The Search for Quasi-Periodicity in Islamic 5-fold Ornament

21 Pages·2008·2.17 MB·English

The Search
for Quasi-Periodicity
in Islamic 5-fold
Ornament
P R. C
ETER ROMWELL
Introduction • abstraction: P knew about X and X is an example of Y
TT he Penrose tilings are remarkable in that they are therefore P knew Y.
non-periodic (have no translational symmetry) but • deduction: P knew X and X implies Y therefore P knew
are clearly organised. Their structure, called quasi- Y.
periodicity, can be described in several ways, including
via self-similar subdivision, tiles with matching rules, and Inbothcases,itislikelythatPneverthoughtofYatall,and
projection ofa slice of a cubic lattice inR5. The tilings are even if he had, he need not have connected it with X.
also unusual for their many centres of local 5-fold and 10- In this article I shall describe a tiling-based method for
fold rotational symmetry, features shared by some Islamic constructing Islamic geometric designs. With skill and
geometric patterns. This resemblance has prompted ingenuity,thebasictechniquecanbevariedandelaborated
comparison, and has led some to see precursors of the in many ways, leading to a wide variety of complex and
Penrose tilings and even evidence of quasi-periodicity in intricate designs. I shall also examine some traditional
traditional Islamic designs. Bonner [2] identified three designsthatexhibitfeaturescomparablewithquasi-periodic
styles of self-similarity; Makovicky [20] was inspired to tilings,usetheunderlyinggeometrytohighlightsimilarities
develop new variants of the Penrose tiles and later, with anddifferences,andassesstheevidenceforthepresenceof
colleagues [24], overlaid Penrose-type tilings on traditional quasi-periodicityinIslamicart.
Moorish designs; more recently, Lu and Steinhardt [17] A few comments on terminology. Many of the con-
observed the use of subdivision in traditional Islamic structions are based on tilings of the plane. A patch is a
design systems and overlaid Penrose kites and darts on subsetofatilingthatcontainsafinitenumberoftilesandis
Iranian designs. The latter article received widespread homeomorphictoadisc.Iuserepeatunitasagenericterm
exposure in the world’s press, although some of the for a template that is repeated using isometries to create a
coverage overstated and misrepresented the actual pattern; it is not so specific as period parallelogram or
findings. fundamental domain. A design or tiling with radial sym-
Thedesiretosearchforexamplesofquasi-periodicityin metryhasasinglecentreoffiniterotationalsymmetry.The
traditional Islamic patterns is understandable, but we must other terminology follows [8] for tilings, supplemented by
take care not to project modern motivations and abstrac- [33] for substitution tilings.
tionsintothepast.Anintuitiveknowledgeofgrouptheory
is sometimes attributed to any culture that has produced Islamic Methods of Construction
repeating patterns displaying a wide range of symmetry AlthoughtheprinciplesofIslamicgeometricdesignarenot
types,eventhoughtheyhadnoabstractnotionofagroup. complicated, they are not well-known. Trying to recover
There are two fallacies to avoid: theprinciplesfromfinishedartworkisdifficult,asthemost
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THEMATHEMATICALINTELLIGENCER(cid:1)2008SpringerScience+BusinessMedia,LLC.
conspicuous elements in a design are often not the design is not yet complete. There are some spikes of each
compositional elements used by the designer. Fortunately motif that are not connected to a neighbouring motif but
medieval documents that reveal some of the trade secrets are free and point into the residual spaces between the
have survived. The best of these documents is manuscript circles. The lines bounding these free spikes are extended
scroll MS.H.1956 in the library of the Topkapi Palace, beyondthecircumcircleuntiltheymeetsimilarlyproduced
Istanbul. The scroll itself is a series of geometric figures lines from nearby stars. This simple procedure bridges the
drawn on individual pages, glued end to end to form a residual spaces and increases the connectivity of the star
continuous sheet about 33 cm high and almost 30 m long. motifs. The same pattern of interstitial filling should be
Itisnota‘howto’manual,asthereisnotext,butitismore applied uniformly to all the residual spaces and the sym-
than a pattern book as it shows construction lines. A half- metry of the design as a whole should be preserved as far
size colour reproduction can be found in [25], which also aspossible.TheresultisshowninFigure 1(c).Inthiscase
includes annotations to show the construction lines and the kites in the interstitial filling are congruent to those in
marks scored into the paper with a stylus, which are not thestar.Thispatternisoneofthemostcommondecagonal
visible in the photographs. References in this article to designs,andweshallnameitthe‘starsandkites’patternfor
numberedpanelsoftheTopkapiScrollusethenumbering reference.
in [25]. This basic approach produces a limited range of peri-
Islamicdesignsoftenincludestarmotifs.Thesecomein odic designs with small repeat units and it only works for
a variety of forms but, in this article, we need only a few stars with an even number of points. A more general
simpleshapesthatcorrespondtotheregularstarpolygons method that can be used with all stars, and also enables
ofplanegeometry.Takingnpointsequallyspacedaround combinationsofdifferentstarstobeusedinasingledesign,
a circle and connecting points d intervals apart by straight isbasedonedge-to-edgetilingscontainingregularconvex
lines produces the star polygon denoted by {n/d}. This, polygons with more than four sides. Figure 1(b) shows a
however, is the star of the mathematician; it is rare for an tiling formed by packing decagons together, leaving non-
artisttousethewholefigureasanornamentalmotif.More convex hexagonal tiles between them. After placing {10/3}
often, the middle segments of the sides are discarded. stars in each decagon tile, we use the same kind of inter-
Many of the early Islamic designs are created by stitial filling procedure as before to develop the pattern in
arranging 6-, 8- or 12-point stars at the vertices of the the hexagons.
standardgridsofsquaresorequilateraltriangles.Themore Thischangefromcircletopolygonmayseemminor,but
general rhombic lattice allows other stars to be used. An itgivesrisetoarangeofgeneralisations.Wearenolonger
example based on {10/3} is shown in Figure 1(a). The restricted to a lattice arrangement of the stars—any tiling
angles in the rhombus are 72(cid:2) and 108(cid:2), both being mul- will suffice. The tiling may contain regular polygons of
tiplesof36(cid:2)—theanglebetweenadjacentspikesofthestar. different kinds allowing different star motifs to be com-
Drawasetofcirclesofequalradiuscentredonthevertices binedinthesamedesign;thetilingnaturallydeterminesthe
ofthelatticeandofmaximalsizesothattherearepointsof relativesizesofthedifferentstars.Wecanevendiscardthe
tangency.Placecopiesofthestarmotifinthecirclessothat regularstarmotifsthatinitiatetheinterstitialfillingandseed
spikes fall on the edges of the lattice. This controls the the pattern generation process from the tiling itself. In this
spacing and orientation of the principal motifs, but the lastcase,weplaceapairofshortlinesinanXconfiguration
at the midpoint of each edge, then extend them until they
encounter other such lines—this is similar to applying
.........................................................................
interstitialfillingtoeverytile.Theanglethatthelinesmake
R PETER CROMWELL may be known to with the edges of the tiling, the incidence angle, is a
HO readers through his contributions to the parameter to be set by the artist and it usually takes the
UT Intelligencer on Celtic art (vol. 15, no. 1) same value at all edges. There is no requirement to termi-
A
nate the line extensions at the first point of intersection; if
andtheBorromeanrings(vol.20,no.1),or
there are still large empty regions in the design, or it is
his books on polyhedra and knot theory,
otherwiseunattractive,thelinescanbecontinueduntilnew
both published by CUP. He is interested in
intersections arise.
anything 3-dimensional with a strong visual
This technique, known as ‘polygons in contact’ (PIC),
element, and also combinatorial and algo-
was first described in the West by Hankin [9–13], who
rithmicproblems.Hewasrecentlyawarded
observedthepolygonalnetworksscratchedintotheplaster
a research fellowship by the Leverhulme
of some designs, while working in India. Many panels in
Trusttowork onthemathematicalanalysis
theTopkapiScrollalsoshowadesignsuperimposedonits
of interlaced patterns.
underlying polygonal network. Although the purpose of
the networks is not documented, it does not seem unrea-
Pure Mathematics Division, sonable to interpret them as construction lines. Bonner [2,
Mathematical Sciences Building 3] argues that PIC is the only system for which there is
University of Liverpool, Peach Street evidence of historical use by designers throughout the
Liverpool L69 7ZL Islamicworld.Themethodisversatileandcanaccountfor
England awiderangeoftraditionalpatterns,butitisnotuniversally
e-mail: [email protected] applicable. An alternative approach is used by Caste´ra [5],
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(cid:1)2008SpringerScience+BusinessMedia,LLC.,Voulme31,Number1,2009
(a) (b)
(c)
Figure1. The ‘stars and kites’ pattern.
who arranges the shapes seen in the final design without tilingcontainingregular9-gonsand12-gons.Ihavechosen
using a hidden grid. an incidence angle of 55(cid:2) to make the convex 12-gon ele-
The PIC method is illustrated in the next four figures. ments in the design into regular polygons and some line
Figure 2 shows two designs produced from a tiling by segments inside the non-convex hexagonal tiles join up
regular decagons, regular pentagons, and irregular convex without creating a corner, but, as a consequence, neither
hexagons.Inpart(b),astarmotifbasedon{10/4}isplaced starmotifisgeometricallyregular.Plates120–122in[4]are
inthedecagontiles,whichgivesanincidenceangleof72(cid:2) traditionaldesignsbasedonthesametiling.Figure 5shows
for the other edges; the completed design is one of the a design with 10-fold rotational symmetry based on panel
most widespread and frequently used of all star patterns. 90a of the Topkapi Scroll, which Necipog˘lu labels as a
Part(c)showsadesignthatiscommoninCentralAsiaand designforadome[25].Theoriginalpanelshowsatemplate
basedon{10/3}withanincidenceangleof54(cid:2).A{10/2}star for the figure containing one-tenth of the pattern with the
and an incidence angle of 36(cid:2) reproduces the stars and design in solid black lines superimposed on the tiling
kites pattern. The design in Figure 3 is from [14] and con- drawninreddotted lines.Noticethatsomeofthetilesare
tains star motifs based on {7/3}; inthe tiling the7-gons are two-tenths and three-tenths sectors of a decagon. Domes
regular but the pentagons are not. Figure 4 is based on a were also decorated by applying PIC to polyhedral
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THEMATHEMATICALINTELLIGENCER
(a) Underlying polygonal network
(b) Incidence angle 72° (c) Incidence angle 54°
Figure2. Atilingandtwostarpatternsderivedfromit.Thepetalsofarosemotif
in each pattern are highlighted.
networks.Patternswithalowerconcentrationofstarswere You can see the PIC method in action and design your
producedbyapplyingPICtok-uniformtilingscomposedof own star patterns using Kaplan’s online Java applet [34]—
regular3-,4-,6-,and12-sidedpolygons—seeplates77,97, you select a tiling and the incidence angles of the star
and 142 in [4] for some unusual examples. motifs,theninferencelogicsuppliestheinterstitialpattern.
The two designs of Figure 2 display another common The tilings used as the underlying networks for the PIC
Islamic motif. In each design, a set of hexagons sur- method of construction often have a high degree of sym-
rounding a star has been highlighted in grey. The metry, and they induce orderlydesigns. Islamicartists also
enlarged star motif is called a rose and the additional produced designs that appear to have a more chaotic
hexagons are its petals. In this case, the rose arises arrangement of elements with local order on a small scale
because the decagon in the underlying tiling is sur- but little long-range structure visible in the piece shown.
rounded by equilateral polygons, but they can also be PanelsintheTopkapiScrollreveal thatthese designs,too,
constructed using a set of tangent circles around the cir- have an underlying polygonal network assembled from
cumcircle of the star [16] and used as compositional copies of a small set of equilateral tiles (see Figure 6)
elements in their own right. whose angles are multiples of 36(cid:2):
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(cid:1)2008SpringerScience+BusinessMedia,LLC.,Voulme31,Number1,2009
(a) (b)
Figure3. Design containing regular 7-point stars.
(a) (b)
Figure 4. Design containing 9- and 12-point stars.
• a rhombus with angles 72(cid:2) and 108(cid:2) decagonhavetwoformsofdecoration.Onedecagonmotif
• a regular pentagon (angles 108(cid:2)) isjustthestar{10/3}anditsconstituentkitesarecongruent
• aconvexhexagonwithangles72(cid:2)and144(cid:2)—thebobbin to those on the bow-tie; the other decagon motif is more
• a convex hexagon with angles 108(cid:2) and 144(cid:2)—the complex and the symmetry is reduced from 10-fold to
barrel 5-fold rotation.
• a non-convex hexagon with angles 72(cid:2) and 216(cid:2)—the Theshapes ofthe tilesarise naturally whenonetries to
bow-tie tile with decagons and pentagons. The bow-tie and barrel
• a convex octagon with angles 108(cid:2) and 144(cid:2) hexagons are familiar from the previous figures. The
• a regular decagon (angles 144(cid:2)). octagonandtheremaininghexagonareshapesthatcanbe
obtained as the intersection of two overlapping decagons.
ThemotifsonthetilesaregeneratedusingthePICmethod The motif on the hexagon resembles a spindle or bobbin
withanincidenceangleof54(cid:2).Thebarrelhexagonandthe woundwithyarn.Thisdistinctivemotifiseasytolocateina
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THEMATHEMATICALINTELLIGENCER
combinationofabobbinwithabow-tieshowninpart(a),
thepositionsofthetilescanbereflectedinaverticallineso
thatthebow-tiesitstop-rightinsteadoftop-left.Thepatch
in part (d) can replace any decagonal tile with a conse-
quentlossofsymmetry,as thebow-tiecanpointinanyof
ten directions. Patches (b) and (c) are another pair of
interchangeable fillings with a difference in symmetry.
Figure 8shows some traditional designs madefrom the
tiles. Parts (a) and (b) are from panels 50 and 62 of the
Topkapi Scroll, respectively; in both cases the original
panelshowsatemplatewiththedesigninsolidblacklines
superimposedonthetilingdrawninreddots.Thedesigns
inparts(b),(c),and(d)areplates173,176,and178of[4].
The designs of (e) and (f) are Figures 33 and 34 from
[16]. The edges of the tilings are included in the figures to
show the underlying structure of the designs, but in the
finished product these construction lines would be erased
to leave only the interlaced ribbons. This conceals the
underlying framework and helps to protect the artist’s
method. The viewer sees the polygons of the background
Figure 5. Designfrompanel90aoftheTopkapiScroll. outlined by the ribbons, but these are artifacts of the con-
struction, not the principal motifs used for composition.
design,anditspresenceisagoodindicationthatthedesign The internal angles in the corners of the tiles are all
could be constructed from these tiles. multiplesof36(cid:2)soalltheedgesinatilingwillpointinone
The promotion of irregular tiles from supplementary offivedirections—theywillalllieparalleltothesidesofa
shapes to compositional elements in their own right pentagon.Fittingthetilestogetherspontaneouslyproduces
marked a significant development in Islamic design. regularpentagonsinthebackgroundoftheinterlacing,and
Regarding the tiles as the pieces of a jigsaw allows a less centresoflocal5-foldor10-foldrotationalsymmetryinthe
formal approach to composition. A design can be grown design. This symmetry can be seen in some of the config-
organically in an unplanned manner by continually urations of Figure 7. However, in patterns generated by
attachingtilestotheboundaryofapatchwithafreechoice translation of a template, this symmetry must break down
among the possible extensions at each step. This new and cannot hold for the design as a whole. This is a con-
approach gave artists freedom and flexibility to assemble sequence of the crystallographic restriction: the rotation
thetilesinnovelwaysandledtoanewcategoryofdesigns. centresinaperiodicpatterncanonlybe2-,3-,4-or6-fold.
It seems to have been a Seljuk innovation as examples Thiswasnotprovedrigorouslyuntilthe19thcenturybutit
started to appear in Turkey and Iran in the 12th–13th must surely have been understood on an intuitive levelby
centuries. The widespread and consistent use of these the Islamic pattern makers. Perhaps these tilings were
decorated tiles as a design system was recognised by Lu appealing precisely because they contain so many forbid-
and Steinhardt [17]; similar remarks appear in Bonner [2] den centres; they give the illusion that one can break free
and the tiles are also used by Hankin [10]. from this law of nature. Unfortunately, when a large
Figure 7 shows small patches of tiles. There are often enoughsectionofatilingisshownfortheperiodicitytobe
multiple solutions to fill a given area. Even in the simple apparent,any(global)rotationcentresareonly2-fold,and
Rhombus Pentagon Barrel (1) Barrel (2) Octagon
Bow-tie Bobbin Decagon (1) Decagon (2)
Figure6. An Islamic set of prototiles.
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(a)
(b)
(f)

(c)
(d)
(g)
(e)
Figure7. Small patches of tiles.
thesymmetrytypeofthe(undecorated)tilingisusuallyone decagonal tile. The star-shaped gaps are filled with the
of pgg, pmm or, more commonly, cmm. five rhombi of Figure 7(b). The design does contain
Figure 9(b) shows the design on one wall of the irregularities and deviations from this basic plan, particu-
Gunbad-i Kabud (Blue Tower) in Maragha, north-west larly in the bottom-left corner of the panel. Also the
Iran; similar designs decorate the other sides of the tower. decagon in the top-left corner is filled with Figure 7(d)
At first sight the design appears to lack an overall rather than a decagonal tile.
organising principle but it fits easily into the framework Figure 9(a) can also be taken as the foundation of the
shown in Figure 9(a). Centred at the bottom-right corner designshowninFigure 10.Thecentresoftherosemotifsin
of the panel is the patch of Figure 7(g) surrounded by a thecentreofthe figureandinthe top-left cornerarediag-
ring ofdecagons. A similar arrangement placed at the top- onallyoppositecornersofarectanglethatisarepeatunitfor
left corner abuts the first, leaving star-shaped gaps. The the design. The underlying framework in this rectangle is
rings of decagons are filled with the patch of Figure 7(d) the same as that of the Maragha panel. The full design is
with the bow-ties facing outwards, except for the one on generated from this cell by reflection in the sides of the
the bottom edge of the panel, which is filled with a rectangle.Notethatitisthearrangementofthetilesthatis
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THEMATHEMATICALINTELLIGENCER
(a) (b) (c) (d)
(e) (f) (g) (h)
Figure 8. Periodic designs.
(a) (b)
Figure9. Design from the Gunbad-i Kabud, Maragha, Iran.
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(cid:1)2008SpringerScience+BusinessMedia,LLC.,Voulme31,Number1,2009
Figure 10. Design from the Karatay Madrasa, Konya, Turkey.
reflected, not the tiles with their decorative motifs; the patch shown in Figure 7(g) but this has been discarded in
interlacing of the full design remains alternating. The favourofalargerosemotif.Adifferentconstructionforthis
boundariesoftheunitrectanglearemostlycoveredbythe patternispresentedbyRigbyin[26].
sides of tiles or mirror lines of tiles, both of which ensure WhenexperimentingwiththetilesofFigure 6,onesoon
continuityofthetilingacrossthejoins.However,inthetop- learns that those in the top row are more awkward to use
right and bottom-left corners (the cell has 2-fold rotational than the others—the 108(cid:2) angles must occur in pairs
symmetry about its centre), the tiles do not fit in the rect- around a vertex and this limits the options. Indeed many
angle but overhang the edges. This is not a problem with designs avoid these tiles altogether and are based solely
thismethodofgeneratingdesigns:theoverhangingtilesare on the three shapes in the bottom row. The design in
simply cut to fit and the reflections take care of the conti- Figure 11 is unusual in that it is largely composed of
nuityoftheribbons.InFigure 10thisismostobviousinthe awkwardtiles(rhombi,pentagons,andoctagons)together
middle near the bottom where pairs of bow-ties and bob- with a few bobbins. The large star-shaped regions in the
bins merge. The centre of the tiling can be filled with the tiling can be filled with the patch shown in Figure 7(f),
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THEMATHEMATICALINTELLIGENCER
Figure11. Design from the Sultan Han, Kayseri, Turkey.
(a) (b) (c)
Figure12. Subdivisionsof threetiles intosmallercopiesofthe same threetiles.
p
The scale factor is 1(cid:2)7þ ffi5ffiffi(cid:3)(cid:1)4:618.
2
continuing the use ofthe same set of tiles, but instead this became known as quasi-crystals and the underlying order
motif is replaced by the star {10/4}. as quasi-periodicity. For crystallographers, the production
Once a design has been constructed, it can be finished of sharply defined points in a diffraction pattern is a
in different ways according to context and the materials defining characteristic of quasi-periodicity. In the study of
used. In some of the accompanying figures, the regions the decorative arts, however, the term ‘quasi-periodic’ is
have been given a proper 2-colouring (chessboard shad- used somewhat informally and does not have an agreed
ing), in others the lines have been made into interlaced definition.Readersshouldbeawareofthispotentialsource
ribbons.Thebasiclinedrawingcanalsobeusedbyitselfas of confusion when comparing papers. For the tilings and
when it is inscribed in plaster. the related geometric designs discussed in this article, one
option is to impose a homogeneity condition on the dis-
What is Quasi-Periodicity? tributionoflocalconfigurationsoftiles(thisisweakerthan
The discovery of crystalline metal alloys with 5-fold sym- the crystallographic definition). This and other properties
metry in their diffraction patterns caused great excitement will be illustrated through the following example.
in the 1980s. Sharp spots in a diffraction pattern are evi- The example is constructed from the patches shown in
dence of long-range order which, at that time, was Figure 12.Thepatchesarechosenonlytodemonstratethe
synonymous with periodicity, but 5-fold rotations are technique and not for any artistic merit—the unbalanced
incompatiblewiththecrystallographicrestrictionsoanew distribution of bow-ties leads to poor designs. Any patch
kindofphenomenonhadbeenobserved.Thenovelsolids tiledbybow-ties,bobbins,anddecagonscanbeconverted
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