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A new method to generate quasicrystalline structures : examples in 2D tilings Jean-François Sadoc, R. Mosseri

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Jean-François Sadoc, R. Mosseri. A new method to generate quasicrystalline structures : examples in 2D tilings. Journal de Physique, 1990, 51 (3), pp.205-221. ￿10.1051/jphys:01990005103020500￿. ￿jpa-00212361￿

HAL Id: jpa-00212361 https://hal.archives-ouvertes.fr/jpa-00212361 Submitted on 1 Jan 1990

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. J. Phys. France 51 (1990) 205-221 1er FÉVRIER 1990, 205

Classification Physics Abstracts 61.40

A new method to generate quasicrystalline structures : examples in 2D tilings

Jean-François Sadoc (1) and R. Mosseri (2)

(1) Laboratoire de Physique des Solides, Université de Paris-Sud et CNRS, 91405 Orsay, France (2) Laboratoire de Physique des Solides de Bellevue-CNRS, 92195 Meudon Cedex, France

(Reçu le 18 juillet 1989, révisé et accepté le 20 octobre 1989)

Résumé. 2014 Nous présentons un nouvel algorithme pour la génération des structures quasi- cristallines. Il est relié à la méthode de coupe et projection, mais il permet une génération directement dans l’espace « physique » E de la structure. La sélection des sites dans l’espace orthogonal est remplacée par un test directement dans une grille de domaines d’acceptance dans l’espace E. Cette méthode montre qu’il y a une sorte de réseau cristallin sous-jacent au quasi- cristal. Nous illustrons la construction dans le cas 4D-2D avec les symétries d’ordre 5, 8, 10 et 12 qui sont obtenues par projection de 4D à 2D. Par la même méthode d’autres types de quasi- cristaux avec une symétrie plus basse, ayant un réseau moyen, sont construits. Nous présentons un exemple de symétrie 4. Les points de ce quasi-cristal sont un sous-ensemble des points du quasi-cristal ayant la symétrie complète d’ordre 8.

Abstract. 2014 We present a new algorithm for the generation of quasicrystalline structures. It is related to the cut and projection method, but allows a direct generation of the structure in the « physical » space E. The orthogonal space site selection is replaced by a direct check in a periodic array of « acceptance » regions in E. This method shows that there is a sort of underlying crystalline lattice in quasicrystals. We illustrate the construction in the 4D-2D cases with the 5-, 8-, 10- and 12-fold symmetries which can be obtained by projection from 4D to 2D. Using this new method we also generate quasicrystals with a lower symmetry which have simple mean lattices. We present for instance a quasicrystal with a 4-fold symmetry. The points of this quasicrystal are a subset of the quasicrystal which has the whole 8-fold symmetry.

1. Introduction.

In this paper we present a new algorithm for the generation of quasicrystalline structures, closely related to the cut and projection (C.P.) method [1], but which allows a direct generation of the structure in the « physical » space E. The orthogonal space site selection is replaced by a direct check in a periodic array of « acceptance » regions in E as will be explain below. This method, which is valid for projection from any , is presented in the next section. We then illustrate the construction in the 4D-2D cases. There is a theorem [2] which allows one to determine the smallest dimension required in order to obtain a 2D quasicrystal with a

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01990005103020500 206 given symmetry. If the symmetry is of order s the number of its prime integers smaller than s give the dimension of the space. We are interested in quasicrystals obtained from 4D.

Consider the different possible symmetries which are not crystalline : 5, 7, 8, 9, 10, 11, 12, ...

Other symmetries give more than 4 numbers. So the 5, 8, 10 and 12 symmetries exhaust the quasicrystalline symmetries which can be obtained by projection from 4D to 2D. In the present paper we shall discuss mainly the octagonal and the dodecagonal quasicrystals with a few comments on the two other cases. We also present in an appendix a method which allows one to determine rapidly the space on which it is interesting to project the 4D structure, and which explains the determination of the projection matrix. In the usual C.P. framework the determination of the physical space is done by considering representations of the symmetry group operations in the n-dimensional space. Here we adopt a more geometrical approach by considering the symmetry of the Petrie attached to the lattice.

2. Description of the algorithm.

Let us first recall a few facts and definitions from the C.P. method. Let L be an n-dimensional lattice embedded in Rn. The « physical » space E, in which the quasiperiodic tiling is to be generated, is a p-dimensional space. For simplicity we shall suppose that n = 2 p. Let f2 be a bounded volume in Rn- n is shifted along E, defining a « strip ». All vertices of L which fall in the strip are selected and orthogonally mapped onto E, giving the vertices of the quasiperiodic tiling. For the « standard » tilings, 5li is the L unit cell or the Voronoi region. In the language of the 2D case, such quasiperiodic tilings are obtained by mapping on E a monolayer surface selected in the strip (this surface is tiled by faces when the lattice L is a cubic lattice). The selection algorithm usually proceeds as follows : the lattice vertices to be selected are precisely those which fall, by orthogonal projection onto the space E’ orthogonal to E, inside a « window » W which is the hull of the projection onto E’ of 5li ., They are therefore obtained by a systematic inspection of all the L vertices. The modified version will proceed differently for the site selection and in most cases will be more efficient. Specifically, the standard C.P. algorithm speed is proportional to the n-th power of the size which is inspected. The modified version runs at only the p-th power of this size. The basic idea is to split L into reticular sub-lattices, i.e. to express L as the product of two complementary p-dimensional lattices, the base B and the fibre F :

B and F are and The respectively generated by {bl, ..., b.p} {f1, ..., fp} . simplest example is, if L is the hypercubic lattice Z", to take as B and F the lattices respectively generated by and where is the canonical basis in Rn. Now the site selection (ei , ... , ep} {ep + 1, ..., en} {ei} is done fibre after fibre. One copy of F, called F lo l is attached.to each Q of the base B 207

(see Fig.1 ) . Let us call W {Q} the intersection of F { Q } the strip S generated by translating llÎ along E :

Fig. 1. - Scheme of the modified projection method. B is the base, F the fibre, E the physical space and A is the centre of an acceptance domain. ’

The points of F {Q} which are to be selected are precisely those which are inside W {Q} . The entire procedure can be done directly on the physical space E. If II is the orthogonal projection onto E, we define :

{Q} The Z form a periodic array of « acceptance domains » (A.D.), which is generally not a tiling. Indeed neighbouring A.D. could .overlap in some cases. To each point of U one attaches a copy of the lattice V {Q} and we select those points of V {Q} which fall inside the A.D. Z {Q} . The quasiperiodic structure X reads formally :

The advantage is that it is easy to calculate a priori which point of V lo l is closest to the center of the A.D. Z {O}. It is then sufficient to test only a small subset T of V lo l surrounding this point. This set T is the maximal subset of V lo l which can fit into the A.D. All these steps will be illustrated below in 2D examples.

3. Lattices, honeycombs and in R4.

It is always possible to characterise the local order of a simple crystalline structure by one, or a few, (a in 4D) [3] : this could be the coordination shell, the Voronoi cell or interstices in the structure. If we are interested in 2D structures mapped from R", such local polytopes are mapped in the plane E inside using the cut and projections method. Therefore it is interesting to consider in R" a polygonal line contained in the local polytope and to study how it could give a polygon with the required symmetry for the quasicrystal when it is projected on a plane. If we want to obtain all the points of the 208

polygonal line in R4 selected by the strip, the plane E must remain close to the polygonal line, so this line gives information on the position of the projection plane in R4. There is a polygonal line which strongly characterizes a polytope : the [3]. The Petrie polygon of a polyhedron is a skew polygon such that every two consecutive sides, but not three, belong to a of the polyhedron. This definition could be extended to polytopes of higher dimension : a Petrie polygon of an n-dimensional polytope is a skew polygon such that any (n - 1 ) consecutive sides, but no n, belong to a Petrie polygon of a cell. The Petrie polygon of a has a symmetry group which is the largest sub-group of the symmetry group of the polytope [3]. In all cases two consecutive edges define entirely a unique Petrie polygon.

3.1 SEARCH FOR THE 4D STRUCTURE LEADING TO THE OCTAGONAL QUASICRYSTAL. - We are looking for a regular in E. So in R4 we search for an octagonal skew polygon. The Petrie polygon of the 4D is a good candidate (see Fig. 2).

Fig. 2. - Schegel diagram of the hypercube with the octagonal Petrie polygon (with black edges).

The cubic 4D lattice Z4, or {4, 3, 3, 4 } in Schlâfli notation, is a structure which is well characterized by the {4, 3, 3 } hypercube which is, in this case, both the interstice configuration and also the Voronoi cell. Consider a Petrie polygon of a given hypercube. It has 8 edges whose mid points are in a single plane. It is clear that if we use this plane as the E plane, with a suitable acceptance region in E’ it will be possible to select vertices of the Petrie polygon, and to reject other vertices of the hypercube. By projection on E and E’ the Petrie polygon gives a regular octagon. This justifies the choice of the 4D cubic lattice in order to derive the octagonal quasicrystal. There are three possibilities for choosing a local polytope characteristic of the local order in Z4 : the cubic cell, the Voronoi cell of a , or the coordination shell of the same vertex. We choose the Petrie polygon of the Voronoi cell because it defines a plane E containing the central vertex. The Voronoi cell is a hypercube with edges parallel to those of the cubic cell. By projection on E, the Petrie polygon gives an octagon. With this choice there is a vertex at the centre of the octagon.

’3.2 THE DODECAGONAL SYMMETRY. - We search for a polygonal line which has twelve vertices. The Petrie polygon of the {3, 4, 3 } polytope has this property. It is represented in figure 3 using the Schlegel projection of the polytope. The {3, 4, 3 } polytope is build from 24 octahedra, three sharing an . There is a regular honeycomb whose coordination 209

Fig. 3. - Schegel diagram of the {3, 4, 3 } polytope with the decagonal Petrie polygon (with black edges). polyhedron (the defined by Coxeter) is a {3, 4, 3 } : the {3, 3, 4, 3 } honeycomb. It is a packing of regular cross polytopes {3, 3, 4 } . This structure could also be described as a lattice : the Leech A4 lattice [4]. It is obtained by begining with a triangular 2D lattice then, in a third dimension, triangular lattices are stacked up leading to an f.c.c. lattice, then in a fourth dimension f.c.c. lattice are stacked up. There are two ways to stack up f.c.c. lattices : sites of the « upper one » could be over tetrahedral interstices, or over octahedral interstices ; it is the latter configuration which leads to the {3, 3, 4, 3 } honeycomb. This lattice could be defined using a cubic crystallographic cell, but there are two possibilities. The first one is built from the cubic cell of the f.c.c. 3D lattice by adding a fourth dimension to obtain a hypercubic cell ; then the lattice is a non-primitive 2-face centered lattice (type F in crystallographic notation). The second possible cubic cell could be derived from the first one. Consider the four diagonals of the hypercube : they form an orthogonal basis. Using this basis another cubic cell is defined ; then the lattice is a body centered lattice (type I). If the E space is chosen such that it contains the mid-points of the edges of the Petrie polygon of a {3, 4, 3 } coordination polytope of the {3, 3, 4, 3 } honeycomb, a dodecagonal quasicrystal results from the cut and projection method.

3.3 THE PENTAGONAL AND DECAGONAL SYMMETRIES. - There is a regular polytope whose Petrie polygon has 5 vertices : the {3, 3, 3 } in 4D space (Fig. 4), which is called the . Unfortunately there is no regular honeycomb whose cells or vertex figure are only {3, 3, 3 } polytopes. Nevertheless, if we consider a crystalline 4D structure in which regular occur periodically, it will be a good candidate in order to give pentagonal quasicrystals. Consider once again the stacking of f.c.c. structures in a fourth dimension. But now we stack up a new f.c.c. structure with its sites over half of the tetrahedral interstices of the first f.c.c. structure, and so on. A honeycomb with two types of cells is obtained. Cells of the first type are simplices and those of the second type are semi- whose vertices are on the mid edges of a simplex. For the same reason which allows in 3D the two dense h.c.p. and the f.c.c. structures, in 4D there are several possibilities of stacking leading also to non- cubic structures, which are not considered in this paper, but which could lead to interesting 210

Fig. 4. - Schegel diagram of the {3, 3, 3 } simplex with the pentagonal Petrie polygon (with black edges). structures by the C.P. method. This honeycomb is also a lattice in a similar way to the 3D example where the f.c.c. lattice is a honeycomb formed by a packing of tetrahedra and octahedra. Notice that this lattice is a reticular 4D space in the cubic lattice Z5 which is used in the standard projection method for obtaining the . There is another simple related lattice defined by a rhombohedral unit cell. It is built by a basis of four vectors joining the centre of a 4D simplex to 4 of its 5 vertices. Adding these 4 vectors gives the opposite of the fifth vector joining the last vertex. So this lattice has the {3, 3, 3 } symmetry. The two lattices are related : the first one consists of a selection of vertices of the second one. The second lattice leads to a decagonal symmetry when the E plane is chosen as the plane of mid points of the edges of the Petrie polygon of the simplex defined by the five vectors (a 5 vector star). This is a consequence of the central symmetry of the lattice. The first lattice could lead to a pentagonal symmetry if the E plane is defined by the Petrie polygon of a simplicial cell.

4. Génération of the octagonal quasicrystal.

The octagonal tiling has already been the subject of many studies (see for example Refs. [5, 6] and has been invoked to describe the structure of CrNiSi alloys [7]. Nevertheless it is a very useful example to present this new method of construction.

4.1 THE PLANE OF PROJECTION E. - In the last section we have concluded that octagonal quasicrystal could result from the projection of a {4, 3, 3, 4 } . In order to define the plane on which we could project the structure we have to consider the Petrie polygon of the coordination shell, a {3, 3, 4 } cross polytope, whose vertices are first neighbours from the origin. It has 8 vertices which are gathered 4 by 4 in two planes Pi and P2. These two planes have only one point in common at the origin ; they are completely orthogonal. They are described in the appendix, in which we determine an « intermediate » plane E between the two planes Pl and P2. This plane E has only the origin in common with Pl and P2. We also show in the appendix how to find a matrix M which changes coordinates in the cubic basis into coordinates in a new basis (ql q2 q3 q4). The first two vectors of this basis characterize the plane E. The other two define an orthogonal plane E’. 211

This matrix could be written

with

Applying this matrix to the 8 vertices of the Petrie polygon and keeping only the first two coordinates we obtain a regular octagon. The two sets of 4 vertices in the two planes Pi and P2 give an octagon drawn by two rotated from ’TT /4. It is this matrix which is used to project the structure : it gives new coordinates, whose first two are coordinates in the plane E of the 2D structure.

4.2 BUILDING OF THE QUASIPERIODIC TILING.

4.2.1 Reticular planes in the 4D crystal. - The {4, 3, 3, 4 } honeycomb is a cubic lattice in R4. So we can apply simple crystallographic principles to it. The plane Pl which contains several vertices of the lattice, contains a whole 2D lattice which is a sublattice of the 4D lattice. The 4 vertices of the Petrie polygon in Pi form a square, the origin being the centre of this square. So the whole 2D lattice is simply a square lattice. Now consider a family of reticular planes parallel to the plane Pi. There are translations which change the plane Pl into another plane of the family. The plane P2 contains also a square lattice which defines these translations. It is possible to associate a of the plane P2 with each plane of the reticular family. Now we can use the formalism presented in section 2. The plane P2 becomes the base B and the reticular plane Pl is a fibre F. Then the lattice Z4 is : Z4 = F E9 B.

4.2.2 Acceptance domain. - Vertices of the 4D lattice, which are mapped on the quasiperiodic tiling, are selected by a strip S. The points in a fibre which are to be selected are those enclosed in an acceptance domain defined by the intersection of the strip with this fibre. The A.D. in F is the oblique projection of the Voronoi cell of the origin onto the F plane. The Voronoi cell is a hypercube of unit side. The projection on E’ is a regular octagon obtained from the common part of two squares relatively rotated from ir/4. (The square edge length is . 1 + B/2/2). The oblique projection on F is also a regular octagon but expended by a factor à ; so the edge length of the squares used to build it are 1 + à.

4.2.3 Mapping on the E plane. - Every lane of the reticular family projected on E gives a square lattice with an edge length 2/2. We call V {Q} the lattice projected from F { Q } , but the origin of these planes are translated by vectors of another square lattice of the same edge length which is rotated by ’TT /4. This lattice is the projection of the base B. As explained in section 2 the A.D. W { Q } in the fibre F { Q } is mapped onto Z {Q} lying on E. The set of Z {Q} forms a periodic array of overlapping as discussed below. Before projection on E, the selection of points consists in taking on each lattice F {Q} only vertices lying within an A. D. W {Q} . The centre of this A. D. is the point A which is common to the planes E and F {Q} . A generic point of the plane F {Q} is given by :

where (fl, f2 ) is the base for the lattice in F and (bl, b2 ) is the base for the lattice in B. If 212 a , {3, f and m are integers then ON is a point of the lattice in R4. If only f and m are integers then N is the current point into the plane F { Q } . Consider in E the two vectors qI and q2 with coordinate in R4 :

which are used to characterize the matrix M (see appendix). Any point C of E is given by : with coordinates in the basis (fl, f2, bl, b2 ) :

Now we write the condition for a point A to be simultaneously in E and in F {Q} :

which gives x and y when f and m are given (f and m determine which plane F {Q} is taken) :

We want to compare the vector OA with the translation OI of the origin of the lattice y0} (Flol mapped on E). The translation in B is expressed in the new basis (ql, q2, q3, q4 ) (see appendix) :

Then in E with the basis (ql, q2 ) it remains

which is just the half of OA, then OA = 2 01.

4.2.4 The algorithm for selecting vertices. - We consider in E a square lattice U, which is the projection of the lattice in B, whose parameter is a = B/2/2. We then consider the periodic array of octagonal A.D. which sits on another square lattice (F) with a parameter 2 a and with the same orientation. This lattice is the lattice of all A 213 points, centres of the acceptance domains. With each translation OP of U we associate a translation 2 OA of r. Surrounding all r vertices we reproduce the A.D. and so we have the periodic array of A.D. Z {Q} . Its in-circle has a (1 + ..Ji/2) or (Nf2 + 1 ) a. As this value in greater than 2 a, A.D. overlap when they are reproduced at each r vertices. The lattice U and the periodic array Z {Q} are shown in figure 5. Then consider a square lattice V {Q} with a parameter a, but rotated by ir /4 with respect to U. We put the origin of this lattice on a vertex P of U and select all its vertices falling within the A.D. associated with OA = 2 OP in r. These selected points are points of the quasiperiodic tiling. The whole quasiperiodic tiling is obtained by repeating this operation for all vertices in U. Figure 6 shows how some points of this tiling appear in the A.D. for an octagonal quasicrystal.

Fig. 5. Fig. 6.

Fig. 5. - Periodic array of octagonal acceptance domains on the physical space E. The U lattice is also drawn.

Fig. 6. - Some points of the octagonal tiling in their acceptance domain and the octagonal tiling.

In practice it is not necessary to test all the positions of the V lol vertices with respect to the acceptance domain Z {Q} . Indeed one can determine explicitly which point in V lol is closest to the A.D. centre (see chapter 7) and then test only a limited part of V {Q } centered on that point. In the present case this set contains 9 points only. This greatly reduces the number of computer steps involved in the tiling generation. More precisely this number of steps grows as the second power of the tiling linear size compared to the fourth power with the usual cut and projection algorithm.

5. Génération of the dodecagonal quasicrystal..

5.1 PROJECTION PLANE. - We have already justified the choice of the {3, 3, 4, 3 } 4D honeycomb. This honeycomb could also be described, using a primitive cell, as a crystal with one point on each node. But this primitive cell is not a cubic one. Nevertheless it is also 214 possible to have a cubic cell with several points in each cell. We use the cell derived from an f.c.c. cell and add one dimension. The four basis vectors of the unit cell expressed in the cubic basis are :

The coordination polytope is a {3, 4, 3} with 24 vertices. This polytope has a Petrie polygon with 12 vertices. These 12 vertices can be gathered 6 by 6 in two planes F and B. These two planes have one common point at the origin (centre of the coordination shell), and the 6 vertices form a regular . We consider an « intermediate » plane E between the two planes and then the Petrie polygon is projected onto it. A regular is obtained. In the appendix we show briefly how to determine the E plane, and how to obtain the matrix which changes the coordinates in the cubic basis into coordinates in a new orthogonal basis (qi, q2, q3, q4 ). The first two vectors of this basis characterize the plane E, the other two define an orthogonal plane E’. This matrix can be written :

Applying this matrix to the 12 vertices of the Petrie polygon and keeping only the first two coordinates we obtain a dodecagon. The two sets of 6 vertices in the two planes F and B give a dodecagon drawn by two rotated by ’TT /6. It is this matrix which is used to project the structure : it gives new coordinates, and the first two are coordinates in the plane E of the 2D structure.

5.2 BUILDING OF THE QUASIPERIODIC TILING.

5.2.1 Reticular planes in the 4D crystal. - The {3, 3, 4, 3} honeycomb is a lattice in R4. The plane F, which contains several vertices of the lattice, contains all a 2D lattice which is a sublattice of thé 4D lattice. The 6 vertices of the Petrie polygon in F form a hexagon the origin being the centre of this hexagon. So the whole 2D lattice is simply a hexagonal lattice. Consider now a family of reticular planes parallel to the plane F. There are translations which change the plane F into other planes of the family. They are in the plane B, and they also form a hexagonal lattice. All points of the quasiperiodic tiling are points of the lattice F mapped on E, or of other planes of the reticular family F lo l , but there are only a small number of points in each plane F {Q} which contribute to the quasiperiodic tiling.

5.2.2 Mapping on the E plane. - Every plane of the reticular family projected on E gives a hexagonal lattice with an edge length a = {(3 + J3)/6} 1/2. They are lattices V {Q} projected from F {Q} , but the origin of these planes are translated by vectors of another hexagonal lattice of the same edge length which is rotated by ’TT /6. This lattice is the projection of the plane B. 215

On each lattice F lol only vertices inside an A.D. are kept. The centre of this A.D. is the point A which is common to the planes E and F lol (Fig. 7).

Fig. 7. - Periodic array of dodecagonal acceptance domains.

With the same procedure used in the octagonal case it is possible to show that 01 = 2 OA. The Voronoi cells of the {3,3,4,3} honeycomb are {3,4,3}. It results that the acceptance domain in F and then in E are limited by a regular dodecagon. This dodecagon is on a circle of d = (1 + J3) a/2 where a is the parameter of the mapped lattice V{Q}.

5.2.3 The algorithm for selecting vertices. - We consider in E hexagonal lattice U, with parameter a. We then consider another hexagonal lattice (r) with parameter 2 a and the

Fig. 8. - A piece of the dodecagonal tiling.

JOURNAL DE PHYSIQUE. - T. 51, N’ 3, 1cr FÉVRIER 1990 216 same orientation. This lattice is the lattice of all A points, centres of the acceptance domains. Surrounding all r vertices we reproduce the A.D., and construct the Z lol array. A.D. overlap when they are reproduced at each r vertices (Fig. 7). Then consider a hexagonal lattice V { Q } with a parameter a, but rotated by ’TT’ /6 with respect to U. We put the origin of this lattice on a vertex P of U and select all its vertices falling within the A.D. associated with OA = 2 OP in Z {Q} . These selected points are points of the quasiperiodic tiling. The whole quasiperiodic tiling is obtained by repeating this operation for all vertices in U. Figure 8 shows this tiling with a choice of edges. In the appendix we describe the different types of tiles encountered in this tiling. Looking at a tiling obtained from the projection of a cubic structure in 4 or 5D, it is usual to see in perspective ; notice that in this case it is possible to see part of the octahedra in perspective, with triangular faces. They are projected from the surface selected in R4 by the strip. In this surface there are also parts of tetrahedra : in fact all four vertices of 3D- tetrahedra, but the surface is formed by two faces of each . Consequently there is an ambiguity on this choice, because it is also possible to consider on the surface the other two triangular faces, and then to have another choice for drawing edges in the quasiperiodic tiling.

6. Génération of the decagonal quasicrystal.

We use the rhombohedral lattice in R4 defined by the simplicial star. This lattice can be considered as the product of two 2D-lattices, the first one is defined by two of its basis vectors, the second by the other two. These four vectors expressed in an orthogonal basis are [5] :

The fifth vector in the simplicial star is :

It is easily shown that a is obtained by projection on the plane defined by the first two coordinates (which is the E plane) of these five vectors (vl, V2, V3, V4, v5 ). The method to obtain a quasicrystal is then the same as that presented for the octagonal and the dodecagonal cases. The two 2D-lattices F and B are mapped on E where they give two different rhombic lattices. The unit cell of the lattice V obtained by projection of the lattice F (basis Vl, V2) is a with an angle 2 ir 15 at the origin of the basis. The unit cell of the lattice U obtained by projection of the lattice B (basis v3 v4) is also a rhombus but with an angle ’TT /5 at the ongin. The lattice of acceptance domains is similar to the lattice V but expended by a factor 5. The acceptance domains is a non-regular deca on (Fig. 9) which could be obtained from a regular inscribed in a circle of radius 2 ( r + 2 )/5 by an affinity of a factor r for the first coordinate, and a factor T-2 for the second coordinate. Figure 10 shows how the 10-fold symmetry appears for the first selected vertices of the tiling. 217

Fig. 9. Fig. 10.

Fig. 9. - Periodic array of decagonal acceptance domains and the first points selected by these A.D. This shows how the local 10-fold symmetry appears.

Fig. 10. - A piece of the decagonal quasilattice.

7. Quasicrystais derived from periodic lattices. In section 2 we showed that the algorithm consists, in a first step, in the determination of which point in V { Q } is closest to the centre of the acceptance domain. If we stop at this step and consider the configuration of points which are then generated, we find a very intersecting new tiling, closely related to the final one. It is also quasiperiodic but it is now very easy to calculate a closed formula for the vertex coordinates. Furthermore it has a very natural underlying average periodic lattice (the centres of the A.D.). In the following we present the case related to the octogonal quasicrystal. Among the vertices of the square lattice V {Q} falling into an acceptance domain, we select only one point : the closest to the centre of the domain. It is clear that the lattice of acceptance domain is an average structure which may be called the labyrinth or the octagonal pivot. In figure 11 the labyrinth is shown : there are 3 kinds of tiles : square, and . This quasiperiodic structure is a subset of the octagonal quasicrystal, but the 8-fold symmetry has been broken into a 4-fold symmetry. It has very interesting geometrical properties which are presented elsewhere [8]. We calculate explicit coordinates for all its vertices. Consider a point 1 of the lattice U (the projection of the base) and the corresponding point A of the lattice of acceptance domain. They have coordinates (L, m ) and (2 l, 2 m ). Let T be the rotation matrix by ?r /4, which transforms the lattice U into the lattice V (the projection of a fibre)

The point 1 expressed in V is T-1. OI (we write as if the vector were a column matrix). In order to attach a V lattice to the point I, one should shift it by a vector s expressed in V by s = frac. (T- 1 - 01) where frac. (x ) is the fractional part of x. s is the vector OI modulo 218

Fig. 11. - The octagonal pivot with three types of tiles : a square, a kite and a trapezoid. translation of V. The point of the shifted V lattice closest to A is Int { T-1. (OA - s)} where Int (x ) is the closest integer to x. In the V basis the point is :

This can be written

This allows a very efficient generation of the octagonal pivot quasicrystal with a computer, but also greatly increases the efficiency of the algorithm used to obtain the octagonal quasicrystal. From the above expression one easily derives the average square lattice by replacing the « int » operator by the identity one. We have presented the example of the octagonal pivot 4-fold quasicrystal, but the derivation of new quasicrystals (with average lattices) derived from the dodecagonal and decagonal quasicrystals proceeds in a similar way.

8. Conclusion.

We have, by this method, an algorithm which generates quasicrystals directly in the physical space. This algorithm is clearly related to crystalline methods using lattices to describe structures. The existence of an underlying lattice (the array of acceptance domains) indicates that one can consider quasicrystals as crystals with an evolving motive in each unit cell. This motive has not a constant number of points in the generic case, and so does not correspond to what is usually called an average lattice. If we restrict to one point in each motive we get new quasiperiodic tilings of lower symmetry. This algorithm also presents some advantages for the computer generation of a quasicrystal. It is clearly more efficient than the classical cut and projection method. There are other 219 algorithms already well known using grid methods which are efficient. For instance the Amman procedure using non-periodic grid, or procedures using periodic N-grid. The comparison of the efficiency between these different procedures is still an open question. Note that the different methods are not fully equivalent, even in their generalized versions. Indeed the grid method only provides tilings with (or rhombohedra) while the C.P. method leads to point configurations which may or may not provide simple tilings. There are several extensions or applications of this new description. Phason fields which are related to a shift of the strip, in the cut and projection method, are for instance in the homogeneous case related to a simple displacement of the origin of A.D. periodic array. More generally it might be fruitful to discuss the relative role of phasons and phonons with respect to the splitting of the lattice as base and fibre. The diffraction patterns of these tilings are also obtained using a similar technique applied to the reciprocal space. The main difference is that in this case acceptance domains have to be replaced by their Fourier transform : in large, small acceptance domains defined in the reciprocal space will select Bragg reflection with high intensities. This method of description may also prove to be interesting for the analysis of excitation spectra in quasicrystals.

Appendix. A plane in R4 passing through the origin can be characterized by its intersection with a sphere centered at the origin. This sphere S3 is cut by a plane along a great circle. The position of this circle defines entirely the plane. Then we systematically use the fibration properties of S3 by great circles [9]. A space can be considered as a fibre bundle if there is a sub-space (the fibre) which can be reproduced by a displacement so that any point of the space is on a fibre and only one. For example the R3 can be considered as a fibre bundle of straight lines, all to the same plane. If fibres are 1-D lines, in a 3-D space, it is possible to determine a point on a fibre by one parameter, then there remain two parameters to characterise the fibre itself. So there is a 2-D space in which a point characterizes a fibre. This space is called the base. Successive toric layers appear naturally if S3 is described by using toroidal coordinates :

a is defined by a constant 4> parameter. Each of these tori could be considered as a 2-D space equivalent to a with opposite sides identified two by two (or a square in the particular case of the spherical torus). All diagonals of these have the same length and give a great circle of 83 after the identification of the sides which close this line. So it is possible to draw on a torus a whole family of great circles, which appear as parallel lines on the equivalent rectangle. All these great circles drawn on a whole family of toric layers defined by their two common axis form a fibre bundle. This is the Hopf fibration of S3. The base is a 2-sphere, but this sphere is not embedded in S3. If it were, it would have two common points with a fibre, as a circle cuts a sphere in two points. This is not possible since only one point on the base characterises a fibre. If a point on the base is defined by spherical coordinates e o and 03A6o, then the toric coordinates of points on the corresponding fibre are : 0 = w + 80 and çb,0/2. So a circle on the base, defined by a constant 00, represents a torus in S3. Consider a regular polytope whose vertices are on a sphere S3. It is possible to gather these 220 vertices into several sets of similar fibres. The 16 vertices of the {4, 3, 3 } cube are gathered into 4 fibres (fc; ) containing 4 vertices. All these fibres are on the same spherical torus so they could be drawn schematically on a square surface whose sides have to be identified. A fibre (f) on the spherical torus which is at equal distance of two fibres (fci fc2) containing cube vertices defines a plane which is the plane E that we consider for projection : the 4 vertices on the two close fibres are mapped on this plane as two squares rotated by Tr/4. It is possible to find two vectors from the origin to two points on f which are orthogonal, and then two other vectors completely orthogonal to this plane. This defines 4 vectors ql, q2, q3> q4° For instance

leading to the matrix M used for the octagonal quasicrystal. The {3, 4, 3 } polytope. In this case there are 4 fibres containing 6 vertices. The 4 fibres are defined by a regular tetrahedron on the base. The plane of projection is defined by a fibre which is at equal distance of two such hexagonal fibres. This leads to the matrix M used to build the dodecagonal quasicrystal. Projection of cube. Figure 12 shows the projection of a {4, 3, 3 } hypercube of the Z4 lattice in R4 on the plane defined above. This hypercube has a vertex at the origin. After mapping the faces are of two types : square or rhombus which are the tiles of the quasiperiodic structure.

Fig. 12. - Projection of a hypercube on the plane which shows the 8-fold symmetry. The two types of tiles of the octagonal tiling appear as projection of 2-faces. 221

Projection of the {3, 3, 4 } polytope. - Figure 13 shows the projection of a {3, 3, 4 } polytope which is the cell of the {3, 3, 4, 3 } honeycomb on the above defined plane. It has a vertex at the origin. After mapping, faces of the {3, 3, 4 } polytope are of three types : equilateral , isocel triangles with a small base, and isocel triangles with a large base. These are the tiles of the dodecagonal non-periodic structure presented here.

Fig. 13. - Projection of a {3, 3,4} polytope on a plane. For clarity of the figure some bonds from the external square vertices to other vertices are omitted.

References

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