Moroccan ornamental quasiperiodic patterns constructed by the multigrid method

13 Pages·2014·4.98 MB·English

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Journalof
Applied
Crystallography
ISSN1600-5767
Moroccan ornamental quasiperiodic patterns constructed by
the multigrid method
Youssef Aboufadil, Abdelmalek Thalal and My Ahmed El Idrissi Raghni
J.Appl.Cryst.(2014). 47, 630–641
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J.Appl.Cryst.(2014).47,630–641 YoussefAboufadiletal.· Moroccanornamentalquasiperiodicpatterns
research papers
Journalof Moroccan ornamental quasiperiodic patterns
Applied
constructed by the multigrid method
Crystallography
ISSN1600-5767
Youssef Aboufadil,* Abdelmalek Thalal* and My Ahmed El Idrissi Raghni*
Received8November2013
Accepted23January2014
DepartmentofPhysics,LSM,FacultyofScience,UniversityofMarrakech-Semlalia,Boulevard
PrinceMyAbdellah,Marrakech40000,Morocco.Correspondencee-mail:
[email protected],[email protected],[email protected]
The similarity between the structure of Islamic decorative patterns and
quasicrystals has aroused the interest of several crystallographers. Many of
these patterns have been analysed by different approaches, including various
kinds of ornamental quasiperiodic patterns encountered in Morocco and the
Alhambra (Andalusia), as well as those in the eastern Islamic world. In the
present work, the interest is in the quasiperiodic patterns found in several
Moroccan historical buildings constructed in the 14th century. First, the zellige
panels (fine mosaics) decorating the Madrasas (schools) Attarine and Bou
InaniainFezaredescribedintermsofPenrosetiling,toconfirmthatbothpanels
have a quasiperiodic structure. The multigrid method developed by De Bruijn
[Proc.K.Ned.Akad.Wet.Ser.AMath.Sci.(1981),43,39–66]andreformulated
byGratias[Tangente(2002),85,34–36]toobtainaquasiperiodicpavingisthen
usedtoconstructknownquasiperiodicpatternsfromperiodicpatternsextracted
fromtheMadrasasBouInaniaandBenYoussef(Marrakech).Finally,amethod
of construction of heptagonal, enneagonal, tetradecagonal and octadecagonal
quasiperiodic patterns, not encountered in Moroccan ornamental art, is
proposed. They are built from tilings (skeletons) generated by the multigrid
#2014InternationalUnionofCrystallography method and decorated by motifs obtained by craftsmen.
1. Introduction crystal diffraction patterns (Fig. 1) shows that Moroccan
‘Inquasicrystals,wefindthefascinatingmosaicsoftheArabic master craftsmen have also built, since the 14th century,
world reproduced at the level of atoms: regular patterns that decorative motifs that are encountered in quasiperiodic
never repeat themselves.’ This is what the Nobel Committee patterns. Several authors have analysed, using different
wrote (Royal Swedish Academy of Sciences, 2011) regarding approaches, various kinds of quasiperiodic patterns encoun-
theNobelprizeinChemistryattributedtoShechtmanin2011 tered in Moroccan–Andalusian decorative art and in the
for his discovery, in an aluminium manganese alloy, of quasi- eastern Islamic world. Makovicky (1992, 2004, 2007, 2008,
crystals with an icosahedral ordered phase, the diffraction
pattern of which exhibits fine well resolved diffraction spots
like in crystals but distributed according to the icosahedral
symmetry that is prohibited by both the two- and the three-
dimensional periodiclattice.
The simplest way of visualizing a two-dimensional quasi-
crystal isthatfollowed inanothercontextbyPenrose (1974),
who invented a nonperiodic tiling of a plane by two types of
tiles,generatedbyadeterministicconstructionusinginflation,
the diffraction pattern of which consists of (cid:2) peaks that are
distributed on figures with pentagonal symmetry. This tiling,
whichquicklybecamethearchetypeofquasicrystals,hasbeen
explained in the multigrid method developed by De Bruijn
(1981) in the plane and generalized by Duneau and Katz
(Duneau & Katz, 1985; Katz & Duneau, 1986), who showed
that a quasiperiodic tiling could be obtained by the cut-and-
Figure 1
projectionmethodfrom ahigher-dimension space.
(a) Zellige panel of the Madrasa Attarine (Fez). (b) The distances
The similarity between the zellige panels (fine mosaics)
betweentwosymmetricalelementsrelativetothecentreofthepattern
found in historical monuments and mausoleums and quasi- areintheratioofthegoldenmean.
630 doi:10.1107/S1600576714001691 J.Appl.Cryst.(2014).47,630–641
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2011),Makovicky&FenollHach-Alı´(1997),Makovickyetal. time a (cid:3) rotation in the plane; N is a generic value greater
n
(1998), Castera & Jolis (1991) and Castera (1996, 2003) than2,anditcorrespondstotherotationorderofthegridin
studied the octagonal and decagonal patterns, Rigby (2005), relationtotheothers.Wethenobtainagrid,calledamultigrid,
Lu & Steinhardt (2007), Saltzman (2008) and Al Ajlouni constitutedbyNfamiliesoflines;eachfamilyischaracterized
(2012) were interested in the decagonal patterns, Makovicky by an orthogonal vector V (Fig.2a).
n
& Makovicky (2011) studied the dodecagonal structure, and Fromacentralpoint,weplacethevectorV anditsopposite
n
Bonner&Pelletier(2012)constructedasevenfoldpattern.In andwetraceorthogonallinespassingbytheirextremities.We
another context, Whittaker & Whittaker (1988) proposed obtainanirregularconvexpolygonandwetracethesegments
heptagonalandenneagonaltiling,andFrancoetal.(1996)also joining the centre point to the vertices of the polygon. By
proposedenneagonal quasiperiodic tiling. drawing a small circle around the central point, we obtain N
Inthisworkweshallfirstshow,usingthemultigridmethod, sectors, each based on the polygon edge (Fig. 2b). In each
thatsomepatternsencounteredinMoroccanmonumentsbuilt sector, we choose a vector W of arbitrary length and direc-
n
by the craftsmen’s method called Hasba (Thalal et al., 2011; tion, and we associateitwiththe correspondingline family.
Aboufadiletal.,2013)areunderlinedbyaquasiperiodictiling. WethenconstructparallelogramsP formedbythevectors
i,j
Wethengiveexamplesofthewellknowndecagonalpatterns W andW takeninpairs.Theseparallelogramswillconstitute
i j
that adorn the Madrasas (schools) Attarine and Bou Inania thepaving tiles.
(Fez), and a variant of the dodecagonal pattern of the We assume that the families of lines are distributed in a
tympanumattheentranceofthemausoleumofMoulayIdriss genericway,sothatthelinesintersectinpairsandthereareno
(Fez). These patterns have been widely analysed by Mako- triple or multiple intersections. With each intersection I
i,j
vicky et al. (1998), Makovicky (2004) and Makovicky & between lines L and L, we associate the parallelogram P
i j i,j
Makovicky (2011). We shall then build new octagonal quasi- engendered by the vectors W and W (Figs. 3a and 3b). The
i j
periodic patterns from periodic patterns that adorn the tilesarearrangedaccordingtothesequenceofintersectionof
Madrasas Ben Youssef (Marrakech) and Bou Inania (Fez). the lines (Fig. 3c) and the whole tiling is obtained by this
Finally, we shall use the multigrid method to construct process (Fig. 3d). The perfectly ordered tiling obtained is a
heptagonal, enneagonal, tetradecagonal and octadecagonal quasiperiodic lattice (except for the case N = 2, which corre-
quasiperiodic patterns not encountered in Moroccan spondsto aperiodic lattice).
geometricart. It is shown (Jaric, 1989) that these tilings are the result of
cutting a periodic N-dimensional object in an irrational
direction. This constructionleadstoremarkableresultswhen
2. The multigrid method
choosing a symmetric multigrid constructed from a single
Several methods are used to construct quasiperiodic tilings. family of lines copied N times by the same angle of rotation
One of the easiest, called the multigrid method and formu- (cid:3) =2(cid:4)/Nandtaking W equalto the vector V .
n n n
lated by Gratias (2002), is discussedin this section. Themultigridmethodissuitableformanyvariationsinthe
We construct a set of equidistant parallel lines, character- choice of grid spacing sequences, their relative positions and
izedbyavectorVorthogonaltothelinesandoflengthequal
to the period. We copy this set N times by performing each
Figure2
(a)AmultigridisthesuperpositionofN>2setsofequidistantparallel Figure 3
linesinNdistinctdirectionsoftheplane.Inthiscase,N=4.Periodsin (a)Intersectionoflines.(b)Eachintersection,I ,ofthelinesL andL of
i,j i j
each direction are not necessarily equal. (b) Vectors V and their themultigridisassociatedwiththeparallelogramP .(c)Thepavingis
n i,j
opposites are traced from the same centre to obtain a convex polygon obtained by placing adjacent parallelograms in the exact order of
generated by lines perpendicular to the vectors and passing by their intersection of the lines in the multigrid. In the case of multiple
extremities.WearbitrarilychooseineachsectoravectorW associated intersections,wemoveafamilyoflinesbyaninfinitesimalamounttolift
n
withthesamefamilyoflines. thedegeneracy.(d)Thecorrespondingtiling.
J.Appl.Cryst.(2014).47,630–641 YoussefAboufadiletal. (cid:2) Moroccanornamentalquasiperiodicpatterns 631
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thevectorsW thatdefinetheshapeofthetiles.Inallcases,we the following, we shall use a symmetric multigrid, and the
n
obtain uniform tilings with beautiful decorative elements. In spacing sequences willbe specified in eachcase.
Figure4
(a)Fibonaccisequence.(b)A5-grid(Ammanngrid).(c)Penrosetiling.
Figure5
(a)ApaneloftheMadrasaAttarine(Sijelmassi,1991).(b)Theinfinitequasiperiodicpattern.(c)Afinitepartofthequasiperiodicpattern.
632 YoussefAboufadiletal. (cid:2) Moroccanornamentalquasiperiodicpatterns J.Appl.Cryst.(2014).47,630–641
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3. Analysis of quasiperiodic patterns of some historical 3.1.1. Madrasa Attarine. The zellige panel (fine mosaic) of
monuments by the multigrid method theMadrasaAttarine(MA)(Fig.5c)hasthesymmetries5and
10.Wedefineinittworhombictileswithvertexangles36and
3.1. Decagonal panels of the Madrasas Attarine and Bou 72(cid:3),asshowninFig.5(a).Ifwearrangethetilesaccordingtoa
Inania
Penrose tiling, we obtain an infinite quasiperiodic pattern
Makovicky et al. (1998) were the first to study and explain (Fig. 5b). The MA panel appears as a finite part of this
the decagonal pattern from Madrasas Attarine and Bou quasiperiodic pattern (Fig.5c).
Inania. The purpose of this section is to show, through these Asalreadymentionedbyseveralauthors,wecannoticethe
two well known examples, that the multigrid method can be similarityofthedecagonalpatterntothediffractionpatternof
adapted to study quasiperiodic patterns. We use the Penrose the quasicrystal AlMn (Shechtman et al., 1984). The ratio of
tiling (Fig. 4c) obtained, by the same process as explained in thedistancesbetweentwosymmetricalelementsinthecaseof
theprevioussection,froma5-grid,whichisasuperpositionof thepanel(Fig.1)isequaltothegoldennumber,(cid:5)=(1+51/2)/
fiveidenticalgridsrotatedbyanangle(cid:3)=2(cid:4)/5tooneanother 2 =1.61803398875....
around the fivefold axis passing through the centre (black 3.1.2. Madrasa Bou Inania. The zellige panel of the
point)(Fig.4b).Theparallellinesofthegridsaredisposedina Madrasa Bou Inania (MBI), which is a variant of the MA
Fibonacci sequence (Fig.4a). panel, is also extracted from an infinite decagonal quasi-
Figure6
(a)ApaneloftheMadrasaBouInania.(b)Theinfinitequasiperiodicpattern.(c)Afinitepartofthequasiperiodicpattern.
J.Appl.Cryst.(2014).47,630–641 YoussefAboufadiletal. (cid:2) Moroccanornamentalquasiperiodicpatterns 633
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periodic pattern obtained from a Penrose tiling. The two another must satisfy the artistic criteria required by the
rhombustileshave the same dimensions asin theMA panel. craftsmen.
The difference between them is the tile decoration (Fig.6).
3.2. Construction of octagonal quasiperiodic patterns from
Both MA and MBI panels are extracted from infinite
known periodic patterns
quasiperiodic patterns. These panels, originally designed by
craftsmen, can be built from decorated tiles arranged in a Inthissection,wepresentquasiperiodicpatternsbuiltfrom
Penrose tiling. The difficult part of such a construction is the known periodic patterns. For instance, we chose the patterns
tile decoration. Indeed, the juxtaposition of the tiles should drawn on the wooden gates of the Madrasas Bou Inania
lead to a harmonious model. The transition from one tile to (MBI) and Ben Youssef (MBY). They have the symmetry
Figure7
Woodengatesat(a)MadrasaBouInania(Fez)and(b)MadrasaBenYoussef(Marrakech).
Figure8
(a)Octagonaltiling.(b)Squareandrhombicdecoratedtiles.(c)Anoctagonalquasiperiodicpattern.
634 YoussefAboufadiletal. (cid:2) Moroccanornamentalquasiperiodicpatterns J.Appl.Cryst.(2014).47,630–641
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groupsp4mmandc2mm,respectively,andtherepeatunitsare
eight- and 16-fold rosettes,respectively(Fig.7).
To build an octagonal quasiperiodic pattern, wereproduce
thesquareandrhombictilestracedinFig.8.Theornamentof
therhombictilesconsists oftwoeightfoldrosettesandpetals
of a 16-fold rosette distributed over the four corners of the
rhombus: two petals at small angles (45(cid:3)) and six petals at
largeangles(135(cid:3)).Thesquaretilehasfoureightfoldrosettes
and four elements of a 16-fold rosette spread over the four
corners of the square. The two types of tile arranged in the
plane according to octagonal tiling give an octagonal quasi-
periodic pattern composed of eight- and 16-fold rosettes
(Fig.8).
4. Construction of new quasiperiodic patterns
Thequasiperiodicpatternsareobtainedfromaquasiperiodic
tiling, which is the skeleton of the pattern, decorated with
elementsextractedfromornamentalpatternsconstructedbya
Figure9
method used by craftsmen, called Hasba.
Atenfoldpattern(MadrasaBenYoussef,Marrakech).
4.1. Ornamental patterns constructed by the craftsmen’s
Hasba method
This method of construction of geometric patterns was
described exhaustively by Thalal et al. (2011). It leads to
patternsconstitutedofn-foldrosettesandn-starshapes,where
n is the number ofrays,forinstance n= 10 (Fig.9).
Patterns obtained by the Hasba method have interlaced
ribbons of constant width, which represents the unit of
measure of the pattern. Even though they are hidden in the
above patterns, interlaced ribbons are naturally present. We
canrevealtheribbonsbyextending,accordingtocertainrules,
the edges of the different shapes that constitute the patterns.
Bymarkingoutthehiddenribbons,amoreartisticinterlaced
pattern emerges from the initial one. The main condition
imposed by the Hasba method is that the ribbons should
alwaysbe continuous throughoutthe pattern (Fig.10).
The n-fold rosettes are the elements used to decorate the
Figure10 skeleton constructed by the multigrid method to obtain a
Continuityofribbonsinatenfoldpattern. quasiperiodic pattern.
Figure11
(a)Stackingofnon-interpenetratingtenfoldrosettes.(b)Atilewithavertexangleof72(cid:3).(c)Stackingofinterpenetratingtenfoldrosettes.(d)Atilewith
avertexangleof36(cid:3).(e)Ribboncontinuityacrossadjacenttiles.
J.Appl.Cryst.(2014).47,630–641 YoussefAboufadiletal. (cid:2) Moroccanornamentalquasiperiodicpatterns 635
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Figure12
(a)Dodecagonaltiling.(b)Squareandrhombustiles.(c)Adodecagonalquasiperiodicpattern.
4.2. Description of the method of n-fold rosettes, where the tile vertices coincide with the
centres ofthe rosettes (Figs.11b and 11d).
ThecombinationoftheHasbaandmultigridmethodsleads The arrangement of the tiles obeys the criterion of conti-
to quasiperiodic patterns that are not encountered in either nuityoftheribbonsrequiredbyHasba,asmentionedinx4.1.
Moroccandecorative art or diffraction patterns. Indeed,theribbonsmustalwaysbecontinuousacrossadjacent
A quasiperiodic pattern is a decorated quasiperiodic tiling tiles over the entire pattern (Fig.11e).
obtained from an N-grid with a rotation angle (cid:3) = 2(cid:4)/N. The Furthermore,theorder pofsymmetryoftherosettesused
decorativeelementsofthetilesareextractedfromastacking todecoratethetilesmustbecompatiblewithitsangles,which
Figure13 Figure 14
(a)Decoratedtiles.(b)Adecagonalquasiperiodicpattern. (a)Decoratedtiles.(b)Anoctagonalquasiperiodicpattern.
636 YoussefAboufadiletal. (cid:2) Moroccanornamentalquasiperiodicpatterns J.Appl.Cryst.(2014).47,630–641
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are multiples of the rotation angle (cid:3) of the N-grid. It follows respectivevertexanglesof30and60(cid:3).Thetilesaredecorated,
that the order pmust satisfy the relation andarrangedaccordingtoadodecagonaltilingobtainedfrom
a 6-grid (Fig.12).
p¼k2(cid:4)=(cid:3); ð1Þ 4.3.2.Decagonalpatterns.Thedecagonaltilingisobtained
froma5-gridwitharotationalangle(cid:3)=2(cid:4)/5.Itconsistsoftwo
where kis an integer(cid:4) 1. rhombic tiles with vertex angles of 36 and 72(cid:3). The order of
symmetryoftherosettecompatiblewiththeseanglesisp=10.
4.3. Application to known quasiperiodic patterns
Thetilesarethenextractedfromtenfoldrosettesarrangedin
The dodecagonal, decagonal and octogonal patterns have two configurations. In the first configuration, the rosettes do
alreadybeendescribedbyseveralauthorswhouseddifferent notoverlap,andthetilewithavertexangleof72(cid:3) isobtained
approaches(Makovicky&Makovicky,2011;Makovicky,1992, by joining the centres of four rosettes (Fig. 11a). The second
2004, 2007, 2008, 2011; Makovicky & Fenoll Hach-Alı´, 1997; configuration, in which the rosettes are interpenetrating,
Makovickyetal.,1998;AlAjlouni,2011;Castera&Jolis,1991; provides the tile with a vertex angle of 36(cid:3) (Fig. 11c). The
Castera, 1996, 2003). In this section, we present a variant of filling of the central gap in the first configuration and the
thesethreepatternsobtainedbythecombinationoftheHasba overlapping arrangement of the second must satisfy the
and multigrid methods. continuity rules imposedby Hasba (Figs.11b and 11d).
4.3.1. Dodecagonal patterns. This quasiperiodic pattern is Aquasiperiodicdecagonalpatternisthenconstructedwith
builtupbyasquaretileandtwotypesofrhombictileshaving the two tiles according to Penrose tiling (Fig. 13). Other
Figure15
(a)Heptagonaltiling.(b)Decoratedtiles.(c)Aheptagonalquasiperiodicpattern.(d)Afinitepartofthequasiperiodicpattern.
J.Appl.Cryst.(2014).47,630–641 YoussefAboufadiletal. (cid:2) Moroccanornamentalquasiperiodicpatterns 637
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configurations are possible, offering a wide variety of deca- Whittaker (1988), who used the method of projection of
gonal quasiperiodic patterns other than those described N-dimensionalspace.Inaddition,anotherwayofconstructing
above. asevenfoldtilingwaspresentedinanothercontextbyBonner
4.3.3. Octagonal patterns. The same process leads to & Pelletier (2012).
octagonalquasiperiodicpatterns(Fig.14).Thedecoratedtiles 4.4.1. Heptagonal and tetradecagonal patterns. Both
are obtained from eightfold rosettes and arranged in an patternsareunderlainbyaheptagonaltilingobtainedfroma
octagonaltiling. 7-grid with arotational angle (cid:3) = 2(cid:4)/7.
Heptagonalandtetradecagonalpatternsareconstructedby
4.4. New quasiperiodic patterns three rhombic tiles with vertex angles of (cid:4)/7, 2(cid:4)/7 and 3(cid:4)/7,
In this section we present the heptagonal, enneagonal, and decorated by seven- and 14-fold rosettes, respectively
tetradecagonal and octadecagonal patterns, which are not (Figs. 15and16).
encountered in Moroccan geometricart. 4.4.2.Enneagonalandoctadecagonalpatterns.Enneagonal
Francoetal.(1996)proposedenneagonaltilingoftheplane and octadecagonal tilings are obtained from a 9-grid with a
using geometric shapes. Seven- and ninefold quasiperiodic rotationalangle(cid:3)=2(cid:4)/9.Theyareconstitutedoffourrhombic
tilings very similar to ours were obtained by Whittaker & tiles with vertex angles of (cid:4)/9, 2(cid:4)/9, 3(cid:4)/9 and 4(cid:4)/9. The
Figure16
(a)Tetradecagonaltiling.(b)Decoratedtiles.(c)Atetradecagonalquasiperiodicpattern.(d)Afinitepartofthequasiperiodicpattern.
638 YoussefAboufadiletal. (cid:2) Moroccanornamentalquasiperiodicpatterns J.Appl.Cryst.(2014).47,630–641
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