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

8/3/2019 J.-F. Sadoc and R. Mosseri- Hierarchical interlaced networks of disclination lines in non-periodic structures

http://slidepdf.com/reader/full/j-f-sadoc-and-r-mosseri-hierarchical-interlaced-networks-of-disclination 1/18

1809

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 polytopepar 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. Laméthode 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 derivedfrom polytopes by an iterative flatteningmethod. 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.

J. Physique 46 (1985) 1809-1826 NOVEMBRE 1985,1

Classification

Physics Abstracts61.40D - 02.40

1. Introduction.

Amorphous systems generally present an appreciableamount of Short-Range Order (SRO). For exampleamorphous metals can be well described by close

packing tetrahedra [1]. A regular tetrahedron is

the densest configuration for the packing of

four equal spheres. The dense random pack-ing of hard spheres problem can thus be mappedon the tetrahedral packing problem. The dihedral

angle of a tetrahedron is not commensurable with 2 n,

consequently a perfect tiling of the Euclidean space

R3 is impossible with regular tetrahedra. Note that,at this local level, the o frustration » (deviation to

perfectness) is of metrical rather than topologicalnature. One of us (J.F.S.) has proposed to define an

ideal (unfrustrated) amorphous structure by allowingfor curvature in the space in order for the local confi-

guration to propagate without defects throughoutthe whole space [2]. It is possible to pave a 3D manifold,the hypersphere S3, by 600 regular tetrahedra arrangedby five around a common edge. The obtained geo-metric object is called the polytope { 3, 3, 5 } usingthe standard notation [3]. Note that the underlying

space S3 is 3 Dimensional although not Euclidean,even if one often thinks of S3 as being imbedded in

R4. Indeed S3 is the locus of points of R4 given byxi + x2 + x3 + x4 = R 2, which shows that only

JOURNAL DE PHYSIQUE. - T. 46, ? 11, NOVEMBRE 1985

3 coordinates are independent. The polytope model,or o Constant Curvature Idealization » (CCI) has

been extended to several other kinds of disordered

materials such as tetracoordinated covalent sys-tems [4]. A simple example is given by the packing of

pentagonal dodecahedra which is forbidden in R3

(as in the tetrahedral case) because of the polyhedrondihedral angle value. Packing these dodecahedra on

S3 leads to the regular polytope { 5, 3, 3 } which is

dual of the above mentioned { 3, 3, 5 } and thus

possesses the same symmetry group. However thevalues of the curvature associated with these two

polytopes are not identical (when scaled to the edgelength) and this reflects the fact that the local angularmismatch (in R3) are not equal. Polytope { 5, 3, 3 }can be useful for modelling the « caged »-like tetra-

valent structures and those which are related like

amorphous ice for example. Several other CCI have

been proposed, like the regular honeycombs in the’

3D hyperbolic space H3 (with constant negativecurvature) [2, 5] and the continuous double twisted

configuration of directors on S3 (as an ideal model

for the cholesteric blue phase) [6].The idea is that the disordered material contains

« ordered >> regions where the local order can be putin correspondence with the ideal model, the comple-mentary regions being the locus of defects. We

expect that a suitable map of the ideal model onto

111

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

http://www.edpsciences.org/

http://dx.doi.org/10.1051/jphys:0198500460110180900

http://dx.doi.org/10.1051/jphys:0198500460110180900

http://www.edpsciences.org/



1810

R3 (minimizing the energy) will provide a realistic

amorphous structure. The mapping introduces dis-

tortions and topological defects, among which thedisclination lines play an important role. The final

structure consists in mixing of regions with positivecurvature (where the local order is that of the

poly-tope) and regions with negative curvature (the locusof defects), arranged such as to give zero curvature

on the average. In this Corrugated Space Approach [7],also called «variable curvature idealization », thelocal order is still perfect within coherence regions. As has already been mentioned, disclination lines

have proved to be natural defects for non-crystallinestructure. S. N. Rivier [8] proposed that one type oflinear defects, characterized by oddness rather than

intensity, is stable in real glasses as a result of thedouble connectedness of the rotation group SO(3).In the curved space approach, line defects can also

be classified using the homotopy theory of defects [9].Let us call Y the icosahedral group (subgroup of

SO(3)) and Y’ its lift in SU(2) (the covering group

of SO(3)). The full symmetry group G* of polytope{3,3,5}, with 14 400 elements, is described in

Appendix B. G, the subgroup of G* containing onlythe direct symmetry operations (preserving orien-

tation), is given by : G = Y’ x Y’/C2.C2 is the two-

element group. As shown by Nelson and Widom [10]the defect lines that can be generated in the polytope{ 3, 3, 5 } belong to the conjugacy classes of R =

ni (SO(4)/G) = Y’ x Y’. More recently Trebin [11]has

proposeda « coarser » classification for

theline

defects in polytope { 3, 3, 5 } by considering the first

homology group of the order parameter space. SinceY’ is a « perfect group » (it is isomorphic to its owncommutator subgroup), the homology group is trivial

which expresses the fact that any line can be transform-ed into any other by a suitable combination process.One might hope to generate increasing numbers

of disclination lines in the { 3, 3, 5 } in order to achievea complete flattening of the polytope. We have

already shown [12] that it is possible to interlace two

such disclination lines and get a polytope containing144 Z 12 vertices and 24 Z 14 vertices with less intrinsic

curvature. We use standard notations [13] to labelthe 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 », denoteeach 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,

and its complexity is encoded in the information

content of the word.

In a recent letter, hereafter referred to as 1 [14], wehave shown how it is possible to bypass the above-

mentioned difficulties and achieve the complete flat-

teningof the polytope. The

keyidea is, at each step,

to introduce a disclination network (instead of a

single disclination line) whose symmetry group iscontained in G. In the present paper we shall give a

complete description of how the method works. In

order for this paper to be self-contained, we shall

give a detailed description of the different tools which

are needed (symmetry group, orthoscheme, bary-centric transformation...).

In section 2, a simple 2-Dimensional example is

described, in which an icosidodecahedron is itera-

tively flattened and gives rise to an asymptotic non-

periodic plane tiling. The main ideas that will be

used later in the 3-Dimensional case are introducedand visualized.

Section 3 contains the application of this method

to the polytope { 3, 3, 5 } case. At each step a « defect »

subnetwork is generated which has the same symme-

try properties as the structure itself. Also the matricial

description of the iterative procedure is introduced.We show how to generate more disorder by combiningtwo different iterative procedures which are compa-tible since their associated defect structure share thesame symmetry group. In the last section (4) we

describe miscellaneous aspects and extensions of themodel.

2. A 2D example : iterative flattening of an icosi-

dodecahedron.

The icosidodecahedron (Fig. 1) is a quasi-regular

or Archimedean polyhedron [15]. It is noted 3 [3]5and shares the same symmetry group with the ico-

sahedron { 3, 5 } and the dodecahedron { 5, 3 }.

Fig. 1. - The icosidodecahedron



1811

Indeed it can be obtained by joining the mid-pointsof the edges of { 3, 5 } or { 5, 3 }. Each vertex belongsto two regular pentagons and two equilateral triangles.The Iterative Flattening Method (I.F.M.) is much

more easily visualized in two dimensions than

in three and almost all of the ideas that we presenthere can be simply generalized to one higher dimen-sion. As will be seen below, the I.F.M. can be under-

stood as an « inflation >> or as a « deflation >> method.

The second one is very easy to visualize (in some sense

it does not need a coordinate system). We shall

however insist on the inflation method since it is the

one which allows to focus on the symmetry relations

between vertices and on the curvature aspect.

Z .1 GEOMETRICAL PROCEDURE. - In appendix B we

introduce the binary icosahedral group Y’ [16] whichis the lift in SU(2) of the icosahedral group Y (thecovering map SU(2) -+ SO(3) is carried on the

discret polyhedral subgroups of SO(3)). Y is a pure

subgroup of Y*, the full icosahedral group (includingindirect transformations) also called the trianglegroup (2, 3, 5)* [17]. This group of order 120 allows

for a division of S2 in a pattern of 120 sphericaltriangles, each of one being a fundamental region of

the sphere tesselation (Fig. 2). Any vertex configura-tion on S2 having Y* as symmetry group is comple-tely defined by giving one fundamental region (theorthoscheme, Appendix C), and the distribution ofvertices inside it. It is analogous to the descriptionof crystals by unit cells and the translation part of

the symmetry group. In the case of spherical tessela-tions, the symmetry operations are reflections in the

sides of the fundamental triangle. The pure rotations

belonging to the direct group Y are products of even

number of such reflections.

In the following we take as a basic icosidodeca-

hedron the 3 which tiles the equatorial >>5

Fig. 2. - Partition of S2 into 120 spherical triangles under

the action of the full icosahedral group Y*. Elements of thepure subgroup Y interchange triangles of the same colour

(either white or shaded). A particular fundamental region

(M6bius triangle) is distinguished.

sphere of polytope { 3, 3, 5 } (see next section). The

vertex coordinates are given by the 30 quatemionsin Y’ whose scalar part vanishes. Indeed to a unit

pure quatemion q = a, i + a2 j + a3 k (see Appen-dix A) there corresponds a point of coordinates

(a1’ a2’ a3) on the sphere S2 of unit radius. The

three orthoscheme vertices Mo, M1,I M2 are vertices

of respectively the { 3, 53 and 5 3 } which{3,5 } { 5{ 5,3 }

share the symmetry group Y*. Indeed it can be veri-

fied that the orbit of Mo (resp. M1, M2) generates a

3 5 resp. a 3 , a 5,3 under successive{ 3,5 ( p ( 5{ )

reflections in the sides of the spherical triangleMo M1 M2. We shall consider that the polyhedraunder construction are either spherical, with bent

faces and geodesic edges, or Euclidean, with flat

faces and straight edges (chords). They are triviallyrelated, the spherical polyhedron being the central

projection of the Euclidean one OIrto-tl1e-surfiice ofthe sphere S2.

Let us now describe the method. In a first step the

original polyhedron is constructed. It will be calledthe «source polyhedron Po. It is characterized bythe orthoscheme MoMiM2 and by the location of a

point M (or several points) in it. Po vertices are theorbit ofM under the Y* symmetries. The second stepconsists in selecting a new triangle MoM1M2 which

shares an angle with the orthoscheme MoMiM2 (herethe angle at vertex M1, see Fig. 3). The triangle

MoM1M2 is chosen as to contain an integral numberof orthoscheme replicas. Thus it contains several Povertices. Note that the two spherical trianglesMOMlM2 and MoMiM2 do not have all their cor-

responding angles identical since they have different

areas (recall that the area of a spherical triangle is

proportional to the sum of its interior angles minus n).

Fig. 3. - The two spherical triangles MOMIM2 and

MoMlM2 which share the vertex Mi. Marks label vertices

on S2 where the symmetry is either 5-fold, 2-fold and 3-fold.The 12 vertices (resp. 30, 20) which are the orbit of Mo(resp. M1,M2) are vertices of an icosahedron (resp. icosido-

decahedron, dodecahedron).



1812

The crucial point consists now in the identification

of the two triangles. In flat space this could be easilydone by homothety when the triangles are similar.

In the curved space S2, homothety is not easy to definebecause of the presence of an internal length scale

(the radius of curvature). It is possible to definehomotheties 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 MOM2requires a continuous set of homotheties. Insteadthere is a very simple and natural way to do this ifwe work with an Euclidean polyhedron (with flat

faces). Now the flat triangles MoMiM2 and MoMlM2are not coplanar (Mo does not belong to the chord

MiMo) and the underlying geometry is no more a

simple 2D geometry but a collection of bounded 2D

flat regions (the faces) glued along the edges. It is

then easier to consider the homogeneous space inwhich the polyhedron is embedded, the 3D Euclidean

space E3. Now any point in E3 can be specified in a

barycentric coordinate system once a particulartetrahedron is given. Let us for instance take thetetrahedron 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 thetetrahedron OMoMiM2 are noted (Sl, ao, al, a2)(with the conditions + ao + ai + a2 = 1). A pointbelongs to the 3D sector bounded by the planesOMOMI, OMOM2, OMlM2 if and only if its last

three coordinates ao, al, a2 are simultaneouslyposi-tive 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 keepthose vertices which have ao, ai and a2 simulta-

neously positive or zero. Finally we re-interpret thosecoordinates (Q’, ao, ai, a2) as being the coordinates

(Q, ao, ai, a2) in the barycentric system based on

the tetrahedron OMoMIM2. The two basic tetra-

hedra are drawn in figure 4. The new points do not

necessarily lie on the flat triangle MoMiM2 but theycan be projected on it by central projection. In terms

of barycentric coordinates it consists in puttingequal 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 ortho-scheme MOMlM2’ It is represented in figure 5a. The

localorder in

P1 presents similarities with that of Po.Its description in terms of order and defects will

be done in the next paragraph. Let us just say thatthe configuration around the point M 1 is identical

Fig. 4. - The two basic tetrahedra OMoMIM2 and

OM’M,M’. 0 is the centre of the sphere S2. The pointsMo, Mo, M1, M2 and M2 have the same location than in

figure 3. But now the edges are straight lines (chords) insteadof geodesic curves.

Fig. 5. - a) The polyhedron Pi (centrally) projected onto

the sphere S2. b) Tentative drawing of the underlying cor-

rugated geometry of Pi. The radius of curvature (scaled to

the edge-length) near Mi and its replicas should be similarto that of Po.



1813

to that of Po (M 1 belongs to 2 pentagons and 2 trian-

gles). In terms of the edge length, the radius R1 of

the P1 circumsphere is much larger than Ro, theradius of the Po circumsphere. In some sense theratio R1 I Ro is characteristic of the «bary centrichomothety ». It is possible to recover some regularityfor the polygons around M1 by supposing that P1vertices lie on the surface of a sphere of a radius R1covered by small domes of radius Ro which centres

are located somewhere on the radii connecting 0with M1 and its replica. This is tentatively drawn in

figure 5b where the whole surface has been smoothed.

It is very easy to iterate again the procedure, builda new polyhedron P2 by keeping the P1 vertices

lying inside M0M1M2, identifying MoMiM2 and

MoMiM2 and then getting P2 with the help of Y*

symmetries. As long as the two triangles keep specificrelations (they have an angle in common and the

bigone contains an

integralnumber of

replicasof

the first one), the successive polyhedron Pn will

present very interesting properties. For example, as

seen below, their defect set presents the same kind of

regularity as the polyhedra themselves. If the same

triangle M0M1M2 is used at each iteration, the

polyhedra Pn belong to the family of deterministicrecurrent sets [18]. From a topological point of view

the asymptotic polyhedron P 00 presents the same

kind of non-crystallinity as the Penrose tiling [19]. A simple algorithm to build iterative polyhedra is

presented in Appendix D.

2.2 THE DEFECT SET. - By definition the source

polyhedron Po is called the «ordered (defect free)configuration ». It can be obtained by the local

building rule : « put two pentagons and two trianglesaround each vertex ». A specific value for the curva-

ture of the underlying 2D manifold is necessarilyassociated with such a rule. This curvature takes a

constant value at each site and is related to the so-

called deficit angle bM at the vertex M :

0 are the internal angles of the flat polygons sharingsite M. Different relations between local configura-tions and curvature has been derived in connection

with the Corrugated Space Approach to disorderedstructures [7]. In 2-Dimensions, owing to the existenceof two famous relations, the Euler-Poincare and theGauss-Bonnet relations, simple and exact correspon-dence can be found between the geometry of the

underlying manifold and the defect density. Here

defects are disclination points whose weight is relatedto the values of polygon angles at the vertices. For

example, when flat space is taken as the ordered

(undefected) state, one has the exact relation

where ( K, is the mean Gaussian curvature perunit area (R is the radius of curvature) and n thenumber of disclination points (each assumed to carrythe same angular deficit 6) per unit area. In the presentcase where the ordered state is defined to be the

polyhedron Po, the defect intensity is measured with

respect to the value of £ 0; at Po vertices (insteadof 2 x). I

The I.F.M. in its inflation >> form yields a very

simple way to locate and measure the defects. Indeed

a disclination point is generated at each vertex of theorthoscheme MOMlM2 (and at all its replicas under

the Y* symmetries) whenever the angle is different

from that of the corresponding triangle M0M1M2.Let us consider the polyhedra as being spherical. Theinternal angles are evidently

In Po the point M2 is surrounded by a pentagon. Atthe first iteration and after identification of the two

triangles MOMlM2 and M0M1M2, the point M2 will

be surrounded in P1 by an hexagon (6 = 5 x eM2/eM2).Sirnilarily, the point Mo in Po is surrounded by a

triangle. Thus the point Mo in P1 is surrounded bya pentagon (5 = 3 x 8Mo/9Mo). Note the important

following remark : in the polyhedron Po, the point Mowas already surrounded by a pentagon, which was

not a defect polygon. At the first iteration this pentagonhas been « pushed >> toward the point M,. The pen-

tagon which now surrounds the point Mo can be

considered as a defect since it is a disclinated form

of the triangle which surrounded Mo before the first

iteration (in Po).Let us call Dp the defect set of a polyhedron Pp.

Dp is the union of several subsets, each of one asso-

ciated with a particular kind of disclination, and all

sharing the symmetry group Y*. For instance D1contains 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 P1.I

- D1 is the set formed by point M2 and all its

replicas under Y*. This point set forms a dodeca-

hedral pattern. Each point is surrounded by an

hexagon.

Both D’ and D1 are represented in figure 6. In

the next iteration it is clear that the disclination

points in D’ and D1 will persist as defects. However

they will be located at different places in the ortho-scheme and the number of points after replicationwill be different. It will happen that two points in

D’ which belonged to two different fundamental

triangles in Pp will be in thesame

triangle Pp+1. Sothe number of defects will increase indefinitely with

the iterations. Also each iteration generates two new



1814

Fig. 6. - The polyhedron Pand its defect set D 1. Disthe union of two subsets D’and D1 which are marked dif-

ferently on the figure. A black q-gon inside a p-gon indicates

that thisp-gon

is a disclinatedq-gon.

Fig. 7. - Local view of the polyhedron P 2 and its local

defect set D2. The large pentagon which limits the figureis a pentagon of the original icosidodecahedron Po.

defect subsets similar to D 1 and D1 . Figure 7 showsa limited region of P2 (mapped onto the plane) with

the different defect points. Note that decagons are

generated in P2 which are disclinated hexagons of P1.*Indeed the angle at Mo equals yc/5 and repeatedreflections in the lines through Mo of a generic pointgive rise to a decagon. The pentagon generated in

P1 came from the special location of a point on an

edge. At each iteration a new type of (larger) polygonwill be generated in addition to the whole set of

previously generated (smaller) polygons. The fact

that the number of types of polygons increases uponiteration is

specificto this 2D

example.This will not

occur in the 3D examples described in the next chapter.It may happen (due to the choice of the triangle

MoM 1 M2) that most ofthe defect points are surround-

ed by p-gons such that

where c is the vertex coordination in the polyhedronPp. These polygons act as neutral charges [7] in that

sense that their underlying manifold is flat. The Pppolyhedra can be called pseudo-crystalline since theycontain larger and larger regions covered by portionsof 2D flat crystals. A simple example is given bygeodesic 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 }.

In this section we show how the method describedabove can be generalized to one higher dimension.

It is applied to polytope { 3, 3, 5 } whose symmetrygroup is presented in Appendix B. Let us first givea 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 perfecticosahedron. A good way to represent in flat spacesome geometrical unit belonging to curved (hyper)spheres consists in making cuts parallel to a tangentflat space (Fig. 8). The intersection of polytope{ 3, 3, 5 } by hyperplanes gives the following succes-

sive polyhedra : an icosahedron (the first neighbourshell of a vertex), a dodecahedron, a larger icosahedronand an icosidodecahedron on the equatorial))

sphere. Here the polytope has been oriented insuch

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 >> firstcoordinate axis). Note that this is slightly different

from what was considered in 1 (Ref [14]) where thenorth pole corresponded to the vertex (0, 0, 0, 1). In

our new notation the north pole coincides with the

identity quaternion in the group Q (Appendix A).

3 .1 GEOMETRICAL PROCEDURE. - As has already beensaid, the method works in a way very similar to the

2D case. G, the total symmetry group of polytope{ 3, 3, 5 } (Appendix B), generates a regular divisionof S3 into 14 400 spherical tetrahedra, a single one

being called an orthoscheme (or fundamental tetra-

hedron). A tesselation is uniquely defined by thelocation of vertices inside an orthoscheme, the othervertices being generated by reflection in the faces of

the orthoscheme. Figure 9 represents one tetrahedralcell of polytope { 3, 3, 5 } and one orthoscheme

inside it. The four vertices of the latter are locatedon one cell vertex, the centre of the cell, the centre

of a face and of an edge. Each { 3, 3, 5 } tetrahedral

cell contains24

copies of theorthoscheme. The

coordinates of the 120 vertices of the { 3, 3, 5 } are

given by the 120 quaternions of Y’. One particulartetrahedral 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 icosido-

decahedron ; the figures are reproduced from Pour la Science, janvier 1985.

Fig. 9. - A tetrahedral cell of the polytope { 3, 3, 5 } with

one particular orthoscheme inside it. The four vertices of

an orthoscheme are a polytope vertex and an edge, face and

cell centres.

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 thenorth pole A. M1 is located at the middle of the

edge AB, M2 is the centre of the face ABC and M3is the centre of the { 3, 3, 5 } tetrahedral cell ABCD

(Fig. 9). Up to this point we consider the polytopeas being spherical. The characteristic simplex Mo,M1,

M2, M3is thus a quadrirectangular

sphericaltetra-

hedron (cut out from a hypersphere S3 by four hyper-planes ; see Ref. [3], p. 139). That is to say, pointsM1, M2, M3 have been radially projected onto theunit radius hypersphere S3. The dihedral angles at



1816

the edges M2M3, MoM3 and MOM, are respectively

the remaining three being right angles.Once the orthoscheme is defined, it is very easy

to generate any tesselation which shares with the{ 3, 3, 5 } the same symmetry group G (Appendix B).It is enough to give the location of vertices inside the

orthoscheme and then to apply the 14 400 symmetry

operations. If the vertex has a generic position, one

ends with 14 400 replicas (including the originalvertex under the identity operation). If however the

vertex has a less general position, its orbit may

contain less points. For instance the image of A = Mogives rise to the 120 vertices of polytope { 3, 3, 5 },while the image of M3 gives rise to the 600 vertices

of the dual polytope { 5, 3, 3 }.Now we can begin the first iteration. We define a

larger tetrahedron which shares the vertex Mo withthe orthoscheme and contains an integer number oforthoscheme replicas. Let us still consider the tetra-

hedra as being spherical. The three new vertices

M 1, M 1, M’ belong to the same great circles as the

geodesic edges MoM!, MoM2 and MoM3. Point Mois located at a vertex of the { 3, 3, 5 } which belongsto a dodecahedral second neighbour shell surroundingthe north pole (Fig. 8). We have seen that point M3is a vertex of the dual polytope { 5, 3, 3 } which has

dodecahedral cells, one of which also surroundingthe north pole. Thus we can anticipate that the

identification of M3and

Mo is equivalentto

theidentification of the large dodecahedron (secondneighbour shell in the { 3, 3, 5 }) lying in the hyper-plane xl = 0.5 with a smaller dodecahedral cell of

polytope { 5, 3, 3 } lying in the hyperplane

In order to overcome the difficulties involved in

the identification procedure in curved space, as in

the 2D case, we consider again the Euclidean versionof the polytopes, with chords as edges and flat (hyper)faces. We compute the coordinates of the { 3, 3, 5 }vertices in the barycentric system based on the

hypertetrahedron OMo M i M 1 Mo with coordinates(01 ao, ai, a2, a3). 0 is the centre of the hypersphere S3.Then only the vertices whose last four coordinatesare simultaneously positive are kept. Let us call Kothe set of such points. Their barycentric coordinatesare now interpreted as being based on the hyperte-trahedron OMoMIM2M3, that is to say the elements

of flo lie now in the small tetrahedron MoMiM2M3.But this tetrahedron is the fundamental region of

group G. It is then easy to generate a new polytopeP1 as the orbit under G of Ao. More precisely JLocontains three points and the first iterated polytope

P1 has 2 160 vertices. It is rather pedagogical to seewhere these 2 160 come from. None of the 3 pointsin To is a generic point of the orthoscheme. Indeed a

generic point has 14 400 replicas (the order of the

group). One element is located at Mo and has 120 repli-cas. A second one is located at M3 and has 600 replicas.The third one is located somewhere on the edge MoM1and has 1 440 replicas. All these add up to the 2 160vertices of P1. The local configurations around each

vertex are not all the same (P1 is not a regular poly-tope). However strong orientational order persists in

P1, which will be discussed in the next section about

the defect set.

It is now easy to iterate again the procedure. Weconsider the new set tÂt1 consisting of points of P 1belonging to the large tetrahedron M o M 1 M 1 Mo.Using the same barycentric homothety as in the first

iteration, the 14 elements of A, are mapped into thefundamental region MoMiM2M3. A new polytope P2is then generated as the orbit of X 1 under G, and

contains 42 480 vertices. The iteration can proceed onand larger and larger polytopes (in term of their num-

ber of vertices) are obtained.3.2 DESCRIPTION OF THE DEFECT SET. - The formal

definition of the defect set is very similar to the 2Dcase while its precise geometry is much more intricate.

The starting polytope Po (the { 3, 3, 5 }) is the defect

free configuration. Defects (disclination lines) are

generated along the edges of the fundamental regionwhenever the dihedral angle differ from that of the

larger tetrahedron MoMiMiMo.Figure 10 represents the two tetrahedraMoMiM2M3

and MOM? M[ Mi and their relation with four tetrahe-

dral cells of polytope { 3, 3, 5 }. Recall that the index i

in Mi (or MD denotes the type of site : i=

0 for a

{ 3, 3, 5 } vertex, i = 1 for a mid-edge { 3, 3, 5 } vertex,i = 2 for the centre of a triangular face and i = 3 forthe centre of a { 3, 3, 5 } tetrahedral cell (e.g. a vertex

of polytope { 5, 3, 3 }). The edges MoMI, MIM3 and

M2M3 belongs to respectively 5-fold, 2-fold and 3-foldrotation axes. But since the basic symmetry operationsare reflections in the orthoscheme faces, the number

of images of a generic point is usually twice as largeas the order of the rotation axis. For example a trian-

gular face is threaded orthogonally by a 3-fold axis.

A generic point on this face has six images on this

triangle (including the original point under the iden-

tity operation). Another way to see it is to remarkthat the orthoscheme dihedral angles are equal to z)

(and not 2 7c) over the order of the rotation axis to

which the associated orthoscheme edge belongs.By inspection of figure 10, it becomes clear that only

dihedral angles at edges M’M", and M3M2 are diffe-rent upon identification of the two tetrahedra, the

respective angles being 2 n/5 and 7r/3. Thus in polytopeP1 the defect lines will be carried by the edge M3M2and its replicas under symmetry group operations.These edges connect the centres of adjacent { 3, 3, 5 }tetrahedral cells (sharing a face). Consequently theyare edges of the dual polytope { 5, 3, 3 }. In the follow-ing, we call Qi the dual of polytope Pi (and so Qo is

the polytope { 5, 3, 3 }). The Qi are easily constructed

by joining the centres of the Pi tetrahedral cells.



1817

Fig.1 Oa.- The two basic tetrahedra ofthe iteration process.The figure shows 4 tetrahedral cells of the { 3, 3, 5 } (heavylines). 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 symmetrygroup they belong. For instance M 1 and M 1 are imagesof Munder given symmetry operations. They are all locatedat the middle of { 3, 3, 5 } edges.

As said before, at the first iteration the orthoschemecontains three vertices. Upon the identification pro-

cedure, only the local order around the vertex at M3is altered. Using standard notations [13] it becomes

a Z16 vertex while the two other elements of Ao (andall 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 becauseZ14 sites will appear in the next iteration. This nota-

tion 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

form a tetrahedral configuration (like the directions

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

three other replicas which intersect at M3.Let us call Di the set of all the defect lines in poly-

tope Pi. We have seen that D1 consists in the edgesof polytope Qo (the { 5, 3, 3 }). The second iteration

generates polytope P2 with its associated defect set

D2. D2 contains two disjoint parts :

- The first one, D2, is introduced at the second

iteration as a result of the identification procedure.Consequently it has the same geometry as Di. But

Fig. lOb. - The 4 triangular faces of the large tetrahedron

M0M0M1M1. The edges and vertices of the orthoscheme

replicas are shown.

Fig. 11. - Coordination shells of Z 12(a), Z I 4(b) and Z 16sites. Sites lying on disclinations are darkened.

it has a larger scale with respect to the first-neighbourdistance. Indeed all the Pi are constructed on a unitradius hypersphere. The intrinsic length ofthe physicalnetwork is the first-neighbour distance between sites,thus a change of scale has to be done when comparing

two different Pi. So D2 has a larger edge lengthcom-

pared to D1. In particular two new Z14 sites are

located between two nodes of D2.- The second subnetwork, D", is the image of D 1

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 dis-

tance 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 charac-

teristic length scales with respect to their nearest node

separation. The local arrangement of DZ and D2 is

tentatively represented in figure 12. When the trans-

formation is iterated again, larger polytopes Pn are

generated. Their defect set Dn contains n interlaceddisclination networks and can be written

where 0153 denotes the union of disjoint sets. Dn has a

hierarchical structure : the intemode separation in

each Qi (in term of the first-neighbour distance in Pn)varies with i. The change of scale between Qi and

Qi + 1 is of the order of the « barycentric homothety »described in the previous paragraph.

3. 3 DEFLATION APPROACH AND MATRIX FORMULATION.

- Table I displays information about polytopes Pnup to n = 5. These quantities can be obtained forsmall n by the direct « inflation » method describedabove which uses the symmetry group operations in

order to build structures of increasing sizes. But thedata of table I can be derived more simply without

explicitly building the polytopes. The main effect of

the iterative transformation can be put in the matrixform :

JV(’) is a 3D vector whose components (N (’), N§"[, N (’))are the total number of Z 12, Z14 and Z16 sites in the

polytope P;. The matrix T can be interpreted as a

transfer matrix for the iteration in a very similar wayto the fractal case [20]. At iteration i, a given vertex,

Fig. 12. - Local view of the disclination network. Heavyline : D2 disclinations (which the same topological struc-

ture as D1). Light line : D2 disclination interlaced with the

previous network.

which belonged to polytope Pi -1’ is replaced by a

collection of new vertices. In the present case theVoronoi cell of a Z12 vertex is filled by 13 new Z12

vertices and 20 new Z16 vertices which are located as

follows : the Z12 Voronoi cell is a pentagonal dode-

cahedron. Its 20 vertices are occupied by Z16 sites.The partition of S3 by the Voronoi cells is such thateach vertex on these cells belong to four such cells.

Thusa

correct count of thenew

vertices requires thatonly 5 Z16 vertices are associated to one « old » Z12vertex. The dodecahedral Voronoi cell is filled by a

centred icosahedron where the 13 vertices are of theZ12 type. This can be written as a formal truncationrelation.

A Z 14 Voronoi cell is filled by 12 Z 12 vertices, 3 Z 14

Table I. - Data corresponding to the

Pn polytopesin the first example ofLF.M. (Sect. 3.1, 2, 3).

Npis 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

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 :

Thus the matrix T is given by :

Note that since iteration begins here with the { 3, 3, 5 }

polytope,one has Y(O) = (120, 0, 0). In its truncation

(deflation) form this transformation can be applied to

any collection of Z 12, Z 14 or Z 16 vertices, even in flat

space. This point will be discussed in the next section.

To the largest eigenvalue of T (the Perron root)corresponds an eigenvector which gives some infor-

mation about the P 00 polytope. Indeed it describes an

asymptotic situation where the relative fraction ofdifferent types of site remains constant under iteration.

The mean coordination number (MCN) can be easilyderived and one sees in table I that the asymptoticvalue 40/3 is closely approached after only very few

iterations. Note that the Perron root A = 20 is equal

to the number of orthoscheme tetrahedra containedin the large tetrahedron MOM’M"M’

Let us now focus on the pattern of points which are

generated at each iteration on the equatorial greatsphere of S3. In the Po case the { 3, 3, 5 }, the great

sphere is tiled by an

3Let ussphere is tiled by an icosidodecahedron 5 .Let us

call this polyhedron Po and Pifor each iterated Pi.The generation of Pi; starting from Pi _ 1 can be done

using a simple decoration procedure : one pentagonin P; - 1 is replaced by 6 pentagons and five triangles(Fig. 13a), one triangle in Pi _ 1 is replaced by one

hexagon and three triangles (Fig. 13b), finally one

hexagon gives rise to one hexagon, 6 pentagons and

6 triangles (Fig. 13c). It is easy to write this procedure

Fig.13. - Deflation on

polygonsupon iteration on the

« equatorial great sphere » of S3. a) 1 pentagon gives 6 pen-

tagons and 5 triangles ; b) 1 triangle gives 3 triangles and

1 hexagon; c) 1 hexagon gives 6 pentagpns, 6 triangles and

1 hexagon.

in the following matrix form :

where N(i) is a 3D vector whose components (NiN’g N’) are the total number of triangles, pentagonsand hexagons in the pol hedron P§. The Perron

eigenvalue is h = (9 + 33)/2 or Ap - 7.37, which

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 similarto P§ is associated to the symmetry group G shared

by all the iterated polytopes Pi [3]. On these 30 asymp-totic P’ oo it is not possible to define a primitive cell

(due to the irrational value of AP). These great spheresare mirrors of the 3D structure and so the lack ofprimitive cell is extended to P 00.

It is a general result that an irrational Perron eigen-value is associated to non-periodicity. For instance in

Penrose non-periodic tiling [19] the Perron eigenvalueof the transfer matrix associated with the deflation

procedure is also irrational.

3.4 A SECOND EXAMPLE OF I.F.M. IN 3-DIMENSIONS.-

Newpolytopes Pn canbe generated ifone takes another

large tetrahedron containing an integral number oforthoschemes. For instance the intersection of the

geodesic lines containing MOM,, MoM2 with the« equatorial » sphere leads to two new vertices. Athird vertex is selected on the geodesic line containingMOM3, which coincides with a { 5, 3, 3 } vertex locatedslightly in the « south » hemihypersphere (very closeto the equator). These three vertices together with Momake up a large tetrahedron which contain 61 ortho-scheme replicas. It is then easy to iterate the proceduredescribed in section 3.2.

Let us give the « deflation » description of this itera-

tion [21]. At the first step, new vertices are placed at

the centre of the { 3, 3, 5 } triangular faces and the

original { 3, 3, 5 } vertices are ignored One gets a

tiling of icosidodecahedra (surrounding the original{ 3, 3, 5 } vertices) and tetrahedra (at the middle ofthe { 3, 3, 5 } cells). Recall that the equatorial sphereof the { 3, 3, 5 } is also an lcosidodecahedron. The

second step consists then in filling the icosidodecahe-dra with half { 3, 3, 5 } polytopes. Inside an icosidode-

cahedron, 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 decom-

position of the 120 icosidodecahedra.These tetrahedra are neither equal nor all of them

regular. In order to make them more regular, one hasto suppose, as in the 2D case, that the curvature is notconstant throughout the polytope and for examplethat the interior ofthe 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 distortionsare generated. At the first iteration only Z12 and Z18 vertices are

generated. The Voronoi cell of a Z18 vertex is repre-sented on figure 14. Three disclination lines are crossingat 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 dis-

clination networks whose hierarchical local arrange-

ment is drawn in figure 15. Table II displays informa-tion about the Pn (up to n = 5). These data have been

derived with the help of the following matrix formula-

tion. 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. 14. - The Voronoi cell of a Z 18 vertex. Note the six

hexagons grouped by three around opposite vertices. The

disclination lines thread these hexagons.

Fig. 15. - Local view of the disclination network in second

I.F.M. example. The hierarchical structure is clearly visible.

The mean coordinationnumber 13.2 is easily derivedfrom 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 definedsome kind of a primitive cell of a crystalline structure.

But if the presence of irrational values for the eigen-vector components (more rigorously for the ratio

between components) is a sufficient condition for non-

periodicity, the lack of irrationality does not necessa-

rily lead to periodicity. In the present case (as in sec-

tion 3.2) the non-periodic character seems obvious

for the following reason. Suppose that, after p itera-

tions, a crystal Pp is obtained Its unit cell necessarily



1821

contains parts of the disclination network Dp (whoseperiodicity is also required). Since the next iteration

will add a new disclination network (interlaced with

those contained in Dp), the size of the Pp + 1 unit cell

is larger when scaled on the first-neighbour distance.

So the size of the unit cell increases upon iteration

which proves the non-periodic character of P. A more

quantitative proof can be given by looking (as in

section 3 . 2) to what happens on one of the 30 symme-

try spheres of the polytope. Triangles, pentagons and

hexagons are generated and, with obvious notations :

imply the non-periodic character of the polytope P 00.

4. Miscellaneous.

4.1 GEODESIC HYPERDOMES. - The deflation-trunca-

tion method can also generate much more regularstructures. By analogy with the generation of B. Ful-

ler’s geodesic domes on S2 (by iterative division of a

triangleinto 4 smaller

triangles)it is

possibleto build

geodesic « hyper » domes on S3. On S2 the originaltriangles of an icosahedron are filled with parts of

triangular lattice. Similarly { 3, 3, 5 } tetrahedral cells

are filled with parts of f.c.c. lattice. The precise proce-dure is described in Appendix E.

4.2 DEFLATION IN EUCLIDEAN SPACE. - Any Eucli-dean crystal composed of tetrahedral cells can be used

as starting structure for the deflation procedure. In

this case the mean coordination number (M.C.N.)varies upon iteration from the original crystal value

toward the asymptotic value associated to the trans-

formation matrix. There is an interesting case when

both values are equal, for instance when the transfor-

mation described in 3 . 2 is applied to the Laves phasestructure (M.C.N. = 13.333). If one applies this trans-

formation to the A15 structure (M.C.N. = 13.5) [22],the first iteration gives a new crystalline structure with

162 atoms in the unit cell and the M.C.N. = 13.358.

Note that this is exactly the same number as in

M932(Al, Zn)49 unit cell as proposed by G. Bergmanet al. [37]. Trying to identify their proposed structure

with a A15 with one iteration meets an unsolved dif-

ficulty : while their description shell by shell of the

atomic structure agrees with our structure, the sym-metry types do not coincide (body centre cubic in

reference and simple cubic for A15 and its iterated

versions).’

Note that the M.C.N. of statistical tetrahedral

honeycomb [23, 24] is 13.39. The successive iterationsdefine a family of crystalline structures with unit cell

of increasing size, inside which the vertex configurationis analysed in terms of interlaced disclination lines.

4. 3 LF.M. IN HYPERBOLIC SPACE.-

The generalmethod, in its «inflation form, applies to any regularstructure whose symmetry group is generated byreflections in the side a fundamental region. Hyper-bolic tesselations, either 2 ou 3-Dimensional, belongto this family. Hyperbolic tilings may also serve as a

template for idealized amorphous solids [5] and thekind of defects that would result from this descriptionhave been classified [25].

It is very easy to devise I.F.M. procedures on 2D

hyperbolic tilings which give rise to geodesic hyper-bolic «domes». More complex non-periodic asymp-

totic

tilings (belongingto the same

familyof recurrent

sets as the spherical ones of section 2) can also be

derived. 3D hyperbolic tilings could also be flattenedwith this method, but the work has not yet been

carried oiNote also that some exceptional hyperbolic tilings

are representations of the Bethe lattices and theHusumi cactus [26]. It is also possible to apply theI.F.M. procedure on these tilings, but some care must

be taken since one angle of the fundamental triangleis then equal to zero.

4.4 DETERMINISTIC OR RANDOM ITERATIONS. - In

chapter 3 we have separately described two differentexamples of I.F.M. for the polytope { 3, 3, 5 }. Sincethe same symmetry group G is associated with the

polytope and the different defect sets, it is possible,at each step of the procedure, to choose, any onebetween these two transformations. Let us define a

two-word alphabet { a, b} where « as (resp. « by)labels the first (resp. the second) type of iteration.

After n iterations the polytope Pn is fully characterized

by a « word », an ordered set of letters belonging to

the alphabet, containing n such letters. The complexityof the structure is encoded in the information content

of the wordSince

each

procedurehas a

specifictrans-

fer matrix, the resulting asymptotic polytope will

depend on the particular sequence of letters in theword. The vector space has to be enlarged in orderto take into account simultaneously the two iterations.

Indeed iteration «a» generates Z12, Z 14 . and Z16

sites while iteration « b » generates Z 12, Z 14 and Z 18

sites, when the starting configuration has Z12 sites.But it is necessary to introduce new relations to des-cribe the truncation of Z 18 sites (resp. Z 16 sites) underiteration a (resp. b). For instance the formal relationis obtained (in iteration a) :

The vector space in now 4 Dimensional and the vector



1822

transfer matrix are :

Let us now consider the two opposite cases :

- The sequence ofa and b is periodic.The polytopes Pn belong to a set of deterministic

recurrent structures. The asymptotic geometrical pro-

perties are given by a new matrix which is the ordered

product of Ta and Tb over a period The simplest cases(short periods) gives rise to structures which are highlyordered in spite of their non crystalline intrinsic cha-

racter. In particular they present a high degree oforientational order. Atomic arrangements with per-

fect icosahedral order become very popular since theirrecent discovery in nature [27] in the case of rapidlyquenched Al-Mn alloys. The general name of o quasi-crystal » has been proposed to label those structures

[28]. Penrose tilings in 2D provide a very nice exampleof patterns with full pentagonal orientational orderand lack of translations. Several kinds of 3D Penrose-

like structures have been proposed [29-32], one ofwhich (using rhombohedral « bricks ») having a theo-retical diffraction pattern very similar to the experi-mental one [32]. Penrose-like models belong to a

larger family of (recurrent-like) structures possessingorientational long-range correlations. The hierarchicalmodels described in this paper belong to this family.Their icosahedral symmetry is a direct consequenceof the I.F.M. procedure where the icosahedral group

(and its extension to SO(4)) is of constant use. The

occurrence of icosahedral long-range order has been

verified by Nelson and Sachdev [33] who calculated

numerically the structure factor of models constructed

(with iteration of type « as) directly in Euclidean

space in the deflation-truncation scheme. The corres-

pondence between the theoretical and experimentalstructure is not exact but they propose that othermetallic alloys with large atoms occupying the Z16

sites could conceivably form such a structure. By sui-tably mixing iterations « a » and « b » (and even otherones to be introduced) we think that a large class of

possible alloys may then be described The advantageof the present description over the Penrose-like

approach is that here we give the precise location ofatoms and disclination defects. The atomic positionsin Penrose-like models are still to be determined. We

conjecture that their defect set will also consists in

interlaced hierarchical disclination networks, perhapsvery similar to the ones presented here [34].

- The sequence of a and b is random.

Much less can be said about the geometrical pro-

perties of the asymptotic polytope. We can only saythat they lie in between those given by pure « a » and

pure « b » iterations. The structure is less regular andis probably more suited for the description of amor-

phous 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 para-

meter when going from a periodic to a randomsequence of a and b type of iteration.

5. Conclusions.

We have described a powerful Method for the gene-ration of non-crystalline structures. The purpose was

to provide a method for flattening 3D polytopes and

get realistic models for non-crystalline materials.Indeed polytopes with prescribed perfect symmetryhave proved to be very powerful templates for amor-

phous solids with similar local order.The inflationform of the I. F. M. is easily implemented

on a computer and yields to simple determination ofthe location and weight for the disclination line defects.

More work remains to be done in this direction. For

example in 2D the relation with the theory of star

polyhedra and Riemann surfaces should be explored[35]. Indeed the large triangle of section 2 is a funda-mental region for the star polyhedra { 2, 3 } and gene-rates a multiple covering of the sphere S2. Perhapsthis could help in understanding the propagation of

orientational order. The analysis of the corrugatedspheres (or hyperspheres) on which the polyhedra(-topes) Pn minimize their distortion will be given in

a

forthcomingpaper in term of

Regge-like analysis[7, 24] and fractal dimension. We have shown how itis possible to generate some disorder in the polytopesby mixing two I.F.M. procedures sharing the same

symmetry group. It could also be interesting to studythe case where one procedure uses a subgroup of the

group used by the other one. Suggestions in thatdirection were already present in an early paper [4].The « deflation-truncation » form of the I.F.M. has

also been described. It yields a very simple matricial

description of the iteration. The geometry of thederived structures is perhaps easier to visualize in this

approach. No metrics is now involved and the deco-

ration procedure can be done directly in Euclideanspace. The fact that orientational correlations are

still present [33], even with a network of disclination

lines, is probably due to the hierarchical nature of thedefect set as well as its symmetry properties.The concept of hierarchical defect structure is

certainly very important in understanding the micro-

scopic properties of the icosahedral « quasicrystals ».

The description ofcomplex systems in term ofordered

regions pierced by defect networks has proved to be

powerful. In a crystal with large unit cells this defect

network is identified with the Frank-Kasper skeleton,while in

amorphous systemsdisclination line networks

play this role. Within the framework of the corrugatedspace description [7], the Frank-Kasper lines and the

disclinations are unified and are the loci of curvature



1823

in the discretized (atomic) underlying corrugated geo-

metry. Any tetrahedral partition of R3 can be analysedin this way. The case in which different kinds of poly-hedra are present (like for instance tetrahedra and

octahedra [36]) could also be treated. Now quasi-

crystalsare intermediate between

crystalsand amor-

phous structures and can also be described in terms

of defect networks. 3D Penrose-like structures give a

good fit to the experimental results on AIMn alloysand the above described hierarchical models are pos-sible models for other structures, yet to be found

Penrose tilings have a self similar geometry and theycan very probably be described in terms ofhierarchicaldefect set (when precise atomic positions are given).

A defect set description could also be done directly onthe rhombohedral cells. Note that we have previouslydone this description on 2D Penrose tilings [34].

*

Finally let us stress that the whole method works

for any kind of local symmetry, tetrahedral and cubicas well as icosahedral. In 3D, the symmetry groupof all the regular polytopes is an extension of theusual point groups and consequently the polytopescan be flattened in a similar way.

Acknowledgments.

It is a pleasure to acknowledge D. P. Di Vincenzo fora critical reading of the manuscript and appropriatesuggestions.

Appendix A.QUATERNIONS. - We briefly recall some propertiesof quaternions. For a more complete description, see

references [16] and [17]. A quaternion a can be written as :

with the following rules

The quaternion a can also be written as a scalar partSa and a vector

partVa :

The conjugate a of the quatemion a is given by :

A quaternion is said to be real if V a = 0 and pure

imaginary if Sa = 0.Unit quaternions (with norm Na = aa = 1) are of

special interest since they form a continuous non

Abelian group Q which is a nice illustration of a topo-logical group. Indeed each element of Q is in one to

one correspondence with the points of a unit radiushypersphere S3. Because Q is a group, it can also label

displacementson S3. This is very similar to the relationbetween the group of unit complex numbers and the

unit circle in a plane, with, in the quaternion case,some added complexity because the non Abelian

nature of Q.

Appendix B.

THE BINARY ICOSAHEDRAL GROUP.-

In Euclidean3D space, the polyhedral symmetry groups are finite

subgroups of SO(3), the continuous group of direct

rotations that leave an origin fixed However SO(3)is not simply connected and is double-covered by the

group SU(2). This 2 : 1 homomorphism is extended

to the finite subgroup, the pre-image of a usual poly-hedral group being called a binary polyhedral group

or a double group. Since the group Q of unit quater-nion is isomorphic to SU(2), it is possible to representthe double group elements by unit quaternions [17].We shall briefly describe the case of the binary ico-

sahedralgroup

Y’

(the prime denotingthe « double

group » case). The knowledge of Y’ is very useful forthe 2D and 3D iterative mappings described in the

text Indeed Y’ is homomorphic to the icosahedral

group Y which is the symmetry group of the icosido-

decahedron (see section 2 of the text). Also Y’ is used

to generate the polytope { 3, 3, 5 } symmetry group

as will be indicated below. For sake of simplicity webegin here with the direct symmetry groups of the

regular bodies (polyhedra, polytopes). Take care that

the full treatment of the iterative process requires the

knowledge of the full symmetry groups (with both

direct and indirect operations) which are subgroups

of 0(3) in the polyhedral case. The quaternion nota-

tion allows a simple determination of the full group

elements (see the polytope { 3, 3, 5 } case below).The image in Q of the dihedral group D2 is the

group V’ (of order 8) consisting of the 8 principal unit

quatemions (noted as ordered quadruplets) :

( ± 1, 0, 0, 0), (0, ± 1, o, 0), (0, 0, ± 1, 0), (0, 0, 0, ± 1 ) .

The lift (image) ofthe tetrahedral group (of order 12)is the group T’ (oforder 24) which contains V’ togetherwith the 16 elements

Consequently T’ can be written in the standard

form [17] :

where Q Ar means the union of n sets Ar which haver=1

no points in common and with the convention that a

quatemion raised to the power zero equals (1, 0, 0, 0),the neutral element of Q.The group Y’ can then be written



1824

Here T is the golden ratio : T = (1 + /5-)/2. The

120 elements of Y’ correspond to the 120 vertices of

polytope { 3, 3, 5 } on a unit radius hypersphere S3.

The (direct) symmetry group of polytope { 3, 3, 5 }is a subgroup of SO(4) and can also be specified withina quaternion representation because of the isomor-

phism SO(4) = SU(2) x SU(2)/Z2. Z2 is the twoelement group with the two quaternions ( ± 1, 0, 0, 0)as elements.

If q E Q denotes a point on S3, the transformation

q --+ I qr-’ with 1, r E Q is an element of SO(4). The

direct symmetry group G’ of polytope { 3,3,5 } is

given by

Since the order of Y’ is 120, the quotient by Z2implies that the order of G’ is 7 200. The total sym-

metry group G of polytope { 3, 3, 5 } also includes

indirectorthogonal transformations, analogous

to

reflections for ordinary 3D discrete groups. These

are given by

This adds 7 200 new elements and gives the group G

of order 14 400.

Appendix C.

THE ORTHOSCHEME. - The polyhedral groups are

particular examples of groups generated by reflec-

tions [3]. The planes of reflection (and their image

under mutual reflection) have acommon

point andtheir intersection with a sphere centred on this pointgives rise to a pattern of spherical triangle calledMobius triangles or orthoschemes (they are right-angled triangles). Any of these triangles is a funda-mental region for the action of the group noted [p, q],where n/p and n/q are the values of the two other

angles in the Mobius triangle. The order g of [p, q]is the number of such triangles that cover the sphere(of area 4 n), that is [3] :

It is verified that the full icosahedral group [3, 5]has order 120. In 4 Dimensions the regular tessela-

tions are also associated with a discrete group with

a fundamental region. These groups are generatedby reflections in the faces of a characteristic tetra-

hedron, the orthoscheme. The orthoscheme is easilyconstructed out of the flag of the polyhedron (orpolytope) [16]. For any regular polytope (in NDimensions), a flag (no, 7rl, ---, nN) is defined to be

the figure consisting of a vertex no, an edge 7r, con-

taining 7Co,a face

7r2 containing 7r,,a

(hyper)face

TEp ( p N ) containing np- 1, and a cell aN contain-

ing 7rN - 1. The orthoscheme is the simplex whose

N + 1 vertices are the centres of the np. For instance

in the case of the dodecahedron, the characteristic

region is the triangle which vertices are a dodeca-

hedron vertex, the mid-point of an edge which ends

at this vertex, and the centre of a pentagonal face.

Note that because they share the same symmetry

group, this triangle is also a fundamental region for

an icosahedron and an icosidodecahedron. Similarlythe two polytopes { 3, 3, 5 } and { 5, 3, 3 } share thesame orthoscheme. Values for the angles in such

characteristic tetrahedra are given elsewhere for any

polytope { p, q, r [16].

Appendix D.

DESCRIPTION OF THE ALGORITHM IN THE 2D CASE. -

An important step of the I.F.M. consists in the iden-tification ofthe two triangles MoM!M2 and MoM1M2.The large triangle MoM1M2 contains seven replicasof the small

triangle.We use two

barycentric(homo-

geneous) coordinate systems based on the tetrahedra

OMOMIM2 and OM’M,M’ (see Fig. 4), where 0is the centre of the sphere.

In the first system, a point M has four coordinates

This can be written in symbolic notation : M = DO

+ao Mo + al M1 + a2 M2. In the second system,the coordinates will be specified by a prime : (D’, ao,a’1, a’2).The algorithm, at a given iteration, is the following :1) Take each point M’, of the polyhedron Pp

lying inside the triangle MoM1M2 with coordinatesI-, I 1 ’B.

0 2

2) Identify the triangles MOMlM2 and MoMlM2.This is done by interpreting the coordinates of M’ as

the coordinates of a transformed point M, in thefirst barycentric system, lying inside the small triangleMOMlM2’ M has coordinates (Q, ao, ai, a2) such

3) Take the orbit of the point M under the group

operations. IfM

is at generic position there will be119 images on the sphere.4) One iteration is finished when this procedure

is done for each point M’ of the first step. The set

formed by the new points gives the vertices of poly-hedron Pp+ 1. Among the vertices of Pp+ 1, those wholie in the triangle MoM1M2 will be kept in the next

iteration where they constitute the set of points M’

in the first step of the algorithm.If one is only interested in building a given poly-

hedron Pp without the full knowledge about the

previous polyhedra, a quicker algorithm can be

used.Since,

at a

given iteration, onlythose

pointwhich lie inside MoM1M2 are used, it is not necessaryto construct the full orbit of points (step 3 above) at

the preceding iteration. It is enough to take a restricted



1825

orbit which maps the triangle MoMl M2 into thesix other replicas which, altogether with MoMlM2,cover the large triangle M0M 1 M2. Let us call Qo, Qiland Q2 the vertices of the ith triangle (i = 0 to 6

andi = 0 refers to the triangle MoMl M2).For sake of simplicity we shall now ignore the

first coordinate Q’ which play no particular role inthe 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 ofM in theseven orthoschemes which cover MoM1M2 (includingM itself for i = 0). If M’ is in generic position (none

of ao, ai, or a2 is equal to zero), the seven matrices /are used. If ao = 0 (resp. al, a2), it remains only 4

(resp. 5, 5) matrices yi.

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-

hedral cells can be filled with part of F.C.C. structure

with a parameter as small as wanted. Defects in thestructure are located on the edges of the primitive{ 3, 3, 5 } which becomes, with the F.C.C. parameteras unit length, highly separated one from each other.The procedure consists in adding new vertices on

the mid-point of edges. A tetrahedron is dividedinto 4 tetrahedra and one octahedron and one octa-

hedron is divided into 8 tetrahedra and 6 octahedra

(see Fig. 16). If the structure at an iteration i is charac-

terized by the vector X(i) whose components NT,No are the number of tetrahedral and octahedralcells the vector X(i+l) can be obtained in matrix

notation by :

Fig. 16. - Derivation of the hyper-geodesic dome.

a) Decomposition of a regular tetrahedron into 4 tetrahedraand 1

octahedron,all

regular. b) Decompositionof a

regularoctahedron into 8 tetrahedra and 6 octahedra, all regular.These divisions give rise to larger and larger F.C.C. portionsinto the original polyhedra.

The largest eigenvalue of the matrix T (the Perron

root) corresponds to an asymptotic structure where

the relative fraction of octahedral and tetrahedralcells remains constant. Here Ap = + 8 and the relative

fraction is two tetrahedra for one octahedron (as in

the F.C.C. structure). A polytope P. (for large n) can

be thought as large tetrahedra filled with F.C.C.structure and glued along the edges of a ( 3, 3, 5 } in

order to cover S3. This is very similar, with one

dimension added, to the case of B. Fuller geodesicdomes (while its usefulness in architecture is less

obvious !). Huge portions of Pn, mapped on the

tangent Euclidean space, can be used to model some

very large metallic or rare gas aggregates after crys-tallization. Note that distance between line defectsincreases upon iteration. This is in contrast with thecase of hierarchical structures described in section 3,in which defects remain close to each other.



1826

References

[1] SADOC, J. F., DIXMIER, J., GUINIER, A., J. Non-Cryst.Solids 12 (1973) 46.

[2] KLEMAN, M., SADOC, J. F., J. Physique Lett. 40 (1979)L-569 ;

SADOC, J. F., J. Physique Colloq. 41 (1980) C8-326 ;

SADOC, J. F., J. Non-Cryst. Solids 44 (1981) 1.

[3] COXETER, H. S. M., Regular polytopes (Dover, New

York) 1973.

[4] SADOC, J. F., MOSSERI, R., J. Physique Colloq. 42 (1981)C4-189 ;

SADOC, J. F., MOSSERI, R., Philos. Mag. B 45 (1982) 467.

[5] KLEMAN, M., J. Physique 43 (1982) 138.

[6] SETHNA, J. P., Phys. Rev. Lett. 51 (1983) 2198.

[7] GASPARD, J. P., MOSSERI, R., SADOC, J. F., in The

structure of Non-Crystalline Materials 1982 (Tay-lor and Francis) edited by P. H. Gaskell, J. M.

Parker and E. A. Davis, 1983 ; Philos. Mag. B50 (1984) 557.

[8] RIVIER, N., Philos. Mag. B 40 (1979) 859.

[9] TOULOUSE, G., KLEMAN, M., J. Physique Lett. 37 (1976)L-149 ;

MERMIN, N. D., Rev. Mod. Phys. 51 (1979) 591.

[10] NELSON, D. R., WIDOM, H., Nucl. Phys. B 240 [FS 12](1984) 113.

[11] TREBIN, H. R., Phys. Rev. B 30 (1984) 4338.

[12] SADOC, J. F., MOSSERI, R., Amorphous Materials,edited by V. Vitek, AIME Conf. Proc. (1983) 111.

[13] FRANCK, F. C., KASPER, J. S., Acta Crystallogr. 11

(1958) 184.[14] MOSSERI, R., SADOC, J. F., J. Physique Lett. 45 (1984)

L-827.

[15] PUGH, A., Polyhedra : a visual approach (Univ. of Calif.)1976.

[16] COXETER, H. S. M., Regular Complex Polytopes(Cambridge Univ. Press) 1974.

[17] Du VAL, P., Homographies, Quaternions and Rotations

(Oxford) 1964.

[18] DEKKING, F. M., Adv. Math. 44 (1982) 78.[19] GARDNER, M., Sci. Am. Jan. 1977, p. 110.

[20] GEFEN, Y., MANDELBROT, B., AHARONY, A., KAPITUL-

NIK, A., J. State Phys. 36 (1984) 827.

[21] MOSSERI, R., Thèse d’état Orsay, 1983.

[22] SADOC, J. F., J. Physique Lett. 44 (1983) L-707.

[23] COXETER, H. S. M., Ill. J. Math. 2 (1958) 746.

[24] SADOC, J. F., MOSSERI, R., J. Physique 45 (1984) 1025.

[25] KLEMAN, M., DONNADIEU, P., to be published.[26] MOSSERI, R., SADOC, J. F., J. Physique Lett. 43 (1982)

L-249.

[27] SHECHTMAN, D., BLECH, I., GRATIAS, D., CAHN, J. W.,Phys. Rev. Lett. 53 (1984) 1951.

[28] LEVINE, D., STEINHARDT, P. J., Phys. Rev. Lett. 53

(1984) 2477.

[29] MACKAY, A. L., Sov. Phys. Crystallogr. 26 (1981) 517.

[30] KRAMER, P., Acta Crystallogr. A 38 (1982) 257.

[31] MOSSERI, R., SADOC, J. F., in The structure of Non-

Crystalline Materials 1982 (Taylor and Francis

Ltd) edited by P. H. Gaskell, J. H. Parker and

E. A. Davis, 1983, p. 137.

[32] DUNEAU, M., KATZ, A., preprint.[33] NELSON, D., SACHDEV, S., preprint.[34] SADOC, J. F., MOSSERI, R., J. Non-Cryst. Solids 61-62

(1984) 487.

[35] This point arose from a discussion with C. Itzykson.[36] NINOMIYA, T., in Topological Disorder in Condensed

Matter (Springer) 1983, p. 40.

[37] BERGMAN, G., WAUGH, J. L. T., PAULING, L., ActaCrystallogr. 10 (1957) 254.

Documents

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