Hierarchical Interlaced Networks of Disclination Lines in Non-Periodic Structures J.-F

Hierarchical interlaced networks of disclination lines in non-periodic structures J.-F. Sadoc, R. Mosseri

To cite this version:

J.-F. Sadoc, R. Mosseri. Hierarchical interlaced networks of disclination lines in non-periodic structures. Journal de Physique, 1985, 46 (11), pp.1809-1826. 10.1051/jphys:0198500460110180900. jpa- 00210132

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Classification Physics Abstracts 61.40D - 02.40

Hierarchical interlaced networks of disclination lines in non-periodic structures

J.-F. Sadoc and R. Mosseri (+) Laboratoire de Physique des Solides, Bât. 510, 91405 Orsay, France (+) Laboratoire de Physique des Solides, CNRS, 1, place A. Briand, 92190 Meudon-Bellevue, France

(Reçu le 3 mai 1985, accepté le 8 juillet 1985 )

Résumé. 2014 Nous décrivons l’ensemble des défauts dans une structure non cristalline déduite d’un polytope par une méthode itérative de décourbure. Les défauts apparaissent comme un ensemble hiérarchisé de réseaux de disinclinaisons entrelacés qui sont le lieu des sites où l’ordre local s’écarte de l’ordre icosaédrique parfait. La méthode itérative est décrite à 2D et 3D. Nous discutons aussi de l’utilité du concept de défaut hiérarchisé pour décrire la structure microscopique des quasi-cristaux icosaédriques.

Abstract. 2014 We describe the defect set in non-crystalline structures derived from polytopes by an iterative flattening method. Defects appear as a hierarchy of interlaced disclination networks which form the locus of sites where the local order deviates from a perfect icosahedral environment. The iterative procedure is fully described in 2D and 3D. We also discuss the usefulness of introducing the concept of hierarchical defect structure for the micro- scopic description of icosahedral quasicrystals.

1. Introduction. 3 coordinates are independent. The polytope model, or o Constant Curvature Idealization » (CCI) has Amorphous systems generally present an appreciable been extended to several other kinds of disordered amount of Short-Range Order (SRO). For example materials such as tetracoordinated covalent sys- amorphous metals can be well described by close tems [4]. A simple example is given by the packing of packing tetrahedra [1]. A regular tetrahedron is pentagonal dodecahedra which is forbidden in R3 the densest configuration for the packing of (as in the tetrahedral case) because of the polyhedron four equal spheres. The dense random pack- dihedral angle value. Packing these dodecahedra on ing of hard spheres problem can thus be mapped S3 leads to the regular polytope { 5, 3, 3 } which is on the tetrahedral packing problem. The dihedral dual of the above mentioned { 3, 3, 5 } and thus angle of a tetrahedron is not commensurable with 2 n, possesses the same symmetry group. However the consequently a perfect tiling of the Euclidean space values of the curvature associated with these two R3 is impossible with regular tetrahedra. Note that, polytopes are not identical (when scaled to the edge at this local level, the o frustration » (deviation to length) and this reflects the fact that the local angular perfectness) is of metrical rather than topological mismatch (in R3) are not equal. Polytope { 5, 3, 3 } nature. One of us (J.F.S.) has proposed to define an can be useful for modelling the « caged »-like tetra- ideal (unfrustrated) amorphous structure by allowing valent structures and those which are related like for curvature in the space in order for the local confi- amorphous ice for example. Several other CCI have ’ guration to propagate without defects throughout been proposed, like the regular honeycombs in the the whole space [2]. It is possible to pave a 3D manifold, 3D hyperbolic space H3 (with constant negative the hypersphere S3, by 600 regular tetrahedra arranged curvature) [2, 5] and the continuous double twisted by five around a common edge. The obtained geo- configuration of directors on S3 (as an ideal model metric object is called the polytope { 3, 3, 5 } using for the cholesteric blue phase) [6]. the standard notation [3]. Note that the underlying The idea is that the disordered material contains space S3 is 3 Dimensional although not Euclidean, « ordered >> regions where the local order can be put even if one often thinks of S3 as being imbedded in in correspondence with the ideal model, the comple- R4. Indeed S3 is the locus of points of R4 given by mentary regions being the locus of defects. We xi + x2 + x3 + x4 = R 2, which shows that only expect that a suitable map of the ideal model onto JOURNAL DE PHYSIQUE. - T. 46, ? 11, NOVEMBRE 1985 111

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198500460110180900 1810

R3 (minimizing the energy) will provide a realistic and its complexity is encoded in the information amorphous structure. The mapping introduces dis- content of the word. tortions and topological defects, among which the In a recent letter, hereafter referred to as 1 [14], we disclination lines play an important role. The final have shown how it is possible to bypass the above- structure consists in mixing of regions with positive mentioned difficulties and achieve the complete flat- curvature (where the local order is that of the poly- tening of the polytope. The key idea is, at each step, tope) and regions with negative curvature (the locus to introduce a disclination network (instead of a of defects), arranged such as to give zero curvature single disclination line) whose symmetry group is on the average. In this Corrugated Space Approach [7], contained in G. In the present paper we shall give a also called «variable curvature idealization », the complete description of how the method works. In local order is still perfect within coherence regions. order for this paper to be self-contained, we shall As has already been mentioned, disclination lines give a detailed description of the different tools which have proved to be natural defects for non-crystalline are needed (symmetry group, orthoscheme, bary- structure. S. N. Rivier [8] proposed that one type of centric transformation...). linear defects, characterized by oddness rather than In section 2, a simple 2-Dimensional example is intensity, is stable in real glasses as a result of the described, in which an icosidodecahedron is itera- double connectedness of the rotation group SO(3). tively flattened and gives rise to an asymptotic non- In the curved space approach, line defects can also periodic plane tiling. The main ideas that will be be classified using the homotopy theory of defects [9]. used later in the 3-Dimensional case are introduced Let us call Y the icosahedral group (subgroup of and visualized. SO(3)) and Y’ its lift in SU(2) (the covering group Section 3 contains the application of this method of SO(3)). The full symmetry group G* of polytope to the polytope { 3, 3, 5 } case. At each step a « defect » {3,3,5}, with 14 400 elements, is described in subnetwork is generated which has the same symme- Appendix B. G, the subgroup of G* containing only try properties as the structure itself. Also the matricial the direct symmetry operations (preserving orien- description of the iterative procedure is introduced. tation), is given by : G = Y’ x Y’/C2. C2 is the two- We show how to generate more disorder by combining element group. As shown by Nelson and Widom [10] two different iterative procedures which are compa- the defect lines that can be generated in the polytope tible since their associated defect structure share the { 3, 3, 5 } belong to the conjugacy classes of R = same symmetry group. In the last section (4) we ni (SO(4)/G) = Y’ x Y’. More recently Trebin [11] describe miscellaneous aspects and extensions of the has proposed a « coarser » classification for the line model. defects in polytope { 3, 3, 5 } by considering the first homology group of the order parameter space. Since 2. A 2D example : iterative flattening of an icosi- Y’ is a « perfect group » (it is isomorphic to its own dodecahedron. commutator subgroup), the homology group is trivial The icosidodecahedron (Fig. 1) is a quasi-regular which expresses the fact that any line can be transformed into any other by a suitable combination process. or Archimedean polyhedron [15]. It is noted 3 [3] One might hope to generate increasing numbers 5 of disclination lines in the { 3, 3, 5 } in order to achieve and shares the same symmetry group with the icosahedron and the dodecahedron a complete flattening of the polytope. We have { 3, 5 } { 5, 3 }. already shown [12] that it is possible to interlace two such disclination lines and get a polytope containing 144 Z 12 vertices and 24 Z 14 vertices with less intrinsic curvature. We use standard notations [13] to label the sites according to their coordination number. In order to annul the curvature, one should iterate this procedure and incorporate the disclination lines step by step. There are up to now unsolved difficulties in doing this which are probably due to the non- commutative character of the required operations (R is non-Abelian). On the other hand, this non- Abelian character is the key to understand why it is possible to model very complex disordered structures starting from a regular polytope and using only a finite collection of defect types. An « alphabet >> can be defined whose elements, the «letters », denote each type of defect (the conjugacy classes of R). A structural model, based on the polytope, is then represented by a « word », an ordered set of letters, Fig. 1. - The icosidodecahedron 1811

Indeed it can be obtained by joining the mid-points sphere of polytope { 3, 3, 5 } (see next section). The of the edges of { 3, 5 } or { 5, 3 }. Each vertex belongs vertex coordinates are given by the 30 quatemions to two regular pentagons and two equilateral triangles. in Y’ whose scalar part vanishes. Indeed to a unit The Iterative Flattening Method (I.F.M.) is much pure quatemion q = a, i + a2 j + a3 k (see Appen- more easily visualized in two dimensions than dix A) there corresponds a point of coordinates in three and almost all of the ideas that we present (a1’ a2’ a3) on the sphere S2 of unit radius. The orthoscheme vertices are vertices here can be simply generalized to one higher dimen- three Mo, M1,I M2 sion. As will be seen below, the I.F.M. can be under- of respectively the {{3,5 3, 5 } 5{ 5,3 3 } which stood as an « inflation >> or as a « deflation >> method. 3{ 5 and The second one is to visualize some sense very easy (in share the symmetry group Y*. Indeed it can be veri- it does not need a coordinate We shall system). fied that the orbit of Mo (resp. M1, M2) generates a however insist on the inflation method since it is the one which allows to focus on the relations p a 5,{ under successive symmetry 3{ 3,5 5 resp. a 35 , 3 between vertices and on the curvature aspect. ( ( ) reflections in the sides of the spherical triangle Z .1 GEOMETRICAL PROCEDURE. - In appendix B we Mo M1 M2. We shall consider that the polyhedra introduce the binary icosahedral group Y’ [16] which under construction are either spherical, with bent is the lift in SU(2) of the icosahedral group Y (the faces and geodesic edges, or Euclidean, with flat covering map SU(2) -+ SO(3) is carried on the faces and straight edges (chords). They are trivially discret polyhedral subgroups of SO(3)). Y is a pure related, the spherical polyhedron being the central subgroup of Y*, the full icosahedral group (including projection of the Euclidean one OIrto-tl1e-surfiice of indirect transformations) also called the triangle the sphere S2. group (2, 3, 5)* [17]. This group of order 120 allows Let us now describe the method. In a first step the for a division of S2 in a pattern of 120 spherical original polyhedron is constructed. It will be called triangles, each of one being a fundamental region of the «source polyhedron Po. It is characterized by the sphere tesselation (Fig. 2). Any vertex configura- the orthoscheme MoMiM2 and by the location of a tion on S2 having Y* as symmetry group is comple- point M (or several points) in it. Po vertices are the tely defined by giving one fundamental region (the orbit of M under the Y* symmetries. The second step orthoscheme, Appendix C), and the distribution of consists in selecting a new triangle MoM1M2 which vertices inside it. It is analogous to the description shares an angle with the orthoscheme MoMiM2 (here of crystals by unit cells and the translation part of the angle at vertex M1, see Fig. 3). The triangle the symmetry group. In the case of spherical tessela- MoM1M2 is chosen as to contain an integral number tions, the symmetry operations are reflections in the of orthoscheme replicas. Thus it contains several Po sides of the fundamental triangle. The pure rotations vertices. Note that the two spherical triangles belonging to the direct group Y are products of even MOMlM2 and MoMiM2 do not have all their cor- number of such reflections. responding angles identical since they have different In the following we take as a basic icosidodeca- areas (recall that the area of a spherical triangle is proportional to the sum of its interior angles minus n). hedron which tiles the equatorial >> the 53

Fig. 3. - The two spherical triangles MOMIM2 and Fig. 2. - Partition of S2 into 120 spherical triangles under MoMlM2 which share the vertex Mi. Marks label vertices the action of the full icosahedral group Y*. Elements of the on S2 where the symmetry is either 5-fold, 2-fold and 3-fold. pure subgroup Y interchange triangles of the same colour The 12 vertices (resp. 30, 20) which are the orbit of Mo (either white or shaded). A particular fundamental region (resp. M1, M2) are vertices of an icosahedron (resp. icosido- (M6bius triangle) is distinguished. decahedron, dodecahedron). 1812

The crucial point consists now in the identification of the two triangles. In flat space this could be easily done by homothety when the triangles are similar. In the curved space S2, homothety is not easy to define because of the presence of an internal length scale (the radius of curvature). It is possible to define homotheties along the geodesic lines M1M0 and MiM2 which makes the points Mo and Mo, Mz and M2 coincide. But to ensure that all the points of the geodesic line M’M’ will lie on the geodesic line MOM2 requires a continuous set of homotheties. Instead there is a very simple and natural way to do this if we work with an Euclidean polyhedron (with flat faces). Now the flat triangles MoMiM2 and MoMlM2 are not coplanar (Mo does not belong to the chord more a the is no - MiMo) and underlying geometry Fig. 4. The two basic tetrahedra OMoMIM2 and simple 2D geometry but a collection of bounded 2D OM’M,M’. 0 is the centre of the sphere S2. The points flat regions (the faces) glued along the edges. It is Mo, Mo, M1, M2 and M2 have the same location than in then easier to consider the homogeneous space in figure 3. But now the edges are straight lines (chords) instead which the polyhedron is embedded, the 3D Euclidean of geodesic curves. space E3. Now any point in E3 can be specified in a barycentric coordinate system once a particular tetrahedron is given. Let us for instance take the tetrahedron with the flat triangle MoMl M2 as its base and the centre 0 of the sphere as its apex. The four barycentric coordinates corresponding to the tetrahedron OMoMiM2 are noted (Sl, ao, al, a2) (with the conditions + ao + ai + a2 = 1). A point belongs to the 3D sector bounded by the planes OMOMI, OMOM2, OMlM2 if and only if its last three coordinates ao, al, a2 are simultaneously positive or zero. The sign of Q says whether the point is in the same side of MoMiM2 as the point 0 (Q > 0) or in the opposite side (Q 0). The identification of the two triangles proceeds as follows : First, we calculate the coordinates (Q’, ao, ai, a2) of Po vertices in the barycentric system based on the tetrahedron OM’M,M’. Then, we only keep those vertices which have ao, ai and a2 simultaneously positive or zero. Finally we re-interpret those coordinates (Q’, ao, ai, a2) as being the coordinates (Q, ao, ai, a2) in the barycentric system based on the tetrahedron OMoMIM2. The two basic tetrahedra are drawn in figure 4. The new points do not necessarily lie on the flat triangle MoMiM2 but they can be projected on it by central projection. In terms of barycentric coordinates it consists in putting equal to zero and rescaling ao, al and a2 (to insure that their sum equals unity). Note that the points can also be projected onto S2 and then belong to the spherical triangle MOMlM2’ The first iteration is almost finished, the first iterated polyhedron P1 is obtained as the orbit under the group Y* operations of the vertices lying in the orthoscheme It is in 5a. The MOMlM2’ represented figure Fig. 5. - a) The polyhedron Pi (centrally) projected onto local order in similarities with that P1 presents of Po. the sphere S2. b) Tentative drawing of the underlying cor- Its description in terms of order and defects will rugated geometry of Pi. The radius of curvature (scaled to be done in the next paragraph. Let us just say that the edge-length) near Mi and its replicas should be similar the configuration around the point M 1 is identical to that of Po. 1813

to that of Po (M 1 belongs to 2 pentagons and 2 trian- where ( K, is the mean Gaussian curvature per gles). In terms of the edge length, the radius R1 of unit area (R is the radius of curvature) and n the the P1 circumsphere is much larger than Ro, the number of disclination points (each assumed to carry radius of the Po circumsphere. In some sense the the same angular deficit 6) per unit area. In the present ratio R1 I Ro is characteristic of the «bary centric case where the ordered state is defined to be the homothety ». It is possible to recover some regularity polyhedron Po, the defect intensity is measured with for the polygons around M1 by supposing that P1 respect to the value of £ 0; at Po vertices (instead vertices lie on the surface of a sphere of a radius R1 of 2 x). I covered by small domes of radius Ro which centres The I.F.M. in its inflation >> form yields a very are located somewhere on the radii connecting 0 simple way to locate and measure the defects. Indeed with M1 and its replica. This is tentatively drawn in a disclination point is generated at each vertex of the figure 5b where the whole surface has been smoothed. orthoscheme MOMlM2 (and at all its replicas under It is very easy to iterate again the procedure, build the Y* symmetries) whenever the angle is different a new polyhedron P2 by keeping the P1 vertices from that of the corresponding triangle M0M1M2. lying inside M0M1M2, identifying MoMiM2 and Let us consider the polyhedra as being spherical. The MoMiM2 and then getting P2 with the help of Y* internal angles are evidently symmetries. As long as the two triangles keep specific relations (they have an angle in common and the big one contains an integral number of replicas of the first one), the successive polyhedron Pn will present very interesting properties. For example, as In Po the point M2 is surrounded by a pentagon. At seen below, their defect set presents the same kind of the first iteration and after identification of the two regularity as the polyhedra themselves. If the same triangles MOMlM2 and M0M1M2, the point M2 will triangle M0M1M2 is used at each iteration, the be surrounded in P1 by an hexagon (6 = 5 x eM2/eM2). polyhedra Pn belong to the family of deterministic Sirnilarily, the point Mo in Po is surrounded by a recurrent sets [18]. From a topological point of view triangle. Thus the point Mo in P1 is surrounded by the the same = asymptotic polyhedron P 00 presents a pentagon (5 3 x 8Mo/9Mo). Note the important kind of non-crystallinity as the Penrose tiling [19]. following remark : in the polyhedron Po, the point Mo A to build iterative is simple algorithm polyhedra was already surrounded by a pentagon, which was in D. presented Appendix not a defect polygon. At the first iteration this pentagon has been « pushed >> toward the point M,. The pen- 2.2 THE DEFECT SET. - By definition the source tagon which now surrounds the point Mo can be polyhedron Po is called the «ordered (defect free) considered as a defect since it is a disclinated form configuration ». It can be obtained by the local of the triangle which surrounded Mo before the first building rule : « put two pentagons and two triangles iteration (in Po). around each vertex ». A specific value for the curva- Let us call Dp the defect set of a polyhedron Pp. ture of the underlying 2D manifold is necessarily Dp is the union of several subsets, each of one asso- associated with such a rule. This curvature takes a ciated with a particular kind of disclination, and all constant value at each site and is related to the so- sharing the symmetry group Y*. For instance D1 called deficit angle bM at the vertex M : contains two subsets D1 and D" 1 - D 1 is the set formed by point Mo and all its replicas under Y*. This point set forms an icosahedral pattern. Each point is surrounded by a pentagon

in I 0 are the internal of the flat P1. angles polygons sharing - is the set formed by point and all its site M. Different relations between local configura- D1 M2 under Y*. This set forms a dodeca- tions and curvature has been derived in connection replicas point hedral Each is surrounded an with the to disordered pattern. point by Corrugated Space Approach hexagon. structures [7]. In 2-Dimensions, owing to the existence of two famous relations, the Euler-Poincare and the Both D’ and D1 are represented in figure 6. In the next it is clear that the Gauss-Bonnet relations, simple and exact correspon- iteration disclination in will as defects. However dence can be found between the geometry of the points D’ and D1 persist underlying manifold and the defect density. Here they will be located at different places in the orthoscheme and the number of after defects are disclination points whose weight is related points replication to the values of polygon angles at the vertices. For will be different. It will happen that two points in which to two different fundamental example, when flat space is taken as the ordered D’ belonged in will be in the same So (undefected) state, one has the exact relation triangles Pp triangle Pp+1. the number of defects will increase indefinitely with the iterations. Also each iteration generates two new 1814

ed by p-gons such that

where c is the vertex coordination in the polyhedron Pp. These polygons act as neutral charges [7] in that sense that their underlying manifold is flat. The Pp polyhedra can be called pseudo-crystalline since they contain larger and larger regions covered by portions of 2D flat crystals. A simple example is given by geodesic domes. In that case it becomes useless to define Po as the ordered configuration.

3. Three-dimensional case : iterative flattening of the polytope { 3, 3, 5 }.

6. - The its defect set Fig. polyhedron P and D 1. D is In this section we show how the method described the union of two subsets D’ and D1 which are marked dif- above can be generalized to one higher dimension. ferently on the figure. A black q-gon inside a p-gon indicates It is to 3, 3, whose that this p-gon is a disclinated q-gon. applied polytope { 5 } symmetry group is presented in Appendix B. Let us first give a more geometrical description of this polytope. It contains 120 vertices which are 12-fold coordinated : each vertex sits (on S3) at the centre of a perfect icosahedron. A good way to represent in flat space some geometrical unit belonging to curved (hyper) spheres consists in making cuts parallel to a tangent flat space (Fig. 8). The intersection of polytope { 3, 3, 5 } by hyperplanes gives the following successive polyhedra : an icosahedron (the first neighbour shell of a vertex), a dodecahedron, a larger icosahedron and an icosidodecahedron on the equatorial)) sphere. Here the polytope has been oriented in such a way that .one vertex is on the arbitrarily defined « north pole » (xl = 1, 0, 0, 0) of the unit radius hypersphere S3 and the hyperplanes are taken to be « horizontal » (orthogonal to the « vertical >> first coordinate axis). Note that this is slightly different from what was considered in 1 (Ref [14]) where the Fig. 7. - Local view of the polyhedron P 2 and its local north pole corresponded to the vertex (0, 0, 0, 1). In set The which limits the defect D2. large pentagon figure our new notation the north pole coincides with the is a of the icosidodecahedron pentagon original Po. identity quaternion in the group Q (Appendix A).

3 .1 GEOMETRICAL PROCEDURE. - As has already been said, the method works in a way very similar to the defect subsets similar to D 1 and D1 . Figure 7 shows 2D case. G, the total symmetry group of polytope a limited region of P2 (mapped onto the plane) with { 3, 3, 5 } (Appendix B), generates a regular division the different defect points. Note that decagons are of S3 into 14 400 spherical tetrahedra, a single one generated in P2 which are disclinated hexagons of P1.* being called an orthoscheme (or fundamental tetra- Indeed the angle at Mo equals yc/5 and repeated hedron). A tesselation is uniquely defined by the reflections in the lines through Mo of a generic point location of vertices inside an orthoscheme, the other give rise to a decagon. The pentagon generated in vertices being generated by reflection in the faces of P1 came from the special location of a point on an the orthoscheme. Figure 9 represents one tetrahedral edge. At each iteration a new type of (larger) polygon cell of polytope { 3, 3, 5 } and one orthoscheme will be generated in addition to the whole set of inside it. The four vertices of the latter are located previously generated (smaller) polygons. The fact on one cell vertex, the centre of the cell, the centre that the number of types of polygons increases upon of a face and of an edge. Each { 3, 3, 5 } tetrahedral iteration is specific to this 2D example. This will not cell contains 24 copies of the orthoscheme. The occur in the 3D examples described in the next chapter. coordinates of the 120 vertices of the { 3, 3, 5 } are It may happen (due to the choice of the triangle given by the 120 quaternions of Y’. One particular MoM 1 M2) that most of the defect points are surround- tetrahedral cell, with the north pole as apex, has 1815

Fig. 8. - « Horizontal » sections of the polytope { 3, 3, 5 1. The section of S2 by a hyperplan is a sphere. With one { 3, 3, 5 } vertex at the « north » pole, the successive section are : b) an icosahedron, c) a dodecahedron, d) an icosahedron, e) an icosidodecahedron ; the figures are reproduced from Pour la Science, janvier 1985.

the four points ABCD as vertices with coordinates : (1, 0, 0, 0), §(1, 0, 1, T-l), i(T9 T- 19 0, 1), t(1", 19 T- 19 0)

T = (1 + /)/2 is the golden ratio. Now a particular orthoscheme inside it has as four vertices Mo, M1, M2, M3. Mo coincides with the north pole A. M1 is located at the middle of the edge AB, M2 is the centre of the face ABC and M3 is the centre of the { 3, 3, 5 } tetrahedral cell ABCD (Fig. 9). Up to this point we consider the polytope as being spherical. The characteristic simplex Mo, M1, M2, M3 is thus a quadrirectangular spherical tetrahedron out from a Fig. 9. - A tetrahedral cell of the polytope { 3, 3, 5 } with (cut hypersphere S3 by four hyper- see Ref. That is to one particular orthoscheme inside it. The four vertices of planes ; [3], p. 139). say, points an orthoscheme are a polytope vertex and an edge, face and M1, M2, M3 have been radially projected onto the cell centres. unit radius hypersphere S3. The dihedral angles at 1816 the edges M2M3, MoM3 and MOM, are respectively group). One element is located at Mo and has 120 replicas. A second one is located at M3 and has 600 replicas. The third one is located somewhere on the edge MoM1 the remaining three being right angles. and has 1 440 replicas. All these add up to the 2 160 vertices of The local around each Once the orthoscheme is defined, it is very easy P1. configurations vertex are not all the same is not a to generate any tesselation which shares with the (P1 regular poly- However strong orientational order persists in { 3, 3, 5 } the same symmetry group G (Appendix B). tope). which will be discussed in the next section about It is enough to give the location of vertices inside the P1, the defect set. orthoscheme and then to apply the 14 400 symmetry It is now easy to iterate the We operations. If the vertex has a generic position, one again procedure. ends with 14 400 replicas (including the original consider the new set tÂt1 consisting of points of P 1 to the tetrahedron vertex under the identity operation). If however the belonging large M o M 1 M 1 Mo. Using the same barycentric as in the first vertex has a less general position, its orbit may homothety the 14 elements of are the contain less points. For instance the image of A = Mo iteration, A, mapped into fundamental A new gives rise to the 120 vertices of polytope { 3, 3, 5 }, region MoMiM2M3. polytope P2 is then as the orbit of under and while the image of M3 gives rise to the 600 vertices generated X 1 G, contains 42 The iteration can on of the dual polytope { 5, 3, 3 }. 480 vertices. proceed and larger and larger polytopes (in term of their num- Now we can begin the first iteration. We define a ber of are obtained. larger tetrahedron which shares the vertex Mo with vertices) the orthoscheme and contains an integer number of 3.2 DESCRIPTION OF THE DEFECT SET. - The formal orthoscheme replicas. Let us still consider the tetra- definition of the defect set is very similar to the 2D hedra as being spherical. The three new vertices case while its precise geometry is much more intricate. M 1, M 1, M’ belong to the same great circles as the The starting polytope Po (the { 3, 3, 5 }) is the defect geodesic edges MoM!, MoM2 and MoM3. Point Mo free configuration. Defects (disclination lines) are is located at a vertex of the { 3, 3, 5 } which belongs generated along the edges of the fundamental region to a dodecahedral second neighbour shell surrounding whenever the dihedral angle differ from that of the the north pole (Fig. 8). We have seen that point M3 larger tetrahedron MoMiMiMo. is a vertex of the dual polytope { 5, 3, 3 } which has Figure 10 represents the two tetrahedra MoMiM2M3 dodecahedral cells, one of which also surrounding and MOM? M[ Mi and their relation with four tetrahe- the north pole. Thus we can anticipate that the dral cells of polytope { 3, 3, 5 }. Recall that the index i identification of M3 and Mo is equivalent to the in Mi (or MD denotes the type of site : i = 0 for a identification of the large dodecahedron (second { 3, 3, 5 } vertex, i = 1 for a mid-edge { 3, 3, 5 } vertex, neighbour shell in the { 3, 3, 5 }) lying in the hyper- i = 2 for the centre of a triangular face and i = 3 for plane xl = 0.5 with a smaller dodecahedral cell of the centre of a { 3, 3, 5 } tetrahedral cell (e.g. a vertex polytope { 5, 3, 3 } lying in the hyperplane of polytope { 5, 3, 3 }). The edges MoMI, MIM3 and M2M3 belongs to respectively 5-fold, 2-fold and 3-fold rotation axes. But since the basic symmetry operations In order to overcome the difficulties involved in are reflections in the orthoscheme faces, the number the identification procedure in curved space, as in of images of a generic point is usually twice as large the 2D case, we consider again the Euclidean version as the order of the rotation axis. For example a trian- of the polytopes, with chords as edges and flat (hyper) gular face is threaded orthogonally by a 3-fold axis. faces. We compute the coordinates of the { 3, 3, 5 } A generic point on this face has six images on this vertices in the barycentric system based on the triangle (including the original point under the iden- hypertetrahedron OMo M i M 1 Mo with coordinates tity operation). Another way to see it is to remark (01 ao, ai, a2, a3). 0 is the centre of the hypersphere S3. that the orthoscheme dihedral angles are equal to z) Then only the vertices whose last four coordinates (and not 2 7c) over the order of the rotation axis to are simultaneously positive are kept. Let us call Ko which the associated orthoscheme edge belongs. the set of such points. Their barycentric coordinates By inspection of figure 10, it becomes clear that only are now interpreted as being based on the hyperte- dihedral angles at edges M’M", and M3M2 are diffe- trahedron OMoMIM2M3, that is to say the elements rent upon identification of the two tetrahedra, the of flo lie now in the small tetrahedron MoMiM2M3. respective angles being 2 n/5 and 7r/3. Thus in polytope But this tetrahedron is the fundamental region of P1 the defect lines will be carried by the edge M3M2 group G. It is then easy to generate a new polytope and its replicas under symmetry group operations. P1 as the orbit under G of Ao. More precisely JLo These edges connect the centres of adjacent { 3, 3, 5 } contains three points and the first iterated polytope tetrahedral cells (sharing a face). Consequently they P1 has 2 160 vertices. It is rather pedagogical to see are edges of the dual polytope { 5, 3, 3 }. In the follow- where these 2 160 come from. None of the 3 points ing, we call Qi the dual of polytope Pi (and so Qo is in To is a generic point of the orthoscheme. Indeed a the polytope { 5, 3, 3 }). The Qi are easily constructed generic point has 14 400 replicas (the order of the by joining the centres of the Pi tetrahedral cells. 1817

Fig.1 Oa. - The two basic tetrahedra of the iteration process. The figure shows 4 tetrahedral cells of the { 3, 3, 5 } (heavy lines). The orthoscheme MOMIM2MI is represented as in figure 9. The large tetrahedron MOM’M’M", contains 20 orthoscheme replicas. Note that the index i in M, or M’ labels the sites according to which orbit in the symmetry group they belong. For instance M 1 and M 1 are images Fig. lOb. - The 4 triangular faces of the large tetrahedron of M under given symmetry operations. They are all located M0M0M1M1. The edges and vertices of the orthoscheme at the middle of { 3, 3, 5 } edges. replicas are shown.

As said before, at the first iteration the orthoscheme contains three vertices. Upon the identification procedure, only the local order around the vertex at M3 is altered. Using standard notations [13] it becomes a Z16 vertex while the two other elements of Ao (and all their replicas) remain Z12 vertices. Coordination shells of Z12 and Z16 vertices are shown in figure 11. The surrounding of a Z 14 site is also displayed because Z14 sites will appear in the next iteration. This notation indexes a vertex according to its coordination number. In term of line defects, a Z12 site is a defect- free site. Its first coordination shell is an icosahedron and its Voronoi cell a dodecahedron. A Z14 site is threaded by one disclination line along a 5-fold axis which transforms it into a 6-fold axis. A Z 16 site is at the

intersection of four « half » disclination lines which Fig. 11. - Coordination shells of Z 12(a), Z I 4(b) and Z 16 form a tetrahedral configuration (like the directions sites. Sites lying on disclinations are darkened. of four sp3 hybridized bonds in diamond for example). In the present case, the Z16 site at M3, the four half defect lines are collinear with the edge M2M3 and its it has a larger scale with respect to the first-neighbour three other replicas which intersect at M3. distance. Indeed all the are constructed on a unit Let us call the set of all the defect lines in Pi Di poly- radius The intrinsic of the We have seen that consists in the hypersphere. length physical tope Pi. D1 edges network is the distance between of polytope Qo (the { 5, 3, 3 }). The second iteration first-neighbour sites, thus a of scale has to be done when comparing generates polytope P2 with its associated defect set change two different So has a larger edge length com- D2. D2 contains two disjoint parts : Pi. D2 pared to D1. In particular two new Z14 sites are - The first one, D2, is introduced at the second located between two nodes of D2. iteration as a result of the identification procedure. - The second subnetwork, D", is the image of D 1 Consequently it has the same geometry as Di. But under the second iteration. Since D1 equals Qo, D2 1818 has the geometry of Q 1 (the dual of P 1 ). The intemode separation in D" is equal tp the first-neighbour distance in P2, up to some fluctuations associated to small distortions during the identification procedure. So D2 is the union of two interlaced disclination networks, D’ and D", which have different characteristic length scales with respect to their nearest node separation. The local arrangement of DZ and D2 is tentatively represented in figure 12. When the transformation is iterated again, larger polytopes Pn are generated. Their defect set Dn contains n interlaced disclination networks and can be written

12. - Local view of the disclination network. where 0153 denotes the union of disjoint sets. Dn has a Fig. Heavy hierarchical structure : the intemode separation in line : D2 disclinations (which the same topological structure as line : disclination interlaced with each term of the distance in D1). Light D2 the Qi (in first-neighbour Pn) network. varies with i. The change of scale between Qi and previous Qi + 1 is of the order of the « barycentric homothety » described in the previous paragraph. 3. 3 DEFLATION APPROACH AND MATRIX FORMULATION. which belonged to polytope Pi -1’ is replaced by a - Table I displays information about polytopes Pn collection of new vertices. In the present case the up to n = 5. These quantities can be obtained for Voronoi cell of a Z12 vertex is filled by 13 new Z12 small n by the direct « inflation » method described vertices and 20 new Z16 vertices which are located as above which uses the symmetry group operations in follows : the Z12 Voronoi cell is a pentagonal dode- order to build structures of increasing sizes. But the cahedron. Its 20 vertices are occupied by Z16 sites. data of table I can be derived more simply without The partition of S3 by the Voronoi cells is such that explicitly building the polytopes. The main effect of each vertex on these cells belong to four such cells. the iterative transformation can be put in the matrix Thus a correct count of the new vertices requires that form : only 5 Z16 vertices are associated to one « old » Z12 vertex. The dodecahedral Voronoi cell is filled by a centred icosahedron where the 13 vertices are of the JV(’) is a 3D vector whose components (N (’), N§"[, N (’)) Z12 type. This can be written as a formal truncation are the total number of Z 12, Z14 and Z16 sites in the relation. polytope P;. The matrix T can be interpreted as a transfer matrix for the iteration in a very similar way to the fractal case [20]. At iteration i, a given vertex, A Z 14 Voronoi cell is filled by 12 Z 12 vertices, 3 Z 14

Table I. - Data corresponding to the Pn polytopes in the first example of LF.M. (Sect. 3.1, 2, 3). Np is the number 2[ sites with coordination number P and N the total number of vertices. T is the total number.of tetrahedral cells. Z is the mean coordination number. 1819 vertices and 24 Z16 vertices. Finally a 216 Voronoi in the following matrix form : cell is filled by 12 Z 13 vertices, 4 Z 14 vertices and (1 + 28) Z16 vertices (1 Z16 vertex in the centre and 28 Z16 vertices on the Voronoi cell vertices). This leads to two new formal relations :

where N(i) is a 3D vector whose components (Ni N’g N’) are the total number of triangles, pentagons and hexagons in the pol hedron P§. The Perron is = which Thus the matrix T is eigenvalue h (9 + 33)/2 or Ap - 7.37, given by : leads to irrational values for the relative number of different polygons in the asymptotic polyhedron P£ . This is another proof of the non-periodicity of the polytope P 00. Indeed a set of 30 great spheres similar to P§ is associated to the symmetry group G shared by all the iterated polytopes Pi [3]. On these 30 asymp- Note that since iteration begins here with the { 3, 3, 5 } totic P’ oo it is not possible to define a primitive cell to the of These polytope, one has Y(O) = (120, 0, 0). In its truncation (due irrational value AP). great spheres are mirrors of the 3D structure and so the lack of (deflation) form this transformation can be applied to any collection of Z 12, Z 14 or Z 16 vertices, even in flat primitive cell is extended to P 00. space. This point will be discussed in the next section. It is a general result that an irrational Perron eigen- To the largest eigenvalue of T (the Perron root) value is associated to non-periodicity. For instance in the corresponds an eigenvector which gives some infor- Penrose non-periodic tiling [19] Perron eigenvalue of the transfer matrix associated with the deflation mation about the P 00 polytope. Indeed it describes an asymptotic situation where the relative fraction of procedure is also irrational. different types of site remains constant under iteration. 3.4 A SECOND EXAMPLE OF I.F.M. IN 3-DIMENSIONS.- The mean coordination number can (MCN) be easily New polytopes Pn can be generated if one takes another and one sees in I that the derived table asymptotic large tetrahedron containing an integral number of value 40/3 is closely approached after only very few orthoschemes. For instance the intersection of the iterations. Note that the Perron root A = 20 is equal geodesic lines containing MOM,, MoM2 with the to the of number orthoscheme tetrahedra contained « equatorial » sphere leads to two new vertices. A in the large tetrahedron MOM’M"M’ third vertex is selected on the geodesic line containing us now focus on are Let the pattern of points which MOM3, which coincides with a { 5, 3, 3 } vertex located generated at each iteration on the equatorial great slightly in the « south » hemihypersphere (very close sphere of S3. In the Po case the { 3, 3, 5 }, the great to the equator). These three vertices together with Mo 3 make up a large tetrahedron which contain 61 ortho- sphere is tiled by an LetLet us icosidodecahedron 5 . scheme replicas. It is then easy to iterate the procedure call this polyhedron Po and Pi for each iterated Pi. described in section 3.2. Let us the « deflation » of this itera- The generation of Pi; starting from Pi _ 1 can be done give description tion At the first new vertices are at using a simple decoration procedure : one pentagon [21]. step, placed in P; - 1 is replaced by 6 pentagons and five triangles the centre of the { 3, 3, 5 } triangular faces and the vertices are One a (Fig. 13a), one triangle in Pi _ 1 is replaced by one original { 3, 3, 5 } ignored gets of icosidodecahedra the hexagon and three triangles (Fig. 13b), finally one tiling (surrounding original and tetrahedra the middle of hexagon gives rise to one hexagon, 6 pentagons and { 3, 3, 5 } vertices) (at the Recall that the 6 triangles (Fig. 13c). It is easy to write this procedure { 3, 3, 5 } cells). equatorial sphere of the { 3, 3, 5 } is also an lcosidodecahedron. The second step consists then in filling the icosidodecahedra with half { 3, 3, 5 } polytopes. Inside an icosidodecahedron, 45 new vertices and 300 tetrahedra are generated The new polytope, called P1, contains : - 600 tetrahedral cells located at the middle of the original { 3, 3, 5 } cells, - 36 000 tetrahedral cells arising from the decomposition of the 120 icosidodecahedra. These tetrahedra are neither nor - all of them Fig. 13. Deflation on polygons upon iteration on the equal In order to make them more one has « equatorial great sphere » of S3. a) 1 pentagon gives 6 pen- regular. regular, tagons and 5 triangles ; b) 1 triangle gives 3 triangles and to suppose, as in the 2D case, that the curvature is not 1 hexagon; c) 1 hexagon gives 6 pentagpns, 6 triangles and constant throughout the polytope and for example 1 hexagon. that the interior of the icosidodecahedra is more curved 1820

than in the remaining 600 tetrahedra. If a constant curvature is needed it is easy to project all the vertices onto the unit radius S3, but then length distortions are generated. At the first iteration only Z12 and Z18 vertices are generated. The Voronoi cell of a Z18 vertex is represented on figure 14. Three disclination lines are crossing at a Z 18 vertex. At the next iterations Z 14 vertices are introduced along the disclination lines. The defect set of polytope Pn is again composed of n interlaced disclination networks whose hierarchical local arrangement is drawn in figure 15. Table II displays information about the Pn (up to n = 5). These data have been derived with the help of the following matrix formulation. A 3D vector X(i) is defined whose components. (N1‘2, N1‘4, N1‘8) are the total number of Z12, Z14 and Z 18 sites. The following relation is satisfied

with T given by

Fig. 15. - Local view of the disclination network in second I.F.M. example. The hierarchical structure is clearly visible.

The mean coordination number 13.2 is easily derived from the knowledge of the eigenvector associated to the Perron root Ap = 61. Here again, as in section 3.2, AP is not irrational and one could hope to have defined some kind of a primitive cell of a crystalline structure. But if the presence of irrational values for the eigenvector components (more rigorously for the ratio between components) is a sufficient condition for non- periodicity, the lack of irrationality does not necessarily lead to periodicity. In the present case (as in sec-

Fig. 14. - The Voronoi cell of a Z 18 vertex. Note the six tion 3.2) the non-periodic character seems obvious hexagons grouped by three around opposite vertices. The for the following reason. Suppose that, after p itera- disclination lines thread these hexagons. tions, a crystal Pp is obtained Its unit cell necessarily 1821 contains parts of the disclination network Dp (whose Note that the M.C.N. of statistical tetrahedral periodicity is also required). Since the next iteration honeycomb [23, 24] is 13.39. The successive iterations will add a new disclination network (interlaced with define a family of crystalline structures with unit cell those contained in Dp), the size of the Pp + 1 unit cell of increasing size, inside which the vertex configuration is larger when scaled on the first-neighbour distance. is analysed in terms of interlaced disclination lines. So the size of the unit cell increases upon iteration 4. 3 LF.M. IN HYPERBOLIC SPACE. - The which proves the non-periodic character of P. A more general quantitative proof can be given by looking (as in method, in its «inflation form, applies to any regular is section 3 . 2) to what happens on one of the 30 symme- structure whose symmetry group generated by try spheres of the polytope. Triangles, pentagons and reflections in the side a fundamental region. Hyper- hexagons are generated and, with obvious notations : bolic tesselations, either 2 ou 3-Dimensional, belong to this family. Hyperbolic tilings may also serve as a template for idealized amorphous solids [5] and the kind of defects that would result from this description have been classified [25]. It is very easy to devise I.F.M. procedures on 2D hyperbolic tilings which give rise to geodesic hyperbolic «domes». More complex non-periodic asymptotic tilings (belonging to the same family of recurrent sets as the spherical ones of section 2) can also be derived. 3D hyperbolic tilings could also be flattened imply the non-periodic character of the polytope P 00. with this method, but the work has not yet been carried oi Note also that some exceptional hyperbolic tilings 4. Miscellaneous. are representations of the Bethe lattices and the Husumi cactus [26]. It is also possible to apply the I.F.M. on these but some care must 4.1 GEODESIC HYPERDOMES. - The deflation-trunca- procedure tilings, be taken since one angle of the fundamental triangle tion method can also generate much more regular is then equal to zero. structures. By analogy with the generation of B. Ful- ler’s domes on S2 iterative division of a geodesic (by 4.4 DETERMINISTIC OR RANDOM ITERATIONS. - In triangle into 4 smaller triangles) it is possible to build chapter 3 we have separately described two different domes on S3. On S2 the original geodesic « hyper » examples of I.F.M. for the polytope { 3, 3, 5 }. Since of an icosahedron are filled with parts of triangles the same G is associated with the triangular lattice. Similarly { 3, 3, tetrahedral cells symmetry group 5 } polytope and the different defect sets, it is possible, are filled with of f.c.c. lattice. The parts precise proce- at each of the to dure is described in E. step procedure, choose, any one Appendix between these two transformations. Let us define a two-word alphabet { a, b} where « as (resp. « by) 4.2 DEFLATION IN EUCLIDEAN SPACE. Eucli- - Any labels the first (resp. the second) type of iteration. dean crystal composed of tetrahedral cells can be used After n iterations the polytope Pn is fully characterized as structure for the deflation In starting procedure. by a « word », an ordered set of letters belonging to case mean coordination number this the (M.C.N.) the alphabet, containing n such letters. The varies iteration from the value complexity upon original crystal of the structure is encoded in the information content toward the value associated to the trans- asymptotic of the word Since each procedure has a specific transformation matrix. There is an interesting case when fer matrix, the resulting asymptotic polytope will both values are for instance when the transfor- equal, depend on the particular sequence of letters in the mation described in 3 . 2 is to the Laves applied phase word. The vector space has to be enlarged in order structure = If one this trans- (M.C.N. 13.333). applies to take into account simultaneously the two iterations. formation to the A15 structure = 13.5) (M.C.N. [22], Indeed iteration «a» generates Z12, Z 14 . and Z16 the first iteration a new structure with gives crystalline sites while iteration « b » generates Z 12, Z 14 and Z 18 162 atoms in the unit cell and the M.C.N. = 13.358. sites, when the starting configuration has Z12 sites. Note that this is the same number as in exactly But it is necessary to introduce new relations to des- unit cell as G. M932(Al, Zn)49 proposed by Bergman cribe the truncation of Z 18 sites (resp. Z 16 sites) under et al. to their structure [37]. Trying identify proposed iteration a (resp. b). For instance the formal relation a one an dif- with A15 with iteration meets unsolved is obtained (in iteration a) : ficulty : while their description shell by shell of the atomic structure agrees with our structure, the symmetry types do not coincide (body centre cubic in reference and cubic for A15 and its iterated The vector in now 4 Dimensional and the vector simple ’ space versions). 1822 transfer matrix are : pure « b » iterations. The structure is less regular and is probably more suited for the description of amorphous solids. However since the IFM procedure uses the icosahedral group, the orientational order should not be lost. It should certainly be very interesting to follow the behaviour of an orientational order parameter when going from a periodic to a random sequence of a and b type of iteration. Let us now consider the two opposite cases :

- 5. Conclusions. The sequence of a and b is periodic. The polytopes Pn belong to a set of deterministic We have described a powerful Method for the gene- recurrent structures. The asymptotic geometrical pro- ration of non-crystalline structures. The purpose was perties are given by a new matrix which is the ordered to provide a method for flattening 3D polytopes and product of Ta and Tb over a period The simplest cases get realistic models for non-crystalline materials. (short periods) gives rise to structures which are highly Indeed polytopes with prescribed perfect symmetry ordered in spite of their non crystalline intrinsic cha- have proved to be very powerful templates for amor- racter. In particular they present a high degree of phous solids with similar local order. orientational order. Atomic arrangements with per- The inflation form of the I. F. M. is easily implemented fect icosahedral order become very popular since their on a computer and yields to simple determination of recent discovery in nature [27] in the case of rapidly the location and weight for the disclination line defects. quenched Al-Mn alloys. The general name of o quasi- More work remains to be done in this direction. For crystal » has been proposed to label those structures example in 2D the relation with the theory of star [28]. Penrose tilings in 2D provide a very nice example polyhedra and Riemann surfaces should be explored of patterns with full pentagonal orientational order [35]. Indeed the large triangle of section 2 is a funda- and lack of translations. Several kinds of 3D Penrose- mental region for the star polyhedra { 2, 3 } and gene- like structures have been proposed [29-32], one of rates a multiple covering of the sphere S2. Perhaps which (using rhombohedral « bricks ») having a theo- this could help in understanding the propagation of retical diffraction pattern very similar to the experi- orientational order. The analysis of the corrugated mental one [32]. Penrose-like models belong to a spheres (or hyperspheres) on which the polyhedra larger family of (recurrent-like) structures possessing (-topes) Pn minimize their distortion will be given in orientational long-range correlations. The hierarchical a forthcoming paper in term of Regge-like analysis models described in this paper belong to this family. [7, 24] and fractal dimension. We have shown how it Their icosahedral symmetry is a direct consequence is possible to generate some disorder in the polytopes of the I.F.M. procedure where the icosahedral group by mixing two I.F.M. procedures sharing the same (and its extension to SO(4)) is of constant use. The symmetry group. It could also be interesting to study occurrence of icosahedral long-range order has been the case where one procedure uses a subgroup of the verified by Nelson and Sachdev [33] who calculated group used by the other one. Suggestions in that numerically the structure factor of models constructed direction were already present in an early paper [4]. (with iteration of type « as) directly in Euclidean The « deflation-truncation » form of the I.F.M. has space in the deflation-truncation scheme. The corres- also been described. It yields a very simple matricial pondence between the theoretical and experimental description of the iteration. The geometry of the structure is not exact but they propose that other derived structures is perhaps easier to visualize in this metallic alloys with large atoms occupying the Z16 approach. No metrics is now involved and the deco- sites could conceivably form such a structure. By sui- ration procedure can be done directly in Euclidean tably mixing iterations « a » and « b » (and even other space. The fact that orientational correlations are ones to be introduced) we think that a large class of still present [33], even with a network of disclination possible alloys may then be described The advantage lines, is probably due to the hierarchical nature of the of the present description over the Penrose-like defect set as well as its symmetry properties. approach is that here we give the precise location of The concept of hierarchical defect structure is atoms and disclination defects. The atomic positions certainly very important in understanding the micro- in Penrose-like models are still to be determined. We scopic properties of the icosahedral « quasicrystals ». conjecture that their defect set will also consists in The description of complex systems in term of ordered interlaced hierarchical disclination networks, perhaps regions pierced by defect networks has proved to be very similar to the ones presented here [34]. powerful. In a crystal with large unit cells this defect network is identified with the Frank-Kasper skeleton, - The sequence of a and b is random. while in amorphous systems disclination line networks Much less can be said about the geometrical pro- play this role. Within the framework of the corrugated perties of the asymptotic polytope. We can only say space description [7], the Frank-Kasper lines and the that they lie in between those given by pure « a » and disclinations are unified and are the loci of curvature 1823 in the discretized (atomic) underlying corrugated geo- unit circle in a plane, with, in the quaternion case, metry. Any tetrahedral partition of R3 can be analysed some added complexity because the non Abelian in this way. The case in which different kinds of poly- nature of Q. hedra are present (like for instance tetrahedra and octahedra [36]) could also be treated. Now quasi- Appendix B. are intermediate between and amor- crystals crystals THE BINARY ICOSAHEDRAL GROUP. - In Euclidean phous structures and can also be described in terms 3D space, the polyhedral symmetry groups are finite of defect networks. 3D Penrose-like structures give a subgroups of SO(3), the continuous group of direct good fit to the experimental results on AIMn alloys rotations that leave an origin fixed However SO(3) and the above described hierarchical models are pos- is not simply connected and is double-covered by the sible models for other structures, yet to be found group SU(2). This 2 : 1 homomorphism is extended Penrose tilings have a self similar geometry and they to the finite subgroup, the pre-image of a usual poly- can very probably be described in terms of hierarchical hedral group being called a binary polyhedral group defect set (when precise atomic positions are given). or a double group. Since the group Q of unit quater- A defect set description could also be done on directly nion is isomorphic to SU(2), it is possible to represent the rhombohedral cells. Note that we have previously * the double group elements by unit quaternions [17]. done this description on 2D Penrose tilings [34]. We shall briefly describe the case of the binary ico- Finally let us stress that the whole method works sahedral group Y’ (the prime denoting the « double for any kind of local symmetry, tetrahedral and cubic group » case). The knowledge of Y’ is very useful for as well as icosahedral. In 3D, the symmetry group the 2D and 3D iterative mappings described in the of all the regular polytopes is an extension of the text Indeed Y’ is homomorphic to the icosahedral usual point groups and consequently the polytopes group Y which is the symmetry group of the icosido- can be flattened in a similar way. decahedron (see section 2 of the text). Also Y’ is used to generate the polytope { 3, 3, 5 } symmetry group Acknowledgments. as will be indicated below. For sake of simplicity we here with the direct of the It is a pleasure to acknowledge D. P. Di Vincenzo for begin symmetry groups bodies Take care that a critical reading of the manuscript and appropriate regular (polyhedra, polytopes). suggestions. the full treatment of the iterative process requires the knowledge of the full symmetry groups (with both direct and indirect operations) which are subgroups A. Appendix of 0(3) in the polyhedral case. The quaternion nota-

- tion allows a determination of the full QUATERNIONS. We briefly recall some properties simple group the case of quaternions. For a more complete description, see elements (see polytope { 3, 3, 5 } below). references [16] and [17]. The image in Q of the dihedral group D2 is the of the unit A quaternion a can be written as : group V’ (of order 8) consisting 8 principal quatemions (noted as ordered quadruplets) : ( ± 1, 0, 0, 0), (0, ± 1, o, 0), (0, 0, ± 1, 0), (0, 0, 0, ± 1 ) . with the following rules The lift (image) of the tetrahedral group (of order 12) is the group T’ (of order 24) which contains V’ together with the 16 elements The quaternion a can also be written as a scalar part Sa and a vector part Va :

Consequently T’ can be written in the standard The conjugate a of the quatemion a is given by : form [17] :

A quaternion is said to be real if V a = 0 and pure imaginary if Sa = 0. means the union of n sets which have = where Q Ar Ar Unit quaternions (with norm Na = aa 1) are of r=1 special interest since they form a continuous non no points in common and with the convention that a Abelian group Q which is a nice illustration of a topo- quatemion raised to the power zero equals (1, 0, 0, 0), logical group. Indeed each element of Q is in one to the neutral element of Q. one correspondence with the points of a unit radius The group Y’ can then be written hypersphere S3. Because Q is a group, it can also label displacements on S3. This is very similar to the relation between the group of unit complex numbers and the 1824

Here T is the golden ratio : T = (1 + /5-)/2. The in the case of the dodecahedron, the characteristic 120 elements of Y’ correspond to the 120 vertices of region is the triangle which vertices are a dodeca- polytope { 3, 3, 5 } on a unit radius hypersphere S3. hedron vertex, the mid-point of an edge which ends The (direct) symmetry group of polytope { 3, 3, 5 } at this vertex, and the centre of a pentagonal face. is a subgroup of SO(4) and can also be specified within Note that because they share the same symmetry a quaternion representation because of the isomor- group, this triangle is also a fundamental region for phism SO(4) = SU(2) x SU(2)/Z2. Z2 is the two an icosahedron and an icosidodecahedron. Similarly element group with the two quaternions ( ± 1, 0, 0, 0) the two polytopes { 3, 3, 5 } and { 5, 3, 3 } share the as elements. same orthoscheme. Values for the angles in such If q E Q denotes a point on S3, the transformation characteristic tetrahedra are given elsewhere for any q --+ I qr-’ with 1, r E Q is an element of SO(4). The polytope { p, q, r [16]. direct symmetry group G’ of polytope { 3,3,5 } is given by Appendix D.

DESCRIPTION OF THE ALGORITHM IN THE 2D CASE. - Since the order of Y’ is 120, the quotient by Z2 An important step of the I.F.M. consists in the iden- implies that the order of G’ is 7 200. The total sym- tification of the two triangles MoM! M2 and MoM1 M2. metry group G of polytope { 3, 3, 5 } also includes The large triangle MoM1M2 contains seven replicas indirect orthogonal transformations, analogous to of the small triangle. We use two barycentric (homo- reflections for ordinary 3D discrete groups. These geneous) coordinate systems based on the tetrahedra are given by OMOMIM2 and OM’M,M’ (see Fig. 4), where 0 is the centre of the sphere. In the first system, a point M has four coordinates This adds 7 200 new elements and gives the group G of order 14 400.

Appendix C. This can be written in symbolic notation : M = DO THE ORTHOSCHEME. - The are polyhedral groups +ao Mo + al M1 + a2 M2. In the second system, of reflec- particular examples groups generated by the coordinates will be specified by a prime : (D’, ao, tions [3]. The planes of reflection (and their image a’1, a’2). under mutual have a common and reflection) point The algorithm, at a given iteration, is the following : their intersection with a sphere centred on this point Take each of the gives rise to a pattern of spherical triangle called 1) point M’, polyhedron Pp inside the with coordinates Mobius triangles or orthoschemes (they are right- lying triangle MoM1M20 2 I-, I 1 ’B. angled triangles). Any of these triangles is a fundamental region for the action of the group noted [p, q], 2) Identify the triangles MOMlM2 and MoMlM2. where n/p and n/q are the values of the two other This is done by interpreting the coordinates of M’ as angles in the Mobius triangle. The order g of [p, q] the coordinates of a transformed point M, in the is the number of such triangles that cover the sphere first barycentric system, lying inside the small triangle (of area 4 n), that is [3] : MOMlM2’ M has coordinates (Q, ao, ai, a2) such

3) Take the orbit of the point M under the group operations. If M is at generic position there will be 119 images on the sphere. It is verified that the full icosahedral group [3, 5] 4) One iteration is finished when this procedure has order 120. In 4 Dimensions the regular tessela- is done for each point M’ of the first step. The set tions are also associated with a discrete group with formed by the new points gives the vertices of poly- a fundamental region. These groups are generated hedron Pp+ 1. Among the vertices of Pp+ 1, those who by reflections in the faces of a characteristic tetra- lie in the triangle MoM1M2 will be kept in the next hedron, the orthoscheme. The orthoscheme is easily iteration where they constitute the set of points M’ constructed out of the flag of the polyhedron (or in the first step of the algorithm. polytope) [16]. For any regular polytope (in N If one is only interested in building a given poly- Dimensions), a flag (no, 7rl, ---, nN) is defined to be hedron Pp without the full knowledge about the the figure consisting of a vertex no, an edge 7r, con- previous polyhedra, a quicker algorithm can be taining 7Co, a face 7r2 containing 7r,, a (hyper) face used. Since, at a given iteration, only those point TEp ( p N ) containing np- 1, and a cell aN contain- which lie inside MoM1M2 are used, it is not necessary ing 7rN - 1. The orthoscheme is the simplex whose to construct the full orbit of points (step 3 above) at N + 1 vertices are the centres of the np. For instance the preceding iteration. It is enough to take a restricted 1825 orbit which maps the triangle MoMl M2 into the six other replicas which, altogether with MoMlM2, cover the large triangle M0M 1 M2. Let us call Qo, Qil and Q2 the vertices of the ith triangle (i = 0 to 6 and i = 0 refers to the triangle MoMl M2). For sake of simplicity we shall now ignore the first coordinate Q’ which play no particular role in the whole procedure. The relation ao + al + a2 = 1 is satisfied by a simple rescaling. The coordinates of these vertices can be written :

or in matrix symbolic notation :

where the M’ (i = 0 to 6) are the replicas of M in the seven orthoschemes which cover MoM1M2 (including Fig. 16. - Derivation of the hyper-geodesic dome. M itself for i = 0). If M’ is in generic position (none a) Decomposition of a regular tetrahedron into 4 tetrahedra of ao, ai, or a2 is equal to zero), the seven matrices / and 1 octahedron, all regular. b) Decomposition of a regular octahedron into 8 tetrahedra and 6 octahedra, all are used. If ao = 0 (resp. al, a2), it remains only 4 regular. (resp. 5, 5) matrices These divisions give rise to larger and larger F.C.C. portions yi. into the original polyhedra. Appendix E.

HYPER GEODESIC DOMES. - We have described in 4.1 an example of iterative deflation which can generate very regular structures. {3, 3, 5} tetra- The largest eigenvalue of the matrix T (the Perron hedral cells can be filled with part of F.C.C. structure root) corresponds to an asymptotic structure where with a parameter as small as wanted. Defects in the the relative fraction of octahedral and tetrahedral structure are located on the edges of the primitive cells remains constant. Here Ap = + 8 and the relative { 3, 3, 5 } which becomes, with the F.C.C. parameter fraction is two tetrahedra for one octahedron (as in as unit length, highly separated one from each other. the F.C.C. structure). A polytope P. (for large n) can The procedure consists in adding new vertices on be thought as large tetrahedra filled with F.C.C. the mid-point of edges. A tetrahedron is divided structure and glued along the edges of a ( 3, 3, 5 } in into 4 tetrahedra and one octahedron and one octa- order to cover S3. This is very similar, with one hedron is divided into 8 tetrahedra and 6 octahedra dimension added, to the case of B. Fuller geodesic (see Fig. 16). If the structure at an iteration i is charac- domes (while its usefulness in architecture is less terized by the vector X(i) whose components NT, obvious !). Huge portions of Pn, mapped on the No are the number of tetrahedral and octahedral tangent Euclidean space, can be used to model some cells the vector X(i+l) can be obtained in matrix very large metallic or rare gas aggregates after crys- notation by : tallization. Note that distance between line defects increases upon iteration. This is in contrast with the case of hierarchical structures described in section 3, in which defects remain close to each other. 1826

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