12b) using the chiral vector . The color fixing group is given by and N_{K}(L) = . Note that N_{K}(L) also does not contain a translation perpendicular to x^{4}y^{2}. This nanotube is also incommensurate and its symmetry group K^{*} belongs to line group family 5.
 Figure 12 (a) A vertex 2coloring of the rectangular tiling given in Fig. 11(a). (b) A 2coloring of the [4,2] nanotube in Fig. 11(c). 
In this work, a method to determine symmetry groups of structural analogs of singlewall carbon nanotubes has been presented. We have characterized the symmetry groups of these nanotubes according to line groups. Our approach is to analyze the symmetry group structure of a kcolored hexagonal tiling of the plane which, when folded along a chiral vector, will result in an orbit space model of a singlewall nanotube consisting of k different atoms. Based on the chirality of the nanotube, particular color fixing symmetries will give rise to cylindrical symmetries that will define the line group symmetry structure of the nanotube.
We have determined that the symmetry group of a nanotube arises from one of the 13 line group families and that the type of symmetry group obtained for a particular type of nanotube varies, depending on the chirality. We have derived the symmetry groups of BN, BC_{3}, BCN and BC_{2}N nanotubes.
The color symmetry approach presented here in studying the symmetry groups of nanotubes in a twodimensional setting suggests a convenient and accessible way of analyzing the line group symmetries. It also facilitates the characterization of the possible line group structures alongside the various arrangements of atoms that exist and are theoretically possible on a nanotube, made possible by a coloring framework (De Las Peñas et al., 2006, 2011) applied to construct vertex colorings of the hexagonal tiling.
The method presented to characterize symmetry groups of carbon nanotubes and their structural analogs may also be applied to nanotubes with other symmetries by studying their corresponding isogonal tilings on the plane. These nanotubes include the incommensurate ones, which may arise from tilings with underlying parallelogramic (oblique), rectangular or rhombic (centered rectangular) lattice structure, depending on the tiling's color fixing symmetries.
A possible next step in the study is to determine the symmetry groups of doublewall and multiwall nanotubes using a similar method.
Another potential problem that can be addressed for future work is to characterize the symmetry group structures of nanotori and investigate other types of symmetric arrangements of various atoms theoretically possible on a torus.
MLANDLP and AMB would like to acknowledge funding support from the National Research Council of the Philippines (NRCP). MLANDLP is also grateful to Kenichi Shinoda and Yasushi Gomi for helpful discussions and to the Mathematics Department of Sophia University, Tokyo Japan, where part of the research was carried out. MLL thanks the Office of the Dean of the School of Science and Engineering (SOSE), Ateneo de Manila University, for research support.
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