RELAXATIONS OF PERIODICITY IN THE TOPOLOGICAL THEORY OF TILINGS

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RELAXATIONS OF PERIODICITY IN THE TOPOLOGICAL THEORY OF TILINGS

A. Dress

To cite this version:

A. Dress. RELAXATIONS OF PERIODICITY IN THE TOPOLOGICAL THEORY OF TILINGS.

Journal de Physique Colloques, 1986, 47 (C3), pp.C3-29-C3-40. �10.1051/jphyscol:1986304�. �jpa- 00225714�

JOURNAL DE PHYSIQUE

Colloque C3, supplement au n" 7, Tome 47, juillet 1986

RELAXATIONS OF PERIODICITY IN THE TOPOLOGICAL THEORY OF TILINGS

A.W.M. DRESS

Department of Mathematics, University of Bielefeld, P.O. Box 8640, 0-4800 Bielefeld, F.R.G.

Abstract - The foundations of the topological theory of tilings in terms of the associated chamber systems are explained and then they are used to outline various possibilities to study the phe- nomena of "quasi-periodicity" of tilings in this context.

1 - INTRODUCTION

The discovery of Shechtmanite, i-e. of solid state phases of Mn-A1 alloys with discrete icosahedral diffraction spectra / I / , has had a stimulating effect in several areas of mathematics.

In particular, the speculation about possible relations between the microscopic geometric structure of such compounds with 3D-analogues of the famous Penrose patterns /2-6/ have aroused quite some interest.

By now, it seems likely that the evolving definition of "quasi-crystals"

will be based mainly on properties related to the existence of discrete Fourier transforms /7-8/ and to higher dimensional periodicity / 9 - 1 1 / . Still, it might be worthwhile to review the mathematical theory of tilings as developed in ref. 12-19 from the point of view suggested by the speculations about quasi-crystals, i.e. by studying properties of tilings with relaxed periodicity requirements. Such an enterprise may base its justification on the view - expressed one way or the other by many physicists - that the whole world is nothing but a decoration of

some basic geometric structure and that tilings are a paradigm of such structures, their phenomenology exhibiting a rather intricate and some- times confusing interplay between metric, affine and topological pro- perties. A justification may also be based somewhat broader on the fact that the study of tilings is a particularly important and interesting case of the study of the relations between local and global properties of d i s c r e t e structures, a program which appears to become a central issue in recent and future developmentsof mathematics and its applications. And finally, one may also just refer to the fact that the study of tilings has always been and will probably remain a rather en- joyable part of recreational mathematics.

In the following note we will first define and study tilings from a purely topological point of view, encoding their topological structure by some algebraic ggdgets, called (thin) chamber systems (of rank 2).

Then, using this encoding, we will consider various relaxations of the concept of periodicity, suggested by (quasi-)crystallography, including a new and, I hope, rather useful definition of displacements of perio- dic tilings, and discuss a number of questions which arise in this con- text.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986304

Chapter I - Topological aspects of planar tilings and their algebraic encoding.

2 - Some basic definitions concerning tilings

Let us define a tiling T of the euclidean plane E = x2 to consist of a closed connected subset T of E such that (a) the connected components of the complement E x T are bounded and (b) for any x E T there exists a neighbourhood U = U(x) , a natural number

r = r(x) €IN := {1,2,3, ...I and homeomorphism cp : U -t B2 :=

:= { z E (I:

I

= 19-e 2nni'r 1 0 5 ~ 5 1 ; n ='0,1, ..., r-11 (cf. fig. I).

For a tiling T 5 1 E we define its set To of vertices to consist of all x E T with r(x) + ² , the set T I of edges is defined to consist of all connected components e r e of T\TO and the set T2 of faces is defined to consist of all connected components f 1 f 1 , . . of

E \ T

.

morphism cp : (0,l) e from the open interval1 (0,l) onto e and that any such homeomorphism extends uniquely to a continuous map

: [0,1] --t 5 e U To of the closed interval1 onto the closure

e

of e and that for any face f there exists some r(f) E N , uniquely 2 -

determined by f and T , and some continuous map q = wf: B -+ f 5 c_ f U T of B~ onto the closure f of f - , also uniquely determined by f and T up to (admissible) homeomorphisms of B~ , such that

(a) cp induces a homeomorphism between the interior Iz E E 1 lzl < 1)

- ¹

of B2 and f , (b) cp (To) = { Z E (I: / zr = 1) and (c) w , re- stricted to the various connected components of _Cz E E

I

-

-

any two faces f, , ^{f E T2 the} intersection f l n f2 is connected (and, hence, it is either empty or a vertex or an edge of T). T is called non-degenerate, if r(x) 2 3 for all x E To and r(f) 2 3 for all f E T2

.

Two tilings T I T' c_lE

-

.

A subdivision of a tiling T is defined to be any tiling T ' contai- ning T as a subset. For a non-degenerate tiling T we define a sub- division T' of T to be a barycentric subdivision if for any f E T2 there exists some gf : B2

-

- 7

ties cpf (TI) = { z E C 1 z2-r(f)>0) - U { z E C

I

-

9 f(TA) = CO} U ^{{ Z} E E 1 z 2 r ( f ) = l ) . One can show t h a t any non-degene- r a t e t i l i n g T h a s a b a r y c e n t r i c s u b d i v i s i o n T' , t h a t any two b a r y - c e n t r i c s u b d i v i s i o n s T ' and TI' a r e ( i n a more o r l e s s c a n o n i c a l way) i s o m o r p h i c , t h a t any such b a r y c e n t r i c s u b d i v i s i o n T ' o f T i s c e l l u l a r , t h a t it i s pseudo-convex i f and o n l y i f T i s c e l l u l a r , t h a t r ( f l ) = 3 f o r any f ' € T i and t h a t t h e boundary a £ ' := - f ' \ f ' o f any f ' E T i c o n s i s t s of p r e c i s e l y t h r e e e d g e s e; = e ; ( f 1 ) ,

e; = e ; ( f 1 ) and e i = e ; ( f l ) i n Ti and t h r e e v e r t i c e s v& = v & ( f l ) , v i = v i ( f l ) and v i = v i ( f 1 ) i n TA such t h a t v& € T o v; E T \ T o and v i E E \ T , w h i l e f o r i = 0 , 1 , 2 one h a s ae; := - e ! \ e T =

- ^{1 1}

{ v ~ , v ; , v ~ ) ~ { v ; } ( c f . f i g . 2 ) . Note t h a t f ' n T = ? ₂

Given a b a r y c e n t r i c s u b d i v i s i o n T' of T we may a l s o d e f i n e t h e d u a l

T ^{o f T} ( r e l a t i v e t o T' ) by

One c a n e a s i l y show t h a t T ' i s a l s o a b a r y c e n t r i c s u b d i v i s i o n o f T and t h a t T i s t h e d u a l o f T r e l a t i v e t o T' , t h a t t h e r e a r e cano- n i c a l b i j e c t i o n s between To and T2 and between T2 and To , ^{t h a t}

T i s c e l l u l a r ( o r pseudo-convex o r n o n - d e g e n e r a t e ) i f a n d o n l y i f T i s c e l l u l a r ( o r pseudo-convex o r non-degenerate, r e s p e c t i v e l y ) a n d t h a t a n y two d u a l s o f T a r e ( i n a more o r l e s s c a n o n i c a l way) isomorphic.

3 - Chamber systems

L e t C = <a O , a l , a 2 1 af = Ir d e n o t e t h e f r e e C o x e t e r group, g e n e r a t e d by t h e t h r e e i n v o l u t i o n s a o , o l , a 2 . So, by d e f i n i t i o n , an element T

of E i s j u s t a 'word' a

i l Oi2

-

i n t h e t h r e e symbols oo,al and a 2 such t h a t i , i i2

...

and f o r two such e l e m e n t s a - . . u i and a

...

I n j1 jm

d e f i n e d t o be oi

...

...

I ' n j l Jm

t h e p r o d u c t i s d e f i n e d r e c u r s i v e l y a s t h e p r o d u c t o f u i,

...

n- I a

...

.

1, 5 ,

A ( t h i n ) chamber s y s t e m C (of r a n k 2 ) i s d e f i n e d t o b e j u s t a s e t C

on which E a c t s from t h e r i g h t . An isomorphism between two chamber systems C and C ' i s a b i j e c t i o n q~ : C + C' such t h a t ~ ( C T ) =

= w(C)r f o r a l l C E C and T E C

.

On any chamber system C we have a w e l l - d e f i n e d m e t r i c by d e f i n i n g d ( C , C 1 ) := i n f ( l ( r ) I ^{r E C ; Cr = C ' )} f o r any C , C ' E C ( - a l l o w i n g f o r d ( C , C 1 ) = , ^{i f f C and C '} a r e i n d i f f e r e n t E - o r b i t s ) .

To establish the relation between tilings and chamber systems, we de- fine the topological realization IIC I1 of a chambersystem C as the topological space one gets from the disjoint union C x A of as many copies of the standard simplex _{A :=} {p = (porpl ,p2) E IR 2 ] 1 pi=l ,pi+) as there are chambers C in C by identifying two elements (C,p) and (C',pa) in this disjoint union if p = p ' and if for some i E {0,1,2} one has pi = 0 and C' = Coi

.

Hence, I I c I ~ comes always with a canonical triangulation, its cells being the images of the various copies C x A 5 C A of A in l l C l l and the action of 1 on C encodes how these triangles have to be glued together along their edges to build up llCll

.

4 - Chamber systems and tilings

Concerning the relation between chamber systems and tilings we have the following

Theorem 1: There exists a 1-I-correspondence between isomorphism clas- ses of non-degenerate tilings T of the euclidean plane and isomor- phism classes of infinite chamber systems C satisfying the following conditions:

(i) 7, acts transitively on C ;

(ii) each oi acts fixed point free on C ;

(iii) for each i,j E {0,1,2) and each C E C the number rij(C) := inf(r E N I ^{C(o.U.)r = C)} is finite;

1 3

(iv) for all C E C one has rO1 (C) = 2 < min(rol (C) ,rI2(C)) ;

(v) for one (or

-

1 j ( T E Z ; O z i < j ~ 2 ) .

The correspondence between tilings and (such) chamber systems is de- fined in terms of barycentric subdivisions and topological realiza- tions: if T is a tiling, choose some barycentric subdivision T' of T , ^{put C = CT :=} Ti and for f' = C E C = Ti and i E E0,1,2} let Cai = £'ai denote the uniquely determined face f" E Ti with

el(£') = ej(f")

.

{(C,p) E llCll I ^{p2 = 01} is mapped under any such homeomorphism onto a tiling TC c_E , whose barycentric subdivision is just the image of C x a A := i ^{(C,p) E C x A} I ^{po = 0 or pl = 0 or p2 = 01,} which implies in turn that C is isomorphic to C

.

then CT and C$ coincide except that a, acts on CT as a2 does on C+ and vice versa.

5 - An illustrative example: the Penrose tiling

Following de Bruijn /3/, we define a Penrose tiling for any sequence ( y o r y 1 , ~ 2 , ~ 3 r ~ 4 ) of real numbers with yo+y1+yZ+y3+y4 = 0 such that with 5 = e 2ni/5

for any z E (1: theenumber

#{k E {0,1,2,3,4) 1 ^{R e k 2 yk (Z) }} is at most two. The associated chamber system C = C can be defined as follows: as a set

(~,r

- .

C consists of all symbols C

(krn")

n,m E % ; ~ , q E { + I }

.

5 : : : : ) u

0 = c (krnr-E) 3 rmr rl and C

(1:~::)~~

(2:;;:) ^.

k n ~

(j :m:n)u2 let z?:: denote the unique z € Z with Re(z. c - ~ ) = n+y k and Re(z.5-j) = m + y , ^{choose 1 € 0 , .4 k and q € 2 such}

j -k j

that il.sgn(Im(< ) l m kz - ^z is positive, but minimal k n , ~

among all such expressions and put C

(

( )

s' = - n - ~ g n ( 1 m ( ~ - ~ - g ~ ) ) - s ~ n ( 1 m ( 5 - ~ - ~ ) ) . It is an immediate conse- quence of de Bruijn's analysis of Penrose tilings that the chamber

system C defined in this way is just the chamber system (yo,.

. -

associated with the Penrose tiling with the parameters yoryl,...,y4

.

Hence the whole intricate geometry of such a tiling is reflected on the chamber system level exclusively in form of the rather complicated and comparatively difficult definition of the action of o2 on

Chapter I1 - The concept of periodicity and some of its various re- laxations.

6 - Equivariant tilings

An equivariant tiling is defined to be a pair (T,r) consisting of a tiling T 5173 and a discrete group r of homeomorphisms of E such that y (T) = T for all y E r

.

( T I are defined to be isomorphic if there exists some homeomor-

- ¹

phism cp : E + E with cp(T) = T ' and r' = {cpycp ( y E r)

.

( T , r ) is an equivariant tiling, then P acts in a natural way and

faithfully on the chamber system C = CT by C-automorphisms, so in particular it acts freely on C , ^since z acts transitively on C :

if yC = C for some y E r and C E C , then for all T E C one has y(Cr) = (yC) r = Cr , i.e. yC' = C' for all C' E C and therefore y = IdC

.

- 1

such t h a t - a s above - r ' = Q [ y E r } ( - b u t n o w r and r '

a r e c o n s i d e r e d a s subgroups o f AutC ( C ) and AutC ( C ' ) , r e s p e c t i v e l y )

.

Hence one can deduce from Theorem 1 t h e f o l l o w i n g consequence:

Theorem 2: Given an e q u i v a r i a n t t i l i n g ( T , r ) , l e t D = D ( T , T ) de -

n o t e t h e 1 - s e t o f r - o r b i t s r \ C T o f C T and f o r i E 0 1 2 and

D = r - c E D p u t p i j ( ~ ) : = A

r . . ( C ) which i s w e l l d e f i n e d s i n c e 1 3

r ( Y C ) = r i i ( C ) f o r a l l Y E r . ^Then ( T , r ) and T a r e i s o - i i

mokphic i f and o n l y i f t h e r e e x i s t s a x-isomorphism

Q : ~ T , T ) a D ( T ~ , T I ) such t h a t p i j ( Q ( D ) ) = p i j (Dl f o r a l l D E D . p r o o f : One can r e c o n s t r u c t C and r 5 AutC ( C ) from (D;pOI ' p q 2 1P02) by c o n s i d e r i n g f o r some D E 0 t h e subgroup

1 / p i j ( D = ) -1

E D := < ~ ( o . u . ) T

I

from Theorem 1 t h a t C i s i s o m o r p h i c t o t h e C - s e t E D \ E of c o s e t s of ED i n C and it i s a l s o e a s i l y e s t a b l i s h e d t h a t E~ i s normal i n t h e s t a b i l i z e r group x D of D and t h a t r 5 A u t C ( z D \ 1) i s j u s t isomor- p h i c t o t h e f a c t o r group _{c ~ / E}. _~

For a g i v e n e q u i v a r i a n t t i l i n g (T, F) we d e f i n e (D;pol ,po2 , p l 2 ) t o c o n s t i t u t e i t s "Delaney symbol".

7 - P e r i o d i c t i l i n g s

A n e q u i v a r i a n t t i l i n g ( T , r ) i s d e f i n e d t o b e p e r i o d i c i f r i s a subgroup of t h e g r o u p I s o m ) o f i s o m e t r i e s o f lE and i f t h e o r b i t s p a c e l'\E i s compact ( o r - e q u i v a l e n t l y - ^{i f} r i s one o f t h e c r y s t a l l o g r a p h i c g r o u p s o f t h e e u c l i d e a n p l a n e ) . A t i l i n g T i s de- f i n e d t o be p e r i o d i c i f t h e r e e x i s t s some r s u c h t h a t (T, T ) i s an e q u i v a r i a n t p e r i o d i c t i l i n g i n which c a s e

( T , l s o ( T ) := {y E I s o m ) I ^{yT = T ) ) i s} n e c e s s a r i l y p e r i o d i c , t o o . Concerning p e r i o d i c t i l i n g s one can prove

Theorem 3 : The Delaney symbols o f e q u i v a r i a n t t i l i n g s a r e p r e c i s e l y t h o s e systems (D;pol , p o 2 , p 1 2 ) where D i s a C-set and p i j : 0 + I R a r e maps from D i n t o IR f o r which t h e f o l l o w i n g h o l d s :

i D i s f i n i t e ;

( i i ) p i j i s c o n s t a n t o n < u , u . > - o r b i t s ; i J

( i i i ) l / p i j (D) E TX f o r a l l D E D and 0 5 i < j 5 2 ; ( i v ) po2(D) = 1/2 > max(pol ( D l , p 1 2 ( D ) f o r a l l D E D ;

1-/pij (D) ( v ) D(0.u )

1 j = D f o r a l l D E D and O z i < j ( 2 ;

Moreover, any such system is the Delaney symbol of an equivariant tiling ( T I T ) with = Iso(T) unless it is the Delaney symbol of one of those twelve'harkednl-isohedral tilings IH 19, 35, 48, 60, 63, 65, 70, 75, 80, 87, 89, and 92 listed in Ref. 13.

The proof of this theorem which depends on a topological classification of the 17 types of 2D-crystallographic groups will appear elsewhere.

It is rather useful in the business of enumeration of periodic tilings with various additional properties /16,17/.

8 - A hierarchy of relaxations of periodicity

In this section we list various properties which are shared by periodic tilings, but in general do not imply periodicity. Very close to being periodic is of course the condition of being "topologically periodic", i.e. of being isomorphic to a periodic tiling. Slightly less restric- tive seems to be the condition for an equivariant tiling (T,r) that the orbit space r\lE should be compact which is of course equivalent with #D(T,r) < . But one can show that the Delaney symbol

(V;pO1 I P ~ ~ I P ~ ~ ) of such a tiling always satisfies the conditions (i)- (v) of Theorem 3 as well as K := (pol (Dl + pO2(D) + pI2(D)

-

DEZ)

so it is either topologically periodic or one has K < 0 in which case one can find a hyperbolic metric on IE such that r consists of isometries with respect to this hyperbolic metric. Another relaxation of periodicity are matching conditions which can be rephrased in our context as follows: given a (generally finite) set V together with three symmetric relations Ro,R1,R2 c_ ^{D x} D and three maps Po1 ,Po21 pI2 : D + IR we say that C satisfies the matching conditions of type

(D;RO,R1,R2;pOl ,p02,p12) if there exists a map ^{q~ :} C ⁺ D such that C E C implies (cp(C) ,cp(Cai)) E Ri for all i E {0,1,2} as well as

-

C Pij(v(Cv)) = 2 for all O s i < j 5 2 .

C'EC<oi,o .>

3

For the Penrose tilings one would choose for instance (see fig. 3):

1 1

p02(X) = - ₂ and pol (X) = for all X E D , ^{pI2 (A1) = pI2 (A4) = 2/10,}

pI2(A2) = pI2(A3) = 3/10 , pI2(B1) = p12(B3) = 1/10 pI2(B2) =

= pI2(B4) = 4/10

.

(v) of Theorem 3 and these matching conditions,is necessarily one of the above chamber systems C described in section 5 and

Yo1.."

vice versa. Y4

Still another way to relax periodicity suggested by the Penrose tilings is the following one: a tiling T is defined to be repetitive, if for any r > 0 there exists some R > 0 such that for any x,y E E there exists some x' EIE and some isometry y E Iso(lE) with y(x) = x' ,

yIt E T I ^llxrtll 5 r) = {t' E T I ^{x ' I r and I l x ' ,y l l 5 R}

.

Periodic tilings are obviously repetitive, but the Penrose tilings show that the converse is not true.

A much weaker notion is, of course, "balancedness", i.e. the property of a tiling T that there exist numbers r > 0 and R > 0 such that for any face f E T2 there exists some x E f with B(x,r) :=

{y E E I Ilx,yllz r) 5 f _c B(x,R). For a repetitive tilinq there exists always some r as above,but not necessarily some R

.

Still weaker is the concept of being "topologically balanced", i.e. of being isomorphic to a balanced tiling, while this in turn is closely related to "quadratic growth" for CT , ^{i. e. to the}

existence of two constants cl and c2 such that for any C E C and n E N one has C ~ - ~ ~ ~ # C ~ ( C ) cc2-n2 with Cn(C) :={C1 EC 1 d(C,Ct)~n}.It

implies also boundedness of the coordination numbers, i.e. the existence of some k E N such that r(x) 5 k for all x E To and r(f) ( k forall f E T 2 .

It is worth noting that an equivariant tiling (T,r) with compact orbit space always has bounded coordination numbers, but it is topologically periodic if and only if the associated chamber system CT is of quadratic growth, while it seems to be an open question what kind of matching conditions imply quadratic growth and whether quadra-

tic growth and bounded coordination numbers implies already topological balancedness. It might also be worth noting, that boundedness of co- ordination numbers implies already that there are only finitely many topological types of bounded regions for a tiling T . More precisely, we may define two chambers C E C and C' E C' in two chamber systems

C and C' (which may or may not coincide) to be n-equivalent if there exists a bijective map q : Cn(C) A Cn(C1)

such that for r E C with ( T ) ( n one has q(Cr) = C' r

.

among all chamber systems of tilings with coordination numbers bounded by k there exist only finitely many n-equivalence classes of

chambers.

The concept of n-equivalence of chambers can also be used to define topologically locally isomorphic tilings: two tilings T and T' are topologically locally isomorphic if for any n €IN and any C E CT there exists some n-equivalent C' E CT, and vice versa. It may also be used to define topologically repetitive tilings as tilings T such

that for any n E LJ there exists some N E M such that Eor any C,C' E T' there exists some C" E T which is n-equivalent to C and satisfies d(C',C1') 5 N

.

which are n-equivalent to some given chamber C E CT divided by

#CN(Cq) whenever such lirnit exists. It seems worthwhile to point out, that in tilings with #Cn(C1) = c.n2 + o(n 2 ) for some c such a limit exists for all C' and is independent of C' if it exists at least for one C' E CT

.

9

-

Finally, one may also use this set-up to study displacements of perio- dic tilings: given a periodic tiling (T,r) with Delaney symbol

(D;pol,p02,p12) and a chamber system C = CT, of some other tiling T' , we will define a D-structure of C to be any map q : C + D

.

Given a D-structure q : C + D of C , we define any <ai>-orbit (C,Coi) to be a first-order-singularity of cp , if q(C) +@(C)oi and we define any <oi , ^o .>-orbit C<oi, o . > to be a second-order-singula-

3 3

rity of cp if it contains no first-order-singularity CB<oi> or C'<o.> , ^{but if Z}

3 pij (q(C')) 9 2 . Note that T' must be C'EC<oi,a .>

isomorphic to T if it has a D-structure without singularities. 3 For any D-structure cp of C let Sl (rp) 5 11 C l l denote the set of elements of type (C,p) E C x A with (C,Cai) a first-order-singula- rity and p(i) = 0 and let S 2 ((P) 5 11 C 11 denote the set of elements of type (C,p) with C<oi,o.> a second-order-singularity and

p(i) = p(j) = 0

.

,

,

r which can be considered as a far reaching analogue of the Burgers vector construction (see /20,21/ for a special case) and seems to be the essential topological invariant of the displacement. A detailed study of this approach towards displacements will appear elsewhere

(unless as claimed by J. Taylor, all this is well known and has been worked out already by J. von Neumann or J. W. Cahn or

...

Analogously, one may of course consider displacements of tilings with matching conditions in a similar fashion.

10 - Conclusion

The above remarks indicate that

- there are many fascinating questions in the topological theory of tilings, which are motivated by the study of icosahedral crystals

a n d P e n r o s e p a t t e r n s a n d whose s t u d y m i g h t h e l p t o c l a r i f y some o f t h e p r o b l e m s c l o s e l y r e l a t e d t o t h i s f i e l d ,

and t h a t

- t h e t h e o r y o f Delaney symbols o f t i l i n g s , o r i g i n a l l y d e v e l o p e d t o s t u d y p e r i o d i c t i l i n g s , i s w e l l s u i t e d t o d e a l w i t h t h e s e q u e s t i o n s a n d t o p r o v i d e a r e a s o n a b l e b a s i s f o r d e v e l o p i n g a t h e o r y o f t i l i n g s s a t i s f y i n g r e l a x e d t o p o l o g i c a l p e r i o d i c i t y c o n d i t i o n s a s d e s c r i b e d a b o v e . By r e p l a c i n g t h e g r o u p i = E 2 = < u o , u l , 0 2 I ^{u: = 1'} by t h e g r o u p C n = < u o , . . . , u n 1 ^{u .} 2 = 1 > it s h o u l d a l s o be p o s s i b l e t o d e a l w i t h t h e h i g h e r d i m e n s i o n a l c a s e s .

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F i g u r e 1

F i g . I . ( a ) shows a f i n i t e s e c t i o n o f a t i l i n q T 5 l E 2 . One h a s r ( x ) = 5 , r ( y ) = l , r ( z ) = 3 , r ( f ) = 2 , r ( f 1 ) = 7 , r ( f l ' ) = l . The d o t t e d c i r c l e around x c i r c u m s c r i b e s a neighbourhood U o f x such t h a t ( U , U f t ~ , { x ) ) i s homeomorphic t o ( b ) .

F i g u r e 2

F i g u r e 2 shows a t i l i n g T c ~( i n d i c a t e d 2 by ^I'-" ) and one o f i t s b a r y c e n t r i c s u b d i v i s i o n s ^{T I} c o n s i s t i n g of t h e d u a l T ( i n d i c a t e d by

11- - - ^{" )} and t h e a d d i t i o n a l e d g e s , i n d i c a t e d by

"...".

Figure 3

Figure 3: De Bruijn's matching conditions for the thin and the thick rhombus can easily be translated into matching conditions for the eight inequivalent triangles in the barycentric subdivision of these rhombi:

a tiling constructed from these two rhombi is a Penrose pattern if and only if each Ai or Bi borders only some other Ai or Bi along its "outer" edge.