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

Jean-François Sadoc, R. Mosseri

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

https://hal.archives-ouvertes.fr/jpa-00212361https://hal.archives-ouvertes.fr

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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 générationdirectement dans l’espace « physique » E de la structure. La sélection des sites dans l’espaceorthogonal est remplacée par un test directement dans une grille de domaines d’acceptance dansl’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 12qui 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 présentonsun exemple de symétrie 4. Les points de ce quasi-cristal sont un sous-ensemble des points duquasi-cristal ayant la symétrie complète d’ordre 8.

Abstract. 2014 We present a new algorithm for the generation of quasicrystalline structures. It isrelated 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 periodicarray of « acceptance » regions in E. This method shows that there is a sort of underlyingcrystalline 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 thisnew method we also generate quasicrystals with a lower symmetry which have simple meanlattices. We present for instance a quasicrystal with a 4-fold symmetry. The points of thisquasicrystal are a subset of the quasicrystal which has the whole 8-fold symmetry.

J. Phys. France 51 (1990) 205-221 1er FÉVRIER 1990,

Classification

Physics Abstracts61.40

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 directgeneration of the structure in the « physical » space E. The orthogonal space site selection isreplaced by a direct check in a periodic array of « acceptance » regions in E as will be explainbelow. This method, which is valid for projection from any dimension, is presented in the nextsection.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

http://www.edpsciences.orghttp://dx.doi.org/10.1051/jphys:01990005103020500

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given symmetry. If the symmetry is of order s the number of its prime integers smaller thans 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 thequasicrystalline symmetries which can be obtained by projection from 4D to 2D.

In the present paper we shall discuss mainly the octagonal and the dodecagonalquasicrystals 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 ofthe projection matrix. In the usual C.P. framework the determination of the physical space isdone by considering representations of the symmetry group operations in the n-dimensionalspace. Here we adopt a more geometrical approach by considering the symmetry of the Petriepolygon 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-dimensionallattice embedded in Rn. The « physical » space E, in which the quasiperiodic tiling is to begenerated, is a p-dimensional space. For simplicity we shall suppose that n = 2 p. Letf2 be a bounded volume in Rn- n is shifted along E, defining a « strip ». All vertices of Lwhich fall in the strip are selected and orthogonally mapped onto E, giving the vertices of thequasiperiodic tiling. For the « standard » tilings, 5li is the L unit cell or the Voronoi region. Inthe language of the 2D case, such quasiperiodic tilings are obtained by mapping on E amonolayer surface selected in the strip (this surface is tiled by square faces when the lattice Lis a cubic lattice). The selection algorithm usually proceeds as follows : the lattice vertices tobe 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’ of5li ., 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-thpower of the size which is inspected. The modified version runs at only the p-th power of thissize.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 respectively generated by {bl, ..., b.p} and {f1, ..., fp} . The simplest exampleis, if L is the hypercubic lattice Z", to take as B and F the lattices respectively generated by(ei , ... , ep} and {ep + 1, ..., en} where {ei} is the canonical basis in Rn. Now the site selectionis done fibre after fibre. One copy of F, called F lo l is attached.to each point Q of the base B

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

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

’

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

The Z {Q} form a periodic array of « acceptance domains » (A.D.), which is generally not atiling. Indeed neighbouring A.D. could .overlap in some cases. To each point of U oneattaches a copy of the lattice V {Q} and we select those points of V {Q} which fall inside theA.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 thecenter of the A.D. Z {O}. It is then sufficient to test only a small subset T ofV lo l surrounding this point. This set T is the maximal subset of V lo l which can fit into theA.D. All these steps will be illustrated below in 2D examples.

3. Lattices, honeycombs and polytopes in R4.

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

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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 Petrie polygon [3].The Petrie polygon of a polyhedron is a skew polygon such that every two consecutive sides,but not three, belong to a face of the polyhedron. This definition could be extended topolytopes of higher dimension : a Petrie polygon of an n-dimensional polytope is a skewpolygon such that any (n - 1 ) consecutive sides, but no n, belong to a Petrie polygon of a cell.The Petrie polygon of a regular polytope has a symmetry group which is the largest sub-groupof the symmetry group of the polytope [3]. In all cases two consecutive edges define entirely aunique Petrie polygon.

3.1 SEARCH FOR THE 4D STRUCTURE LEADING TO THE OCTAGONAL QUASICRYSTAL. - Weare looking for a regular octagon in E. So in R4 we search for an octagonal skew polygon. ThePetrie polygon of the 4D cube 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 honeycomb {4, 3, 3, 4 } in Schlâfli notation, is a structure whichis well characterized by the {4, 3, 3 } hypercube which is, in this case, both the intersticeconfiguration and also the Voronoi cell.

Consider a Petrie polygon of a given hypercube. It has 8 edges whose mid points are in asingle plane. It is clear that if we use this plane as the E plane, with a suitable acceptanceregion in E’ it will be possible to select vertices of the Petrie polygon, and to reject othervertices of the hypercube. By projection on E and E’ the Petrie polygon gives a regularoctagon. This justifies the choice of the 4D cubic lattice in order to derive the octagonalquasicrystal.There are three possibilities for choosing a local polytope characteristic of the local order in

Z4 : the cubic cell, the Voronoi cell of a vertex, or the coordination shell of the same vertex.We choose the Petrie polygon of the Voronoi cell because it defines a plane E containing thecentral 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 atthe centre of the octagon.

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

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Fig. 3. - Schegel diagram of the {3, 4, 3 } polytope with the decagonal Petrie polygon (with blackedges).

polyhedron (the vertex figure 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 bedescribed as a lattice : the Leech A4 lattice [4]. It is obtained by begining with a triangular 2Dlattice 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 octahedralinterstices ; it is the latter configuration which leads to the {3, 3, 4, 3 } honeycomb. Thislattice could be defined using a cubic crystallographic cell, but there are two possibilities. Thefirst one is built from the cubic cell of the f.c.c. 3D lattice by adding a fourth dimension toobtain a hypercubic cell ; then the lattice is a non-primitive 2-face centered lattice (type F incrystallographic 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 basisanother 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 Petriepolygon of a {3, 4, 3 } coordination polytope of the {3, 3, 4, 3 } honeycomb, a dodecagonalquasicrystal results from the cut and projection method.

3.3 THE PENTAGONAL AND DECAGONAL SYMMETRIES. - There is a regular polytope whosePetrie polygon has 5 vertices : the {3, 3, 3 } in 4D space (Fig. 4), which is called the simplex.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 regularsimplexes occur periodically, it will be a good candidate in order to give pentagonalquasicrystals.

Consider once again the stacking of f.c.c. structures in a fourth dimension. But now westack up a new f.c.c. structure with its sites over half of the tetrahedral interstices of the firstf.c.c. structure, and so on. A honeycomb with two types of cells is obtained. Cells of the firsttype are simplices and those of the second type are semi-regular polytopes whose vertices areon 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

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Fig. 4. - Schegel diagram of the {3, 3, 3 } simplex with the pentagonal Petrie polygon (with blackedges).

structures by the C.P. method. This honeycomb is also a lattice in a similar way to the 3Dexample where the f.c.c. lattice is a honeycomb formed by a packing of tetrahedra andoctahedra. Notice that this lattice is a reticular 4D space in the cubic lattice Z5 which is used inthe standard projection method for obtaining the Penrose tiling.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 4vectors 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 ofvertices of the second one. The second lattice leads to a decagonal symmetry when the Eplane is chosen as the plane of mid points of the edges of the Petrie polygon of the simplexdefined by the five vectors (a 5 vector star). This is a consequence of the central symmetry ofthe lattice. The first lattice could lead to a pentagonal symmetry if the E plane is defined bythe 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 veryuseful example to present this new method of construction.

4.1 THE PLANE OF PROJECTION E. - In the last section we have concluded that octagonalquasicrystal could result from the projection of a cubic honeycomb {4, 3, 3, 4 } . In order todefine the plane on which we could project the structure we have to consider the Petriepolygon of the coordination shell, a {3, 3, 4 } cross polytope, whose vertices are firstneighbours from the origin. It has 8 vertices which are gathered 4 by 4 in two planesPi and P2. These two planes have only one point in common at the origin ; they are completelyorthogonal. 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 withPl 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 basischaracterize the plane E. The other two define an orthogonal plane E’.

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This matrix could be written

with

Applying this matrix to the 8 vertices of the Petrie polygon and keeping only the first twocoordinates we obtain a regular octagon. The two sets of 4 vertices in the two planesPi and P2 give an octagon drawn by two squares rotated from ’TT /4. It is this matrix which isused to project the structure : it gives new coordinates, whose first two are coordinates in theplane 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 inR4. So we can apply simple crystallographic principles to it. The plane Pl which containsseveral vertices of the lattice, contains a whole 2D lattice which is a sublattice of the 4Dlattice. The 4 vertices of the Petrie polygon in Pi form a square, the origin being the centre ofthis square. So the whole 2D lattice is simply a square lattice. Now consider a family ofreticular planes parallel to the plane Pi. There are translations which change the planePl into another plane of the family. The plane P2 contains also a square lattice which definesthese translations. It is possible to associate a translation of the plane P2 with each plane of thereticular 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 thequasiperiodic tiling, are selected by a strip S. The points in a fibre which are to be selected arethose 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. TheVoronoi cell is a hypercube of unit side. The projection on E’ is a regular octagon obtainedfrom 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 factorà ; 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 asquare lattice with an edge length 2/2. We call V {Q} the lattice projected fromF { Q } , but the origin of these planes are translated by vectors of another square lattice of thesame edge length which is rotated by ’TT /4. This lattice is the projection of the base B. Asexplained 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 octagons as discussed below.

Before projection on E, the selection of points consists in taking on each latticeF {Q} only vertices lying within an A. D. W {Q} . The centre of this A. D. is the point A whichis 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

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a , {3, f and m are integers then ON is a point of the lattice in R4. If onlyf 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 planeF {Q} is taken) :

We want to compare the vector OA with the translation OI of the origin of the latticey0} (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 theprojection 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

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points, centres of the acceptance domains. With each translation OP of U we associate atranslation 2 OA of r. Surrounding all r vertices we reproduce the A.D. and so we have theperiodic array of A.D. Z {Q} . Its in-circle has a diameter (1 + ..Ji/2) or (Nf2 + 1 ) a. As thisvalue in greater than 2 a, A.D. overlap when they are reproduced at each r vertices. Thelattice U and the periodic array Z {Q} are shown in figure 5. Then consider a square latticeV {Q} with a parameter a, but rotated by ir /4 with respect to U. We put the origin of thislattice on a vertex P of U and select all its vertices falling within the A.D. associated withOA = 2 OP in r. These selected points are points of the quasiperiodic tiling. The wholequasiperiodic tiling is obtained by repeating this operation for all vertices in U. Figure 6 showshow 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 alsodrawn.

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 tothe acceptance domain Z {Q} . Indeed one can determine explicitly which point inV lol is closest to the A.D. centre (see chapter 7) and then test only a limited part ofV {Q } centered on that point. In the present case this set contains 9 points only. This greatlyreduces the number of computer steps involved in the tiling generation. More precisely thisnumber of steps grows as the second power of the tiling linear size compared to the fourthpower 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 } 4Dhoneycomb. This honeycomb could also be described, using a primitive cell, as a crystal withone point on each node. But this primitive cell is not a cubic one. Nevertheless it is also

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possible to have a cubic cell with several points in each cell. We use the cell derived from anf.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 Petriepolygon 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), andthe 6 vertices form a regular hexagon.We consider an « intermediate » plane E between the two planes and then the Petrie

polygon is projected onto it. A regular dodecagon 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 orthogonalbasis (qi, q2, q3, q4 ). The first two vectors of this basis characterize the plane E, the other twodefine 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 twocoordinates we obtain a dodecagon. The two sets of 6 vertices in the two planes F and B give adodecagon drawn by two hexagons rotated by ’TT /6. It is this matrix which is used to projectthe structure : it gives new coordinates, and the first two are coordinates in the plane E of the2D 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. Theplane F, which contains several vertices of the lattice, contains all a 2D lattice which is asublattice of thé 4D lattice. The 6 vertices of the Petrie polygon in F form a hexagon the originbeing 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 whichchange the plane F into other planes of the family. They are in the plane B, and they also forma hexagonal lattice.

All points of the quasiperiodic tiling are points of the lattice F mapped on E, or of otherplanes of the reticular family F lo l , but there are only a small number of points in each planeF {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 ahexagonal 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 anotherhexagonal lattice of the same edge length which is rotated by ’TT /6. This lattice is theprojection of the plane B.

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On each lattice F lol only vertices inside an A.D. are kept. The centre of this A.D. is thepoint 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 that01 = 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 ison a circle of radius d = (1 + J3) a/2 where a is the parameter of the mapped latticeV{Q}.

5.2.3 The algorithm for selecting vertices. - We consider in E hexagonal lattice U, withparameter 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

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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 hexagonallattice V { Q } with a parameter a, but rotated by ’TT’ /6 with respect to U. We put the origin ofthis lattice on a vertex P of U and select all its vertices falling within the A.D. associated withOA = 2 OP in Z {Q} . These selected points are points of the quasiperiodic tiling. The wholequasiperiodic 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 differenttypes 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 cubes in perspective ; notice that in this case it is possible to see part of the octahedra inperspective, with triangular faces. They are projected from the surface selected in R4 by thestrip. 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 tetrahedron. Consequently there isan ambiguity on this choice, because it is also possible to consider on the surface the other twotriangular 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 beconsidered 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 pentagon is obtained by projection on the plane defined by the firsttwo 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 twodifferent rhombic lattices.The unit cell of the lattice V obtained by projection of the lattice F (basis Vl, V2) is a

rhombus with an angle 2 ir 15 at the origin of the basis. The unit cell of the lattice U obtainedby 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 expendedby a factor 5. The acceptance domains is a non-regular deca on (Fig. 9) which could beobtained from a regular decagon inscribed in a circle of radius 2 ( r + 2 )/5 by an affinity of afactor r for the first coordinate, and a factor T-2 for the second coordinate. Figure 10 showshow the 10-fold symmetry appears for the first selected vertices of the tiling.

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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 ofwhich point in V { Q } is closest to the centre of the acceptance domain. If we stop at this stepand consider the configuration of points which are then generated, we find a very intersectingnew tiling, closely related to the final one. It is also quasiperiodic but it is now very easy tocalculate 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 thecase related to the octogonal quasicrystal. Among the vertices of the square latticeV {Q} falling into an acceptance domain, we select only one point : the closest to the centre ofthe domain. It is clear that the lattice of acceptance domain is an average structure which maybe called the labyrinth or the octagonal pivot. In figure 11 the labyrinth is shown : there are 3kinds of tiles : square, kite and trapezoid.

This quasiperiodic structure is a subset of the octagonal quasicrystal, but the 8-foldsymmetry has been broken into a 4-fold symmetry. It has very interesting geometricalproperties which are presented elsewhere [8]. We calculate explicit coordinates for all itsvertices.

Consider a point 1 of the lattice U (the projection of the base) and the corresponding pointA 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 (theprojection of a fibre)

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

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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)} whereInt (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, butalso 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 anddecagonal quasicrystals proceeds in a similar way.

8. Conclusion.

We have, by this method, an algorithm which generates quasicrystals directly in the physicalspace. This algorithm is clearly related to crystalline methods using lattices to describestructures. The existence of an underlying lattice (the array of acceptance domains) indicatesthat one can consider quasicrystals as crystals with an evolving motive in each unit cell. Thismotive has not a constant number of points in the generic case, and so does not correspond towhat is usually called an average lattice. If we restrict to one point in each motive we get newquasiperiodic 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

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algorithms already well known using grid methods which are efficient. For instance theAmman procedure using non-periodic grid, or procedures using periodic N-grid. Thecomparison 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 rhombuses (or rhombohedra) while theC.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 thehomogeneous 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 withrespect to the splitting of the lattice as base and fibre. The diffraction patterns of these tilingsare also obtained using a similar technique applied to the reciprocal space. The maindifference is that in this case acceptance domains have to be replaced by their Fouriertransform : in large, small acceptance domains defined in the reciprocal space will selectBragg reflection with high intensities. This method of description may also prove to beinteresting 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 spherecentered at the origin.

This sphere S3 is cut by a plane along a great circle. The position of this circle definesentirely 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. Forexample the Euclidean space R3 can be considered as a fibre bundle of straight lines, allperpendicular 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 oneparameter, then there remain two parameters to characterise the fibre itself. So there is a 2-Dspace 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 torus is defined by a constant 4> parameter.Each of these tori could be considered as a 2-D space equivalent to a rectangle with

opposite sides identified two by two (or a square in the particular case of the spherical torus).All diagonals of these rectangles have the same length and give a great circle of83 after the identification of the sides which close this line. So it is possible to draw on a torus awhole family of great circles, which appear as parallel lines on the equivalent rectangle. Allthese great circles drawn on a whole family of toric layers defined by their two common axisform 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 sinceonly 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 toriccoordinates of points on the corresponding fibre are : 0 = w + 80 and çb,0/2. So a circleon 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

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vertices into several sets of similar fibres. The 16 vertices of the {4, 3, 3 } cube are gatheredinto 4 fibres (fc; ) containing 4 vertices. All these fibres are on the same spherical torus so theycould 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 verticesdefines a plane which is the plane E that we consider for projection : the 4 vertices on the twoclose 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 vectorsql, 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 aredefined by a regular tetrahedron on the base. The plane of projection is defined by a fibrewhich 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 Z4lattice in R4 on the plane defined above. This hypercube has a vertex at the origin. Aftermapping the faces are of two types : square or rhombus which are the tiles of thequasiperiodic structure.

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

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Projection of the {3, 3, 4 } polytope. - Figure 13 shows the projection of a {3, 3, 4 } polytopewhich is the cell of the {3, 3, 4, 3 } honeycomb on the above defined plane. It has a vertex atthe origin. After mapping, faces of the {3, 3, 4 } polytope are of three types : equilateraltriangles, isocel triangles with a small base, and isocel triangles with a large base. These arethe 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 theexternal square vertices to other vertices are omitted.

References

[1] DUNEAU M., KATZ A., Phys. Rev. Lett. 54 (1985) 2688 ;ELSER V., Acta Cryst. A 42 (1985) 36 ;KALUGIN P. A., KITAEV A. Y. and LEVITOV L. C., J. Phys. Lett. 46 (1985) L601.

[2] See in Du cristal à l’amorphe, Ed. C. Godrèche (Editions Françaises de Physique) 1988,Introduction à la Quasi-Cristallographie by D. Gratias.

[3] COXETER H. S. M., Regular Polytopes Dover.[4] CONWAY J. H. and SLOANE N. J. A., Sphere packings, lattices and groups (N. Y. Springer) 1988.[5] BENKER F. P. M., Report S2-WSK-04 sept. 1982 Dept. of Math. University of Technology,

Eindoven.

[6] SOCOLAR J., Phys. Rev. B 39 (1989) 10519.[7] WANG N., CHEN F. H. and Kuo K. Y., Phys. Rev. Lett. 59 (1987) 1010.[8] SIRE C., MOSSERI R. and SADOC J. F., J. Phys. France (déc. 1989).[9] NICOLIS S., MOSSERI R. and SADOC J. F., J. Phys. France 49 (1988) 599.