Islamic Design and Its Relation to Mathematics

Brian Wichmann and David Wade

Contents

The Geometric Mode in Islamic Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Theories, Problems, and Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Symbolic Meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Early Islamic Art: The Emergence of an Islamic Aesthetic Sensibility . . . . . . . . . . . . . . . . . . . 6Islam’s Greek Inheritance: Mathematics, Science, and Philosophy . . . . . . . . . . . . . . . . . . . . . . 8Theoretical Geometry and Artisanal Practice in the Islamic World . . . . . . . . . . . . . . . . . . . . . . 12Mathematics in the Islamic World and Its Involvement in Geometric Ornament . . . . . . . . . . . 14Conclusion of Historical Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Modern Mathematical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Computer Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Abstract

Complex geometrical designs are a familiar feature of Islamic art. Here weconsider the historical context of this genre, tracing it back to the influenceof the Platonism and Hellenistic Neo-Platonism that Islam encountered in itsearly conquest of Byzantine territories (in particular the enormous contributionof Euclid).

David Wade is the primary author of sections 1–8, and Brian Wichmann for sections 9–10.

B. Wichmann (�)Independent Scholar, Woking, UKe-mail: brian.wichmann@bcs.org.uk; brian.a.wichmann@gmail.com

D. WadeIndependent Scholar, Llanidloes, UKe-mail: davidgwade@gmail.com

© Springer Nature Switzerland AG 2019B. Sriraman (ed.), Handbook of the Mathematics of the Arts and Sciences,https://doi.org/10.1007/978-3-319-70658-0_91-1

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In addition, the modern analysis is presented showing the relationship inmathematical terms. Such an analysis naturally leads to the presentation of thepatterns using computer graphics.

KeywordsIslamic design · Geometry · Historic influences · Modern graphics

The Geometric Mode in Islamic Art

Complex geometrical patterns are a well-known feature of Islamic art and archi-tecture, together with its proclivity for highly symmetrical arrangements. But whatare the origins of this mode? And what aspects of the Islamic ethos does this broadstylistic predisposition actually reflect? There is a natural reluctance among serioushistorians of Islamic Art to jump to facile conclusions on the evolution of Islamicdecorative forms – which is understandable given the paucity of archaeological evi-dence. But clearly there must be a reason, and a compelling one, for the continuinguse of strong symmetrical arrangements and highly geometric modes of decorationby the many nationalities and ethnicities within the Islamic sphere for well over athousand years. The following observations revisit various aspects of this subject.

Theories, Problems, and Evidence

The main consideration in any discussion of “Islamic geometrical design” shouldsurely be the sheer breadth of this topic. With its extraordinary range of examples,drawn from centuries of development throughout the Islamic world, this is clearly asubject on which it is perilous to make sweeping generalizations. But appreciationof the intrinsic qualities of this art form has not always been accompanied withthe sort of careful evaluation that it obviously deserves. In fact, it has to be saidthat the “geometric mode,” as a theme in Islamic decorative art, has endured morethan its fair share of gratuitous interpretations and misconceptions. Among the morecommon of these are:

1. That the resort to geometric and “arabesque” patterns are entirely due to theQur’anic prohibition on image-making;

2. The belief that these designs are constructed according to some underlyingsymbolic meaning;

3. The notion that theoretical mathematicians were involved in their creation.

The first of these assertions clearly does have some basis as an influence, derivingfrom the strict Islamic adherence to the 2nd commandment, Exodus 20 (“You shallnot make unto you any graven image . . . ” etc.). But this explanation, by itself, isentirely inadequate in accounting for the particular forms that have been adopted.As for the other propositions, there is no evidence at all for the use of symbolism

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of any kind in Islamic geometrical or vegetal designs and precious little for anyconnection with the discipline of “pure” mathematics in the former.

It should also be said that there are other important aspects of this broad subjectthat remain obscure (and contentious) and are likely to remain so, such as preciselywhen and where the distinctive Islamic decorative canon first comes into being;the extent to which it drew on late-Classical forms; and whether it had doctrinalassociations, as some have claimed – or, indeed, was influenced by religious orphilosophical ideas at all. Obviously, the principal sources of evidence will alwayslie in the surviving work itself, but even here there are problems, not least thefact that so much in the architectural record has been destroyed. In addition, it isfrequently unclear whether a specific example of decorative art is wholly attributableto the artists and artisans involved in making the work or had been determined by thetastes of those who commissioned it. Moreover, there is seldom any good evidenceto indicate the realities of the creative processes involved, especially in large-scaleprojects where, for instance, it may be difficult to assign the relative contributionsof the architect of a monument and those employed to decorate it.

In all probability, the setting of aesthetic priorities and the contributions ofthe different agencies involved in grander schemes are likely to have variedenormously. However, it seems most likely that innovations in the particulars ofvegetal arabesque and geometrical pattern were usually developed from the artisanlevel upward. This is an art of decorative playfulness, largely deriving from thecreative genius of individual artist/craftsmen, who were keen to impress patrons(and their peers) with their creative skills, both in the particular medium that theywere working in and by the sophistication and complexity of their designs. InIslamic arts and crafts generally, technical virtuosity was always highly regarded.

As for speculation regarding the extent to which mathematicians proper advisedon the creation of geometric patterns, again, there is some small evidence ofcontributions of this kind in the form of geometrical manuals that were intendedto be read by craftsmen (see below), and even of occasional cross-disciplinarymeetings, but it seems far more probable that those who created these formsgenerally relied on their own informal, nonacademic working knowledge of planegeometry, which was clearly quite considerable. There are no biographical accountsof architects from the formative periods of Islamic art and architecture, let aloneartisans – and unsurprisingly, the latter have left few documents detailing their workpractices. Moreover, there are no indications at all that either were influenced by anoverarching theory of any kind. The geometric patterns, arabesques, and calligraphythat have played such an important role in this art were usually created withinworkshop traditions that were handed down through generations.

The unwritten rules of this decorative canon, and the skills involved in creating it,would have become second nature to the artist/craftsmen involved and the means oftheir construction almost amounting to a trade secret in many cases. Even the extentof specialization in these different decorative components (geometric, floriatedarabesque, and calligraphic) and the methods of organizing these into a finishedproduction are simply not known. It is unlikely though that the Islamic ornemanisteswho created geometric designs were at all familiar with the more technical

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Fig. 1 The “decorative canon” of Islamic art is comprised of three distinct elements (calligraphic,pure geometric pattern, and floriated arabesque), which all appear in endless varieties“One would enquire in vain for the masters who brought this system to its flowering, or those whoopened up new ways for its development. This art is totally anonymous and it would contradictthe artist’s noblest charge, which was the liberation of the spirit from the transitoriness of worldlyties.” (Kühnel, 1949)

geometrical knowledge required by architects or those involved in civil engineering,town-planning, land surveying, etc. Their artistic geometry was primarily concernedwith elegant appearance, not mathematical exactitude; approximations were oftenacceptable here, if the final effect justified it (Fig. 1).

However, the paucity of evidence on these and other matters bedevils Islamicart-historical studies. Islam’s long cultural heritage has endured more than its shareof the ravages of time, and the many terribly destructive events in Islamic historyhave meant that important parts of the archaeological record have been destroyedor seriously diminished. Apart from the terrible human costs, this presents seriousdifficulties in determining the artistic evolution of Islamic decorative art, particularlythe paths of its developments from its earlier, more derivative, styles through to themature forms.

Islamic Art, in common with the artistic productions of other great cultures, hasstylistic features that render it instantly recognizable – but the precise qualities thatcreate these distinctive cultural qualities are often not so easy to pin down (Wade,2018). Islam’s proclivity for geometric ornament is of course very much part ofits character and part of what makes so instantly recognizable – but this form ofartistic expression is essentially a component of a broader theme, namely, that ofsymmetry. Many other cultures use symmetries of course, but in Islam, it is a sinequa non. Islamic art is not simply suffused with geometrical symmetries – these areits central principle. The overriding question in this topic has to be, Why and Howdid this come about?

At the risk of being accused of the sort of generalizations riled at earlier, our owninstincts regarding the origins and continued use of the geometrical mode in Islamicart and architecture are that it somehow reflects the aesthetic ideals of a Classicalphilosophical tradition that had been thoroughly absorbed into mainstream Islamicthought at an early stage of its development. We refer, of course, to the intellectualspeculations that began with Pythagoras, were consolidated by Plato, and continuedby the “Neo-Platonists” of Late Classicism. The very fact that Islam had expanded

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into this sphere of Hellenised Late Antiquity meant that Neo-Platonism was the firstmajor philosophy that it encountered, and it later incorporated many of its themesinto its own broad philosophical outlook. In fact, Classical Greek philosophies ingeneral were to become very much part of the Islamic tradition. In its formativeperiod, much of Plato and Aristotle in particular were found to be acceptable, indeedperfectly compatible with core Islamic beliefs.

That a central concept of the Platonic stream of thought involved the associationof Ideal Forms and Beauty and that it was preoccupied with the purity of geometricforms as an expression of these concepts would seem to have a strong bearing on theprinciple themes of Islamic art. The Greek author Plutarch, long before the advent ofIslam, and speaking like a true Pythagorean, epitomized this association of geometryand spirituality within Platonic tradition – “The function of geometry is to draw usaway from the world of the senses and of corruption to a world of the intellectand the eternal. For the contemplation of the eternal is the end of philosophy asthe contemplation of the mysteries is the end of religion.” Could there be a moreperfect description of the underlying intentionality behind the extraordinary varietyof geometric designs, and the continuous process of artistic exploration of thismode for hundreds of years within the Islamic sphere? With its pure geometries,perpetually playing out their symmetrical dances on the stage of the Euclideanplane this genre, in our view, is quintessentially Platonic. Precisely how these loftyphilosophical speculations were transmitted to those living at the lower, artisanallevel of society is of course another matter entirely . . .

Symbolic Meaning

As indicated in the previous passage, it is tempting to ascribe symbolic meaningto the traditional repertoire of forms that we have characterized as the “Islamicdecorative canon,” i.e., the repertoire of calligraphic, arabesque, and geometricelements that can be found in one form or another throughout the art and architectureof the Islamic world. Tempting, but misconceived. Despite various assertions to thecontrary, these forms have rarely carried any symbolic or doctrinal associations.In fact, there is a consistent avoidance of symbolism throughout Islamic art –butreluctance to accept this fact has led to various recent misinterpretations of the genre,particularly in regard to its geometric aspects. These include mystical/religiousinterpretations, including kitsch “New Age” explanations involving cryptic “cos-mological” and astrological symbolism; the unlikely use of magic squares ingeometrical constructions; and the assignation of specific doctrinal associations– all of which are attributions for which there is no real evidence. Wasma’a K.Chorbachi, an Arabic speaker, having examined hundreds of original workshopdrawings, has found no reference whatsoever to any mystical, astrological, orreligious symbolism in them (Chorbachi, 1989, p771). In fact, the more recentmisattributions of symbolic meaning to this genre seem to have arisen preciselybecause it has no real equivalent in Western European (or any other) art.

As for doctrinal associations, according to Jonathan Bloom, the taste for purearabesque and geometry reigned supreme by the later Fatimid (Shi’ite) period,

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as it did everywhere else in the Islamic lands. He expresses a confidence thatthese motifs and modes of decoration had no doctrinal associations (Bloom, 2007).Similarly, Terry Allen writes that there is no evidence to support the claim thatgeometric designs were somehow understood to be “emblematic” of the AbbasidCaliphate and Sunnism (Allen, 2004). And Sheila Blair has declared that in her view,the canonization of round scripts and geometry had nothing to do with religioussectarianism in the tenth century (Blair, 2006).

The fact is that by the tenth-century geometric patterns, floriated arabesques andcalligraphy were established features of the repertoire of Islamic art and that thisdecorative canon had developed and persisted through all the fractious differencesin the formulation of Islamic doctrine. “Sunni Revival” or not, these forms werewell into the process of becoming an international Islamic style. The particularart-historical problems concerning the mapping of the development of this styleare complicated to an impossible degree by the lack of archaeological evidenceof this period resulting from the disaster of the Mongol invasions, during whichenormous swathes of Abbasid art and architecture were destroyed. But there willalways be more general difficulties in entering the mindset of the artist/craftsmenand architects that actually produced the work of centuries past. Styles that resonatewith cultural attitudes or weltanschauung invariably spring into being from belowrather than as a calculated imposition from above and the “intentionality” of thosewho produced these designs throughout the Islamic world are bound to remainlargely inaccessible. Who knows what any individual craftsman actually had inmind? In the final analysis, it is the work itself that counts. The many geometrical,arabesque, and calligraphic variations in Islamic decoration are as unique as theIslamic ethos itself and have an overarching cultural resonance and meaning of theirown.

Early Islamic Art: The Emergence of an Islamic AestheticSensibility

The first impressions of surviving examples of Islamic art and architecture fromthe earliest period, that is to say from the time of the first Islamic dynasty, theUmayyads, can appear unfamiliar and have few of the features that most wouldassociate with later Islamic Art. This is because the cultural and artistic values ofIslam were far more gradual in their development than its purely religious aspects.In the early years of its conquests, the Umayyads had moved north from the Al-Hejaz region in Arabia, to Syria, and established Damascus as their capital andcenter of administration (661 CE). Here, they soon adopted the administrative andcivil practices of the previous, Byzantine, authority, and in a very short time madethe transition from rough conquerors to a sophisticated military aristocracy. Con-temporary accounts indicate that the new Muslim regime, confident though it wasin its own religious mission, was deeply impressed by the civilized sophisticationof the Byzantine territories that had fallen into its possession – in particular by theopulence and beauty of the Christian art and architecture of the region.

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It is not surprising then that the earliest important Islamic architectural projects,notably the Dome of the Rock in Jerusalem and the Great Mosque in Damascus,were intended to stand comparison, even outshine, the most impressive of Christianholy places. They were, of course, also intended as symbols of Islamic ascendancy,and in building these monuments, the Muslims were clearly developing their ownideas concerning the function and appearance of sacred precincts. But the newrulers were still a small minority at this time, so the architects, builders, andartist/craftsmen who were commissioned to build and decorate these monumentsare likely to have been predominantly Christian. The decorative aspects of theseand other buildings in the first decades of Muslim rule are therefore particularlyinteresting for indications, such as they are of the emergence of distinctive Islamicaesthetic preferences (Al-Khalili, 2010; Freely, 2009; Kennedy, 2005).

One of the most striking features of both of the monuments mentioned above istheir use of large areas of finely detailed wall mosaics. The significance and meaningof the subject matter of these panels, which in large part consists of luxuriant plantforms, is not entirely clear, but in both cases, the decoration probably refers toMuslim notions of the Paradise promised to Believers in the afterlife. The elaborate,stylized vegetal forms in the Dome of the Rock are drawn from the repertoire oflate antique forms and are enclosed within extended bands of jewel-like ornamentsand calligraphy. The mosaics of the Great Mosque, which was built some 15 yearsafter the Dome of the Rock, are somewhat more representational and include exoticbuildings among resplendent gardens, with flowing rivers and graceful trees. Butthere is no iconography, in the Christian sense, in either monument – in fact there areno portrayals at all of humans or animals, subjects that Muslims clearly consideredas entirely inappropriate in religious settings such as these.

The impression that is conveyed by this art and architecture is, however, distinctlyspiritual, essentially that of an orderly otherworldliness. This sense of repose isachieved by the use of architectural and decorative symmetries which, pointedly,are not reliant on overt symbols of any kind. The mosaic panels of the Dome ofthe Rock conform to the buildings strong overall symmetries, and together withthe deliberate architectonic duplications in the building as a whole, might seemcalculated to induce a sense of dissociation in the pilgrims ambulating aroundits interior. Here, as in later Islamic architectural decoration, there is a feeling ofthe dissolution of the buildings physicality. The Damascus Great Mosque mosaicsproject a different vision, but again, its panels seem to depict an ideal, paradisiacalstate of being. Both monuments present distinctive artistic approaches but also clearevidence that there were conscious attempts at an Islamic self-definition, and what isparticularly interesting is that these aesthetic preferences, particularly in their use ofstrong symmetries, seem to be influenced by the sort of Neoplatonic attitudes thathad already been absorbed by Christian thought in this region (see below). Thereare other interesting architectural details, again based on earlier forms, that appearto reflect the new Islamic sensibilities, the window grills of the Great Mosque ofDamascus, for example (Fig. 2). It is intriguing, to say the least, that the possibilitiesof purely geometric decorative effects should appear at such an early stage in Islamicarchitecture, particularly as these examples seem to be rather more adventurous in

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Fig. 2 Window grilles from the Ummayad Great Mosque of Damascus featuring patterns that areclearly derived from late-Classical models, but which seem to point to the more dynamic use ofpattern in the Islamic sphere

their design than the late antique models from which they obviously derive. OlegGrabar has observed of these and other examples of the early use of ornament inIslam that they appear to show a conscious selection of previous forms, togetherwith an avoidance of obvious, rigid constructions and a preference for decorativemotifs that are capable of infinite extension by the use of symmetrical arrangements(Grabar, 1973).

Islam’s Greek Inheritance: Mathematics, Science, and Philosophy

Pythagoras is the presiding genius of mathematical study in Islam. Greek and Indianelements are mingled in it, it is true, but everything is regarded from a neo-Pythagoreanpoint of view. (De Boer, 1903)

In the second Islamic century, by the time the Abbasid Caliphate had built theirnew capital in Baghdad, Muslim administrators and scholars were able to draw on anextraordinary range of knowledge from the vast territories that were now under theircontrol. Classical, Romano-Greek, Chinese, Indian, Persian, and Egyptian learningof many kinds became available and were investigated, but the most importantsource of general and practical knowledge remained the extensive body of Greekworks that the Muslims had first encountered as a result of their conquest of theByzantine territories of Egypt and Syria in the first flush of their conquests.

The early Muslims, who, under the Umayyad Caliphate, had established Dam-ascus as their administrative center, were as impressed by the level of Greekerudition as they were by the general level of civilized life in Egypt and Syria.This part of the world had been under Roman rule for seven centuries, was heir

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to Hellenistic civilization, and had been Christian since the fourth century CE. Atthe time Islam entered this region, in the late seventh century CE, Greek science andliterature from the earlier Classical and Hellenistic eras was still being taught (albeitfiltered through a Christian lens), as were the philosophies of Plato and Aristotle.Neoplatonism, with its roots in these philosophies, was an abiding influence in thismilieu (Morewedge, 1992).

From their first encounters with this huge body of Classical knowledge, however,Muslims tended to be divided in their responses. Classicism, with its infidel, paganassociations, was regarded with great suspicion by those of a strictly religious dispo-sition, but since the administration of a rapidly increasing Empire had now to be con-ducted in Arabic, there was a pressing need for the translation of huge amounts ofwritten material, including official documents, from their original Greek. The Mus-lims had much to learn and Byzantine sources had much to teach, and although theoriginal impetus for the translation of Greek material may have been for purely prac-tical reasons, increasing familiarization with this wealth of information inevitablyled to a broader interest in Greek thought. Translations during this early periodtended to be unofficial, i.e., by private sponsorship, but since these sources clearlyhad so much to offer in many fields, this was to change. As the numbers and qualityof translations from Greek originals accumulated their value became ever moreapparent. Medical treatises were of particular interest at this stage, as were thosedealing with Astrology and Alchemy, but as more and more subjects were translated,they became part of a general cultural influence that was being absorbed by Muslimsociety. The eventual outcome of this cultural osmosis was that Islam accepted late-Classical Greek attitudes to such an extent that, almost by default, they became animportant part of its own tradition. In many ways, the templates for Islamic civil lifewere set here and permanently incorporated into its own, Islamic, ethos.

The so-called Golden Age of Baghdad, in the second great Islamic Caliphate,the Abbasids (in the Third Islamic century), was characterized by an extraordinaryenthusiasm for the translation of not only Greek but Parthian and Indian knowledge.But even by the time the renowned House of Wisdom (Bayt al-Hikma) was set upin Baghdad (830 CE), such important figures as Plato, Aristotle, and Plotinus wereas familiar to Islamic intellectuals as they had long been to those in the Byzantineworld. This process was facilitated by the earlier Christian and Jewish adjustmentsto Classical thought, which, in a manner of speaking, had “sanitized” its paganassociations, effectively making it more acceptable to Islamic sensibilities.

The transference of the Court from Damascus to Baghdad had meant that the“Byzantinization” of the Caliphate had to a great extent been superseded by a“Persianization” of its religiopolitical ideas, attitudes, and manners (even thoughthe rulers still tended to be Arab). But the general appetite for Greek texts wasundiminished. The Bayt al-Hikma was originally founded as a center for translationbut naturally developed into a library, and then to a university, where Greek andother works were made available to scholars. The establishment of this institutionalso meant that original manuscripts on a whole range of academic subjects weresoon being actively sought from all available sources, including Byzantium, Persia,

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and India. As in Damascus, the first texts to be translated were those of subjectsthat were deemed to be of most immediate use to the Rulers, including Medical,Astrological, Agricultural, and other technological treatises, but as the translationmovement gathered pace, particularly during the reign of Caliph Harun ar-Rashid,scholars turned their attention to scientific and philosophical works. By the timeof the rule of his son Caliph al-Mamun in the first half of the ninth century, thetranslation movement had become highly fashionable, involving many differentteams of translators and scribes, funded both by the Royal Court and by private,wealthy individuals.

The increasing familiarity with Greek ideas in Baghdad court circles engendereda genuine sense of admiration, particularly for texts those dealing with philosoph-ical/scientific and mathematical subjects. The works of Euclid, Apollonius, andArchimedes in particular came to be treated with enormous respect. Euclid whosefamous Elements established the basis of plane geometry, was translated into Arabicduring Harun ar-Rashid’s reign (786–809 CE) by the mathematician al-Hajjaj ibnMatar (who later provided his son, the future Caliph Ma’mun, with an abbreviatedbut improved version).

During the ninth century CE, Baghdad, as the capital of the Caliphate, was theintellectual, religious, and commercial center of the Islamic world. This was a periodof extraordinary scientific and medical innovation – developments that were oftenassociated with philosophical speculation. Increased understanding of the principlesof Classical mathematics and geometry led naturally to a sense of self-confidenceamong Islamic intellectuals, allowing them to advance on and, where necessary,correct the assertions of their illustrious predecessors. During Caliph al-Mamun’srule (813–833 CE), there was a marked increase in the commissioning of new workin a range of areas – including mathematics, astronomy, geography, and medicine.These often went far beyond mere translation. A new class of Islamic scholarshipcame into being that was capable of making commentaries on Greek works andwas increasingly involved in original research. In astronomy, for instance, the studyof Ptolemy’s Almagest encouraged Al-Mamun to set up a program to verify theaccuracy of existing star charts. This in turn led to the appointment of officialastronomers and the building of observatories in Baghdad and Damascus – layingthe foundations of Islam’s 700-year involvement with this science (the considerableachievements of which contributed much to modern astronomy). The translationmovement itself more or less came to an end around the end of the tenth centuryCE, largely because everything that was available had already been translated butalso because by this time Islamic science was well and truly established on its ownaccount and had produced a great deal of original work.

One of the more remarkable accomplishments of the era of Baghdad’s “GoldenAge” and one that gives an indication of the general spirit of scientific enquiry atthis time were the attempts to make an accurate measurement of the circumferenceof the Earth, a project in which Al-Mamun himself took a personal interest. Thisfirst attempt involved an accurate measurement of a stretch of flat desert in Sinjar,northwestern Iraq, and the comparison of the elevation of the Pole Star at thebeginning and end of the process, then a calculation of the curvature of the Earth

Islamic Design and Its Relation to Mathematics 11

from the angular difference. The result, 8000 farsakhs (24,000 miles), which waschecked by a second expedition, was remarkably accurate for its time.

Caliph al-Mamun personally employed the famous mathematician andastronomer al-Khwarizmi who, among many other achievements, was responsiblefor the introduction of Indian numerals into Arabic mathematics. Al-Khwarizmialso developed the procedure we now know as Algebra (al-Jebr) and gave hisname to the term “algorithm.” There was a heady, pro-science atmosphere in theBaghdad court at this time, but in this early-Islamic setting, it was inevitable thatthe problem of reconciling Reason with Faith should arise. The Caliph’s responseto mounting religious criticism of his proto-rationalistic views was to promote thetheology known as Mu’tazilism, which argued that Allah’s moral obligations areaccessible to rational thought and that, because knowledge derives from reason,the latter should be the “final arbiter.” The Mu’tazilites disliked conventionalanthropomorphic interpretations of the Qur’an and went so far as to declare that theHoly Book could not properly be considered as the word of Allah, since He couldhave no separable parts. In this view, the Qur’an was therefore created, not eternal,as popular religious belief would have it.

This was heady stuff and an interpretation of basic Islamic precepts that didnot go down at all well among the religiously orthodox, who bitterly opposedevery aspect of these rationalistic ideas. In the event, this conflict brought aboutan ideological crisis that led the Caliph, al-Mamun, to attempt to force the issueby making the acceptance of Mu’tazilism a condition of official service and, inthe face of continuing opposition, instituted an Inquisition (mihnah) against thosewho refused to accept his ruling. This, in turn, prompted a strong religious andpopular reaction against these moves, so that Caliph’s attempt at imposing a degreeof separation between rational practice and religious observance ultimately failed.With hindsight, this proved to be a fork in the road for Islam, where eventuallyFaith was to take precedence over Reason – as a result of which the influence ofClassical, Hellenistic attitudes decreased. Scientific ideas and sceptical, rationalthought were to continue to play an important role in the Islamic world over thefollowing centuries, and many important discoveries were made but, objectivelyspeaking, Islamic science peaked around 1000 CE and went into a long, slowdecline. Unfortunately, the tendencies toward religious conservatism and dogmawithin the Islamic world after this time increasingly tended to create conditionsthat were less conducive to the pursuit of scientific knowledge.

A key moment in this process occurred in the eleventh century CE whenthe influential theologian Al-Ghazali, in a work entitled The Incoherence of thePhilosophers (Tahafut al-Falasifa), attacked the entire Greek philosophical tradition(in particular the Emanationist principles of Neo-Platonism), denouncing them as aform of heresy. This declaration marked an important shift in Islamic epistemology.However, by this time, Neoplatonic attitudes had been thoroughly assimilatedinto mainstream Islamic thought, and ironically Al-Ghazali himself used Classicaldialectical methods in his refutation of this and other Classical philosophies. Neo-Platonism continued to influence Islamic culture in many subtle ways; however,indeed the otherworldly, geometrical, preoccupations of Islamic art can be seen as

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part of this enduring legacy. The complex ideas of this religious philosophy areunlikely to have been of interest to most artist/craftsmen, but they (along with manyothers in Islamic society) seem to have absorbed many of its broad notions by wayof the cultural osmosis referred to above.

Al-Ghazali (1058–1111 CE) was a towering figure in Islamic cultural historyand is still counted as one of its greatest religious thinkers. He is importantbecause by his time, the question of ultimate authority in matters of law andlegislation (which had its roots in the Mu’tazilism controversy mentioned above)had become part of a cultural crisis. Islamic society had developed in a varietyof somewhat irreconcilable directions – among which was the continuing chal-lenge of the Hellenistic rational/philosophic tradition. Al-Ghazali experienced thisepistemological fracturing at a most personal level, in the form of a crisis offaith and conscience that led to a complete mental and physical breakdown. Hisrecovery from this traumatic experience came in the form of a conversion from hisprevious extreme philosophical skepticism to a more mystical acceptance of therole of prophetic revelation. His Revival of the Religious Sciences (Ihya ulum l-din),amounting as it did to a thorough refocusing of Islamic verities, was enormouslyinfluential. From this and other works, he came to be regarded as the “Restorerof the Faith.” His achievement is reflected in the fact that his ideas managed toachieve widespread acceptance. To the Sufi’s, he remained a mystic; more orthodoxtheologians regarded him as an important religious teacher; and for the legalists, hecontinued to be admired as an eminent jurist.

In a much later, somewhat ironical, development, Al-Ghazali (who was creditedwith producing some 70 books) came indirectly to influence European rationalthought in its slow emergence from the Dark Ages – particularly through thephilosophical works of Thomas Aquinas and Pascal.

Theoretical Geometry and Artisanal Practice in the Islamic World

In view of its origin, carpentry needs a good deal of geometry of all kinds. It requires eithera general or specialized knowledge of proportion and measurement in order to bring formsfrom potentiality into actuality in the proper manner, and for the knowledge of proportionsone must have recourse to the geometrician. Therefore the leading Greek geometricianswere all master carpenters. Euclid, the author of the Book of Principles, was a carpenter,and known as such. The same was the case with Apollonius, the author of the book onConic Sections, and Menelaus and others. Ibn Khuldun from his Muqaddimah (Khaldun,1967).

In the Introduction to his Classic, encyclopedic History of the World (theMuqaddimah, finished 1377 CE), the Islamic philosopher and social historian IbnKhuldun makes various observations about geometry and the crafts, from whichthe above quotation is drawn. His comments are interesting, partly because of theirapparent familiarity with various important figures of Classical mathematics, butalso because they seem to reflect a rapport between the geometry practiced byarchitects and craftsmen and the tradition of academic geometry that Islam had

Islamic Design and Its Relation to Mathematics 13

inherited from the Greeks. This apparent common interest in matters Geometric hasled many in more recent times to assume a rather closer connection than the evidenceseems to support. There is certainly plenty of evidence for both Geometries in Islambut, when it comes down to it, little, actually, for their interconnection.

The continuity of the Greek tradition of mathematics in the Islamic sphere, partic-ularly of its Geometry, is certainly impressive. Geometry was a favored subject fromthe beginning of the translation movement, largely because it had many obviouspractical applications. Euclid’s Elements, which provided the foundations of planegeometry, was itself translated at an early stage and came to be greatly admired forits orderly, logical presentation of the subject. In fact, this work came to be valuednot only as the foundation of theoretical Geometry but also as an exemplar of logicalmethodology. With its steady progression through a series of axioms, it set a bench-mark for rigor in Islamic mathematical and scientific investigations of all kinds.

Most importantly for the subject under discussion here, the new spirit of enquiryamong the early Islamic translators and scholars, and their enthusiasm for Greeklearning, seems to have exerted a broader influence on Islamic cultural attitudes.The well-known Islamic penchant for complex geometrical patterns, which canlay claims to being the most “scientific” of all art forms, can be seen as anexpression of this shared enthusiasm and an indication of the extent to whichClassical knowledge of mathematics had been absorbed in the Islamic world. Thishas led some commentators to assume a degree of collaboration between Islamicmathematicians and artisans in decorative panels that demonstrate a high degreeof geometric complexity; in reality, however, there is little evidence that cross-disciplinary collaborations of this kind actually took place. There was certainlya familiarity with Geometry among mathematicians and among skilled artisansof various kinds. But their interests, skills, and aims were different (as was theirstanding in society). As the art historian Terry Allen has pointed out a propos of thissubject, we cannot simply lump together every manifestation of interest in geometry(Allen, 2004). The theoretical geometry that is a branch of mathematics and the“hands-on” geometry used by artisans and architects were equally specialized,but quite distinct disciplines. The notion that complex geometries in decorativepatterns derived from advanced understanding of geometrical theory is, for the mostpart, misconstrued. Moreover, although the “geometric mode” certainly became aubiquitous form of decoration in Islamic art, this was not a manifestation of a moregeneral interest in geometry in the Islamic world. There are no accounts of what theordinary Muslim citizen thought of geometric ornament – (we can imagine that theywould have been as impressed and intrigued as we are today) – but it doesn’t followthat they would have had any better idea than a modern non-Muslim with regard totheir underlying principles of construction. These are more likely to have belongedto realm of specialized, insider knowledge.

In fact, there are almost no references to decorative ornament in any survivingmathematical text from Islam’s medieval period. The geometric forms in its art andarchitecture art were created by artist/craftsmen, and in actuality, even the mostcomplex decorative panels are relatively trivial from a purely mathematical point

14 B. Wichmann and D. Wade

of view. It would of course have been perfectly natural for mathematicians of thetime to take an interest in geometric designs – they still do today. And it is true thatthere are a few tantalizing glimpses of exchanges between what one might describeas “academic” geometers and artisans that used geometry, but again, there is noevidence of collaboration on actual projects. As far as can be judged (because theevidence of such encounters tends to be from the academic side), there was never acomplete meeting of minds between the parties. The “advice to artisans” literature,such as it is, tends to consist of finger-wagging attempts to get the artist/craftsmento construct their designs “properly,” i.e., according to purely Euclidean geometricprinciples. As indicated earlier, the inclusion of “approximations” (i.e., the inclusionof “not-quite” regular figures into a design) would have been anathema to anyacademic geometrician but were occasionally acceptable to designers.

The widespread use of complex patterning in the Islamic world, over manycenturies, clearly indicates that this mode of expression satisfied something integralto the Islamic ethos – not least because the use of the geometric mode is notexpressed in any single unbroken tradition. Technically speaking, there are anynumber of ways to create high levels of complexity in Islamic geometric patterns,and in the end, it is only the broad geometric theme itself that is constant. That isto say that although there is tendency toward greater geometric complexity throughtime in the various Islamic regions, this is expressed in a wide range of differentstyles and approaches. It is however intriguing that these humble Muslim craftsmen,over time, managed to uncover and exploit for decorative purposes many of thepossibilities allowed of formal symmetrical arrangement (see section “ModernMathematical Analysis” below).

There is little doubt that artist/craftsmen traveled (voluntarily or otherwise)across widely separated regions and that at different times and places pattern booksand working drawings would have facilitated transmission. Given Islam’s fracturedhistory, it is likely that traditions were broken, sometimes to be resumed in subtlydifferent ways and that patterns would have been adapted from one medium tobe used in another. Taken as a whole, this long tradition indicates that over time,experimentation vied with adherence to established forms. There is a thread ofcontinuity in the use of decorative ornament in Islamic, but the real constant was athorough appreciation of the underlying geometry of plane division and a mastery ofwhat can only be described as artistic geometry. This involved a keen awareness ofan unspoken set of rules which involved a strong sense of symmetry and a preferencefor the careful, balanced distribution of elements within framing panels. In the end,it is these criteria, rather than any mathematical formulae, that go a long way tomake Islamic art, in all its diversity, so recognizable, coherent, and distinctive.

Mathematics in the Islamic World and Its Involvement inGeometric Ornament

In recent times, there has been a welcome degree of recognition of the enormouscontribution made by early Islamic scientists and mathematicians to world knowl-

Islamic Design and Its Relation to Mathematics 15

edge; this is long overdue. In particular, the role of Islamic learning in the recoveryand development of Classical texts, and the subsequent transmission of this hugebody of scholarship to Medieval Europe, is now widely acknowledged. It is nolonger believed that there was an unbroken chain of learning in Europe from theClassical period to the early-Modern – and it is now properly seen that the greatrekindling of interest in all the sciences that occurred during the Renaissance waslargely fueled by Islamic erudition (Montgomery Watt, 1972; Saliba, 2011).

We have already referred to the “Golden Age” of Islamic science that flourishedin the early Abbasid period (in Baghdad between the eighth and tenth centuries CE),which produced such outstanding scholars as al-Khwarizmi, al-Kindi, and OmarKhayyam. But Islamic science, building on the foundations of Classical, Parthian,and Indian knowledge, was to continue making important advances in variouscenters around the vast Islamic dominions for some centuries to come. Whereverthe social, intellectual, and economic conditions were conducive, advances weremade – particularly in such fields as mathematics, astronomy, optics, and medicine.Nevertheless, Science, whose pagan philosophical associations were never entirelyforgotten, continued to be regarded with some suspicion by the orthodox. Scien-tifically minded thinkers under royal patronage were usually afforded a measureof protection from the more zealous religious critics – who were in any case lessconcerned with such abstract fields of study as mathematics. It can be said thenthat, at least in higher intellectual levels, the study of mathematics (particularlyof geometry) was well established and widely taught, throughout the medievalIslamic world and that the Alexandrian Platonists, Euclid and Ptolemy, retainedtheir positions as the revered progenitors of geometry and astronomy, respectively.

However, as indicated earlier, there is little evidence of interaction between“theoretical” mathematicians and artisan geometricists in this Islamic society –although there are a small number of surviving texts indicating that this didoccasionally occur. The authors Gülru Necipoglu (1995) and Alpay Özdural (2000)single out a couple of examples – one originating in the tenth century involvingthe mathematician and astrologer Abu al-Wafa al-Buzjani, and a later, anonymousPersian text from the fourteenth century CE on “Interlocking Figures.”

Al-Wafa al-Buzjani (940–998 CE) was the author of a manual of practicalgeometry On the Geometric Constructions Necessary for the Artisan (Risâla fimâyahtâju al-sâni’u min a’mâl al-handasa), of which four known hand-written versionssurvive – one in Arabic and three in Persian. The original work was written inBaghdad, in Arabic, but no longer exists, and each of the later copies has somemissing information and chapters. The surviving Arabic version (which is kept inthe Library of Aya Sofya, Istanbul), although not original, is more complete thanthe others. The best known of the three Persian manuscripts is kept in the NationalLibrary in Paris, France. Although clearly directed at artist/craftsmen, the generaltone of this work is of a somewhat petulant criticism of their reluctance to adopt“proper” (i.e., Euclidean) methods of geometrical construction.

The On interlocking similar or congruent figures (Fî tadâkhul al-ashkâl al-mutashâbiha aw al-mutawâfiqa) is a geometric manual by an unknown author,probably originating in Tabriz during the Ilkhanid period. There seems to be as

16 B. Wichmann and D. Wade

much interest here in the “puzzle” aspect of the figures involved as in their use indecorative ornament, indicating that it may not have been directed exclusively atartist/craftsmen.

There are a few other, even less specific, allusions to the involvement ofmathematicians with decorative ornamental schemes. The polymath/inventor Al-Jazari drew and described an elaborate plan for a door in his book on IngeniousMechanical Devices (although the author George Saliba has pointed out that thetranslation on which this is based may be faulty). And there is a tantalizing referenceby Omar Khayyam (no less) to Meetings of Artisans and Geometers – but thisallusion too is vague and open to misinterpretation. In summary, it is difficult on thebasis of this and other available evidence to make any substantial claims whatsoeverfor the notion of sustained academic involvement in the genre of Islamic geometricornamentation in any period.

Conclusion of Historical Perspective

By beauty of shape I want you here to understand not what the multitude understands by thisexpression, like the beauty of living things or of paintings resembling them, but somethingalternatively rectilinear and circular, and the surface and solids which one can produce fromthe rectilinear and circular, with compass, set-square and rule. Because these things are not,like the others, conditionally beautiful, but are beautiful in themselves.

Plato, from the dialogue Philebus

One of the more interesting aspects of the geometric mode in Islamic design,deriving from years of familiarity with this genre, is precisely that there is no single“Key to Islamic Pattern” but rather a whole variety of ways of achieving geometriccomplexity. Regional variations have in fact resulted in a whole range of distinctmodes and families of pattern and symmetry types, not to mention differing attitudestoward geometric “purity.” This is perhaps the most fascinating aspect of this artbecause it seems to beg the question, “What, exactly, are the constant criteria inthis broad aesthetic tradition?”. That is to say, what was it that kept the Islamicpreoccupation with geometric pattern going as an essential artistic expression forsuch a long period, across the whole range of craft productions, over such a vastgeographical area?

The proposition here is that the Islamic dedication to both symmetrical andgeometrical forms of ornament, both derived ultimately from the aesthetic idealsof a particular Classical philosophical tradition, namely, that of Platonism/Neo-Platonism, which had been thoroughly absorbed into mainstream Islamic thoughtat a very early stage of its cultural evolution, but which was later the subject ofreligious opposition. In short, we claim that Islamic geometricism, with its delightin the endless play of abstract geometrical shapes on the Euclidean plane, is anessentially Platonic art form.

We are conscious that, from an academic perspective, this assertion could beviewed as hopelessly generalized. It suffers as much from an insufficiency ofcontinuant examples of geometric design as it does from a lack of documentary

Islamic Design and Its Relation to Mathematics 17

Fig. 3 Symmetries present in the Alhambra

Fig. 4 Kasehgaran Madrasah, Isfahan (Al-Naqsh, 1983), *5• (d5) symmetry, data181/F111. Forthis pattern reference and subsequent ones, see (TI, 2019)

evidence – however, all investigations into this subject are confronted by these sameproblems. As indicated above, there is little real evidence for the involvement ofacademic mathematicians in the elaboration of Islamic geometric art (which doesn’tmean that there was none), and the notion of the primacy of doctrinal influences(as expressed in the “Sunni Revival” theory) is at the very least questionable.

18 B. Wichmann and D. Wade

Fig. 5 Royal Palace, Fez, Paccard (1980), data203/S32A

Nevertheless, as stated at the beginning of these notes, there must be a compellingreason for the adherence to strong symmetrical arrangement and highly geometricmodes of decoration in the art and architecture of the Islamic world. This aesthetictendency seems to have been fastened on at an early stage of Islamic culturalconfidence and was indeed an expression of it. And once this particular aesthetictrajectory had been settled upon, in which highly symmetrical decorative forms hadbecome a primary means of cultural expression, it was natural that the inheritedlate-Classical decorative elements would be embellished, developed, and made evermore complex over the generations.

Modern Mathematical Analysis

In the late nineteenth century, Fedorov, Schoenflies, and Barlow classified the 17wallpaper groups (two-dimensional crystallography groups) and the 230 three-dimensional crystallography groups. For the authoritative modern analysis, seeGrünbaum and Shephard (1987). A question that has been asked several times is

Islamic Design and Its Relation to Mathematics 19

how many of the repeating types appears in the Alhambra, one of the greatestof Islamic geometric sites. These are Müller (1944), Jaworski (2006), Montesinos(1985), Grünbaum (2006), and Du Sautoy (2008). In our case, we only considertiling patterns with straight edges whose repetition is clear. This gives Fig. 3.

This implies that our strict criteria only have 8 of the 17 cases, while Du Sautoyhas all 17. For the frequency of the symmetry groups over all the Islamic world, seetable (Wichmann and Wade, 2017, Page 208). This table shows a very nonuniformdistribution with two symmetries having no known Islamic example: ×× (pg) and22× (pgg).

Actual buildings have a patterned area which can usually be presented as a repeatpattern of the types given above. Some patterns are more faithfully displayed withcircular symmetry, as in Fig. 4. See also Wichmann and Wade (2017, Page 207,Fig, A31). This analysis was first undertaken by Bourgoin; see Bourgoin (1873,1879). Interest in this area was partly due the work of Owen Jones (Jones, 1856;Jones and Goury, 1837).

Another modern analysis is the constructibility of patterns by ruler and compassalone. Gauss showed that a regular 17-sided polygon is constructible (none appearto exist in Islamic art), while a 9-sided regular polygon is not constructible (at leastthree examples in Islamic art); see Jagy (2000).

The analysis of the line drawing of Islamic patterns can be straightforward inmany cases. For instance, in Wichmann and Wade (2017, Chapter 8), a patternfrom the Alhambra is shown to be completely determined which implies that any

a b

Fig. 6 Decagonal patterns. (a) Two star types. (b) Star and diamonds

20 B. Wichmann and D. Wade

divergence from the mathematical formulation is an error. For analysis of starpatterns, see Lee (1987) and Lee and Soliman (2014).

One property of many Islamic patterns is the repeated use of specific tile shapes.In the case of ceramic tiles, this could be achieved by a copying process. A commonset of tiles are those that include a khatem, the 8-pointed star-polygon-shaped tilewith a vertex angle of 90◦. Such a set of tiles is given in Wichmann and Wade (2017,Fig. 9.10).

An important feature of many tiling patterns is a large central star. In Morocco,a good example is shown in Fig. 5. The black band around the central star canbe seen to have six-sided tiles of varying sizes, which is a consequence of themathematical analysis in this computer drawing. A hand-produced pattern wouldtend to be adjusted to avoid such problems. Stars with 16 points have only smalldiscrepancies in this area, but for larger stars, the situation is visually clear. For thedetails, see Wichmann and Wade (2017, Chapter 10).

An important class of tiling patterns are those whose angles are always multiplesof 36◦ (Wichmann and Rigby, 2009). This angle ensures that a 10-pointed star

Fig. 7 Fez, data194/FEZ1

Islamic Design and Its Relation to Mathematics 21

polygon with angles of 72 or 108 degrees are allowed. The variety of such patternsis huge which is well represented by mosques in Turkey.

In Fig. 6, we illustrate the essential features of this class of patterns. In Figure(a), the two main stars are shown, both 10-pointed. Figure (b) is a common designusing diamonds. Both patterns have a 10-pointed star surrounded by kites and thenblack petals – an example of a rosette. Note that the size of the diamonds can varyby means of adjusting the distance between the rosettes.

Another class of patterns are those with sixfold symmetry: *632 (p6m) or 632(p6). Figure 7 shows a 24-pointed star, but such sixfold patterns may well have6-pointed or 12-pointed stars.

For two further examples, see Fig. 8. Figure (a) has a hexagon rather than a star,but the unusual feature is the three-way interlacing. This could be avoided (as insome variants) but introducing a small triangle. Figure (b) is an early pattern fromthe Eastern tomb tower at Kharraqan in Iran. The coloring used here is from Turkey.

The final class of pattern are those for which a mathematical analysis is eitherdifficult or so time consuming that it would not be worth undertaking (since it wouldnot clarify the design). An example of such a pattern appears in the first part ofWichmann and Wade (2017, Chapter 16) where a tricky analysis is undertaken andthe alternative by a manual construction is given. The existence of this class is hardlysurprising since the original artists were designing for an effect and were not usingmathematics.

a b

Fig. 8 Two examples of sixfold patterns patterns. (a) Hexagons and interlacing. (b) Early example

22 B. Wichmann and D. Wade

Date Frequency Site

1000

1200

1400

1600

Cairo

Isfahan

Central Asia

Turkey/Iraq

Iran

Alhambra

Cairo

Fatehpur Sikri/Morocco

Iran

Fig. 9 Date analysis

Although it is clear the Islamic world communicated their ideas of design, thevarious areas had a distinctive style. This is summarized in Wichmann and Wade(2017, Fig. 17.4). Another analysis that has been undertaken is to produce datesfor the various designs so that a histogram can be produced. This was produced inWichmann and Wade (2017, Fig. 17.3) but is revised here due to the availability ofmore data.

Islamic Design and Its Relation to Mathematics 23

Fig. 10 Part showing theborder to the central star,data0/S48P1

Figure 9 shows the analysis of the date when a pattern first appeared plottedagainst frequency. The text on the right shows the area or site which contributedmost to the larger frequency shown in the histogram. The large number of patternsfrom the Alhambra and Fatehpur Sikri is very clear. It is thought that the readyavailability of paper was helpful for the dissemination of patterns, Bloom (2001).

Computer Usage

The attraction to drawing one’s own Islamic-style patterns using modern graphictools on a computer is clear (Wichmann, 2001). Hence, many sites of such materialare available on the Internet.

Three books also consider this issue with more care than many Internet sites.In Castéra (1999), the author considers only patterns from Morocco. This providesbeautiful and colorful patterns which are drawn to perfection. One issue which needscare is the boundary of a large star which is analyzed mathematically in Wichmannand Wade (2017, Chapter 10) as noted above. Consider part of an elaborate pattern inthe Paris mosque which was produced by Moroccan craftsmen, in Fig. 10. The tilesin the majority of this figure are based on Khatems with related shapes. However,one of the brown stars has seven points in order to fit round the 48-pointed star. Incontrast, the 16-pointed stars at the bottom of this figure provided no difficulty.

Another tricky design which would be virtually impossible to analyze mathemat-ically is from the Regent’s Park mosque in London. Here the 7-pointed stars are anintegral part of the design rather a means of overcoming a potential problem; seeFig. 11.

24 B. Wichmann and D. Wade

Fig. 11 Regent’s Park mosque, border under dome, data0/REG1

The second book to consider is Broug (2013). The main text includes someexcellent photos, many of which are not easily found. The text has small drawingsgiving clue about producing the complete pattern on the computer. The appendixconsists only of drawings, mainly produced using the technique described in Broug(2008). Some of the drawings appear to be in Islamic style rather than based uponactual artifacts. The drawing method does not always produce a pattern in theproportions used in Islamic examples.

The third book, Bonner (2016), aims to be comprehensive and appears to havethe largest collection of Islamic geometric patterns available in book format. Oddly,the book omits lattice patterns; see http://www.tilingsearch.org/special/lattice.html.About 100 photos are provided, but many more have clearly been used to producethe numerous diagrams. The source of such patterns is given when it is not aninvention, although the source of many photos is not specified. A concluding chapteris provided by Craig Kaplan concerned with algorithms for producing Islamic

Islamic Design and Its Relation to Mathematics 25

patterns. The book is an impressive collection which will remain a key source ofpatterns for many years to come.

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