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QUASIPERIODIC PATTERNS
A. Katz, M. Duneau
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C8, supplkment a u n012, Tome 46, dkcembre 1985 page C8-31
QUASIPERIODIC PATTERNS
A. Katz and M. DuneauCentre de Physique Thdorique, EcoZe Polytechnique, Plateau de PaZaiseau,
91 128 PaZaiseau Cedex, France
Resum6 - Noun CLXpVbVMn une ththeohie den pawagen e t pLu4 giin&cLeemevLt den o&uc- &en quai-pCkiodiyueh pkL6entartt un ohdtre ohievLtationnd gLobd. NUUA pk&- oevLtoMn no&e "muhode de La bande" (une wa4iauzte
d a
miithoden de pkoje&on)n u t
un modPLe unLdimeMnionneR. Nou4 expooom emLLite Le can de La o y m W e icooathedhiyue en vue de L1appficication aux q u a i - d t u u x du t y p e ALwnivLium- Mangan2ne. La muhode popobiie donne uccPb aux phophiEtk% tan2 LoccLea(danoi6iccLtion den enwLtonneme& poooibLu de chayue oLte) yue g L o b d a (ccLecul e x p f i c i t e de La &aMndohm&e de Founien].
A b s t r a c t
-
We phaevLt a t h ~ v t r y06
y u a n i p d o d i c Xilingo and mohe genend quani;o&odic pcLttehuzcl wLth Long hange ohievLtcLtiond okdeh. We explain out " b a k i p method" ( a vahiartt od t h e p o j e o t i o n me~oo!A) on a one dimenbiond mod&. Then t h e icooahehcLe cane AA inwuLLgigcLted i n v i w0 6
fithe quanictyotahod
t h e type AL-Mn. OWL method &VWA t h e bRudy03
t h e d o c d p k o p ~ e ~ (ouchan
fie &oi6ic&on od t h e u&ex n e i g h b o u t h o o ~ ) an weRe an t h e ydobd onen, and an a n d y i i c exphenoionad
t h e FowLieh &at~a~ndotun AA ddehived, exkibi- LLng f i t h e quanipiyJeh-iodicay.INTRODUCTION
T h i s l e c t u r e d e a l s w i t h t h e t h e o r y o f q u a s i p e r i o d i c t i l i n g
,
and more g e n e r a l l y w i t h q u a s i p e r i o d i c p a t t e r n s . We p r e s e n t t h e i d e a s i n t r o d u c e d i n C91,
w h i c h we c a l l t h e " s t r i p method".The i n t e r e s t f o r non p e r i o d i c t i l i n g s came f i r s t f r o m problems o f mathematical l o g i c However, s i n c e t h e i n v e n t i o n b y R. Penrose o f h i s w e l l known a p e r i o d i c t i l i n g o f t h e p l a n e [ 1 , 7 1
,
t h e m o t i v a t i o n has changed t o t h e s t u d y o f t h e g e o m e t r i c a l p r o p e r t i e s o f t h e s e p a t t e r n s . A f t e r t h e f i r s t t h e o r e t i c a l work b y R. Penrose, J. Conway [71 and N.G. de B r u i j n [ 2 1 , A.L. Mackay [ 3 ] and Mosseri and Sadoc [ 4 ] have drawn a t t e n t i o n t o t h e p o s s i b l e i m p l i c a t i o n s o f these s t r u c t u r e s f o r s o l i d s t a t e p h y s i c s . A 3-dimensional general i z a t i o n o f t h e Penrose t i 1 in g ( u s i n g two rhombohedra i n s t e a d o f two rhombs) has been d e v i s e d by R. Amrnann and t h e o r e t i c a l l y d e s c r i b e d by Kramer and N e r i [5,61 i n t h e c o n t e x t o f t h e de B r u i j n approach.The s p e c i a l p r o p e r t i e s o f t h e F o u r i e r t r a n s f o r m o f t h e Penrose p a t t e r n s , w h i c h a r e now i d e n t i f i e d as t h e i r q u a s i p e r i o d i c i t y , have been e m p i r i c a l l y observed b y
A. L. Flackay, u s i n g o p t i c a l methods. D. L e v i n e and
P. S t e i n h a r d t have suggested i n r e f .
[81 t o t a k e q u a s i p e r i o d i c i t y as t h e b a s i s t o b u i l d i c o s a h e d r a l q u a s i c r y s t a l s .The e x p e r i m e n t a l d i s c o v e r y o f i c o s a h e d r a l symmetry t o g e t h e r w i t h some k i n d of l o n g range s p a t i a l o r d e r i n s o l i d s by D. Schechtnan, I. Blech, D. G r a t i a s , and J.W. Cahn
[ I 0 1 has aroused an i n t e n s e a c t i v i t y among t h e o r i s t s , as w e l l about o n t h e s t r u c t u r a l
problem [6,8,9,11,12,13,14,151
,
and a b o u t s t a b i l i t y q u e s t i o n s w h i c h w i l l n o t beJOURNAL
DE
PHYSIQUEdiscussed here.
The theory described here, which takes i n t o account both t h e icosahedral symmetry and the q u a s i p e r i o d i c i t y o f the t i l i n g
,
i s based on an idea independantely introduced i n t h i s c o n t e x t by V. E l s e r [12, 131 and us [9,15] ( f o l l o w i n g N.G. de B r u i j n ) .A OWE DIMENSIONAL MODEL
L e t us begin by a d e s c r i p t i o n o f t h i s method i n t h e s i m p l e s t one-dimensional case (see F i g . 1) : t o t i l e o f a s t r a i g h t l i n e w i t h two types o f i n t e r v a l s , we s t a r t w i t h the square l a t t i c e 2' i n the plane IR', i n which the l i n e E t o be t i l e d i s drawn. Consider now t h e s t r i p o b t a i n e d by s h i f t i n g t h e u n i t square K o f t h e l a t t i c e along t h e l i n e . The c l a i m i s t h a t , f o r almost a l l p o s i t i o n s o f the l i n e , t h i s s t r i p contains a unique broken l i n e (made o f edges o f t h e l a t t i c e ) , which j o i n s e x a c t l y a l l the v e r t i c e s f a l l i n g i n s i d e t h e s t r i p . I t j u s t remains t o p r o j e c t t h i s broken l i n e o r t h o g o n a l l y on t h e given d i r e c t i o n t o g e t the p r e d i c t e d t i l i n g , the two t i l e s a and b being c l e a r l y t h e p r o j e c t i o n s o f t h e v e r t i c a l and h o r i - zontal edges o f t h e u n i t square K.
One can understand t h e e s s e n t i a l f e a t u r e s o f o u r t i l i n g w i t h t h i s simple example. Consider f i r s t t h e choice o f the u n i t square t o b u i l d t h e s t r i p . I t i s c l e a r t h a t one has t o take a u n i t c e l l o f the l a t t i c e i n order t o o b t a i n a t i l i n g (by t h e p r o j e c t i o n s o f the edges o f t h e c e l l ) : w i t h a narrower s t r i p , one would g e t "holes" i n t h e broken l i n e , and w i t h a l a r g e r s t r i p , one would g e t e x t r a p o i n t s . However, any u n i t c e l l o f 712 i s possible, and n o t o n l y the canonical square. This i s c l o s e l y r e l a t e d t o the s e l f - s i m i l a r l y p r o p e r t i e s o f these t i 1 ings.
Secondly, consider t h e slope o f t h e l i n e E w i t h r e s p e c t t o t h e canonical b a s i s of IR'. I t i s c l e a r t h a t t h e t i l i n g i s p e r i o d i c i f and o n l y i f t h i s slope i s a r a t i o n a l number : i n f a c t , t h e slope i s j u s t the r a t i o o f t h e r e l a t i v e abundances of each type o f t i l e s , and t h i s r a t i o i s c e r t a i n l y r a t i o n a l f o r a p e r i o d i c t i l i n g .
Fig. 1
Now, g i v e n a n i r r a t i o n a l slope, c o n s i d e r what happens when t h e s t r i p i s t r a n s l a t e d i n IR' : each t i m e a p o i n t l e a v e s t h e s t r i p , a n o t h e r p o i n t e n t e r s t h e s t r i p . These two p o i n t s a r e t h e v e r t i c e s o f t h e d i a g o n a l o f some square i n t h e l a t t i c e , i n such a way t h a t t h e b r o k e n l i n e "jumps" f r o m one s i d e o f t h e square t o t h e o t h e r . T h i s i s t h e o n l y e l e m e n t a r y change o f t h e t i 1 in g , b u t f o r any f i n i t e t r a n s l a t i o n o f t h e s t r i p ( t r a n s v e r s e t o i t s d i r e c t i o n ) such jumps o c c u r i n f i n i t e l y many times, i n such a way t h a t f o r each i r r a t i o n a l s l o p e , t h i s c o n s t r u c t i o n g e n e r a t e s a non coun-2 t a b l e s e t o f d i f f e r e n t t i l i n g s ( o f course, each t i m e t h e t r a n s l a t i o n belongs t o Z
,
one g e t s m e r e l y a t r a n s l a t i o n o f t h e i n i t i a l t i l i n g ) .N o t i c e t h a t we can s l i g h t l y g e n e r a l i z e t h e c o n s t r u c t i o n and d i s t i n g u i s h between t h e l i n e a l o n g which t h e s t r i p i s b u i l t and t h e l i n e on which t h e broken l i n e i s p r o j e c t e d
t o o b t a ~ n t h e t i l i n g . It i s c l e a r t h a t t h e o r i e n t a t i o n o f t h i s l a s t l i n e ( t h e t i l e d one) m e r e l y d e f i n e s t h e r e l a t i v e s i z e s o f t h e t i l e s , and t h a t t h e o r i e n t a t i o n o f t h e s t r i p d e f i n e s a l l t h e i m p o r t a n t p r o p e r t i e s , f o r i n s t a n c e t h e r e l a t i v e abundance o f each t y p e o f t i l e . I f we v a r y t h e o r i e n t a t i o n o f t h e s t r i p w h i l e m a i n t a i n i n g unchanged t h e t i l e d l i n e , we g e t d i f f e r e n t t i l i n g w i t h t h e same p a i r o f t i l e s . Each t i m e t h e s l o p e i s r a t i o n a l , t h e t i l i n g i s p e r i o d i c , i n such a way t h a t t h e non p e r i o d i c t i l i n g o b t a i n e d f o r i r r a t i o n a l s l o p e s can be t h o u g h t o f as " i n t e r p o l a t i o n s " b e t w e e n . p _ e ~ i o d i c ~ t t i c e s .
L e t u s now e x p l a i n why any f i n i t e p a t c h o f t i l e s t h a t belongs t o a t i l i n g appears i n - f i n i t e l y many t i m e s i n any t i l i n g d e f i n e d t h r o u g h a s t r i p w i t h t h e same g i v e n i r r a - t i o n a l s l o p e (Conway's 1 o c a l isomorphism p r o p e r t y , see [7
1
) . Consider such a t i 1 ing, p i c k any f i n i t e p a t c h of t i l e s i n i t . and c o n s i d e r t h e f i n i t e b r o k e n l i n e t h a t p r o - ~ e c t s i n t h i s p a t c h . S i n c e t h e s l o p e - i s i r r a t i o n a l , i t i s c l e a r t h a t t h e p r o j e c ' t i o n o f t h i s b r o k e n l i n e on E' (see f i g . 1 ) i n s t r i c t l y s m a l l e r t h a n t h e p r o j e c t i o n o f t h e whole s t r i p , i n such a way t h a t t h e i r e x i s t s an open s e t (non empty ! ) o f t r a n s l a t i o n s i n E ' t h a t keep t h e p r o j e c t i o n o f t h e f i n i t e b r o k e n l i n e i n s i d e t h e s t r i p . Now con- s i d e r t h e a u x i l i a r y s t r i p t h a t p r o ' e c t s on E ' o n t o t h i s s e t o f t r a n s l a t i o n s : t h e r e a r e i n f i n i t e v e r t i c e s o fZ3 i n s i d e i t , and each o f them d e f i n e s a t r a n s l a t i o n
t h a t maps t h e f i n i t e broken l i n e i n s i d e o u r main s t r i p o n t o a copy o f i t , which i s i n d i s t i n g u i s h from a p a r t o f t h e i n f i n i t e b r o k e n l i n e t h a t p r o j e c t s on t h e t i l i n g . T h i s shows t h a t any f i n i t e p a t c h which appears i n a t i l i n g , appears i n f i n i t e l y many t i n e s . One can show w i t h t h e same t y p e o f argument t h a t , g i v e n any two p a r a l l e l s t r i p s and a f i n i t e b r o k e n l i n e i n one o f them, t h e r e always e x i s t s a t r a n s l a t i o n i n22 t h a t
maps t h e f i n i t e b r o k e n l i n e i n s i d e t h e o t h e r s t r i p , which means t h a t any f i n i t e p a t c h t h a t appears i n one t i l i n g appears i n a l l t h e t i l i n g s a s s o c i a t e d w i t h t h e same s l o p e . There a r e i n t e r e s t i n g q u e s t i o n s c o n c e r n i n g t h e d i s t a n c e between two c o p i e s o f a g i v e n p a t c h . I t can be shown t h a t t h e r a t e o f " r a r e f a c t i o n " o f t h e c o p i e s o f a p a t c h when i t s s i z e i n c r e a s e s depends on a r i t h m e t i c a l p r o p e r t i e s o f t h e slope. F o r a l g e b r a i c numbers, t h e mean d i s t a n c e between two c o p i e s i s p r o p o r t i o n a l t o t h e s i z e o f t h e patch, b u t f o r L i o u v i l l e numbers, t h e mean d i s t a n c e can grow a r b i t r a r y q u i c k l y .F i n a l l y , l e t us d e s c r i b e t h e c a l c u l u s o f t h e F o u r i e r t r a n s f o r m o f t h e measure d e f i - ned b y p u t t i n g one D i r a c d e l t a a t each v e r t e x o f t h e t i l i n g ( f i g . 2 ) . F o l l o w i n g t h e s t r i p method, t h i s measure i s o b t a i n e d i n t h r e e s t e p s : t a k e t h e square l a t t i c e o f D i r a c d e l t a s i n t h e p l a n e , m u l t i p l y b y t h e c h a r a c t e r i s t i c f u n c t i o n o f t h e s t r i p , and t h e n p r o j e c t on t h e l i n e . To compute t h e F o u r i e r t r a n s f o r m , we have j u s t e t o o p e r a t e t h e " F o u r i e r images" o f t h e s e t h r e e s t e p s i n t h e r e c i p r o c a l space. The f i r s t one i s t o t a k e t h e F o u r i e r t r a n s f o r m o f t h e square l a t t i c e , which i s o f course a n o t h e r square l a t t i c e o f D i r a c d e l t a s . The second s t e p i s t h e c o n v o l u t i o n b y t h e F o u r i e r t r a n s f o r m o f t h e c h a r a c t e r i s t i c f u n c t i o n o f t h e s t r i p , w h i c h i s easy t o conpute s i n c e i t i s a p r o d u c t : i n t h e d i r e c t i o n E* dual t o t h e d i r e c t i o n o f t h e s t r i p , t h i s F o u r i e r t r a n s f o r m i s a D i r a c d e l t a and i n t h e o r t h o g o n a l d i r e c t i o n
C8-34 JOURNAL
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PHYSIQUEL e t us make a few comments about the r e s u l t . The most important p o i n t i s t h a t the F o u r i e r transform thus obtained i s a sum o f weighted D i r a c d e l t a s , c a r r i e d by
2
the p r o j e c t i o n o f the whole l a t t i c e Z
,
i . e . by a two generators Z-module, and t h i s p r o p e r t y o f the support o f i t s F o u r i e r t r a n s f o r m i s t h e v e r y d e f i n i t i o n o f the quasi p e r i o d i c i t y o f a measure. Observe t h a t t h e q u a s i p e r i o d i c i t y f o l l o w s d i r e c t l y from the s t r i p being b u i l t on a s t r a i g h t 1 in e : one could produce more general t i 1 in g s w i t h a s t r i p obtained by s h i f t i n g t h e u n i t c e l l along a curved o r u n d u l a t i n g l i n e ( w h i l e m a i n t a i n i n g the p r o j e c t i o n on a s t r a i g h t l i n e ) and they would n o t be quasi- p e r i o d i c . The second p o i n t i s t h a t a t r a n s l a t i o n o f the s t r i p changes t h e F o u r i e r transform o n l y by a phase f a c t o r ( a d i f f e r e n t one f o r each D i r a c d e l t a ) i n such a way t h a t a l l the t i l i n g s associated w i t h t h e same i r r a t i o n a l slope g i v e the same d i f f r a c t i o n p a t t e r n s , as f a r as o n l y i n t e n s i t i e s are measured.F i n a l l y , observe t h a t although t h e support o f t h e F o u r i e r transform i s a dense set, the i n t e n s e peaks correspond t o v e r t i c e s which are near the l i n e E* (see f i g . 2 ) , and t h a t t h e s e t o f peaks whose i n t e n s i t y i s g r e a t e r than any s t r i c t l y p o s i t i v e t h r e s h o l d i s always d i s c r e t e . I n f a c t , t h e q u a n t i t a t i v e a n a l y s i s o f the i n t e n s i t i e s i n t h e d i f f r a c t i o n p a t t e r n may be used t o r e c o n s t r u c t the geometry i n the d i r e c t space (up t o a t r a n s l a t i o n o f the s t r i p ) i n t h e same sense and w i t h the same l i m i t a t i o n s as i n o r d i n a r y c r y s t a l l o g r a p h y .
THE GENERAL CASE
We g i v e now the g e n e r a l i s a t i o n o f t h i s method t o a r b i t r a r y dimensions. We generate a q u a s i p e r i o d i c t i l i n g o f a p-dimensional space E as t h e p r o j e c t i o n , from a h i g h e r dimensional space IR", o f a p-dimensional surface made up o f a s u i t a b l e union of p-facets o f a r e g u l a r n-dimensional l a t t i c e L i n 1 ~ ~ .
F i g . 2,
A c t u a l l y , l e t
L=
Z n be t h e n-dim l a t t i c e i n lRn generated by the n a t u r a l basis [el,..
.
,en], and l e t yn be the u n i t cube. L e t E c l R n be a p-dimensional subspace o f lRn, and assume t h a t E does n o t c o n t a i n any p o i n t o f t h e l a t t i c e , except the o r i g i n .There are (") d i f f e r e n t p-facets ( t h e p-dimensional analog o f an edge) of yn con- P
t a i n i n g t h e o r i g i n . These f a c e t s p r o j e c t on E on a - p r i o r i (n) d i f f e r e n t p-volumes.
D
A t i l i n g o f E by mean o f these volumes i s obtained i n the f o i l o w i n g way : L e t S=E+y be the open s t r i p generated by s h i f t i n g yn along E . Then the c l a i m i s t h a t
n
t h e union o f a l l p - f a c e t s e n t i r e l y contained i n S i s e x a c t l y a p-dimensional surface o f I R ~ , t h e p r o j e c t i o n o f which on E gives t h e announced t i l i n g .
I f E' i s t h e orthogonal complement o f E i n lRn, t h e p r o j e c t i o n K o f the s t r i p on E' i s j u s t t h e p r o j e c t i o n o f t h e u n i t cube yn. Moreover, i f E n L = 10
1
,
t h e p r o j e c t i o n of t h e l a t t i c e L i n E' i s one t o one. Thus t h e s e t o f v e r t i c e s o f t h e q u a s i p e r i o d i c t i l i n g corresponds t o the s e t o f p o i n t s o f L t h a t p r o j e c t i n E' i n s i d e K.Now, i f E i s i n v a r i a n t under the a c t i o n o f a subgroup G o f t h e p o i n t group o f the l a t t i c e L, the a - p r i o r i
(i)
d i f f e r e n t t i l e s f a l l i n t o classes, i n such a way t h a t the t i l e s o f each c l a s s have the same shape, and are permuted by G.The s e t o f p-volumes around a given v e r t e x x o f t h e t i l i n g i s completely s p e c i f i e d by the s e t o f corresponding p - f a c e t s f a l l i n g i n K around t h e corresponding p o i n t x ' i n E l . Note t h a t no ( p + l ) , (p+2),
...,
n - f a c e t o f L can p r o j e c t i n E ' s t r i c t l y i n s i d e K.THE ICOSAHEDRAL CASE
L e t us now s p e c i a l i z e t o t h e case o f the icosahedral symmetry. The s i m p l e s t cons- 6
t r u c t i o n i n v o l v e s I R endowed w i t h an orthogonal r e p r e s e n t a t i o n o f t h e icosahedral 6
group G , which permutes Z6 and such t h a t I R f a 1 1s i n two G - i n v a r i a n t 3-spaces, E and E ' equipped w i t h non e q u i v a l e n t i r r e d u c i b l e r e p r e s e n t a t i o n s o f G.
The 20 d i f f e r e n t 3-facets o f
L=
ZL6 fa71 i n E (and E ' ) on d i f f e r e n t rhombohedra, w i t h t h e same f a c e t s ( w i t h angles Atn(2) and n - A t n ( 2 ) ) , each o f which being repeated 10 times. These are the rhombohedra considered by A.L. Mackay i n [31,
P. Kramer and R. N e r i i n [5,61, and us i n [91 ( F i g . 3b and c ) .6 The open s t r i p i s d e f i n e d by S=E+y6 where y6 i s the open u n i t cube o f I R
.
It can be seen t h a t t h e p r o j e c t i o n s n ( L ) on E and n l ( L ) on E' are Z-modules6 which a r e dense i n E and E '
.
I f E ~ , . ,. ,
s6,
i s the n a t u r a l b a s i s o f IR,
n ( k ~ ~ ) , . . . , n ( c ~ ~ ) p o i n t t o t h e 12 v e r t i c e s o f a r e g u l a r icosahedron centered a t t h e o r i g i n o f E and so do n1(?c1),..
.
,
n 1 ( + s 6 ) i n E ' .JOURNAL
DE
PHYSIQUEF i g . 3
The p r o j e c t i o n ( a ) o f t h e u n i t cube o f i n E and E ' i s a triacontahedron. A 5 - f o l d a x i s , a 3 - f o l d a x i s and 2 - f o l d a x i s are represented. I n ( b ) a f l a t rhombohe- dron and i n ( c ) a t h i c k rhombohedron, which a r e the 2 t i l e s o f t h e icosahedral t i l i n g s .
f a l l i n s i d e an o t h e r rhombohedron, spanned by el ,e ,e
,
i n such a way t h a t ( i , j ,k,l ,m,n) i s a permutation o f (1,2,3,4,5,6), an9 l o e a t e d i n t h e triacontahedron i n one o f e i g h t canonical p o s i t i o n s according t o the v e r t e x considered. Looking f o r the i n t e r s e c t i o n s o f these 160 rhombohedra y i e l d s a c e l l decomposition o f t h e triacontahedron i n which each c e l l corresponds t o a v e r t e x neighbourhood. One thus can see t h a t , up t o the symmetry operations, t h e r e a r e o n l y i n these t i l i n g s 24 p o s s i b l e p a t t e r n s o f rhombohedra around a p o i n t . I n p a r t i c u l a r the c e n t r a l c e l l o f the triacontahedron corresponds t o a 20-pronged s t a r w i t h t h e icosahedral symmetry. We present i n F i g . 4,5 and 6 t h r e e sections o f t h i s t i l i n g , which are associated t o axes o f o r d e r 5, 3 and 2 r e s p e c t i v e l y . The c u t s , which are made along 2-dimensional supfaces taken from 2-facets o f the t i 1 in g , y i e l d q u a s i p e r i o d i c t i 1 in g s o f the plane. A c t u a l l y the f i r s t section, c a r r i e d o u t o r t h o g o n a l l y t o a 5 - f o l d a x i s , p r o j e c t s on a g e n e r a l i s e d Penrose t i l i n g . The two o t h e r s e c t i o n s p r o j e c t on q u a s i p e r i o d i c ti- l i n g s o f t h e plane associated w i t h 3 - f o l d and 2 - f o l d symmetries.As i n t h e one dimensional case, t h e s t r i p can be t r a n s l a t e d , which y i e l d s an uncoun- t a b l e s e t o f non-isomorphic t i l i n g s .
v e r t e x , i s q u a s i p e r i o d i c : i t s F o u r i e r t r a n s f o r m i s a sum o f w e i g h t e d D i r a c measures w i t h s u p p o r t i n t h e Z-module generated b y t h e p r o j e c t i o n s o f t h e b a s i s v e c t o r s o f t h e l a t t i c e . Moreover, a l l t i l i n g s have t h e same F o u r i e r t r a n s f o r m , up t o a phase a t each p o i n t , i n such a way t h a t t h e y a r e i d e n t i c a l f r o m a " d i f f r a c t i o n a l " p o i n t o f view.
I n f a c t , t h e c a l c u l u s o f t h e F o u r i e r t r a n s f o r m r u n s e x a c t l y as i n t h e one dimensio- n a l case. Here i s a b r i e f a c c o u n t o f t h e p r o o f : l e t v =
x
s
x
6~ ~ 6 66 be t h e measure a s s o c i a t e d t o S "Z16, where i s t h e c h a r a c t e r i s t i c f u n c t i o n o f t h e s t r i p i n 1R6. Then n = iCE z6 ( < ) 6 n ( Z ) i s t h e measure a s s o c i a t e d t o t h e t i l i n g i n E. I f t ; = ( x , x 1 ) i s t h e o r t h o g o n a l decomposition o f ~ E I R ~ i n E and E l , and i f ~ = ( k , k ' ) i s t h e c o r r e s p o n d i n g decomposition i n t h e dual space, t h e F o u r i e r t r a n s f o r m o f n i s g i v e n b y n * ( k ) =v*(k,O). Now v* = x z*
C A EZ6 6 A , and s i n c e x S ( x , x l ) = x T R ( x L ) where xTR i s t h e c h a r a c t e r i s t i c f u n c t i o n o f t h e t r i a c o n t a h e d r o n i n E l ,x z ( k , k ' ) = 6 ( k ) xTR ( k ' ) . F i n a l l y n * ( k ) = iAE ~ 6 6 ( k - 1 ) xTR ( - 1 ' ) where A =(1,11) i s t h e d e c o m p o s i t i o n o f X i n t h e d u a l space.
We g i v e i n F i g . 7;8 and 9, t h e computed F o u r i e r t r a n s f o r m s o f a t i l i n g , i n t h e 3 p l a n e s r e s p e c t i v e l y o r t h o g o n a l t o symmetry axes of o r d e r 5,3 and 2. These p a t t e r n s a r e a s t o n i s h i n g l y s i m i l a r t o t h e e l e c t r o n d i f f r a c t i o n images c b t a i n e d by D. Shechtman, I. Elech, D. G r a t i a s and J.W. Cahn, r e p o r t e d i n [ I 0 1
,
on r a p i d l y c o o l e d a l l o y s o f A1 and Mn.The g e n e r a l framework p r e s e n t e d h e r e can be g e n e r a l i s e d i n many ways. F o r i n s t a n c e i f t h e c o n d i t i o n which i n s u r e s an e x a c t t i l i n g o f t h e space i s removed, more g e n e r a l q u a s i p e r i o d i c p a t t e r n s a r e o b t a i n e d .
On t h e o t h e r hand, t h e p r o j e c t i o n space and t h e s t r i p may have d i f f e r e n t o r i e n t a - t i o n s : t h e f i r s t one s p e c i f i e s t h e t i l e s w h i l e t h e second one d i c t a t e s t h e i r r e l a t i v e abundance. As i n t h e one dimension model, t h e q u a s i p e r i o d i c t i l i n g t h u s o b t a i n e d can be seen t o i n t e r p o l a t e between p e r i o d i c ones, w h i c h correspond t o r a t i o n a l o r i e n t a t i o n s o f t h e s t r i p .
N o t i c e t h a t t h e q u a s i p e r i o d i c i t y o f t h e t i 1 in g s i s independant o f t h e f a c t t h a t n e i t h e r t h e pentagonal n o r t h e i c o s a h e d r a l symmetries a r e c o m p a t i b l e w i t h p e r i o d i c o r d e r i n g .
I n f a c t t h e s e c t i o n s c o r r e s p o n d i n g t o t h e 2 - f o l d and 3 - f o l d axes, ( g i v e n i n F i g . 5 and 6 ) , show q u a s i p e r i o d i c t i l i n g s o f t h e p l a n e w h i l e these symmetries a r e o f c r y s t a l t y p e .
As a f i n a l remark, l e t us s t r e s s on t h e i d e a t h a t t h e c o n s t r u c t i o n p r e s e n t e d h e r e i n works i n any dimension and w i t h any l a t t i c e . F o r i n s t a n c e , i t i s easy t o see t h a t an o t h e r t y p e o f q u a s i p e r i o d i c t i l i n g o f t h e space w i t h i c o s a h e d r a l symmetry
? n
JOURNAL DE PHYSIQUE
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Submitted t o J. Non C r y s t . S o l i d s .
