Islamic Geometric Patterns: Their Historical Development and Traditional Methods of Construction

616 Pages·2017·91.01 MB·English

by  Jay Bonner

Jay Bonner
Islamic
Geometric
Patterns
Their Historical Development
and Traditional Methods
of Construction
Foreword by Sir Roger Penrose
Islamic Geometric Patterns
Jay Bonner
Islamic Geometric Patterns
Their Historical Development
and Traditional Methods
of Construction
with a chapter on the use of computer algorithms to generate
Islamic geometric patterns by Craig Kaplan
JayBonner
BonnerDesignConsultancy
SantaFe,NewMexico,USA
Withcontributionsby
CraigKaplan
UniversityofWaterloo
Waterloo,Ontario,Canada
ISBN978-1-4419-0216-0 ISBN978-1-4419-0217-7 (eBook)
DOI10.1007/978-1-4419-0217-7
LibraryofCongressControlNumber:2017936979
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For my wife, Shireen, without whose support this book would never
have seen the light of day.
Jay Bonner
For Reza Sarhangi, who convinced me to cross a bridge
and never look back.
Craig Kaplan
Foreword
Foroverelevencenturies,Islamicartistsandarchitecturaldesignershavedevelopedextraor-
dinaryskillstoproducemanyamazingformsofdecoration.Thissuperblyillustratedbook,by
JayBonner(withafinalchapterbyCraigKaplan),isawonderfullycomprehensiveanddeeply
thoughtfulaccountofthesehighlydistinctivedesigns.Ifeelsurethatthisbookwillrepresenta
landmarkinthestudyofIslamicdesign.
These remarkable artworks take various forms, but they are almost always of an entirely
nonrepresentational character, only a very few showing indications of recognizable natural
objects, such as leaves or flowers. Yet there is a distinctive beauty in these designs, most of
these patterns being of a particular geometrical character, demonstrating a keen and subtle
knowledge and interest in geometry and a profound skill in using geometrical motifs to
produce some incredibly intricate patterns. These would normally be repetitive, in a planar
arrangement,thoughsometimesbenttocoveracurvedsurface,suchasforadomeorceiling.
They gain some of their beauty from the intricacy and ingenuity of the designs, which
frequently involve sophisticated geometrical ideas in unexpected ways. These artists had
clearly developed some significant understanding of mathematical notions that were not
properly developed by professional mathematicians until the early twentieth century, when
the plane Euclidean symmetry groups (“wallpaper symmetries”) were finally classified into
17 distinct types. Representations of these different symmetries are copiously exhibited in
Islamicpatternsand,remarkably,all17aretobefoundinthesinglelocationoftheAlhambra
PalaceinGranada,Spain,constructedmorethanhalfamillenniumearlierthanthetimethat
these different symmetries were explicitly distinguished by mathematicians. In the early
1930s, these Alhambra designs had a seminal inspirational influence on the extraordinary
workofthewell-knownDutchartistM.C.Escher.
Yet, there is much more than the illustration of different Euclidean symmetries in the
ingeniousandintricateuseofgeometryintheseIslamicdesigns.Thereisnodoubtaboutthe
skill and artistry that have gone into the production of these creations. There is much, also,
which is of mathematical interest in the local symmetries that have been frequently
incorporated in this way, where we find symmetrical star-shaped regions of all sorts of
improbable-seeming symmetry, such as combinations of stellar shapes with 13-fold and
9-foldsymmetry,whicharequiteruledoutasoverallsymmetriesofcrystal-likearrangements,
forwhichonly2-fold,3-fold,4-fold,and6-foldsymmetriesareallowed.
The question has sometimes been raised, as to whether these ancient Islamic geometers
mighthavediscoveredandmadeuseofquasisymmetricalfeatures,suchasthefivefoldones
that I found myself in the mid-1970s. These would give rise to patterns that “almost” (in a
technicalsense)repeat,butwhichneverquitedoso,andthefivefoldsymmetryislikewisea
featurethat“almost”holdsbutnotquite.WhenIfoundsuchdesignsthatarisenaturallyasthe
only arrangements in which one can assemble a pair of “jig-saw” shapes, I was particularly
struck by the simplicity of these particular shapes, which could indeed force such fivefold
quasisymmetric patterns. I thought it not unlikely that things of this nature might well have
been made use of by the Islamic geometric artists of antiquity. Indeed, various later
researchers have claimed that there is indeed evidence that some ancient Islamic buildings
vii
viii Foreword
mightwellemploysuchquasisymmetricfeatures.Inthisbook,JayBonnerarguesotherwise,
findingstrongindicationsthatthedesireforexactperiodicitywasanoverridingdrivingmotive
fortheIslamicpatterns,andthatthereisnoevidencethatquasisymmetricfeatureswereever
madeuseof.
Itis,ofcourse,stillpossiblethatconvincingevidencemighteventuallycometolightthat
theideasofquasisymmetrydidplayaroleforsomeoftheancientIslamicdesigners.However,
IaminclinedtoagreewithJayBonnerthatstrictperiodicityseemstohaveplayedsuchakey
rolefortheseancientgeometricartiststhatquasisymmetricconsiderationswouldbeunlikely.
It is, to me, quite extraordinary how some of these Islamic designers were able to fit such
improbablesymmetriesas13-foldsymmetricstarscombinedwith9-foldonessoastocreatea
design with such a natural-looking elegance, and a periodicity that appears at a distance not
muchgreaterthantheextentofthesestellarshapesthemselves.Mathematicalquasisymmetric
patterns with, say 11-fold or 13-fold quasisymmetry, can now be produced in computer-
generatedpictures,butitwouldonlybeatextremelyremoteplaceswherelocalregionswith
thissymmetrywouldbeevident.Ifoneislookingforbeautyofdesign,withsuchregionsof,
say,11-foldor13-foldsymmetry,thentheancientIslamicdesignswinhandsdown!
RogerPenrose
EmeritusRouseBallProfessorofMathematics
MathematicalInstitute
UniversityofOxford
Preface and Acknowledgements
More than with any other cultural heritage, the work of Muslim artists over the centuries
revealsthevisualbeautythatisinherenttogeometry.Drawinginspirationfromgeometryled
toanabundanceofaestheticinnovationswithinthetraditionofIslamicgeometricdesignthat
were inexorably associated with methodological practices. Of particular importance was the
discoverythatgeometricpatternscouldbeextractedfromunderlyingpolygonaltessellations,
herein referred to as the polygonal technique. Over time, and in the hands of skilled and
dedicated practitioners, this design methodology engendered the extraordinary breadth of
diversity that characterizes this artistic tradition. As architectural ornament, Quranic illumi-
nation, and its more limited use within the applied arts, geometric designs were enthusiasti-
cally embraced by succeeding Muslim cultures, and along with calligraphy and the floral
idiom,becameanintegralaspectofIslamicaesthetics.Assuch,itsrolewithinIslamicartasa
wholeisparamount.Yetscholarshipofthisvastsubjectremainsunderrepresentedintwokey
areas: historical development and design methodology. Related to this relative lack of
attentionisthesurprisingabsence ofacomprehensive approachtocategorizing therangeof
patterns within this tradition. Additionally, while the exceptional achievements of past
geometric artists serve as a source of inspiration for many contemporary artists, designers,
craftspeople,andarchitects,thelossofmethodologicalknowledgeassociatedwiththisartistic
tradition has substantially thwarted those with an interest in incorporating such designs into
theirwork.Inconsiderationofthesegapsthisbookhasseveralintentions:toprovideagreater
understandingofthedevelopmentalhistoryofthisremarkableartistictradition;toemphasize
a more nuanced attribution of the geometric and design diversity that is a hallmark of this
distinctive form ofart; and toprovidea detailed elucidation of the methodological practices
thatareresponsibleforthediversity,beauty,andlongevityofthisartisticdiscipline.Whatis
more, it is my hope that the focus upon design methodology, including the use of computer
algorithms,willempowerthosewithasincereanddedicatedinterestinapplyingthecreative
processesofthepasttotheirownoriginalworks.
Thecombinedfocusuponhistoricaldevelopmentanddesignmethodologywithinthiswork
requiresanapproachtothesubjectofIslamicgeometricpatternsthatisbothchronologicaland
analytical. As such, there is an inevitable repetition of information concerning geometric
characteristics and the dates and location of specific examples when bridging between these
twoapproaches.Thisisnecessarytoclarifyhistorical,geometric,andmethodologicalcontext
throughout the book, and I request the reader’s patience and indulgence when encountering
such repetition. As a practical consideration, all dates accord with the Gregorian calendar
rather than either the Islamic or Persian calendars. In organizing the terminology employed
throughoutthisworkIhaveerredonthesideofclarityandsimplicityandhavegenerallytried
toavoidbeingoverlytechnical.Whereverapplicable,unlessthereisacompellingreasontodo
otherwise,Ihaveadoptedpriorterminology.Theglossaryprovidesbriefdefinitionsofmany
of the terms used throughout this work, including those that are of foreign origin (Arabic,
Persian,Urdu,Turkish,orSpanish),thosethataretechnicalandassociatedwiththescienceof
tiling and geometry, and those that pertain specifically to design methodology. In this latter
category,muchofthenomenclatureisofmyowninvention.Thisisduetothefactthatmany
ix
x PrefaceandAcknowledgements
significant features of this artistic discipline have not been previously identified as such and
are therefore without name or title. I argue that the polygonal technique was the principle
designmethodologyemployedbyMuslimgeometricartists.Thistechniquehasbeenreferred
tovariouslyastheHankinmethod(indeferencetoErnestHanburyHankinwhofirstidentified
thehistoricaluseofthismethodology),orthePICmethod(polygons-in-contact).Ipreferthe
termpolygonaltechniqueforitssimplicityanddescriptiveaccuracy.Inpreviouspublications
I have referred to the polygonal mechanism that characterizes the polygonal technique as
subgrids.However,intheinterestsofdescriptiveclarity,inthisbookIrefertothisimportant
methodological feature as the underlying generative tessellation, or alternatively as the
underlying polygonal tessellation. The polygonal technique was employed in two very
different modalities: systematically and nonsystematically. My identification of patterns as
beingeithersystematicornonsystematicresultsfromtherebeingnosuchpreviousdifferenti-
ationbypriorscholars.Mydescriptivetitlesforthefivehistoricaldesignsystemsthatemploy
repetitivemodules tocreate geometricdesigns stemfrom their nothavingbeen identifiedas
systems, per se, by other specialists prior to my work. I have titled these as the system of
regular polygons, the fourfold system A, the fourfold system B, the fivefold system, and the
sevenfold system. My classification of the four historical varieties of dual-level design has
changedslightlyfrommy2003accountofthisdiscipline(inwhichIhadonlyidentifiedthree
types)andresultsfromtherebeingnopreviousdifferentiationwithinthepublishedliterature.
RatherthanemployingadescriptivetitleIhavesimplyusedthemoreprosaictermsTypeA,
TypeB,TypeC,andTypeD.Similarly,mynamesforthefourprinciplepatternfamiliesthat
areubiquitoustothistraditionstemfromtheabsenceofprioridentifyingclassificationsfrom
previous sources in the English language. These are descriptively named the acute, median,
obtuse, and two-point pattern families. In writing about a tradition that encompasses many
distinct cultures with separate languages and artistic terms, I have chosen to refrain from
employing terms that are specific to select Muslim cultures in writing about this discipline
more generally. For example, despite being in common usage, I do not use the Farsi word
girih, meaning “knot,” when referring to geometric designs. I have sought to keep my
geometric terminology as nontechnical as possible, while following convention to maintain
clarity. For the most part my terminology corresponds with Craig Kaplan’s in Chap. 4.
He prefers the term translational unit for what I typically refer to as repeat unit. Similarly,
he employs the phrase template tiling for what I refer to as the underlying generative
tessellation,oralternativelyastheunderlyingpolygonaltessellation.
Chapter1chroniclesthehistoricaldevelopmentofIslamicgeometricpatternsfrominitial
influences and early manifestations through to full maturity. In identifying broad stylistic
trends, geometric characteristics, and diverse varieties of design, I have referenced multiple
individualpatternexamplesfromsuccessiveMuslimcultures.Mychoiceofexamplesreflects
thechronologicaldevelopmentfromsimplicitytocomplexityandwillfrequentlybuildupon
thefamiliartointroducethelesswellknown.Thischoiceofhistoricalexamplescannothelp
butbesubjective,buteveryattempthasbeenmadeinaligningmyaestheticpreferencesand
value judgments with impartial historical significance. Similarly, the many photographs
includedwithinthischapterprovideasenseofthebroadaestheticdiversitycontainedwithin
this tradition. In discussing the contributions of successive Muslim dynasties, the general
structure within each section flows from the more basic patterns to more complex designs.
Emphasisisalwaysplaceduponinnovationsthatoccurredduringagivenepoch,asthesewere
primary vehicles for the advancement of this ornamental tradition. Considerable attention is
giventothedevelopmentofdual-leveldesignswithself-similarcharacteristics.Thiswasthe
last great outpouring of creative innovation, and despite the relatively small number of
examples,theirbeautyandgeometricingenuityplacethemintoahighlysignificantcategory
oftheirown.Attentionisalsogiventotheapplicationofgeometricpatternstonon-Euclidean
surfaces of domes and domical niche hoods. There are two varieties of this form of Islamic
geometric ornament: those that utilize radial gore segments as their repetitive schema and
PrefaceandAcknowledgements xi
those that employ polyhedra as their repetitive device. The latter are far less common, and
almost all of the significant historical examples are included within this study. To a very
limited extent, each new dynasty is placed into a brief historical context that primarily
describes their rise to power, and what set them apart from their predecessors. This will be
redundant to historians, but many readers may benefit from the placement of the geometric
idiomwithinabroaderculturalandpoliticalmilieu—howeverbrieflyoutlined.Attentionhas
been given to the relatively few examples of Islamic geometric design that were created for
non-Muslim clients and in some cases by non-Muslim artists. The influence of Islamic art
uponnon-Muslimculturesisbeyondthescopeofthisstudy,buttheexamplescitedareworthy
of inclusion due to their geometric character as well as their historical circumstance. This
opening chapter tracks the history of Islamic geometric patterns through to achieving full
maturityandforthemostpartleavesthelater,morederivativemanifestationofthistradition
forfutureconsideration.
Chapter 2 explores the varied discrete features that characterize this multifaceted disci-
pline. Previous works have concentrated primarily on the variety of star types and regular
polygons within specific patterns when categorizing geometric designs, and more recent
studies have classified patterns according to their crystallographic plane symmetry group.
Yet there are many more distinguishing characteristics that help to broaden an overall
understanding and appreciation of this artistic tradition. Emphasis is given to the variety of
repetitive stratagems employed by Muslim geometric artists. Of course the simple square,
equilateraltriangle,andregularhexagonwerecommonlyemployedasrepeatunits,andmany
verycomplexpatternsemploythesewellunderstoodstructures.Fromasearlyastheeleventh
century patterns with alternative repetitive structures entered the artist’s repertoire. These
included rectangular, rhombic, and irregular hexagonal repeats, as well as designs with
rotationalsymmetry.Therepetitivestratagemsincludedinthischapteralsoincludeoscillating
square designs and rotating kite designs. These are essentially orthogonal, but by
incorporatingalternatingrhombiandsquaresintheformer,andalternatingkitesandsquares
in the latter, it ispossible to produce designs with seemingly incompatible regions of n-fold
local symmetry into an otherwise fourfold structure. Considering the early period of origin,
and the fact that there was no historical precedent that these Muslim geometric artistscould
have borrowed from, their familiarity with these diverse repetitive structures is surprisingly
sophisticated and predates analogous repetitive structures from other cultures by many
centuries. The intrinsic relationship between the n-fold symmetry of a given pattern and the
proportionsofitsrepeatunitareexaminedindetail.Aspatternsbecameincreasinglycomplex,
withmultipleregionsoflocalsymmetryincorporatedintoapatternmatrixthatwasbasedon
neither a square nor a triangular repeat, these regions of local symmetry were critical
proportional determinants of the overall repetitive structure: be it rectangular, rhombic, or
irregular hexagonal. This chapter also differentiates between patterns according to their
numeric qualities and postulates an abbreviated descriptive nomenclature that is based upon
rather basic geometric and numeric analysis. In discussing Islamic geometric patterns it is
sometimes difficult to express concisely and with precision the qualities of a given pattern.
This is especially true of the more complex designs. The approach to identifying the salient
featuresofagivendesignthatisadvocatedinthischapterisintendedtopromotebothcogency
indialogueandclarityinunderstanding.Onceagain,thissectionmovesfromthesimpletothe
complex, beginning with examples that employ single star forms or regular polygons into
simple orthogonal or isometric repetitive structures, and ending with complex designs with
multiple star forms within a single pattern matrix that repeat upon the less common grids
mentioned above. In classifying these more complex structures, I have identified the
conventions for including added regions of local n-fold symmetry of the primary stars.
These are placed at key locations within the repetitive structure, such as the vertices of the
repetitive grid, the vertices of the dual grid, upon the midpoints of the repetitive edges, and
occasionally upon lines of radius within the field of the pattern matrix. Another category of