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Inflationary character of Penrose tilings
Yuval Gefen, Maurice Kléman, Andrdi Pavlovitch, Jacques Peyrière
To cite this version:
Yuval Gefen, Maurice Kléman, Andrdi Pavlovitch, Jacques Peyrière. Inflationary character of Penrose
tilings. Journal de Physique, 1988, 49 (7), pp.1111-1118. �10.1051/jphys:019880049070111100�. �jpa-
00210794�
Inflationary character of Penrose tilings
Yuval Gefen (1), Maurice Kléman (2), Andrdi Pavlovitch (3) and Jacques Peyrière (4) (1) Department of Nuclear Physics, the Weizmann Institute of Science, Rehovot, 76100 Israel
(2) Laboratoire de Physique des Solides, associé
auCNRS, Université de Paris-Sud, bâtiment 510, 91405 Orsay, France
(3) Section de Recherches de Métallurgie Physique, Centre d’Etudes Nucléaires de Saclay,
91191 Gif
surYvette Cedex, France
(4) Laboratoire de Mathématiques, Université de Paris-Sud, batiment425, 91405 Orsay, France
(Reçu le 3 décembre 1986, révisé le 12 f6v?ier 1988, accepté le 17
mars1988)
Résumé.
2014On décrit les propriétés d’inflation des pavages de Penrose à l’aide des matrices de transfert de fractals (TMF) qui mettent l’accent sur la nature auto-similaire de
cespavages apériodiques. Nous calculons explicitement les valeurs propres et les vecteurs propres correspondants, dans les
casd’un pavage à six tuiles
élémentaires et d’une famille de pavages à neuf tuiles élémentaires, certains étant déterministes, d’autres
nel’étant pas. Nous discutons et interprétons géométriquement leurs propriétés d’itération. Nous étudions aussi
un
type spécial de défauts d’inflation des pavages déterministes, qui sont le résultat d’un défaut d’accolement entre tuiles, introduit à
uncertain niveau d’itération, l’itération étant poursuivie par la suite. En supposant que
l’expression du coût
enénergie de
cesdéfauts comprend
unterme linéaire
enles densités des diverses tuiles,
nous
montrons que l’énergie d’une région fautée de taille R est
enRf, où f 2 est
unexposant non-trivial.
Dans certains cas, f est plus grand que la valeur canonique 1.
Abstract.
2014The inflation of Penrose tilings is described by means of Transfer Matrices of Fractals (TMFs)
which emphasize the self-similar nature of these aperiodic tilings. The related eigenvalues and eigenvectors are explicitly calculated for Penrose tilings employing six elementary shapes and for another Penrose-like family (with nine shapes), which includes both deterministic and non-deterministic tilings. Average iteration properties and their geometrical interpretation
arediscussed. We also study
aspecial type of inflated defects in
a
deterministic Penrose tiling. These defects
arethe result of
amismatch between the tiles, introduced at
acertain iteration stage and then inflated iteratively. Assuming that the expression of the energy cost for such defects includes
aterm which is
alinear function of the shapes densities,
weshow that the energy of
adefective
region of linear size R may scale
asRf, where f 2 is
a nontrivial exponent. In certain cases,
f is larger than the canonical value 1.
Classification
Physics Abstracts
05.50
-61.70
-61.90
1. Introduction.
Penrose tilings (PT) in two dimensions (d
=2), and non-periodic three-dimensional space fillings have
attracted much attention both from the theoretical
[1-8] and experimental [6, 9] view point. Alloys like [6, 9] A186Mn14 are believed to have a new state of
matter characterized by icosahedral orientational correlations and therefore no translational invari-
ance. This type of ordering may be closely related to
the PT and its generalizations.
In this note we concentrate on the self-similar nature of the Penrose tiling. Under dilation trans-
formation a PT maps onto a (generally distinct) PT [1-4]. We study this inflationary property by employ-
ing the TMF (Transfer Matrices of Fractals) method.
The mathematical properties and certain applications
of TMFs have been discussed in references [10,11].
TMFs for the geometries discussed in the present work will be constructed and studied within the
analysis given below. We have tried to present this analysis in a self-contained manner. At this point we only remark that within this approach dilation trans-
formations are described in terms of interscale transfer matrices, which relate the basic geometrical configurations on a large scale to those more elemen- tary shapes which appear on a smaller scale. Some of the eigenvalues and eigenvectors of these mathemati- cally interesting TMFs may also be assigned direct geometrical interpretation. Only short range statisti-
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019880049070111100
1112
cal properties of our tilings (as opposed to long
range correlations) are calculated within this ap-
proach. The same TMF method enables us to study
average statistical properties of non-deterministic
tilings (the latter term refers to tilings whose rules of iteration may be chosen at random at each step of
iteration).
After describing various PTs in terms of TMFs, we study a special family of defects, hereafter denoted
as inflationary defects. These defects are introduced
at a certain stage of the iteration process and are then iterated by means of the TMF until a micro-
scopic scale is reached. We next discuss the energy cost of such defects. It consists of several contri-
butions, including the elastic energy and the energy associated with the mismatches between the defec- tive domain and the surrounding perfect tiling.
Usually one expects that the latter term will scale
linearly with the perimeter R of the domain. Here
we assume that certain energies are associated also with the various microscopic geometrical shapes (which, in turn, may correspond to certain atomic
configurations)
-not only with the boundaries
between microscopic shapes. Following this assump- tion we show that the expression for the energy cost of the defects considered by us now includes a term
which scales up as R f . We demonstrate that
f may satisfy 1 f 2. Physically this means that
due to the inflationary character of the Penrose tiling
the energy associated with the formation of a
domain may scale up faster that the length of the
domain wall. It is straightforward to generalize the present discussion to a hierarchy of inflationary
defects on various length scales. A general
mathematical study of the TMF technique applied to
the PT is presented by one of us elsewhere [8]. In principle it can be generalized to the d
=3 case.
We emphasize that this work is primarily a geo- metrical study of certain aspects of the inflationary
symmetry of PTs. Much is yet to be done before the relevance to realistic systems can be established.
Since we do not have detailed information about the
dynamics involved in the formation of quasi-crystals,
the physical existence of inflationary defects is not
yet proven. We also stress that we do not offer
microscopic justification for the energetics of the tilings discussed here (although the assumption about
a microscopic energy associated with each particular microscopic configuration is not implausible). What
we are trying to do here is to establish a relation between PTs and more conventional fractals, specifi- cally by employing the TMF approach. We present
an eigenanalysis of certain TMFs and study its geometrical interpretation, we distinguish between
deterministic and non-deterministic tilings and also
discuss inflationary, non-topological defects. Since
our purpose here is to demonstrate certain aspects of inflation symmetry we felt comfortable using specific
presentations of the two dimensional 5-fold sym- metry.
The PTs used here are obtained by starting from a large polygon (e.g. a pentagon), decorating it accord- ing to a certain rule, and repeating this decoration iteratively for each of the resulting smaller polygons,
until one reaches the microscopic scale. We consider
two examples. The first, presented in section 2, is adapted from reference [3] and consists of 6 tiles.
We use this example to present the basics of the TMF method. The second one, discussed in sec-
tion 3, is inspired by the original Penrose construc-
tion (Ref. [lb]) and employs 9 basic shapes. It leads
to a family of 9 x 9 TMFs. The last section, deals
with the scaling properties of inflationary defects in
the PT. In the appendix we discuss a relation
between the 6 shape tiling and the 9 shape tiling presented earlier.
2. A Penrose tiling with six tiles.
The iterative inflation process, shown in figure 1,
can be described in terms of linear equations that
count how many offsprings of each type at the
( j + 1 )-st iteration step are formed from a given (large) shape at the j-th step.
In this example, we obtain the following transfer
matrix
This matrix connects two six-vectors before and after
an iteration step. The entries of a six vector are
proportional to the frequency of each basic shape in
the tiling. Taking for example the fourth column of the TMF, we transform (03B4) by iteration into a
(a) :
The remaining 5 obtuse triangles of (6) are not
counted on the right hand side of this equation, as they belong to 5 pentagons of type (e) which are
accounted for in the iteration of the neighbours of (6).
As another example, ( y ) iterates into (6) and (8 )
But in this case we have included along with the
acute triangles the obtuse ones too, the latter coming
from neighbouring tiles. More generally, the careful reader will notice that the matching rules consist of
removing obtuse triangles and sticking them to acute
ones to form pentagons, respecting at the same time
Fig. 1.
-(a) A perfect Penrose tiling obtained by iterative decoration of six shapes (from Ref. [3]). Two iterations
aredrawn ; (b) the six basic shapes. The notations 0, 1,
...is introduced to define matching rules between neighbouring shapes : 0 should match with 0 etc. (c) The decoration process leading to the next iteration step. (d) An iteration of
atile
(y ).
the matching of 0 with 0, 1 with 1 and 2 with
2, at all scales. This process is illustrated in figure Id
for equation (3). The eigenvalues of the transfer matrix are, in decreasing order,
1+-,15-
.where T
= 2
.The components of the right eigenvector associated with A 1 are proportional to
the frequencies of the various tiles a, 03B2, ’Y, ..., according to Perron-Frobenius theorem (see
Ref. [10]). Moreover, since the scaling ratio is
T, the Perron-Frobenius theorem implies that the
number of tiles within a distance r" is of the order of
À i ; hence the corresponding fractal dimensionality,
in the sense of reference [10] (p. 343), is
The other eigenvalues of this matrix provide us with
corrections to scaling. This point will be discussed in
the context of the second example, presented below.
3. A Penrose tiling with 9 tiles.
The second example employs 9 basic shapes which
are shown in figure 2a. The original PT consisting of
these 9 tiles is shown in figure 2b [12]. We use this example to demonstrate the geometrical interpre-
tation of the TMF and also to introduce the notion of non-deterministic tiling.
The inflation rules of the basic shapes are indicated
in figure 2a. For example, a star (f) will be divided,
in the next iteration step, into 5 pentagons (e),
5 triangles of type (c) and one small star (f). This is
described by the equation
Similar equations hold for the other shapes. Note, however, that a subtle problem arises at this stage. A
large pentagon yields, in the subsequent iteration step, 6 smaller pentagons and 5 triangles which may be either of type (a) or (b). One has to specify the
type of these triangles. We choose these triangles
with relative frequencies 5 x and 5 (1- x ) respect-
ively. The equation for the pentagon then reads
1114
Fig. 2.
-The original Penrose tiling obtained by iterative decoration of nine shapes. (a) The nine basic shapes, each
iterated once ; (b) 3 iterations steps. A triangle of type (a) and
apentagon
aremarked with bold lines.
This equation is valid only on the average. Similar
equations are obtained for the shape (h) (with
relative frequencies 3 y and 3 (1 - y ) for triangles (a)
and (b) respectively) and for the shape (i) (with
relative frequencies 2 z and 2 (1- z ) for (a) and (b) respectively). Thus we can write down a set of 9 linear equations for the iteration rules. These are
expressed in terms of a TMF, M (x, y, z ), operating
in a 9-dimensional shape space.
The TMF approach is devised to give information about the frequencies of the various shapes at each
iteration step. This approach can be also applied to
describe random constructions [10] where the iter-
ation rules are chosen at random (and are indepen- dent) for different regions in space and/or for different iteration steps. In that case relative fre- quencies should be replaced by probabilities.
We next discuss in more detail the geometrical significance of the parameters x, y, z. Figure 3 shows
the tiling obtained after three iterations for the case x
=y
=z =1. We observe that only five shapes
appear but one more iteration would give rise to
seven shapes : (a), (c), (e), (f), (g), (h) and (i). Note
that in general the tiling obtained for a set of values x, y,
z(an example being Fig. 3) is a generalization
of the original Penrose tiling allowing, for example,
a matching of two shapes of type (h) (this was
forbidden in the original tiling, Fig. 2b).
For x
=k,15, y
=k2/3, z
=k3/2 (ki, k2, k3 integers, 1, x, y, z > 0) it is possible to construct
deterministic tilings. By this we mean that it is possible to find iteration rules which leave no place
for choice at any given iteration step. For the above particular choice of x, y, z the iteration scheme is
Fig. 3.
-Three iterations employing 7 shapes ; the tiling
obtained is not perfect (note e.g. that tiles of type (h)
arematched with other tiles of type (h) ; such
amatching is
not allowed in the perfect PT, cf. Fig. 2b).
deterministic. For example, we may define five directions in space, parallel to the mediatrices of
triangles (a) and (b), and to each of these directions
attach a triangle of either type (a) or (b). This way
we generate an anisotropic homogeneous tiling (1).
This procedure, leading to a deterministic tiling,
cannot be employed for a general set of x, y, z.
(1) Another possibility is to color the tiles and redefine the rules of inflation and matching in
a mannercompatible
with the given values of x, y and
z.Let us now give a general argument proving that,
for all I > x, y, z , 0, the largest eigenvalue of the
matrix M is A, = T 4. A vector in the shape space
I u) = I U (1), U (2),
...,u (9» describes a configuration
which consists of u (1) tiles of type (a), u (2) tiles of
type (b), etc. (Hereafter we use for convenience the bra-ket notation). Starting from an initial configur- ation luo), after n decoration steps we obtain a configuration described by the vector Mn I uo). If
(S I denotes the left vector the components of which
are the areas of the various shapes, one has the
relation
which expresses the area conservation under in- flation. Since it follows from the Perron-Frobenius theorem that (S I tr I uo> is of the order of A1n, we
have A, = T 4. Furthermore, the Perron-Frobenius theorem also implies that At n tr approaches I AI> (All I when n goes to infinity, if the right and
left eigenvectors I AI> and (All are chosen such that
(All I A1/ =1. Thus
As this last relation holds for any I uo), we obtain
that (S I is a left eigenvector of M associated with
Al.
The characteristic polynomial of M is
Hence, five eigenvalues do not depend on x,
y and z. The left eigenvector (Ai j [ (whose compo-
nents are proportional to the areas of the shapes) is given (up to a normalization factor) by
The equation (Ai (M A, I A, can be written an 9
equalities, each involving numbers of the form
a + bT, with a, b rationals. These equalities should
still hold if we replace each factor of a + bT by its conjugate in Q(J3), a - b 1 , i. e. , if we replace
T
each factor
Tby 1- T. Thus we obtain a left eigenvector associated with A
=T - 4 (= ( 1- T )4 )
by replacing
Tby 1-
Tin (All. From equation (12)
it follows that any product of n matrices M, even
with different sets of values for x, y, z, has
T 4 n as its largest eigenvalue. However the other
eigenvalues of such a product need not be the product of the individual eigenvalues. Note also that the right eigenvector A1 . does depend on x,
y, z.
When the non-zero eigenvalues are non-degener-
ate, the left and right eigenvectors which correspond
to these eigenvalues can be chosen to form a biorthogonal system and we have, for n
>1
As we have already stated, for large n the term corresponding to the largest eigenvalue A, dominates
the above expression [10,11]. It follows [10, 11] that
the entries of A1 ) represent the asymptotic fre-
quency at which various basic shapes appear in the structure. Since for large n the number of tiles
increases by a factor a1 at each iteration (which
involves a rescaling factor Tb, the fractal dimen-
sionality of the structure, D, is given by D
=In At/In T 2. Since the largest eigenvalue A, does not depend on x, y, z and is always equal to T 4, the
fractal dimensionality of the tiling is always 2. This is geometrically evident, since we are dealing here with tiling of the plane. Note once more that the above analysis gives us information on the frequency of the
various shapes, but not on their spatial arrange- ments.
The smaller eigenvalues introduce finite size cor-
rections [11] to the asymptotic behaviour. For
example, after n decoration steps the number of shapes of type j ( j
=(a), (b), ... (i)) will be (pro-
vided that either ( Ai [ uo ) # 0 or (Azl uo> #: 0),
where R is the size of the sample (R = T 2 n ) and
In our notation, I u) j is the j-th entry of the vector
lu). We note that f -- 2 can be interpreted as a secondary fractal dimension [10]. The existence of a
negative eigenvalue means that the approach to asymptotic distribution involves decaying oscillations in the relative number of shapes and in the correc-
tions to the energy (see Sect. 4 below). The exponent
f is larger than 1 when the second eigenvalue A2 is larger than T 2. For the deterministic tiling
discussed above this occurs for the values of
(kl, k2, k3) listed in table I.
Table I.
-Values of (kl, k2, k3) ) for which
A2 :::’ T 2 (see text).
1116
4. Inflationary defects in deterministic tilings.
We shall now study a special family of non-topologi-
cal defects, introduced at a certain iteration step and then inflated. As has been stated in the introduction,
the physical relevance of such inflationary defects is
not clear, and at this stage we consider them because of their geometrical interest. After explaining how inflationary defects are generated, we shall discuss the scaling of an energy term associated with the
frequencies of the microscopic configurations (this is
not the term related to whether matching rules are obeyed or not). Our analysis is fairly general and
should hold for the scaling of other quantities which
are linear .functions of the frequencies of the micro- scopic shapes.
We shall confine the discussion to deterministic
tilings, as discussed in section 3. Let us first consider,
as an example, the pentagon marked by heavy lines
in figure 2b. If we rotate this pentagon by 72° we
shall generate disallowed matching (e.g. two shapes
of type i with a common edge), hereafter denoted as
mismatches. We may associate energies with each
such a mismatch ; consequently the energy cost of such a domain is proportional to the total length of
its boundary. If, rather than rotating this pentagon by 72° (or an integer multiple thereof) we rotate it by
an arbitrary angle, there will be an elastic energy (in
addition to the domain boundary energy) which
should be added to the total energy cost. The point
we are trying to emphasize here is that the core of a
defect, represented by a domain of size R, may involve an additional energy term which scales as
Rf, f > 1. Let us assume that the elementary geometrical shapes represent certain local atomic
configurations, each with a characteristic energy. At this point we ignore energy terms due to shape- shape interactions (mismatches) ; these should be
included separately and will lead to a term pro-
portional to R 1. A microscopic shape of type j (j
==(a), (b),
...(i)) possesses energy Ej. The
energy associated with the perfect tiling is
where
This perfect tiling is assumed to correspond to a local
energy minimum in the global configuration space.
We now replace part of the tiling, e.g. the triangle
marked with bold lines in figure 2b (which is a type
(a) shape iterated twice), by another domain (e.g. a (b)-type triangle iterated twice), keeping the total
area constant. In this particular example no elastic
strains are generated since the former domain is
replaced by a shape of identical boundaries, but in general an elastic energy term should be included.
Not only the mismatch energy, which is proportional
to the length of the boundary of the new domain, has
to be accounted for, but also the statistics of
elementary shapes has been changed, when the new
domain has been introduced. More generally we
shall take R = abp (a being a microscopic length).
The sample size is L = ab n (n > p ). In our specific example the rescaling factor at each iteration step is b
=T 2. In order to construct a structure with a core
defect one starts as before from the scale L, and iterates (n - p ) times, obtaining the vector un _ p ) .
We now introduce some perturbation on this scale
[13].
thus creating a defect whose core size will eventually (at the end of the iterative process) be R. We then continue to decorate down to the microscopic scale.
The energy associated with this structure is
The core energy, dE, is obtained by subtracting the
energy of the perfect tiling (Eq. (16)). Hence
If the defect represented by I du) does not involve
a change in the total area, if follows that
(Alldu) = 0 (in the above example Idu) ’"
1 -1,1,0,0,0,0,0,0,0)).
If (Azldu) 0 then
where,
For Az > TZ (cf. Tab. I), dE scales faster than linearly with the diameter of the domain. The
particular scaling of this energy term is due to the
inflationary character of the defect considered here.
Note that within our picture one may introduce a whole hierarchy of defects (domain) of size abP, abP’, abP"... (p" > p’ > p ) [14]. An intriguing ques- tion that arises here is whether such domains can be formed during a quenching process which lead to a local quasi-periodic structure (of inflationary charac- ter) separated from the rest of the quasi-periodic system by a domain wall.
Acknowledgments.
We thank B. B. Mandelbrot for advice and dis- cussions. Y. G. acknowledges the supports of CNRS for his stay in the Laboratoire de Physique des
Solides at Orsay. This work was partially supported by the U.S.-Israel Binational Science Foundation
(BSF). M. K. and J. P. are happy to use this
opportunity to thank their friends J. P. Allouche, F.
Axel and M. Mendes-France for the pleasant and
fruitful discussions on aperiodic systems they held aperiodically during the Academic Year 1984-1985 and C. Godreche, who drew our attention to the six shapes tilings of reference [3]. Y. G. is a Bat-Sheva
Fellow.
Appendix.
A natural question that arises is the following : is it possible to describe the perfect PT of figure la as a
deterministic version of the tilings described in
section 3 ? In other words, is it possible to split the shapes (a), ({3), ( y ), (61’ ( -- ) and (03B6 ) (see
section 2) into pieces that combine to form shapes (a), (b), (c), (d), (e), (f), (g), (h), and (i) in such a
way that the iterative decoration of the new tiles
((a), (b), (c), ... ) thus obtained will be described in
terms of the matrix M (x, y, z ) ? We thus want to study the relation between two different realizations of the five-fold tiling.
We have to determine 15 quantities a, b, c,
...a , {3, 1’, ... which are proportional to the fre- quencies of the elementary tiles of the two tilings
discussed in this paper. The right eigenvector as-
sociated with the largest eigenvalue of L, T2 is
written as
while the right eigenvector of M associated with the
eigenvalue T 4 is written as
Here we choose the normalization of the vectors such that their entries will correspond to the number
of tiles (of a given type) per unit area. We now can
write down equations for the quantities a, b, c,
...a, (3, y,
...whose derivation is based on the follow-
ing considerations :
(i) there are six entries of I À 1) which satisfies
This provides us with six equations.
(ii) Similarly the relation
provides us with 9 equations.
(iii) We have four other equations that relate
shapes generated by L to shapes generated by M. For example one can see from figure 2 that, in
the M-tiling pentagons correspond to either tiles (e)
or a combination of tiles (h) and (i). Hence the total
number of pentagons in that tiling is equal to the
number of (e)’s plus the number of (h)’s (or (i)’s).
From figure lc we can see that in the L-tiling pentagons are related to either (03B4 ), (6) or (03B6 ).
Hence we can write
From similar considerations we obtain
The factors x, y, z do not appear in 17 of these
equations. This enable us to calculate the relative
frequencies of a, b, c,
...In other words : the first eigenvector of M (x, y, z ) (up to a normalization factor) is independent of
x, y, z ; hence the relative frequencies a, b, c,
...are
independent of x, y, z. The two equations that
contain x, y, z may be reduced to a single compatibi- lity condition :
The last equation has no solution of the form
x
=k1/5, y
=k2/3, z
=k3/2 (kl, k2, k3 being inte- gers). Turning back to the question we have raised in
the beginning of this section, we conclude that it is not possible to describe the perfect PT of section 2 as a deterministic tiling of the type discussed in sec- tion 3. However, equation (A.10) has, naturally,
many solutions. The simplest one is
The geometric interpretation of the general solution
of equation (A.10) remains open at this stage. We only remark that the particular solution, equation (A.11), implies that the statistics of a
decorated (h) shape, matched with an (i), is the
same as that of a pentagon (an (e) shape). Con- versely, each pentagon can be devided into shapes (h) and (i) whose orientation may be chosen arbi-
trarily (provided that it is along one of the five-fold
symmetry directions) without affecting the statistics
of the decorated shapes.
1118
References
[1] (a) PENROSE, R., Bull. Inst. Math. Appl. 10 (1974) n°7/8 ;
(b) PENROSE, R., Math. Intelligencer 2 (1979) 32.
[2] GARDNER, M., Sci. Am. (January 1977) p.110.
[3] GRÜNBAUM, B. and SHEPPARD, G. C., Tillings and
Patterns (Freeman, San-Francisco) 1987.
[4] DE BRUIJN, N. G., Proc. K. Ned. Akad. Wet. A 84
(1981) 39 ; (1981) 59.
[5] LEVINE, D. and STEINHARDT, P. J., Phys. Rev. Lett.
53 (1984) 2477.
[6] An overview of both theoretical and experimental aspects of the problem
canbe found in the Les Houches Conf. Proc. Aperiodic Crystals, Eds.
D. Gratias and L. Michel, J. Phys. Colloq.
France 47 (1986) C3.
[7] For
areview
onquasi-crystals see HENLEY, C. L.,
Comments in Cond. Mat. Phys. B 13 (1987) 59.
[8] PEYRIÈRE, J., J. Phys. Colloq. France 47 (1986) C3-
41.
[9] SHECHTMAN, D., BLECH, I., GRATIAS, D. and CAHN, J. W., Phys. Rev. Lett. 53 (1984) 1951.
[10] MANDELBROT, B. B., GEFEN, Y., AHARONY, A. and PEYRIÈRE, J., J. Phys. A 18 (1985) 335.
[11] GEFEN, Y., MANDELBROT, B. B., AHARONY, A. and KAPTTULNIK, A., J. Stat. Phys. 36 (1984) 827 ;
AHARONY, A., GEFEN, Y., KAPITULNIK, A. and MURAT, M., Phys. Rev. B 31 (1985) 4721.
[12] The backbone of the structure shown in figure 2b is
afractal of dimensionality 1.86 and
wassuggested
as a
possible model for amorphous materials during
aninformal discussion which included AHARONY, A., GEFEN, Y., KLÉMAN, M., MAN- DELBROT, B., ORBACH, R., and PEYRIÈRE, J.
(Les Houches Meeting
onFractals, March 1984, unpublished) ;
see
also Yu, K. W., Thesis, University of Californie,
Los Angeles (1984).
[13] Consider for example du> ~ | 1, -1, 0, 0, 0, 0, 0, 0, 0 ~ . This corresponds to the substitution of
onetype (a) triangle by
atriangle of type (b).
[14] Since
ourprocedure takes into account the number of various shapes but not their spatial arrangement,
we