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Xiangbing Zeng, Goran Ungar
To cite this version:
Xiangbing Zeng, Goran Ungar. Inflation rules of square-triangle tilings: from approximants to do- decagonal liquid quasicrystals. Philosophical Magazine, Taylor & Francis, 2006, 86 (06-08), pp.1093- 1103. �10.1080/14786430500363148�. �hal-00513624�
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Inflation rules of square-triangle tilings: from approximants to dodecagonal liquid quasicrystals
Journal: Philosophical Magazine & Philosophical Magazine Letters Manuscript ID: TPHM-05-May-0261.R1
Journal Selection: Philosophical Magazine Date Submitted by the
Author: 02-Sep-2005
Complete List of Authors: Zeng, Xiangbing; University of Sheffield, Department of Engineering Materials
Ungar, Goran; University of Sheffield, Department of Engineering Materials
Keywords: self-assembly, quasicrystals, liquid crystals
Keywords (user supplied): supramolecular liquid crystals, Frank-Kasper phases, dendrimers
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Inflation rules of square-triangle tilings: from approximants to dodecagonal liquid quasicrystals
XIANGBING ZENG* and GORAN UNGAR
Department of Engineering Materials, University of Sheffield, Sheffield S1 3JD, UK
Telephone: 0114 222 5967 (XZ), 0114 222 5943 (GU) Fax: 0114 222 5943
Email: x.zeng@shef.ac.uk, g.ungar@shef.ac.uk
Total number of words in the main text: 2,860.
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Inflation rules of square-triangle tilings: from approximants to dodecagonal liquid quasicrystals
X. B. ZENG* and G. UNGAR
Department of Engineering Materials, University of Sheffield, Sheffield S1 3JD, UK
Running head: Square-triangle inflation rules
An inflation rule for square-triangle tilings is presented and its properties described. Its relationship to other previously found inflation rules, used to generate models of dodecagonal quasicrystals, is discussed. The rule is applied to generate a series of approximants of dodecagonal quasicrystal, including the well know Pm3n and P42/mnm structures. The bigger the unit cell dimensions, the closer the structures of these approximants are to that of the dodecagonal quasicrystal.
Keywords: supramolecular liquid crystal; Frank-Kasper phases; dendrimers
1. Existing square-triangle inflation rules
Square-triangle tilings with 12-fold symmetry are most commonly used to model dodecagonal quasicrystals found in metal alloys1,2,3,4 and, more recently, in dendrimeric supramolecular liquid crystals5. Normal cut and project method in a four-dimensional space generates tilings with 12-fold symmetry formed by squares, triangles and rhombuses6,7. However, rhomboidal units are seldom found in high resolution
transmission electron micrographs (HREM) of dodecagonal quasicrystals, and they are normally considered as defects. In order to generate a “defect-free” dodecagonal tiling consisting of only squares and equilateral triangles, the acceptance domain in the perpendicular space must be allowed to be disconnected, i.e. fractally shaped8,9.
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An alternative way of generating quasiperiodic tilings uses their self-similar property. A parent patch is first inflated by the inflation constant (a number related to the particular symmetry of the tiling, e.g. the golden ratio for Penrose tiling), then the inflated tiles are decomposed into tiles of original size and a new pattern is thus produced. The rule by which the inflated tiles are decomposed is defined as the “inflation rule”. A quasiperiodic tiling is obtained by repeating the inflation and decomposition process ad infinitum10.
The first such inflation rule to generate square-triangle tilings with 12-fold symmetry was found by Stampfli. The inflation constant is λ = 2 + 3 , and the inflation rule is reproduced in Figure 1a11. The generation of the Stampfli tilings involves random choices at each step of decomposition. This gives rise to the maximally random version of the Stampfli inflation rule12: after inflation, each vertex of the original patch is
replaced by a dodecagon centred at the vertex with its sides perpendicular to the original tile edges; the dodecagons are then decomposed into six squares and twelve triangles, with a choice of two different orientations (Figure 1b); the gaps between dodecagons are tiled with either one triangle, or one square plus four triangles.
[insert Figure 1 here]
An inflation rule which produces deterministic twelve-fold symmetric square- triangular tilings was discovered by M. Schlottmann and described in ref. 13. It can be extended, in what became known later as the extended Schlottmann rule, in a similar way to the maximally random Stampli inflation rule described above. The only difference, but an important one, is that in the new patch the orientation of the tiled dodecagon at each vertex is now determined by the orientations of the edges surrounding that vertex in the original patch (Figure 1c). The original Schlottmann rule is a subset of its extended version.
Although the above inflation rules are formulated for the generation of
quasiperiodic tilings, the random Stampfli rule and the extended Schlottmann rule can be applied to almost any given square-triangle tiling of the plane (the only limitation in the case of the extended Schlottmann rule is that no four squares should share one vertex).
Starting from a periodic tiling of the plane, such inflation rules would generate a series of
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periodic square-triangular tilings of the plane, one for each inflation-decomposition cycle.
With increasing period length, such periodic tilings would become closer and closer to a quasiperiodic tiling. While quasiperiodic tilings are used to model quasicrystals, periodic tilings correspond to crystal approximants of quasicrystals. Closely related to
quasicrystals in structure, crystal approximants often exist at similar conditions as, or even coexist with, quasicrystals. In fact many structural models of quasicrystals have been built based on the motifs found in their approximants.
However, the above mentioned inflation rules are not very helpful in generating periodic square-triangular tilings. Some of them, e.g. the random Stampfli rule, are not defined well enough to allow precise control over the resulting tilings. Others, on the other hand, e.g. the Schlottmann rule, are so constrained that they can only produce a limited number of periodic tilings. In addition, the inflation constant λ used in all these rules is 2 + 3 . Consequently, after one inflation-decomposition step, the unit cell
parameter of the periodic tiling increases by a factor of ~4, and the area of the unit cell by a factor of ~14. It would be desirable to generate periodic tilings in as small steps as possible.
2. The half-step rule
A new set of inflation rules, which have a smaller inflation constant ξ = λ = 2 + 3 , are shown in Figure 2. It should be noted that inflation rules with the same constant have been found for dodecagonal quasiperiodic lattices, e.g. in ref. 6, albeit not for square- triangle tilings. In the new set of rules, the edges of the squares and triangles are coloured blue (b) or red (r), and only edges of different colour can be joined together in the tiling.
The triangles always have one red and two blue edges, while the squares must have either two blue and two red edges, or four red edges. After inflation, a triangle is decomposed into one square and two triangles of the original size, a square with blue and red edges into two squares and four triangles, and a square with red edges only into one square and four triangles. The child tiling, ignoring the colouring of its edges, is dictated by the
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the colouring of the child tiling, each vertex in the parent tiling is marked with either an empty or a solid circle. Following the rules laid out in Figure 2, after two consecutive inflation-decomposition steps, each vertex in the original tiling will be replaced by one of the two dodecagons as shown in Figure 1b. The orientation of the dodecagon is
determined by the type of the circle at each vertex. After decomposition, circle types at new vertices of the child tiling are left undecided in Figure 2. These can either be assigned at random, or can follow a predefined rule such as that used in the extended Schlottmann inflation rule.
[insert Figure 2 here]
It can be easily demonstrated that the Stampfli and Schlottmann decomposition rules described above can all be generated by two consecutive operations of the new inflation rule. For this reason we refer to the new rule as the half-step rule.
3. Applications of the half-step rule - generation of tiled patches, periodic and quasiperiodic tilings
The random Stampfli rule has the property that it can be used to inflate any given square- triangle tiling of the plane. In contrast, the half-step rule presented here is only applicable when all tile edges can be coloured according to the colour matching rules. This adds some limitations to the type of tilings that can be inflated this way. In the following, we consider some basic conditions that these tilings must satisfy.
We will start by considering the patches of triangles that are surrounded by
squares. Firstly these patches must be convex, as any concave angle at a vertex formed by triangles can only be tiled further by triangles. The simplest way n triangles can tile together is to form a straight row, which can be labeled {∆, n}. {Note to typesetter: First symbol in brackets is an empty triangle or upper case Greek delta.} For such a row of triangles, all but one of the outer edges of the patch must be coloured blue. Row-like patches with any number of triangles are allowed but this will impose certain constraint on the surrounding tiles. For example, for {∆, 6} either the patch of squares directly above, or below, must contain at least two rows (Figure 3). Patches of triangles
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containing two rows can be labelled {∆, n, m}, where n and m are the numbers of triangles in each row. The number of such patches is limited, mainly because of the condition that two hexagonal (36) vertices are not allowed to be adjacent to each other.
All of the two-row triangular patches allowed are shown in Figure 3. No patches of triangles which have more than two rows are allowed (Since the direction of rows in a two-dimensional patch can be ambiguous, we choose it to be the direction of the longest row).
[insert Figure 3 here]
Patches of squares can only be rectangular in shape and can be labelled by their number of rows and columns as {□, n, m}. {Note to typesetter: First symbol in brackets is an empty square.} A square with all edges coloured red can only share a side with another square which has two blue and two red sides. As the blue side in a square must be opposite a red one (see Figure 2), the red edges will propagate along the row or the column direction in a patch of squares. One of the consequences is that there can be no more than one square tile with all red edges in any row or column of a patch of squares.
For a square patch {□, n, m}, there will be at least |n–m| columns or rows lacking any all- red tile. For such rows or columns, the edges at the opposite ends must be different in colour. This, combined with the fact that triangular patches can have no more than one red outside edge, implies that |n–m| ≤ 2. Some of the allowed patches of squares are shown in Figure 4, together with their surrounding tiles, which are determined by the colour matching rule.
[insert Figure 4 here]
Using the half-step inflation rule and starting from a periodic tiling of the plane by squares, a series of periodic square-triangle tilings can be generated (Figure 5a-d). The colouring scheme of the squares in Figure 5a is given in Figure 5e. For successive tilings, the assignment of the circle type at each vertex follows that used in the extended
Schlottmann rule, in order to ensure that the periodic tilings converge towards the deterministic quasiperiodic tiling with increasing generation.
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[insert Figure 5 here]
4. Tile decoration - structure of dodecagonal quasicrystal and its approximants The square and triangular tiles used in the tilings above are mathematical abstractions, while in reality crystals are generated by packing of spherical objects, i.e. atoms for alloys, and micelles for liquid quasicrystals. The link between tilings and realistic models of quasicrystals or their approximants can be established by tile “decoration”. The tile decoration rule establishes positions of a group of atoms (or micelles) in relation to each tile, and models of crystals in three-dimensional space are generated by replacing each tile with such a group of atoms or micelles. Since dodecagonal quasicrystals are periodic in z direction, a tile is actually decorated with an infinite periodic column of atoms normal to the tiling plane. The decoration rule (Figure 5e) used in this paper has been first proposed by Gähler7 who compared the crystal structures of several approximants of dodecagonal quasicrystals in metal alloys. These include the A-15 structure (space group
n 3
Pm ) and the σ-phase (space group P42/mnm). Not surprisingly, these two structures have also been found in thermotropic (solvent-free) “micellar” liquid crystals in the vicinity of the dodecagonal liquid quasicrystal14,15. The structures of the approximants, generated by decoration of the tilings in Figure 5, are shown in Figure 6. The first two generations of periodic tilings produce the crystal structure of Pm3n and P42/mnm phases respectively. The simulated diffraction patterns of these approximants are also shown in Figure 6. With increasing generation, their pseudo-twelve-fold symmetry tends toward true twelve-fold symmetry of the quasicrystal.
[insert Figure 6 here]
In dodecagonal quasicrystal and its approximants there are four layers of atoms or micelles in one period along the z axis: two sparsely populated layers at z = ¼ and z = ¾, and two densely populated layers, one at z = 0, and the other at z = ½ (cf. Figure 5e). In all the models the periodicity along the z axis is equal to the edge length of the tiles. For the Pm3n phase it is a condition of cubic symmetry, and for the P42/mnm and LQC phases it is found experimentally to hold almost exactly. The projection along the z-axis
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of atoms (or micelles) in the sparsely populated layers sit exactly at the vertices of the square-triangular tiling. Around each atom in the sparsely populated layers, there is a hexagonal antiprism formed by twelve neighbouring atoms, six in the dense layer immediately above, the other six in the layer immediately below. The atoms or micelles in the densely populated layers, on the other hand, can be viewed as vertices of a network of distorted equilateral triangles and hexagons. The adjacent dense layers are rotated in plane by 90° with respect to each other.
5. An alternative method of generating a model dodecagonal quasicrystal The fact that atoms in the densely populated layers forms a network of triangles and hexagons gives rise to another method of generating a model of dodecagonal
quasicrystals. The x-y plane tiled by equilateral triangles of edge length 1 would have six- fold symmetry. We now make a copy of such a tiling, rotate it in plane around the origin by π/2, and displace it by z0/2 along the z-axis. The pattern thus generated will have a single twelve-fold (12) rotational axis through the origin. If we place an atom at each vertex of the two tilings, there will be pairs of atoms whose projections onto the x-y plane are very close. We now replace these pairs of atoms with two atoms, whose horizontal position is the average of the two original ones, and whose vertical positions are z0/4 and 3z0/4 respectively. Thus alternating sparsely and densely populated layers are generated, and each atom in the sparse layers is surrounded by six atoms above and six below in the densely populated layers, as in the models constructed in section 4.
For a pair of atoms to be replaced, their projections along the z-axis can be written as m1e1 + m3e3 and m2e2 + m4e4 respectively, where en = cos(nπ/6)i + sin(nπ/6)j. Thus the projected position of the replacement atoms can be written as m1e1/2 + m2e2/2 + m3e3/2 + m4e4/2. At the same time the difference in the projected positions of the two original atoms equals to m1e1 – m2e2 + m3e3 – m4e4. The two formulae can be rewritten as m1e1//
+ m2e2// + m3e3// + m4e4//, and 2(m1e1⊥ + m2e2⊥ + m3e3⊥ + m4e4⊥) respectively, where en// = (cos(nπ/6)i + sin(nπ/6)j)/2 and en⊥ = (cos(-5nπ/6)i + sin(-5nπ/6)j)/2. The criterion used to determine the closeness of two atoms from the two hexagonal networks respectively is
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cut and project method.6 This provides a possible explanation of the physical origin of the acceptance domain in the cut and project method, and may help understand the existence and the stability of dodecagonal quasicrystals.
Figure 7 shows how, in such a construction of dodecagonal quasicrystal, the positions of atoms in the sparsely populated layer can be chosen to be identical to those of the
quasiperiodic square-triangular tiling generated by the Schlottmann rule. The edge length of the tiles in the sparsely populated layer, which should be equal to the periodicity z0 along the z-axis, is 1 + 3 /2.
[insert Figure 7 here]
The alternative model of dodecagonal quasicrystal is essentially the same as that
constructed by decoration of the dodecagonal square-triangular tiling, except now that the atoms in the densely populated layers fall on a network of regular triangles without distortion. Such regularity within each densely populated layer may be more energetically favoured. However, as a consequence of such regularity in the densely populated layers, the “decorations” of the trianglar and square tiles of the sparsely populated layer vary from place to place in the new model. The question of which structure is ultimately more stable would be an interesting problem for further investigation.
References
[1] T. Ishimasa, H.-U. Nissen and Y. Fukano, Phys. Rev. Lett. 55 511 (1985).
[2] H. Chen, D. X. Li and K. H. Kuo, Phys. Rev. Lett. 60 1645 (1988).
[3] C. Beeli, F. Gähler, H.-U. Nissen and P. Stadelmann, J. Phys. (Paris) 51 661 (1990).
[4] M. Conrad, F. Krumeich and B. Harbrecht, Angew. Chem. Int. Ed. Engl. 37 1383 (1998).
[5] X.B. Zeng, G. Ungar, Y.S. Liu, V. Percec, A.E. Dulcey, and J.K. Hobbs, Nature 428 157 (2004).
[6] N. Niizeki and H. Mitani, J. Phys. A 20 L405 (1987).
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[7] F. Gähler, in Quasicrystalline Materials, edited by C. Janot and J.M. Dubois (World Scientific, Singapore, 1988), pp 272–284.
[8] M. Baake, R. Klitzing, and M. Schlottmann, Physica A 191 554 (1992).
[9] A.P. Smith, Journal of Non-Crystalline Solids 153&154, 258 (1993).
[10] B. Grünbaum and G.C. Shephard, Tilings and Patterns (W.H. Freeman, New York, 1987).
[11] P. Stampfli, Helv. Phys. Acta 59 1260 (1986).
[12] M. Oxborrow, C.L. Henley, Phys. Rev. B 48 6966 (1993).
[13] J. Hermisson, C. Richard, M. Baake, Journal de Physique I 7 1003 (1997).
[14] V.S.K. Balagurusamy, G. Ungar, V. Percec and G. Johansson, J. Am. Chem. Soc.
119 1539 (1997).
[15] G. Ungar, Y. Liu, X.B. Zeng, V. Percec and W.D. Cho, Science 299 1208 (2003).
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Figure Captions
Figure 1. Inflation rules for the generation of dodecagonal quasiperiodic tilings. (a) Stampfli inflation rule. (b) Random Stampfli inflation rule. (c) Choice of dodecagon orientations in the extended Schlottmann rule.
Figure 2. The half-step inflation rule for square-triangle tilings. The colour of the edges are marked by letters: r, red; b, blue. The vertex types are represented by empty and solid circles, which essentially determine the orientation of the dodecagon at each vertex after two consecutive steps of inflation. In the second and third rows only the representative combinations of vertex types are shown. The combinations not presented can be easily derived from those shown: the change of circle type in the parent tile will induce the colour change of the edges originating at the respective vertex in the child patch. The vertex types can be reassigned after each inflation-decomposition step, and are represented by a circle with broken outline.
Figure 3. Some of the triangle patches allowed by the half-step inflation rule. The outer boundaries of the patches are outlined by thick lines. Surrounding tiles, which are
determined by the patch inside, are also shown. Letters r and b are not shown for brevity, as there are a number of possible colour configurations of the edges for each patch shown.
Figure 4. Some of the square patches allowed by the half-step inflation rule. Outer boundaries of patches are outlined by thick lines. Surrounding tiles, determined by the patches inside, are also shown. Letters r and b are not shown for brevity, as there are a number of possible colour configurations of the edges for each patch shown.
Figure 5. (a-d) A series of periodic square-triangle tilings generated by successive application of the inflation rule given in Figure 2. (e) The colours of edges and circle types used for the tiles in (a). (f) Decoration rule for the generatation of models of crystal approximants from square-triangle tilings. z denotes elevation as fraction of the unit cell.
Figure 6. Structure models viewed along the c-axis (top) and the corresponding simulated diffraction patterns of a series of approximants of the dodecagonal quasicrystal. The cell parameters are a=b=81.4Å (Pm3n), 157.3Å (P42/mnm), 303.8Å (Approx. 3) and 586.9Å
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(Approx. 4). c=81.4Å in all cases. The number of micelles in the unit cell is 8, 30, 112 and 418, respectively.
Figure 7. The alternative construction of dodecagonal quasicrystal. The two triangular tilings (36), one being the rotation by π/2 of the other, are coloured red and black, respectively. The black circles indicate the positions of pairs of atoms which should be replaced and which form a dodecagonal square-triangle tiling, identical to that generated by the Schlottmann rule.
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Figure 1. Inflation rules for the generation of dodecagonal quasiperiodic tilings. (a) Stampfli inflation rule. (b) Random Stampfli inflation rule. (c) Choice of dodecagon orientations in the extended Schlottmann rule.
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Figure 2. The half-step inflation rule for square-triangle tilings. The colour of the edges are marked by letters: r, red; b, blue. The vertex types are represented by empty and solid circles, which essentially determine the orientation of the dodecagon at each vertex after two consecutive steps of inflation. In the second and third rows only the representative combinations of vertex types are shown. The combinations not presented can be easily derived from those shown: the change of circle type in the parent tile will induce the colour change of the edge originating at the respective vertex in the child patch. The vertex types can be reassigned after each inflation-decomposition step, and are represented by a circle with broken outline.
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Figure 3. Some of the triangle patches allowed by the half-step inflation rule. The outer boundaries of the patches are outlined by thick lines. Surrounding tiles, which are
determined by the patch inside, are also shown. Letters r and b are not shown for brevity, as there are a number of possible colour configurations of the edges for each patch shown.
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Figure 4. Some of the square patches allowed by the half-step inflation rule. Outer boundaries of patches are outlined by thick lines. Surrounding tiles, determined by the patches inside, are also shown. Letters r and b are not shown for brevity, as there are a number of possible colour configurations of the edges for each patch shown.
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Figure 5. (a-d) A series of periodic square-triangle tilings generated by successive application of the inflation rule given in Figure 2. (e) The colours of edges and circle types used for the tiles in (a). (f) Decoration rule for the generatation of models of crystal approximants from square-triangle tilings. z denotes elevation as fraction of the unit cell.
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Figure 6. Structure models viewed along the c-axis (top) and the corresponding simulated diffraction patterns of a series of approximants of the dodecagonal quasicrystal. The cell parameters are a=b=81.4Å (Pm3n), 157.3Å (P42/mnm), 303.8Å (Approx. 3) and 586.9Å (Approx. 4). c=81.4Å in all cases. The number of micelles in the unit cell is 8, 30, 112 and 418, respectively.
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Figure 7. The alternative construction of dodecagonal quasicrystal. The two triangular tilings (36), one being the rotation by π/2 of the other, are coloured red and black, respectively. The black circles indicate the positions of pairs of atoms which should be replaced and which form a dodecagonal square-triangle tiling, identical to that generated by the Schlottmann rule.
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Inflation rules of square-triangle tilings: from approximants to dodecagonal liquid quasicrystals
XIANGBING ZENG* and GORAN UNGAR
Department of Engineering Materials, University of Sheffield, Sheffield S1 3JD, UK
Telephone: 0114 222 5967 (XZ), 0114 222 5943 (GU) Fax: 0114 222 5943
Email: x.zeng@shef.ac.uk, g.ungar@shef.ac.uk
Total number of words in the main text: 2,860.
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Inflation rules of square-triangle tilings: from approximants to dodecagonal liquid quasicrystals
X. B. ZENG* and G. UNGAR
Department of Engineering Materials, University of Sheffield, Sheffield S1 3JD, UK
Running head: Square-triangle inflation rules
An inflation rule for square-triangle tilings is presented and its properties described. Its relationship to other previously found inflation rules, used to generate models of dodecagonal quasicrystals, is discussed. The rule is applied to generate a series of approximants of dodecagonal quasicrystal, including the well know Pm3n and P42/mnm structures. The bigger the unit cell dimensions, the closer the structures of these approximants are to that of the dodecagonal quasicrystal.
Keywords: supramolecular liquid crystal; Frank-Kasper phases; dendrimers
1. Existing square-triangle inflation rules
Square-triangle tilings with 12-fold symmetry are most commonly used to model dodecagonal quasicrystals found in metal alloys1,2,3,4 and, more recently, in dendrimeric supramolecular liquid crystals5. Normal cut and project method in a four-dimensional space generates tilings with 12-fold symmetry formed by squares, triangles and rhombuses6,7. However, rhomboidal units are seldom found in high resolution
transmission electron micrographs (HREM) of dodecagonal quasicrystals, and they are normally considered as defects. In order to generate a “defect-free” dodecagonal tiling consisting of only squares and equilateral triangles, the acceptance domain in the perpendicular space must be allowed to be disconnected, i.e. fractally shaped8,9.
Formatted: Superscript
Formatted: Superscript
Deleted: Multiply twinned Pm3n
domains, which show a pseudo twelve- fold symmetry, are occasionally found to coexist with the dodecagonal liquid quasicrystal in a dendrimeric compound.
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An alternative way of generating quasiperiodic tilings uses their self-similar property. A parent patch is first inflated by the inflation constant (a number related to the particular symmetry of the tiling, e.g. the golden ratio for Penrose tiling), then the inflated tiles are decomposed into tiles of original size and a new pattern is thus produced. The rule by which the inflated tiles are decomposed is defined as the “inflation rule”. A quasiperiodic tiling is obtained by repeating the inflation and decomposition process ad infinitum10.
The first such inflation rule to generate square-triangle tilings with 12-fold symmetry was found by Stampfli. The inflation constant is λ = 2 + 3 , and the inflation rule is reproduced in Figure 1a11. The generation of the Stampfli tilings involves random choices at each step of decomposition. This gives rise to the maximally random version of the Stampfli inflation rule12: after inflation, each vertex of the original patch is replaced by a dodecagon centred at the vertex with its sides perpendicular to the original tile edges; the dodecagons are then decomposed into six squares and twelve triangles, with a choice of two different orientations (Figure 1b); the gaps between dodecagons are tiled with either one triangle, or one square plus four triangles.
[insert Figure 1 here]
An inflation rule which produces deterministic twelve-fold symmetric square- triangular tilings was discovered by M. Schlottmann and described in ref. 13. It can be extended, in what became known later as the extended Schlottmann rule, in a similar way to the maximally random Stampli inflation rule described above. The only difference, but an important one, is that in the new patch the orientation of the tiled dodecagon at each vertex is now determined by the orientations of the edges surrounding that vertex in the original patch (Figure 1c). The original Schlottmann rule is a subset of its extended version.
Although the above inflation rules are formulated for the generation of
quasiperiodic tilings, the random Stampfli rule and the extended Schlottmann rule can be applied to almost any given square-triangle tiling of the plane (the only limitation in the case of the extended Schlottmann rule is that no four squares should share one vertex).
Starting from a periodic tiling of the plane, such inflation rules would generate a series of 3
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periodic square-triangular tilings of the plane, one for each inflation-decomposition cycle.
With increasing period length, such periodic tilings would become closer and closer to a quasiperiodic tiling. While quasiperiodic tilings are used to model quasicrystals, periodic tilings correspond to crystal approximants of quasicrystals. Closely related to
quasicrystals in structure, crystal approximants often exist at similar conditions as, or even coexist with, quasicrystals. In fact many structural models of quasicrystals have been built based on the motifs found in their approximants.
However, the above mentioned inflation rules are not very helpful in generating periodic square-triangular tilings. Some of them, e.g. the random Stampfli rule, are not defined well enough to allow precise control over the resulting tilings. Others, on the other hand, e.g. the Schlottmann rule, are so constrained that they can only produce a limited number of periodic tilings. In addition, the inflation constant λ used in all these rules is 2 + 3 . Consequently, after one inflation-decomposition step, the unit cell parameter of the periodic tiling increases by a factor of ~4, and the area of the unit cell by a factor of ~14. It would be desirable to generate periodic tilings in as small steps as possible.
2. The half-step rule
A new set of inflation rules, which have a smaller inflation constant ξ = λ = 2 + 3, are shown in Figure 2. It should be noted that inflation rules with the same constant have been found for dodecagonal quasiperiodic lattices, e.g. in ref. 6, albeit not for square- triangle tilings. In the new set of rules, the edges of the squares and triangles are coloured blue (b) or red (r), and only edges of different colour can be joined together in the tiling.
The triangles always have one red and two blue edges, while the squares must have either two blue and two red edges, or four red edges. After inflation, a triangle is decomposed into one square and two triangles of the original size, a square with blue and red edges into two squares and four triangles, and a square with red edges only into one square and four triangles. The child tiling, ignoring the colouring of its edges, is dictated by the colours of the edges of the parent tiling. To determine the tiling of another generation or
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the colouring of the child tiling, each vertex in the parent tiling is marked with either an empty or a solid circle. Following the rules laid out in Figure 2, after two consecutive inflation-decomposition steps, each vertex in the original tiling will be replaced by one of the two dodecagons as shown in Figure 1b. The orientation of the dodecagon is
determined by the type of the circle at each vertex. After decomposition, circle types at new vertices of the child tiling are left undecided in Figure 2. These can either be assigned at random, or can follow a predefined rule such as that used in the extended Schlottmann inflation rule.
[insert Figure 2 here]
It can be easily demonstrated that the Stampfli and Schlottmann decomposition rules described above can all be generated by two consecutive operations of the new inflation rule. For this reason we refer to the new rule as the half-step rule.
3. Applications of the half-step rule - generation of tiled patches, periodic and quasiperiodic tilings
The random Stampfli rule has the property that it can be used to inflate any given square- triangle tiling of the plane. In contrast, the half-step rule presented here is only applicable when all tile edges can be coloured according to the colour matching rules. This adds some limitations to the type of tilings that can be inflated this way. In the following, we consider some basic conditions that these tilings must satisfy.
We will start by considering the patches of triangles that are surrounded by squares. Firstly these patches must be convex, as any concave angle at a vertex formed by triangles can only be tiled further by triangles. The simplest way n triangles can tile together is to form a straight row, which can be labeled {∆, n}. {Note to typesetter: First symbol in brackets is an empty triangle or upper case Greek delta.} For such a row of triangles, all but one of the outer edges of the patch must be coloured blue. Row-like patches with any number of triangles are allowed but this will impose certain constraint on the surrounding tiles. For example, for {∆, 6} either the patch of squares directly above, or below, must contain at least two rows (Figure 3). Patches of triangles
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containing two rows can be labelled {∆, n, m}, where n and m are the numbers of triangles in each row. The number of such patches is limited, mainly because of the condition that two hexagonal (36) vertices are not allowed to be adjacent to each other.
All of the two-row triangular patches allowed are shown in Figure 3. No patches of triangles which have more than two rows are allowed (Since the direction of rows in a two-dimensional patch can be ambiguous, we choose it to be the direction of the longest row).
[insert Figure 3 here]
Patches of squares can only be rectangular in shape and can be labelled by their number of rows and columns as {□, n, m}. {Note to typesetter: First symbol in brackets is an empty square.} A square with all edges coloured red can only share a side with another square which has two blue and two red sides. As the blue side in a square must be opposite a red one (see Figure 2), the red edges will propagate along the row or the column direction in a patch of squares. One of the consequences is that there can be no more than one square tile with all red edges in any row or column of a patch of squares.
For a square patch {□, n, m}, there will be at least |n–m| columns or rows lacking any all- red tile. For such rows or columns, the edges at the opposite ends must be different in colour. This, combined with the fact that triangular patches can have no more than one red outside edge, implies that |n–m| ≤ 2. Some of the allowed patches of squares are shown in Figure 4, together with their surrounding tiles, which are determined by the colour matching rule.
[insert Figure 4 here]
Using the half-step inflation rule and starting from a periodic tiling of the plane by squares, a series of periodic square-triangle tilings can be generated (Figure 5a-d). The colouring scheme of the squares in Figure 5a is given in Figure 5e. For successive tilings, the assignment of the circle type at each vertex follows that used in the extended
Schlottmann rule, in order to ensure that the periodic tilings converge towards the deterministic quasiperiodic tiling with increasing generation.
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[insert Figure 5 here]
4. Tile decoration - structure of dodecagonal quasicrystal and its approximants The square and triangular tiles used in the tilings above are mathematical abstractions, while in reality crystals are generated by packing of spherical objects, i.e. atoms for alloys, and micelles for liquid quasicrystals. The link between tilings and realistic models of quasicrystals or their approximants can be established by tile “decoration”. The tile decoration rule establishes positions of a group of atoms (or micelles) in relation to each tile, and models of crystals in three-dimensional space are generated by replacing each tile with such a group of atoms or micelles. Since dodecagonal quasicrystals are periodic in z direction, a tile is actually decorated with an infinite periodic column of atoms normal to the tiling plane. The decoration rule (Figure 5e) used in this paper has been first proposed by Gähler7 who compared the crystal structures of several approximants of dodecagonal quasicrystals in metal alloys. These include the A-15 structure (space group
n 3
Pm ) and the σ-phase (space group P42/mnm). Not surprisingly, these two structures have also been found in thermotropic (solvent-free) “micellar” liquid crystals in the vicinity of the dodecagonal liquid quasicrystal14,15. The structures of the approximants, generated by decoration of the tilings in Figure 5, are shown in Figure 6. The first two generations of periodic tilings produce the crystal structure of Pm3n and P42/mnm phases respectively. The simulated diffraction patterns of these approximants are also shown in Figure 6. With increasing generation, their pseudo-twelve-fold symmetry tends toward true twelve-fold symmetry of the quasicrystal.
[insert Figure 6 here]
In dodecagonal quasicrystal and its approximants there are four layers of atoms or micelles in one period along the z axis: two sparsely populated layers at z = ¼ and z = ¾, and two densely populated layers, one at z = 0, and the other at z = ½ (cf. Figure 5e). In all the models the periodicity along the z axis is equal to the edge length of the tiles. For the Pm3n phase it is a condition of cubic symmetry, and for the P42/mnm and LQC phases it is found experimentally to hold almost exactly. The projection along the z-axis
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