Representation of certain self-similar quasiperiodic tilings with perfect matching rules by discrete point sets

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Representation of certain self-similar quasiperiodic tilings with perfect matching rules by discrete point sets

Richard Klitzing, Michael Baake

To cite this version:

Richard Klitzing, Michael Baake. Representation of certain self-similar quasiperiodic tilings with

perfect matching rules by discrete point sets. Journal de Physique I, EDP Sciences, 1994, 4 (6),

pp.893-904. �10.1051/jp1:1994234�. �jpa-00246953�

J. Phys. I IFance 4

(1994)

893-904 JUNE1994, PAGE 893

Classification Physics Abstracts

61.50E 02.10 02.40

Representation of certain self-similar quasiperiodic tilings with

perfect matching rules by discrete point sets

Richard

Klitzing

and Michael Baake

Institut für Theoretische Physik, Universitàt Tübingen, Auf der Morgenstelle 14, D-72076 Tübingen, Germany

(Received

24 August 1993, re~4sed 21 Ja~luary1994, accepted 8 March

1994)

Abstract. Simple quasipenodic tilings with 8-fold and 12-fold symmetry _are presented that possess local

de-linflation

symmetry and perfect matching rules. The special feature of these tilings is that the fuit information is already derivable from the set of vertex sites atone. This

means that the latter _{is a} valid representative of the corresponding equivalence class of mutual

local derivability.

l. Introduction.

The existence of

perfect matching

rules for several

quasiperiodic tilings

_[1-5j is

interesting

both

mathematically

and

physically.

On the _one

hand,

_one obtains so-called

aperiodic

sets

(like

the two rhombi of the Penrose pattern

il, _6]),

_on the other hard

they

ofler

possible geometric approaches

to certain types of

long

_range orientational

order,

_as it _occurs in

quasicrystals,

compare [1]. ^{While trie former is} a well established branch of discrete

geometry,

^{the latter} needs both further

explanation

and

exploration.

Several types ^of

matching

rules bave been discussed [8], ⁱⁿ

particular

_strong

matching

rules

(enforcing apenodicity)

and

perfect matching

rules

(uniquely specifying

_a local

isomorphism dass).

Perfect

matching

rules for

nonperiodic tilings

are

automatically

strong while the _con-

verse is not

necessarily

true. In this article _we focus _on

perfect matching

rules which have

the

advantage

that

they

_are

applicable

to the

periodic

_{case as} well:

penodic tilings

_possess

perfect matching

rules in _a trivial way via the

repetition

rule of the fundamental cell. This _con

simultaneously

be used _{as a} local

growth

rule

whereby

_one obtains

larger

and

larger patches

from _a suitable seed.

The

aperiodic

_case is _more

complicated though: perfect matching

rules _are not

automatically

local

growth

rules. Even _worse, there is to _our

knowledge

no

example

known where

they

are.

Consequently,

_{m companson} with the

penodic

_case,

possessing matching

rules _now is

a weaker property ^{from the}

physical

point of _view.

Nevertheless,

_one important property

remains:

perfect matching

rules _ensure the

energetic

stabilization of

quasiperiodic

structures

as trie

ground

state of _a suitable

Hamiltonian, by favouring finitely

_many local

configurations

relative to

others,

_compare the discussion in [9].

In _view of the local nature of

physical

interactions, this statement requires some care: If

matching

rules _are

significant, they

must follow

locally

from the

physical

_structure,

ideally

from the set of atomic

positions,

_say. In

geometric

terms, the

simplest

such sets of

points

are the vertex sets of

tilings.

The

question

then is whether

perfect matching

rules _can be

derived

locally

from trie set of _vertex sites. This is the _case for the Penrose and the

decagonal triangle tiling il, loi,

but not for the

Ammann-Beenker,

the

Socolar,

and the Gàhler-Niizeki

tiling

[2,

II, 3, 12, 4]. (We

restrict trie discussion to

planar tilings here;

for _some

general

results

we refer _to [5, 13] ^{and the remarkable article}

by

Lunnon and Pleasants

[14]).

Until

recently,

several researchers believed that

perfect matching

rules cannot be obtained for 8- _or 12-fold

symmetric tilings

without

introducing

decorations which _are Dot

locally

derivable

from the bare

tilings.

This lias been

disproved by

two

counterexanlples [loi.

For trie

tilings Tj(~

^and

Tj(~~

^{the existence of local de-}

linflation

_symmetry and

perfect matching

rules _was

shown

[15, loi,

the latter

by

the

composition Idecomposition

method [4]. ^One

important

result is that these

properties automatically

extend to ail members of trie

equivalence

dass of mutual local

derivability

with symmetry preservation [16], ^{called SMLD dass from} now on. For trie concept and its

properties

_we refer to

iii].

To be _more

explicit:

trie existence of

perfect matching

rules is _a property ^of an SMLD

dass,

and _can trier be formulated for _any of its members

(being

_a dass of local

isomorphism

LI

dass).

Now it is obvious that _{we can} choose

a point set

representative

out of the SMLD

dass,

and the

matching

rules _are then

given

_{as a} finite list of

possible patches

_up to a certain

(finite)

size.

Though

this

might

not _seem most

practical,

it is

certainly

_very close _to the idea to stabilize _a

quasicrystal by

the selection of

finitely

_many dusters of _atoms. Such _an

approach

has been tried for _a

decagonal T-phase by

a decoration of the

decagonal triangle tiling iii. There,

the decoration enforces the

matching rules,

and it is therefore _an

explicit example

^of

physical relevance,

_compare the discussion in [9].

To match

physical applications,

the selection of _a point set

representative

of _an SMLD dass will prove useful and it is

always possible

_{[9, 18].}

Consequently,

the formulation of

matching

rules, ^if existent, con be dore in this

framework,

and

quite

_a bit is known about the icosahedral [19] ^{and the}

decagonal

_case

[la,

_{1, 9].} In this article _{we are} interested in 8- and 12-fold symmetry ^{where still much less is known. For the} 8- and 12-fold

tilings

in

[15],

^trie set

of _vertex sites atone does net represent the SMLD dass.

Therefore,

it _{was an} obvious exercise

to find and describe other members of the SMLD dass which _are derivable from their _vertex sites atone. A short

presentation

of two such

tilings

is the aim of the present

article, though

our

generation procedure applies

to _a much broader dass of _cases. We thus _continue

[loi,

in

particular

the

appendix

of it.

2. The

eightfold

_case: P(~~

Several different 8-fold

tilings by

_squares and rhombi _are known.

Firstly,

there is the standard

octagonal square-rhombus tiling,

found

independently by

Ammann [2] and Beenker

iii]

and also

by Lück,

in the latter _case with various other

exanlples

obtained

by

substitution [20].

Recently,

this

tiling

_was put on a broader basis

by investigation

of the

possible

2-tile substi- tutions with _square and rhombus [21].

Thereby,

two _more

tilings

of the mentioned kind _were found. Dur

tiling

_is still another _one,

because,

in terms of the substitution

formalism,

different

decompositions

for both tiles occur, 1-e- trie

generation by

_means of substitution is _no

longer

one

by

two tiles

only (although only

two

shapes

of tiles

occur).

But the way we have found it

differs,

wherefore _we invite those who _are interested in the

N°6 QUASIPERIODIC TILINGS IIITH PERFECT MATCHING RULES 895

Fig. l. The quasiperiodic tilings

Tj(~

^and Tj(~~

Fig. 2. The quasipenodic tilings Pl~~ and P°~~. _The patches shown _are exact deflations of octag- onally _resp. dodecagonally shaped start configurations.

substitutional

point

of view to have _a look into the

Appendix.

Let _us start here with another 8-fold

tiling,

the

tiling Tj(~

^{which is shown} in

figure

1. This

tiling

_can be derived from the 4D root lattice D4

by

the

projection

method [15]. ^{It has} a local deflation and inflation and possesses

locally

derivable

matching

rules

[loi.

On the other

hand,

the full local information

Fig. 3. The local denvation of P~~~ from

Tj(~.

Fig. 4. The decoration fuie for

Pl~l

_{See text for additional} explanation.

requires the

knowledge

of the

edges

in the

tiling,

the set of _vertex sites atone is insuliicent.

In the present context, our task is to introduce additional

points

which

complete

trie local information encoded in trie

points.

Let _us describe _one

possible procedure.

Firstly,

_one gets rid of the

edge-inherent

^additional information

by introducing

_new vertices.

Especially hexagonal patches,

_compare

figure

1,

consisting

of _{one acute}

triangle

in the middle surrounded

by

two

oblique

_ones at the cathedes and _a flat

triangle

at trie

base,

bave to be decorated

by

_new vertices _in such _a way that trie

decomposition by

those

edges

_is

uniquely

reflected within their sites.

Therefore,

_we introduce

quite

near the base of the _acute

triangle

_an additional _vertex. To get more handsome tiles _we decorate the cathedes of the _acute

triangle

with two additional vertices. The

resulting

derivation _is shown _in

figure

3, ^its

locality

is obvious.

Next,

_{we prove} trie local

invertibility

of this derivation. In order to do

this,

_we introduce _an orientation of trie squares.

Clearly,

this orientation must be _a local _one: this is shown in

figure

4. Just _one

ambiguous

_case remains: The

patches

of the lower _row of

figure

4

(without

the lateral

rhombus)

_{may occur}

succeedingly

_up to three times in _a line. In that case, it suliices to demand that the _muer

patch

should be directed in the _{same way as} the outer _ones.

Now, having

decorated trie squares, trie backward direction becomes _easy: it is

just

trie rule shown in

figure

5. It is obvious that trie fiât

triangles

_are

regained by

these rules

as well.

The

resulting

8-fold

tiling P(~)

_is

again

face-to-face and consists of _squares and 45°-rhombi

only.

As _an immediate consequence, this

tiling

_is

locally

derivable from its vertex sites alone.

Put

together

^{with trie} results of

[loi

_one finds: trie

tiling P(~)

bas

perfect matching

rules _as well. Furthermore, trie

complete

information _is

localy

denvable from trie set of vertex sites.

N°6 QUASIPERIODIC TILINGS WITH PERFECT MATCHING RULES 897

1

-

_

~

~

-

~ ~

Fig. 5. The remaining derivation from P~~l back _to

Tj(~

~~~

A B C

Fig. 6. The acceptance domains for trie tiling P'~. _The tetrasymmetric _{one occurs in} two orienta- tions. Domains A and B _are situated around interstitials

l'holes',

_{more precise:} A around those with integer ^Euclidean

coordinates),

whereas C is the _one

aroqnd

lattice points. The projection of _a unit

vector _ez would point from the lower of trie left vertices of A, for example, towards the central point.

Especially:

The set of vertices of

P(~)

_bas

perfect matching

rules.

For trie sake of

completeness,

we will

give

trie acceptance domains for the _new

tiling

described

above,

where _{we use an}

embedding

into the root lattice

D4,

the 4D checkerboard lattice [15].

Starting

from the

primitive hypercubic

lattice with the standard orthonormal basis _{vectors e~,}

we can write trie lattice _as

D4 ₌

L

_xiei

l'xi

₊

°12)

1.

In order to preserve trie 8-fold symmetry within trie 2-dimensional

subspaces,

trie

projector

into trie

physical

_space is chosen _as

£

1

~

2 2 ~

2

II 1

£

^{1 '}

~

2 2

~

and for trie

projector

into trie intemal space trie second and fourth column bave to be

multiphed by

-1.

Figure

6 shows trie acceptance domains of trie

tiling

P(~~ The small octagon

IA)

and the inner squares of trie tetrasymmetric _ones

(B),

defined

by

the _concave vertices of

them,

correspond

to the

original tiling Tj(~,

compare [15].

The

corresponding

vertex

configurations

_are shown in

Figure

1. Trie vertices Al A3 stem from trie small

octagonally shaped

acceptance domain

A,

and _are,

therefore,

deducible from

~~

Al A2

~

~

Bl

B~ B~ B~

cfic~

Fig. 7. The 9 vertex configurations of the tiling Pl~l.

Table I. Relative

frequencies

^of occurrence for _vertex

configurations

vithin the

octagonal tiling P(~),

= 1 +

và

_is the silver _mean.

Vertex ^[

Frequencyofoccurrence

Al

1/2À~

₌

(29

12À)

/2

m 1. 47187 %

A2

(1

+ À)

/2À5

₌

(-41 +1?1) /2

_m 2. 08153 % A3 (1 + À)

/2À~

₌

(Ii il) /2

m 5. 02526 %

Bl

1/2

=

1/2

_m 50. ooooo %

82

1/À~

= -12 ₊ 5À _m 7. 10678 %

83

1/2À~

₌

(29

12À)

/2

m 1. 47187 %

84

1/2À~

=

(-12

+ 5À)

/2

m 3. 55339 %

Cl

(1+ _À)/2À~

=

(Ii 7À)/2

_m 5. 02526 %

C2 ₊

= -7 + 3À _m 24. 26407

%

trie

corresponding

vertex

figures

of

Tj(~,

^{Bl 84} are

mostly

in _one-to-one

correspondence

to those of

Tj(~, ^just

some vertices of type ^Bl _occur

additionally, nanlely

those _new

vertices,

in

comparison

to

Tj(~,

^which are located _at trie

edges

^{of trie} acute

triangles.

Their sites do _occur in the deflations of

Tj(~

^This is

why

trie fourfold symmetric acceptance ^{domains B do} appear

enlarged

with respect to

Tj(~

^Worth

^noticing

^{is the fact that the relative}

^frequency

^{of the}

vertices of type ^{Bl within}

tiling P(~)

is

exactly 50%,

cf. table I. The last _{two vertex}

config-

urations _are those which _occur, with respect to

Tj(~, additionally

within the acute

triangles.

Their acceptance domain _is the

larger octagon

C.

3. The twelvefold _case:

P(~~)

Just as

before,

_we transformed _a

D4-tiling by introducing

new vertex sites.

Again

we started with _a

hexagonal patch,

this time

consisting

of _an acute

triangle

_m the

middle,

two flat _ones at its cathedes and _a small equilateral

triangle

ail of them

dearly

from the twelvefold

tiling

N°6 QUASIPERIODIC TILIIQGS IIITH PERFECT MATCHING RULES 899

Fig. 8. The local derivation of Pl~~l from

Tj(~l

Fig. 9. The acceptance domains of the tiling Pl~~l. The _{square occurs} in three orientations. It

is situated around interstitials

l'holes'),

whereas the other _{one occurs} around lattice places. The projection of _a unit lattice vector would point, for example, from

a vertex of the _square towards its center.

T)(~~ [15],

compare

figure

1.

Here,

_a

single

additional vertex site at the circumcenter of the acute

triangles

_was

enough

to

complete

trie local information. Trie vertex points of the

tiling, P(~~),

obtained this _{way tum out} to be the union of the vertex sites of

Tj(~~

^{and those of}

a

tiling

of

Niizekilmitani

and

(independently)

Gàhler [12]. ^{Therefore the} vertex acceptance domains _are those of these

tilings,

^cf.

figure

9.

Up

to this

point,

^this was

already shortly

mentioned in trie

appendix

of reference

[loi.

Now _we have to prove mutual local

derivability.

Trie direction from

Tj(~~

^to

P(~~)

_is

an easy

tile-tc-tile derivation shown _m

figure

8. Due _to the fact that the

tiling Tj(~~

^is

already

defined

by

^the set of _acute

triangles atone,

1-e- all the

edges

of trie other tiles _are

given

^from those of the acute

triangles, only

these

triangles

need to

by regained

for trie other direction. This looks

diflicult,

because

figure

8 indicates that the acute

triangles

_as well _as the small

equilateral

_ones

result in the saule tiles. But

again,

^due to the mentioned property ^of

Tj(~~,

^{it is dear that}

the small

triangles

^of

_Tj(~~

^have a

unique surrounding by

three _doute

triangles.

This '3-star'

patch

will be transformed into _a

'pyramid'

of four

(small) triangles.

So it becomes dear that,

whenever such _a

pyramid configuration

_{occurs m}

P(~~),

trie central _one does _not stem from _an acute

triangle.

Like in trie 8-fold case, we thus get _a

tiling

with

only

_one

edge-length.

The latter is the second shortest distance between vertex sites.

Only

the small

diagonal

of the rhombus is

shorter,

but

~~~

l 2 3

4

[6

î

~ ~

~ ~

~

~~

i~ii

Fig. lo. The vertex configurations of tiling P°~l. Vertices 1 through 3 correspond to the dodecago-

nal acceptance domain, whereas the others

belong

to the quadratic

one(s).

no bond. On the other hand, this point distance _occurs

only

between two vertex sites if these

are connected

by

_an

edge,

except ^{for trie}

single

_case that two rhombi _are

joined together.

From trie hst of vertex

configurations,

shown _in

figure

la, it is obvious that _more than _two rhombi

are not allowed to

join. Therefore,

_we bave to handle just this

single exceptional patch.

Here, trie _presence of trie _common rhombus vertex in between is

enough

to rule out that _case.

Thus,

trie

complete

information _on trie

tiling ^P(~2)

_is contained in its vertices.

Having

shown trie

equivalence

to trie

tiling Tj(~~

as well and

using again

trie results of

[loi,

_we find:

The

tiling P(~~)

bas

perfect matching

rules and its

complete

information is

locally

deriv- able from trie set of _{vertex sites.}

We present, ⁱⁿ

figure

9, ^trie acceptance ^domains for P(~~~. Trie

quadratic

domains

correspond

to

Tj(~~,

^trie

dodecagonal

_one to trie

tiling

of references

[12,

4]. In

figure

10, trie

corresponding

vertex

configurations

_are shown. Trie first _three

configurations correspond

to trie _new vertices,

I.e., their acceptance ^domain is trie

dodecagonal

_one, trie _one of trie

(sub-)tiling

shown _in [12, 4], whereas trie others

correspond

one-to-one to those of

Tj(~~

[15]. ^{From trie relative} size of trie

dodecagonal

acceptance domain with respect to trie

quadratic

_ones trie circumradius of trie

dodecagon equals

trie

edge

of trie _sqare it becomes clear that trie vertices 1 3 bave _a total

frequency

of _occurrence of 50%.

(This

correlation between relative _size of acceptance ^domains and

frequencies

is due to trie minimal dimension of

embedding).

N°6 QUASIPERIODIC TILINGS MITH PERFECT MATCHING RULES 901

Table II. Relative

freque~lcies

of _occurrence of vertex

configurations

with1~l the

dodecago~lal

til1~lg

P(~~~,

₌ 2 +

Và

_{is trie}

_platinum

number.

Vertex ^[

Frequencyofoccurrence

15 _m

2

(-26

₊

7Q)/2

m 6. 21778 %

3

3(4 Q)/2

_m 40. 19238 %

4

(4 Q)/2

m 13. 39746 %

5 -26 +

7g

_m 12. 43556 %

6

(56 15Q)/2

m 0. 96190 %

7 -26 + 7Q)

/6

_m 2. 07259 %

8

(-41

₊

llg)/2

_m 2. 62794 %

9

(45 12g) /2

m 10. 76952 %

10 -26 ₊

7Q)/6

m 2. 07259 %

II _m 5. 66253 %

4.

Concluding

remarks.

For

eight-

and twelvefold symmetry, _we

presented

_a construction of

relatively simple quasiperi-

odic

tilings

which _possess

perfect matching

rules

inherently,

1-e-, in such _{a way} that these rules

ca~l be

locally

derived from trie set of vertex sites atone. Trie concept of mutual local derivabil-

ity

was central in trie chain of arguments,

giving simultaneously

trie relation _{to other}

tilings

that _are

already

known and trie method _{to construct} trie acceptance domains

systematically,

We bave _not focussed

explicitly

_on

deflationlinflation

properties of these

tilings

and their local _nature. But this is another feature _we get ^{for free from trie}

equivalence

to trie

D4- tilings

_[15]. Stated

explicitly:

The

tilings

^P(~~ and P(~~~ both _possess local deflation and inflation property, ^{1-e- trie} deflated _as well _as trie inflated

tilings

_are

locally

derivable from trie

original

_ones.

A discussion of that will be given in trie

Appendix.

Constructively,

_we know that local

deflationlinflation

symmetry ^and

locally

derivable match-

ing

^rules _are

compatible

with 8-,

10-,

and 12-fold symmetry

(cf. [loi

for trie 10-fold

case).

It is

possible

to construct

simple

discrete point set representatives.

Now,

one obvious

question

is whether this is _an accidental coincidence hnked to

quadratic

irrationalities.

Preliminary

inves-

tigations

of

tilings

with 7-fold symmetry [22] ^{and several}

general

results _on

arbitrary

symmetry groups and their relation to

algebraic integers

_[14] indicate that this _{is not} trie _{case: one may} expect ^trie

tripel "quasiperiodicity

^local

deflationlinflation perfect matching

rules" _to be much _more

pertinent

than it was

expected

_so

far, although

_we admit that trie consideration of

suitably

closed

tiling

ensembles

might

be necessary.

Acknowledgements.

This article is part of _a

study

_on trie

theory

of

quasicrystals, performed

at trie Institute of Theoretical

Physics, Tübingen University.

^It is _a

pleasure

to thank P. Kramer for

helpful

advice

and continuous support, M. Schlottmann for

carefully reading

trie manuscript and

suggesting

vanous

improvements,

D.

Joseph

for

providing

important materai and

figure

1, and last but

Fig. Il. Local substitution rule for Pl~l. _It

can be shown that the decoration used is locally derivable from the bare tiling. For

P°~l

see text.

not least F. Gàhler and P-A-B- Pleasants for useful discussions. This work _was

supported by

Deutsche

Forschungsgemeinschaft.

Appendix.

Let _{us comment on} how trie

global

^inflation operates on trie _set of _vertex sites, which in turn can be read from trie vertex acceptance ^domains.

It lias been shown that all these domains _are

star-shaped

with respect to their centers.

IA

set is called

star-shaped

if it contains _a

point

such that each line from this point to any other

point

of trie set lies

completely

inside trie

set).

Trie centers,

however,

_are trie

possible

fixed

points

of

inflation/deflation

transformations.

Consequently,

_a

global

inflation of these patterns,

corresponding

to a shrink of trie domains towards trie fixed

points

_up to a

permutation

between congruent ^domains

yields

smaller domains

lying completely

inside trie

original

ones, I.e.,

considering

trie

physical

_space of trie

tilings,

all vertex sites of trie inflated

tilings

coincide with _some sites of trie

original tiling.

N°6 QUASIPERIODIC TILINGS MITH PERFECT MATCHING RULES 903

Now,

let _us

give

_a ^brief discussion of trie local

(de-)composition rules,

1-e- trie substitutional aspect of trie

tilings P(~~/P(~~~.

In reference

[loi

trie local

(de-)composition

rules bave been shown for trie tiles of

Tj(~

^and

Tj(~~. Hence, starting

^with some

patch

in

P(~)/P(~~),

_one

applies

trie local derivation rule towards

Tj(~ /2j(~~ ^first,

^{then trie}

(de- )composition

rule

there,

and

finally

trie local derivation rule back

again.

Because of

being

tile to

tile,

trie derivation rule

yields,

in trie _case of

P(~~),

_a rather trivial transformation of trie substitution rules of

Tj(~~ ^given

^{in reference}

[loi:

The

triangle

which

corresponds

to trie small

triangle

of

Tj(~~decomposes

^{into trie}

^shield;

^{trie other} one which

corresponds

to the acute

triangle

of

Tj(~~decomposes

^into two

shields,

two

triangles

^{of trie}

same kind and _an additional

rhombus;

trie shield

decomposes

into _one

triangle

of trie first kind and three of trie

latter; and,

due to trie fact that this

decomposition

rule is Dot

shape preserving,

trie rhombus will vanish.

An

analogous procedure

^for trie 8-fold _case

results,

if formulated _as

simple

_as

possible,

in

a local

decomposition

rule for _two

locally distinguishable

kinds of oriented rhombi and two

(again distinguishable) rectangular triangles,

_as shown in

figure

II.

Therefore, P(~) yields

_a four tile substitution rule.

Thereby,

trie squares are to be devided

along

trie

orientating

_arrows,

introduced in section 2.

By

^trie _way, ^that _arrow stems from trie decoration of this

dividing

line. Thus, part ^{of trie}

proof

for trie

locality

of trie decoration introduced here has

already

been

given.

^What remains is _{an as} easy exercise.

Clearly,

in trie 8-fold _case trie deflation factor is À; ⁱⁿ trie 12-fold it is

fi.

References

iii

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32jis

a repnnt ^{of: Eureka 39}

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16.

[2] ^{R. Ammann's}

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1.

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dodecagonal

quasicrystals, Phys. Reu. 839

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10519.

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160.

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263.

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(World

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1991)

185.

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L405. Gàhler F., Quasicnstalline Materials, Ch.

Janot and J-M- Dubois Eds.

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627.

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217. Pleasants P-A-B-, priv. _commun.

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ils]

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1927.

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il

_9] Danzer L., Three-dimensional analogs of the planar Penrose tilings and quasicrystals, Dgscr. Math.

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1.

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