Inflationary character of Penrose tilings

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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

CNRS, Université de Paris-Sud, bâtiment 510, 91405 Orsay, ^France

(3) Section de Recherches de Métallurgie Physique, Centre d’Etudes Nucléaires de Saclay,

91191 Gif

Yvette 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}

1988)

Résumé.

On 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

pavages apériodiques. ^{Nous calculons} explicitement les valeurs propres ^et les vecteurs propres correspondants, ^{dans les}

^d’un pavage à ^{six tuiles}

élémentaires et d’une famille de pavages à neuf tuiles élémentaires, certains étant déterministes, ^d’autres

l’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

certain niveau d’itération, l’itération étant poursuivie par la suite. En supposant que

l’expression ^{du coût}

énergie ^de

^défauts comprend

^{terme linéaire}

les densités des diverses tuiles,

nous

montrons que l’énergie ^d’une région fautée de taille R est

Rf, ^où f ^{2 est}

exposant non-trivial.

Dans certains _cas, f ^est plus grand que la valeur canonique ^1.

Abstract.

The 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

discussed. We also study

special type ^of inflated ^{defects in}

a

deterministic Penrose tiling. These defects

the result of

mismatch between the tiles, introduced at

certain iteration stage and then inflated iteratively. Assuming ^{that the} expression of the energy ^cost for such defects includes

term which is

linear function of the shapes densities,

show that the energy of

defective

region of linear size R may scale

Rf, ^where f ^{2 is}

^trivial 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

drawn ; (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

tile

(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

pentagon

marked 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,

(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)

matched with other tiles of type (h) ; ^such

matching ^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

compatible

with the given ^{values of x, y and}

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

by ^{1- T. Thus we obtain a left} eigenvector associated with A

T - 4 (= ( 1- T )4 )

by replacing

by ^1-

ⁱⁿ (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

be found in the Les Houches Conf. Proc. Aperiodic Crystals, ^Eds.

D. Gratias and L. Michel, ^J. Phys. Colloq.

France 47 (1986) ^C3.

[7] ^For

^review

quasi-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}

fractal of dimensionality ^{1.86 and}

suggested

as a

possible model for amorphous ^materials during

informal discussion which included AHARONY, A., GEFEN, Y., KLÉMAN, M., MAN- DELBROT, B., ORBACH, R., ^and PEYRIÈRE, J.

(Les ^Houches Meeting

^{Fractals, 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

type (a) triangle by

triangle ^of type (b).

[14] ^Since

procedure takes into account the number of various shapes ^{but not their} spatial arrangement,

we

ignore here the effect of core-core

lations

the energy. One may, however, ^im- prove the present approach systematically by considering larger configurations

^{our basic} shapes, for which at least short-range spatial

correlations may by accounted for explicitely

(see ^Ref. [8]).