Islamic Art and the Argument from Academic Geometry

by Terry Allen

published by Solipsist Press, Occidental, California, 2004

Copyright © 2004 by Terry Allen

An electronic publication     ISBN 0-944940-06-4

Table of Contents

Geometry's Connection with the Arabesque
Philosophy and the Argument from Academic Geometry
Necipoğlu's Argument
Neo-Platonism and Geometry in Baghdad: an Anecdote
The `Abbâsid Translation Movement
The Supposed Spread of Geometric Design
Late Antiquity and Early Islamic Geometric Design
Complexity in Early Islamic Geometric Design
Increase in Complexity
Novel Radial Symmetry
Use of Approximations
Patterns of Diffusion
Spirituality and Art
A Muqarnas in Constantinople
Abbot Suger, Neo-Platonism, and Anagogical Phenomenology
A. The Practicalities of Applied Geometry
Railroads and Skew Vaults
Georgian Builders
Laying out Octagons
Crop Circles

Geometry's Connection with the Arabesque

The great error lies in supposing that even the truths of what is called pure algebra, are abstract or general truths. And this error is so egregious that I am confounded at the universality with which it has been received. Mathematical axioms are not axioms of general truth.

— Edgar Allan Poe, “The Purloined Letter”

In connection with my study of the origins of the arabesque, by which I mean the geometrized vine scroll that arose apparently in tenth-century Baghdad,1 I have been interested in the cultural context of its origin, which leads naturally to an interest in the cultural context of geometry at the same time and place. In this article I shall explore certain work to date, add a little of my own, and suggest the most fruitful direction to pursue regarding geometry in Islamic art and architecture.

Philosophy and the Argument from Academic Geometry

The scholarship on the cultural context of geometry that is most directly relevant to the arabesque links the development of increasingly complex geometric designs with an intellectual interest in the science of geometry on the part of artisans, architects, geometers, and others. I call this the argument from academic geometry.

Perhaps the fullest exposition of the argument from academic geometry is found in Gülru Necipoğlu's book, The Topkapı Scroll—Geometry and Ornament in Islamic Architecture: Topkapı Palace Museum Library MS H. 1956. 2 While the topic of this study is a scroll of geometric designs from somewhere in the Persian-speaking world, dating to the late fifteenth or early sixteenth century A.D., which is not relevant to this discussion, inter alia, Necipoğlu surveys the literature on Islamic geometric ornament, and collects and extends the interpretion I am examining.

I believe Necipoğlu confuses her argument by adopting a very wide and archaic definition of the arabesque,3 thus conflating the geometric vegetal ornament usually (and here) called the arabesque with the quite different geometric designs she studies. This confusion may be set aside in the current context, as it does not affect her exposition of the argument from academic geometry.

Necipoğlu's Argument

In this section I outline Necipoğlu's argument, with quotations and paraphrases of supporting material from her text. I have had to supply some of the argument in plain words: although Necipoğlu is generally quite clear, at many points crucial to this discussion her writing turns vague. The argument is summarized at the end of the section.

In summary, then, Necipoğlu's argument regarding the practical and intellectual background of the development of geometric decoration (and calligraphy, and the arabesque) in Islamic art is composed to support four claims:

Of these claims, the first is broadly likely. By reading or discussion, master artisans should have had access to formal geometry in tenth-century Baghdad, as they did later.25 But we have no reason to suppose that such discussion sessions were devoted to the design of new geometric ornament. They may rather have provided orientation to geometry and solutions for various problems occurring in the construction of existing schemes of geometric ornament—or problems not concerning decoration at all.

The third claim, about the supposed appreciation of art for spiritual reasons, and fourth claim, about the diffusion of the new class of geometric design, I shall take up later.

The second claim raises questions about the roles of various participants in the development of geometric ornament.

Neo-Platonism and Geometry in Baghdad: an Anecdote

Before exploring the importance of roles, I think it useful to introduce an interesting and perhaps revealing anecdote, from or recalling the period when geometry was being popularized, that purports to contain an exchange of letters describing the introduction of geometry in a literary gathering. It is a fragment attributed to Ahmad b. al-Tayyib al-Sarakhsî (d. 899) by Abû Hayyân al-Tauhîdî (ca. 310–after 400/922–after 1009) in his Kitâb al-wazîrayn and quoted by Shihâb al-Dîn Yâqût, the geographer, (1179?–1229) in his Irshâd al-arîb, or Dictionary of Learned Men. It is evidently a fiction and a parody, which Franz Rosenthal, who translated it, thought ridiculed “narrow-minded orthodoxy.” I think it also, though incidentally, ridicules unbounded admiration of geometry, and thus gives us the obverse of the philosophical viewpoints discussed by Necipoğlu, something in the manner of treatises against heresies. I have relied on Rosenthal's translation, which I abridge, slightly repunctuated.26

A friend of the Secretary Abû l-`Abbâs [Ahmad b. Muhammad] b. Thawâbah27 by the name of Abû `Ubaydah, once said to Ibn Thawâbah,

“You are, thank God, a man of high culture and eloquence. How about rounding out your great qualities by adding to them knowledge of the method of syllogistic proof and of geometrical figures which lead to the knowledge how things really are? You ought to read and interpet Euclid.” Here Ibn Thawâbah asked, “What and who is Euclid?” and Abû `Ubaydah explained that he was one of the Greek scholars who was called by this name. Euclid (he continued) was the author of a book which contains many different figures leading to the knowledge of how things, both known and hidden, really are. This books sharpens the mind, refines understanding, and gives thorough knowledge. It clears the sensory perception and establishes intellectual insight. Handwriting was developed from it, and the qualities of the letters of the alphabet were recognized. Abû l-`Abbâs b. Tawâbah asked how this was possible, and Abû `Ubaydah answered that he [Ibn Thawâbah] could not know how it was possible unless he saw the figures and looked at the syllogistic proof. Whereupon he (Ibn Thawâbah) said, “Do what seems best to you.” And Abû `Ubaydah brought him a well known scholar by the name of Qwyry [Rosenthal: perhaps Cyrus], but he never again returned.

Ahmad b. al-Tayyib said, “I found that humorous and astonishing. Therefore I wrote to Ibn Thawâbah a letter of the following contents [inquiring about the affair; omitted here].

Ibn Thawâbah's reply is the heart of the parody, running five and a half printed pages. In it he says that

Abû `Ubaydah … tried to mislead me. He intended to do harm to my religion without my knowledge, to divert me from my firm belief and faith in God … and in His prophet …, and, in his bad intention, to deliver me over to heresy by means of geometry.

He continues on, recounting visits to his salon from two exponents of geometry. The first is evidently the aforementioned Cyrus.

[Abû `Ubaydah] came with a shaykh, a monk, with staring look, with enlarged nerve of the eye, as tall and thin as a string bean.… I said, “I understand that you know something of geomtery and that you possess a knowledge which is so excellent that the man who studies it is enables to attain wisdom and progress in every science ….”

There ensues a comical exchange in which Cyrus attempts (poorly) to explain the geometrical concept of a point of infinitesimal dimension and his host completely fails to understand him, becoming irascible and bombastically pietistic, at length accusing him of heresy. Ibn Thawâbah has him thrown out, but others present defend him.

So Ibn Thawâbah has another man produced, a Muslim brought by one of the attendees at his salon.

He brought a man who was very short, with a dark brown complexion, swollen face, and feeble eyes. He had no hair at the temples, and he had a flat nose, a bad appearance, and an ugly attire … .

After some discourse about Cyrus and geometrical points, the visitor, one Abû Yahyâ, perceives that Cyrus failed to explain himself intelligbly. Abû Yahyâ then has a strangely shaped “box” (evidently a tablet covered with wax or the like) brought and produces from his sleeve a “large probe” (evidently a scribe). In answer to a pious objection, he says, “I draw geometrical figures upon it [the tablet] and establish proofs upon it by means of syllogism and philosophy.” He proceeds to draw a line and tells Ibn Thawâbah that it has no width, at which Ibn Thawâbah flies off the handle again, accuses Abû Yahyâ of attempting to lead him into religious error, and has him thrown out. Then, says Ibn Thawâbah,

I took a piece of paper and wrote down an oath with my own hand. In it I swore with all strong and repeated expressions of binding obligations and with an oath for which, when broken, there is no expiation, that I would never study or investigate or learn geometry from anyone, whether secretly or openly, in any way or under any conditions. I took a similar oath on behalf of my offspring and the offspring of my offspring.

Yâqût remarks that this story is probably fictitious, but adds, “But perhaps Ibn Thawâbah said something similar to what was said by Ibn `Abbâd [decades later], who was the reason for Abû Hayyân's mentioning Ibn Thawâbah. Abû Hayyân said about Ibn `Abbâd that he used to abuse geometricians” and to react similarly to Ibn Thawâbah.

What can we learn from this passage?

As for its authorship and date, Rosenthal thought that in style it might be attributable to al-Tauhîdî. On the other hand, one might speculate that Al-Tauhîdî, a contemporary of Ibn `Abbâd, composed it himself and attributed it to the earlier writer, placing it in an earlier period to avoid being charged with criticizing his powerful contemporary, who was an enemy;28 Abu'l-Qâsim Ismâ`îl b. `Abbâd (326–85/938–95), a famous and powerful Bûyid wazir, was also a patron of literature, and just the sort of figure who might have been influential in the patronage of the arts.29 The remark in the passage that handwriting was developed from Euclid suggests a date of composition after Ibn Muqla's (272–328/885–940) invention of proportioned script—too late for al-Sarakhsî.

On the other hand, Ibn Thawâbah was also a notable patron and the portrait of him given in the passage is apt. S. Boustany, author of the Encyclopaedia of Islam article on him, writes, “Ibn Thawâbah presided over a circle in which a number of poets and men of letters met regularly. His generosity, sometimes ostentatious, led some poets of his time (such as al-Buhturî and al-Rûmî) to write of him very elegant panegyrics, which still survive. But the disagreements which he had with some of them, and notably with Ibn al-Rûmî, earned him a series of epigrams full of irony and persiflage. Some writers of the following centuries, and notably al-Tawhîdî, retained the image of him which is given in these satires and present him in some of their anecdotes as a grotesque, narrow, and pretentious bore.” The passage, then, may have been written primarily to paint a critical picture of the character of Ibn Thawâbah, with geometry of only secondary interest; still, read with caution, the views of geometry held by the participants may be enlightening.

We can imagine, then, that at a top salon of ninth- and tenth-century Baghdad there were litterateurs who knew of and perhaps even understood geometry, but also those who did not. Both of the visitors to Ibn Thawâbah's salon are described as ill-proportioned and ugly, as (implicit) satirical references to neo-Platonic notions about the earthly reflection of beautiful heavenly prototypes, and both are so mired in an academic approach to geometry that they cannot explain it to a skeptical audience. They are not to be taken as real. But the claims that geometry leads to knowledge of “how things really are,” that it “sharpens the mind” and “establishes intellectual insight” are of the period, as Necipoğlu's sources show. This passage shows geometry being promoted as a universal aid to true knowledge at such a top salon.

It is worth noting, though, that both visitors to Ibn Thawâbah's salon were brought in by regulars who were not capable of explaining Euclid on their own, and that the visitors make no explicit mention of neo-Platonic emanations; the claim that geometry leads to the knowledge of how things really are is made by the instigator of the interviews, Abû `Ubaydah, who connects it with “syllogistic proofs and philosophy.” That is, there is nothing in this parody of over-the-top claims regarding geometry about its use in design aside from the reference to the proportioned script, and no indication that the regulars at the salon would regard it as anything other than an intellectual matter that a well educated man should know something about—to “round out [his] great qualities.” There is certainly no indication that neo-Platonism a topic of discussion among this audience.


We need to be able to imagine how real people participated in the development of geometry in art and architecture in tenth-century Baghdad. Given the nature of our literary evidence, it is reasonable to consider not particular individuals but groups playing particular roles. We need to define the possible roles of:

Necipoğlu identifies theoretical geometricians who may have been involved in the development of star-and-polygon designs, and claims that they supervised the “formulation” of “the girih mode,” but that assertion stops well short of specifying the respective roles of artisans and geometers. In arguing that star-and-polygon designs spread in the eleventh century among Sunnî “local courts” at which even some rulers, such as Mas`ûd I the Ghaznavid, were skilled in geometry (or wished to be thought so), she presumably intends Sunnî dynasts to have been the patrons of artisans working in “the girih mode.” But these men lived later than the crucial time, and by Necipoğlu's own argument were following a trend in taste already developed earlier in Baghdad. Özdural's study of the anonymous Fî tadâkhul al-ashkâl al-mutashâbiha au al-mutawâfiqa, makes it clear that the discussions between geometers and artisans were focussed on construction geometric figures, not getometric patterns.

Similarly, the artisans who are said to have interacted—somehow—with the Baghdad geometers are anonymous. While it is true that the recording of names and biographical information of artisans and architects varied widely by place and period in the premodern Islamic world, the tenth century is not entirely obscure.30 The evidence of the encyclopaedias, which do not name artisans, shows only that encyclopaedists felt that practice should be governed by theory, not that it was—and there is no visual evidence whatsoever that practice was governed by theory instead of practicality.

We know the identities of some of the patrons and customers of artisans and architects, such as the caliphs and their top military commanders, who are named in the sources. But these men may have relied upon artistic advisors we are ignorant of. There is scope for more detailed research on the topic. As for the audience of tenth-century `Abbâsid art, we can only assume that it was composed of the social and business circles of the patrons.

The `Abbâsid Translation Movement

Given the clear existence of emanationist thought in tenth-century Baghdad but the difficulty of placing it in a milieu that includes patrons and artisans, it is essential to understand how emantionist texts arrived in Islamic culture in the first place. This process has been convincingly clarified by Dmitri Gutas, who identifies a “Graeco-Arabic translation movement” sponsored by the early `Abbâsid caliphs, in particular al-Mansûr, as part of a strategy of coöpting or appropriating an aspect of Zoroastrianism that had been championed by the Sâsânians.31 The `Abbâsids were motivated to do so in order to cut the ground from under various revolutionary causes in the former Sâsânian empire, appeal to Persians of whatever religion, and stake a panreligious claim to universal kingship.32

The early `Abbâsid caliphs tried to legitimize the rule of their dynasty in the eyes of all factions in their empire … by promulgating the view that the `Abbâsid dynasty, in addition to being the descendants of both Sunnî and Shî`î Muslims, was at the same time the successor of the ancient imperial dynasites in `Irâq and Iran, from the Babylonians through the Sasanians, their immediate predecessors.33

The Zoroastrian view of the “origins and transmission of knowledge and the sciences,” Gutas argues on the basis of several texts (and much further solid evidence), was as follows:

Zoroaster received from Ohrmazd the Good God the texts of the Avesta, which include all knowledge. The destruction wrought upon Persia by Alexander the great, however, caused these texts to be dispersed throughout the world. The Greeks and the Egyptians derived their knowledge from these Zoroastrian texts which Alexander had translated into Greek and Coptic. Subsequently Sasanian emperors took it upon themselves to collect all these texts and the knowledge that was derived from them from the various places where they had been scattered …. Then Chosroes I Anûširwân promulgated all these texts, which collectively form the Zoroastrian religion, and decreed that they be studied and discussed for the benefit of mankind. 34

Gutas shows that there was in fact a significant translation movement under the Sâsânians, and that it continued after the Arab conquest, preserving knowledge by translating Pahlavi texts into New Persian and later from New Persian into Arabic.35

The essential points here are that the `Abbâsid translation movement was not a new invention in an intellectual vacuum, that it had a clear religiopolitical purpose entirely aside from the intellectual content of what was translated, that it did not originate in Harran or among Harranians in Baghdad, and that in fact it did not at first involve translation from Greek sources at all.36 Above all, it arose centuries before the development of more complex geometry in art. The earliest `Abbâsid translations are “overwhelmingly astrological in nature,” and the importance of astrology is shown in the background to the foundation of Baghdad, which also has its geometrical aspect.37 Other early `Abbâsid uses of geometry in apparently astrological contexts include Hiraqlah near Raqqah,38 which of course did not involve any geometrical means of construction not known since Antiquity, and was conceived and built before the rise of academic geometry as a topic of study by the Arabs.

According to Gutas, it was also for the study of astrology that Pahlavi and New Persian sources were first depleted, and recourse was had to Greek texts, and it was also in the field of astrology that Arabic contributions to Antique sciences were first made. Demand led to supply.

With the demand [for astrological source material] unabated, there eventually developed an extensive astrological literature in Arabic of pseudepigraphic and anonymous works. The pattern that was set by astrology was to be repeated, in general terms, with all the other translated sciences. Political considerations, ideological or theoretical orientations, or practical need would initially occasion translations, their study and use would result in original Arabic compositions in that particular field, and the development of research on the particular subject in this way would further generate a need both for more accurate translations of texts already available and for translations of new texts.39

Thus the “translation movement” had its own motors, but in any given field, it was the uptake of the initially translated material that led to further and better translations.

It is important to emphasize this point. It was the development of an Arabic scientific and philosophical tradition that generated the wholesale [as opposed to initial] demand for translations from the Greek (and Syriac and Pahlavi), not, as is commonly assumed, the translations which gave rise to science and philosophy.40

Now it is fairly clear that the Ibn Thawâbah anecdote is meant to point up Ibn Thawâbah and his circle as philosophically unsophisticated, for the entertainment of other circles in which neo-Platonism was appreciated or at least known. Gutas points out that “if, as Yâqût claims, as-Sarahsî concocted the correspondence [quoted in the above anecdote] in order to please al-Mu`tadid, this means that the caliph had an appreciation of the foreign sciences.”41 These circles, and those with which al-Fârâbî and the members of the Ikhwân al-Safâ' associated (both Shî`î, incidentally), were interested in geometry for philosophical reasons (and covertly, for some, religious reasons). Necipoğlu's survey shows us three other clearly distinct reasons to be interested in geometry: the need to carry out practical administrative tasks, such as measuring land or estimating the volume of brick required for a building; the desire to understand the intellectual and scientific basis of mathematics and contribute to its development; and the practical need to construct geometric designs more complexly, more accurately, or more quickly. All these four motives may have and probably did overlap with some of the others, not least in connection with actual translations or those who could explain them; in the case of artisans this would have been the geometers.

What this diversity of motive for understanding geometry means is that Islamic interest in geometry was not one single thing. We cannot simply lump together every manifestation of interest in geometry, from the Ikhwân al-Safâ' to decorated buildings, and generalize across the whole set of data. And this feature of the evidence means that the academic study of geometry, the practical interests of administrators, or the speculations of the philosophically inclined need not have had anything to do with the development of star-and-polygon designs in art and architecture, for we have yet to find a nexus of appreciation of geometry, neo-Platonism, and the application of academic geometry to design. Nor should we expect to: the association of geometry and neo-Platonic ideas that Necipoğlu's argument depends on occurred well before Muhammad's lifetime, yet produced no star-and-polygon decoration.42 Indeed, if patrons had encouraged the application of emanationist philosophy to art and architecture that Necipoğlu argues for, one would expect to find in the Arabic sources attacks on them for irreligious practices.43

While we cannot rule out patrons, such as the high-ranking bureaucrats who figure in the passage about Ibn Thawâbah, or indeed the `Abbâsid caliphs themselves, as motive forces in the increasing use of geometric designs in art and architecture, we have to ask whether they demanded it (and if so, why) or were only pleased by it when they saw it. In the feedback loop between artisan and patron that produces trends in taste, it may be impossible to tell. But there is no evidence that any patron, or for that matter any theoretical geometer, urged any artisan to use geometry to a greater extent, or to use more complex designs. The artisans who are said to have met with geometers may have come to such meetings to learn how to construct geometric designs, but, for example, in Abû Bakr al-Khalîl al-Tâjir al-Rasadî, A`mâl,44 it is clear that the “master craftsmen (ustâdân)” had come to the meeting with their problems already in hand: the geometers were responding to, rather than creating, demand. At the other end of the spectrum of roles, the anecdote concerning Ibn Thawâbah shows academic geometry and neo-Platonic and perhaps emanationist ideas being discussed but not being linked with geometric design in art or architecture. (The mirror image of the scenes in the anecdote would be the telling of the anecdote in an appreciative setting, but here too there is no connection with art or architecture.)

If emanationism is not a plausible motive for the increase in complexity of geometric design, perhaps another notion connected with neo-Platonism is: the idea that, as the Ikhwân al-Safâ' wrote, ratio and proportion (which are of course the same) underlie all good craft and art. The Ikhwân al-Safâ' gave as examples melodic sounds in music, prosody in poetry, letters in proportioned script, and harmonious colors and proportionally joined figures in painting and mechanical devices.45 Thus proportionality would have been introduced into art, perhaps at the behest of a patron, where it was lacking before in order either to produce a better effect or to indicate (by its obviousness) to the viewer that a better effect existed and that the viewer ought to strive to appreciate it. Unfortunately this line of argument by the neo-Platonists is circular: only beautiful things are thought to be well proportioned, and no rules of proportion (except perhaps for the script) or for obtaining harmonious color combinations are provided. One may imagine that this aesthetics based on inherited neo-Platonic thought was believed in by artisans and patrons, but as it would have been inadequate to guide their design processes and experience of art, there is no reason to think that it was other than an intellectual construction.

The earliest examples of “the girih mode” that Necipoğlu produces are clearly drawn from the Late Antique heritage (see below). It is notable that neither the Late Antique designs nor the later star-and-polygon patterns share anything with the works of the supposedly geometrically and neo-Platonically inclined Sabians, whose citadel at Harran invoked geometry by the device of towers with odd numbers of sides, not by geometric ornament, nor with manifestations of neo-Platonism in Late Antique Christianity.46 So the argument that geometry in art springs from emanationist yearnings fails to add up.

The straightforward explanation for the development of complex geometrical decoration in Islamic art is that:

It is perfectly reasonable to think that no one expected the resulting works to be effective in some emanationist manner; that theoretical geometers did not participate in the feedback loop as instigators; and that the artisans may well have been entirely ignorant of neo-Platonist philosophy and to some degree unappreciative of the academic rather than practical solutions prescribed by the geometers. Neither patron nor geometer ever said to an artisan, “Ahmad, I like that design, but if you made it more geometrical it would help me hear the music of the spheres better (or feel closer to Allâh).” This imaginary remark may seem tendentious, but if patrons and artists communicated about geometric design, we must imagine how they did so. If it is too much to ask for literary evidence of such exchanges, we should at least expect literary articulations of the underlying ideas. But no specific building or work of art is mentioned in connection with increased use of new geometric designs. No literary source I know of says, “When I gaze upon the decoration of the interior of the palace of the Caliph, I imagine that I hear the music of the spheres better (or feel closer to Allâh).”

Nor is there any parallel in the other arts for philosophical theory governing practice: the structure of Arabic poetry does not derive from philosophy, nor is there any indication that, despite the neo-Platonic enthusiam for the “music of the spheres,” philosophy affected music at all.47

Necipoğlu's argument puts the cart before the horse. The evocation of a transcendental experience of heaven was not the place of art, nor of artisans. As I remarked in Five Essays, art does not seem to have been nearly as important as literature to the educated classes of the tenth and eleventh centuries. Even the most worldly figures in the cultural life of tenth-century Baghdad are not recorded as having close relationships with artists.48 While Necipoğlu says that geometry “elevated the status of architecture and the crafts by giving them a respectable scientific foundation,”49 such elevation appears to have occurred in the hierarchies of encyclopaedias rather than in the real world of social relationships. If architects and artisans enjoyed a significant status in this age one would expect there to have survived much more biographical data on architects and artisans than is extant.

I conclude that the argument from academic geometry fails to provide an explanation for the increasing use of geometry in Islamic art and in particular for the geometrization of the vine scroll as the arabesque; admirable as it may seem as an example of hypothesis formation, it is unfounded in real life and unsupported by evidence. In fact, the argument from academic geometry is a reflection or complication of the wrongheaded notion that Islam must have a religious art, if only we could find it: it casts geometry in a quasireligious role and attributes artistic developments to spiritual abstractions. This is not how art happens. The enthusiasm for geometry evident in the art of the tenth century and later must have grown not from book learning but from an appreciation of designs that in fact were derived from Late Antique prototypes.

The Supposed Spread of Geometric Design

Necipoğlu's fourth claim, that star-and-polygon designs spread in the eleventh century among Sunnî “local courts” following the spread of interest in academic geometry, is actually two claims. One is that there is a clear and significant distinction between early Islamic geometric design and the star-and-polygon designs found in the album which is her proper concern. The other is that star-and-polygon designs did in fact spread first to cities governed by Sunnî rulers.

Late Antiquity and Early Islamic Geometric Design

The art and architecture of the seventh and eighth centuries shows that both the Arab and Persian halves of the new Islamic empire were suffused with geometric designs, often of considerable complexity, drawn from Late Antique sources. For datable monuments of sufficient complexity we must look primarily to architecture. The “interlaced geometric patterns” found in window grilles, floor mosaics, and wooden ceilings are dismissed by Necipoğlu as “hardly a prominent feature of early Islamic architectural decoration before the late tenth century.”50 This is not so. These elements are visually strong parts of architectural compositions, just as they had been in Late Antiquity. I shall cite a few prominent examples.

The window grilles of such monuments as Qasr al-Hayr al-Gharbî and the Great Mosque of Damascus are eye-catching and possessed of considerable subtlety: for example, they include octal and sextal radial symmetries, multiple levels of pattern, and studies in fitting the pattern to the opening as well as cropping it.51

Khirbat al-Minya has some dazzling wall-to-wall mosaic floors, as of course does the bath at Khirbat al-Mafjar. One floor at Khirbat al-Minya has a design constructed on an unacknowledged net of construction lines of sextal radial symmetry 52

I cannot think of a full early wooden ceiling with a geometric design, although there must have been such. We have essentially no evidence for wooden ceilings from the Umayyad and Abbasid palaces. The fragments of woodwork from the Aqsâ Mosque include some geometric patterns of the sort found in floor mosaics, though they do not predominate. Where wood was available in sufficient quantity, it may well have been used to create coffered ceilings of the sort known in Late Antiquity, as these were part of the inherited suite of types of architectural decoration that included the window grilles and the mosaic floors.53

Other parts of buildings boasted loud geometric or geometrized ornament in the early Islamic period, such as a group of columns in the prayer hall of the Great Mosque of Wâsit (a device that must have been common, as decorated columns appear in early Persian mosques). The walls of the entry hall to the palace at Khirbat al-Mafjar, as well as those of the apse of the throne room in the bath and other areas, were covered with carved stucco in patterns drawn in part from the mosaic floor repertoire, and the stucco items identified as “balustrade panels” even include two examples of pental radial symmetry, although neither is extended beyond a single pentagram. One of these panels is unfinished and shows the net of construction lines used to lay out the pattern.54 The entry to the palace at Qasr al-Hayr al-Gharbî exhibits large panels of geometric interlace. 55

These are not isolated examples, or elements of decoration easily overlooked. They testify to a love of geometric design as part of an overall scheme of rich ornament—from so early a date that it is clear the Umayyads were attempting to carry on Late Antique practices of luxurious decoration.

In the ninth century, the stucco of Samarra, of course, similarly includes designs formed on a geometric basis, and such designs also appear in the stucco of the Mosque of Ibn Tûlûn in Cairo, a reflection of Iraqi style, and in the imported woodwork and stonework of the Great Mosque of Qayrawân, though less prominently. The geometric heritage of Late Antiquity was persistent and as widespread as there is evidence for the decoration of the earliest Islamic architecture.

The first monuments that Necipoğlu will admit as having decoration in “the girih mode” are in the Persian-speaking east.

The earliest formulation of [the girih mode's] design principles probably took place in late tenth-century Baghdad from which it spread to other courts [sic]. The appearance of predominanatly geometric revetments in Islamic architecture is usually traced to the tenth-century “brick style” of northern-northeastern Iran and Central Asia, from where it is thought to have migrated westward [no sources are cited for this assertion]. However, this hypothesis is largely based on the accidental survival of the earliest brick buildings with geometric revetments in those regions, in contrast to the disappearance of monuments from the period of Buyid and Seljuq tutelage in the Abbasid capital Baghdad.56

The notion that Khurasan was the hotbed of Abbasid artistic innovation is by now dead and buried, but it is not therefore obvious that brick monuments in Baghdad must have had brick decoration similar to the early survivals in the east.57

In any event, Necipoğlu finds star-and-polygon patterns first in the `Arab-Ata mausoleum in Tim.58 But there is nothing in the facade of the Tim mausoleum that is not drawn directly from the Late Antique repertoire of geometric design (for the more ambitious pattern, in the tympanum of the entry, compare the Umayyad window grilles). Her next example is one of the mausolea at Uzgand, from the early eleventh century; here too there simply is no star-and-polygon, but rather a simple interlace of octagons with some sides cunningly omitted to allow the formation of octagrams.59 I do not know without conducting a wider search whether this particular variation of the overlapping-octagon theme was novel; even if it was, there is nothing at all novel about the laying out of the underlying net of construction lines: that is ancient. Necipoğlu's next examples are from the two tomb towers at Kharraqan.60 The first is a tympanum from the eastern tomb, and is constructed in the same way as that seen at Uzgand, although in sextal rather than octal symmetry (both, it will be recalled, were popular in the Umayyad period), and with rather more verve. Again, there is nothing new about the underlying net of construction lines; what separates this design from certain Umayyad window grilles61 is that in the Umayyad designs many of the lines between construction points are bent into curves (creating circles, segments of circles or sinuous lines), whereas the Kharraqan design is entirely composed of straight line segments. The second Kharraqan example, a rectangular lower panel from the western tomb tower, is an example of a venerable Late Antique design, in which skewed swastikas are set at angles to the surrounding design. Robert Hamilton published a very close relative from Khirbat al-Mafjar (his fig. 192); he analyzed the construction net for it, drew it out, and described it:

Designs like Figs. 198 and 199 [two variants in which the swastikas are joined by curves] were drawn on a lattice of diagonal squares superimposed on horizontal lines dividing the height of the field into four equal parts. … To draw the swastikas the draghtsman subdivided the diagonal squares into thirty-six, and the dividing lines gave him the axes of the straps (Fig. 236).

The most complex of the patterns based on squares were Figs. 192 and 200. The basis of the former is given in Fig. 237 [which is essentially the same as Necipoğlu's catalogue no. 59 from the Topkapı scroll], drawn on squares one-eighth of the height of the panel.62

It is hard to understand how such a classic piece of analysis of Islamic geometric ornament could have been overlooked.

Amazingly, Necipoğlu calls these Persian buildings examples of a “new geometric idiom,” though for monuments of this date (well into the twelfth century) it is probably correct to say that they are regional reflections of a more central `Abbâsid style—although what she actually says is that they are “regional echoes of innovations originating in the late Abbasid capital,” (emphasis added) and insofar as geometric design is concerned, this is false. There are no innovations in design, and these designs should have been spread throughout the east by the end of the ninth century, if not earlier.

With the tympanum of the portal of the Gunbad-i Surkh (mid-twelfth century) and the all-over raised pattern covering the lower part of the Gunbad-i Qâbûd, both at Marâgha (late twelfth century), Necipoğlu finally cites an extended star-and-polygon pattern (or something similar—there are no stars in the Gunbad-i Qâbûd pattern) that does not appear to have a direct Late Antique prototype.

Better examples of the application of more sophisticated geometric design are found in the late eleventh-century and early twelfth-century parts of the Great Mosque of Isfahan. The vault of the northern dome chamber is not a star-and-polygon pattern, but a pentagonal arrangement warped across the curved inner surface of the dome. This design must have been achievable using some relatively simple practical aids, and it must have taken considerable experience, imagination, and flair to devise them (see Crop Circles, below). Star-and polygon (and related) patterns may be found in the tympana of the arches under the springing of the squinches in the same dome chamber, and in the fragmentary tympanum of the portal of 515/1121–22.63 Thus we might take the mid-eleventh century as the horizon of new-model star-and-polygon designs in architecture.

In any event, now that we have located a dated example of them, it is useful to examine more closely the innovations in the development of star-and-polygon designs.

Complexity in Early Islamic Geometric Design

There are two kinds of innovation in the development of Islamic star-and-polygon designs, as compared with the geometric decoration of Late Antiquity. One is the increase in their complexity; the other is their utilization of radial symmetries that are not sextal or octal.

Increase in Complexity

The increase in complexity of Islamic star-and-polygon patterns has been studied by Antonio Fernández-Puertas, who provides diagrams and narrative analyses of the construction of many designs in the Alhambra. In a section prefacing a more detailed description of one family of design, he summarizes exactly the first of the two aforementioned innovations:

The lazo-of-eight [octally symmetric geometric design] is based on the proportional ratio between the side of a square and its diagonal …. The square is the basic unit or generating polygon of this lazo. If the side of a square is given the conventional value of unity (= 1), its diagonal will have the value of √2 ….

The origins of the practical knowledge of this ratio, and its use, are lost in antiquity, but Pythagoras expressed it mathematically in his famous theorem. Artists and artisans nonetheless constructed their figures in a purely empirical way, passing down the knowledge of their craft from master to apprentice through the workshops. The technique of constructing in this manner survived into the seventeenth century, when the carpenter Diego López de Arenas, realising that the practical knowledge was fading, wrote a book on it which was published in Seville. He also left a marvellous manuscript illustrated with drawings which was later discovered and published by Manuel Gómez-Moreno.

From what I have seen of pre-Islamic designs, chiefly Roman ones, I observe that they never advanced beyond constructing eight-pointed stars of 45° angled points, and occasionally linked the vertices of the stars with some decoration. The novelty of the Muslim lazo, therefore, was to expand the star system by prolonging the star points with parallel lines [producing a net of construction lines] and intercrossing them to produce various polygons that form a complete rueda (wheel) of externally defined geometric shapes encircling the central star. …

The proportional ratio between the side of the square (= 1) and its diagonal (= √2) cannot be expressed exactly in mathematical terms, because √2 = 1.4142… which is an infinite progression, and therefore an incommensurable number that cannot be measured. Artisans, however, believed that this ratio between the sides and the diagonal of a square was 7/5 (= 1.4)—to judge from López de Arenas' instructions on how to compose a lazo decoration … [further fractional approximations omitted].

The designers were thus always able to draw on a basis of whole numbers or fractions, though never using the decimal system because this was unmanageable in practice. Nor, evidently, did artists and artisans understand irrational numbers, which would have been incomprehensible to them [circular reasoning, but probably true]. They simply adjusted or manipulated the mathematical ratios slightly, and were happy to take √2 to be the equivalent of 7/5 [more correctly, they were happy to take 7/5 to be the diagonal of a square the side of which measured 1], though there is a tiny error of 14 or 15 thousandths. [Quoting A. Prieto y Vives:] “This has been called ‘a geometry of approximation’, where a tiny geometric error, carefully hidden [more correctly, simply ignored because its existence was unknown to the artisans], allows the construction of designs that would be impossible with the geometry of Euclid [more correctly, would require different techniques of construction].”64

Fernández-Puertas gives instructions for constructing various kinds of star-and-polygon design in this fashion; his fig. 165, no. 24, shows a perfectly acceptable design composed in this manner, which consists of a central octagram surrounded by pentagons, in turn surrounding by pentagrams and irregular octagons. While it is not as sophisticated as some of the diagrams in the Topkapı scroll, it is more complex than any Late Antique design, which illustrates my point exactly: one kind of innovation was simply to increase the complexity of inherited geometric constructions. Doing so would have required no new geometric knowledge, only a desire to achieve novel results. The Uzgand pattern (apparently) achieves novel results by “connecting the dots” of the net of construction lines in an interesting way (Fernández-Puertas discusses some of these ways). I suspect that the design of the lower section of the Marâgha tomb is similarly just an inventive scheme of building a pattern on a well known net of construction lines (though I have not analyzed it).

Novel Radial Symmetry

At Khirbat al-Mafjar, and again in Isfahan, pentally symmetrical designs occur. In neither case is anything very complex done with them, but designs based on nonoctal and nonsextal symmetries are the very acme of Islamic geometric patterning. Ernst Herzfeld analyzed and published what must be among the most advanced of such constructions, a now-missing early thirteenth century panel of woodwork from the Lower Maqâm Ibrâhîm in the Citadel of Aleppo, which had ten-, eleven-, and twelve-pointed stars; the pattern was shown only in part, and would have required substantially more space to play out completely.65 But this was hardly the earliest example of such complex geometry. According to Özdural, Abu'l-Wafâ' al-Bûzjânî provided solutions for the constructions of such difficult geometric figures as the pentagon and heptagon,66 though no example of nearly so early a date is known in art. While the geometric construction of a field of odd-sided figures (to arrive at a net of construction lines with nonsextal, nonoctal symmetry) is more complex than the construction of a square lattice, developing a design upon a grid of construction lines remains an Antique invention. Designs with such novel symmetries advance the complexity of simpler, octally or sextally radially symmetric constructions, without constituting a new class of decoration.

Use of Approximations

Fernández-Puertas raises the issue of the use by artisans of numerical approximations to irrational numbers when laying out a pattern by calculation (according to Özdural, Abu'l-Wafâ' al-Bûzjânî offered more elaborate approximations by use of a sequence of fractions)67. Another kind of approximation is the use of an approximate method of geometric construction. Necipoğlu indicates that this approach must have been in use. To quote again, “the A`mâl wa ashkâl often provides very complicated, scientifically correct geometric constructions that are accompanied by simpler methods for constructing the same patterns.”68 These simpler methods must not have yielded exactly correct results, for if they did, they would have been more elegant mathematically than the “very complicated, scientifically correct” methods, which would therefore have been of no interest. According to Özdural at least some of the alternate methods were unnecessarily complex but technical less demanding, as they relied on the use of a compass fixed open at a constant angle throught the exercise.69 The use of approximations directly undermines the notion that there was an emanationist motive in the use of geometric decoration: if craftsmen were happy to use approximations, how could they have been implementing an emanationist aesthetic?

Patterns of Diffusion

Necipoğlu's assertion that geometric design spread to Sunnî courts that had been softened up by the earlier spread of academic geometry is part of a larger argument, in which she asserts in a somewhat convoluted way that star-and-polygon patterns were promoted as symbols of `Abbâsid legitimacy and were shunned by the religiopolitical opponents of the `Abbâsids. There is an ancillary argument, which has been expounded by others as well, that geometric design reflects certain Sunnî, specifically Ash`arî, theological attitudes, but it is really only a special case of the argument from academic geometry, and I will not explore it separately. It is the supposedly Sunnî bias to the distribution of star-and-polygon patterns that is more interesting. Its flavor can be had from the following few extracts:

It was in this context of Sunni revival [patronage of the Ash`arîs after the end of the Bûyid dynasty] during the hegemony of the Great Seljuqs that the girih mode suddenly flourished.70

Besides spreading throughout Iran and Iraq during the eleventh and twelfth centuries the geometric mode extended further east beyond the Seljuq domains, appearing in regions ruled by other Sunni dynasties who acknowledged the suzerainty of the Abbasid caliphs. … After the demise of the Great Seljuqs in 1194 predominantly geometric architectual revetments continued to spread in regions ruled by Sunni dynasties allied with the rejuvenated Abbasid caliphate.71

The notable resistance to the geometric mode in areas dominated by the legacy of the Fatimid and the Spanish Umayyad caliphs provides support for the close association of its emblematic forms with the rival Abbasid caliphate and its vassals or allies.72

Geometric strapwork and the muqarnas, not widely used in the Fatimid territories of Syria and Egypt before, suddenly proliferated in those regions under Zangid, Ayyubid, and Mamluk rule.73

In short: star-and-polygon patterns based on a new “mode” of geometric design that began in tenth-century Baghdad were politically significant; the distribution of surviving monuments demonstrates that only the `Abbâsids and those who derived legitimacy from them used these designs; and after the fall of opposition dynasties star-and-polygon decoration flooded into the areas in question.

This argument falls apart on brief inspection. Its conflation of intellectual interest with political symbolism is contradicted by the Ibn Thawâbah anecdote, which shows high-level interest in geometry to derive from interests other than art. There is the further difficulty that the Shî`î Buyids were in control of Baghdad and western Iran at the very time Necipoğlu believes that geometry was first being promoted as a basis for artistic design, for intellectual reasons.

It is certainly true that there are not many fields of geometric ornament in the art and architecture of the Spanish Umâyyads. But as the geometric designs seen in the Syrian Umâyyad palaces were part of their artistic heritage, this lack must be seen as part of a change in artistic direction in the west, which we still do not understand well. And in fact the Maghrib adopted many architectural themes that originated in the `Abbâsid central lands, and at an early date.74 The intersecting lobed arches of the Great Mosque of Cordoba are serious exercises in geometry, and the intersecting ribs of the domes next to the mihrab there not only reflect the methods of construction of geometric field designs, they are also connected with the same methods used to develop the muqarnas (the intersecting ribs are part of the net of construction lines for both Cordoban-style vaults and muqarnases). And anyway, the Spanish Umayyads avidly copied `Abbâsid court styles in music and attire.

As for the Persian-speaking east, aside from the Bûyids, there simply is no substantial evidence for the art and architecture of any Shî`î dynasty. The art of the area is essentially a whole, more or less provincial, regardless of whether its patrons were friends of the `Abbâsids or politically opposed to them.

In Syria and the Fertile Crescent there were local Shî`î dynasties, such as the Hamdânids, but practially nothing remains from their rule.

The Fâtimids have left us much more, though the corpus of relevant evidence is not all that large. Necipoğlu attempts to discredit this evidence as it does not support her view: “Geometric patterns that hesitantly trickled into Egypt from Syria were not typical of the predominantly vegetal Fatimid decorative vocabulary.”75 But this will not do. There is no call for saying that geometric patterns in Fâtimid art “hesitantly trickled in”: patterns cannot be anthropomorphized, and their occurrence is what is significant, not some judgement of typicality. If they were politically objectionable, they should not have appeared at all. I think the most prominent examples might be these:

Necipoğlu makes play with the late arrival of the muqarnas in Egypt, but, likewise, there are in fact Fâtimid muqarnases, though in truth not many. What all this shows is not any ideological choice of ornament, but the consistent pattern of Cairene artisans: they tended to carry on in their own tradition, absorbing exotic, especially Syrian, motifs at a deliberate rate and executing them in their own local style and building methods.

Finally, Necipoğlu asserts that “the geometric mode and the muqarnas only reached Muslim Spain after its unification with North Africa under the Berber dynasties between the late eleventh and thirteenth centuries.”82 There is very little evidence for the late eleventh century, and what evidence there is for the later occurrence of geometric field patterns and muqarnases is no earlier than the appearance of the same devices in Cairo.

Thus there is insufficient evidence for connecting the elaboration of Late Antique geometric ornament, inherited by the Umayyads and developed over three centuries into something we recognize as novel, with religious or political trends. There is apparently a greater use of such ornament in the east in the eleventh century than in the west, but given the fragmentary state of the evidence it is difficult to say more. In particular, we lack an understanding of the connections, if any, between western and eastern tilework (for which an underappreciated link is perhaps the Qal`ah of the Banû Hammâd).

Spirituality and Art

I have already presented arguments against Necipoglu's third claim, that geometric designs were intended to evoke spiritual experiences. Here I wish to cite as comparative material a more realistic description of aesthetic appreciation of a geometric design (the muqarnas) by a Byzantine, and to review the proper interpretation of the best-known account thought to relate to the spiritual experience of architecture in the West. I feel justified in citing these non-Islamic examples, in light of Necipoğlu's assertions:

In the Islamic world, Byzantium and western Europe, Classical philosophy was applied similarly as a guide to the creation of transcendental aesthetic experience, resulting in the use of geometric patterns and bright colors [emphasis added]. … No doubt the similarities of medieval aesthetic theories in the Islamic and Christian contexts were conditioned by a shared classical heritage. Similar aesthetic theories resulted in differentiated [differing] visual idioms in the Latin West, Byzantium, and in the Islamic lands, yet these idioms were united by a transcendental ethos expressed through the predominant use of abstract geometric schemes and bright colors. Despite striking stylistic differences an underlying similarity remained, unified by a medieval orientation toward metaphysical splendor and the wonderful marvels of creation.83

A Muqarnas in Constantinople

The account of a muqarnas is by Nikolaos Mesarites, guardian of the Pharos shrine where the Mandylion was kept, and involves the late twelfth-century hall in the Great Palace of the Byzantine Emperors in Constantinople known as the Mouchroutas.84

The Mouchroutas is an enormous building adjacent to the Chrysotriklinos, lying as it does on the west side of the latter. … This building is the work … of a Persian hand, by virtue of which it contains images of Persians in their different costumes. The canopy of the roof, consisting of hemispheres joined to the heavenlike ceiling, offers a variegated spectacle; closely packed angles project inward and outward; the beauty of the carving is extraordinary, and wonderful is the appearance of the cavities which, overlaid with gold, produce the effect of a rainbow more colorful than the one in the clouds. There is insatiable enjoyment here—not hidden, but on the surface [emphasis added]. Not only those who direct their gaze to these things for the first time, but those who have often done so are struck with wonder and astonishment.

This nonspiritual appreciation of beauty and the wonder it provokes is paralleled by many other accounts of real-world things (as opposed to nonspecific “beautiful things”) in Islamic sources.

Abbot Suger, Neo-Platonism, and Anagogical Phenomenology

Abbot Suger, head of the Abbey of St. Denis near Paris from 1122 until 1151, supervised the construction of and probably participated in designing a new church for the abbey. He wrote extensively about this project, thus qualifying as a rare source for mediaeval architecture, and he is well known for having expressed an anagogical view of man's experience of the world—essentially the same as the emanationist views of the Ikhwân al-Safâ'. Necipoğlu cites Suger, quoting the locus classicus for the modern view that he appreciated his new church anagogically.85 The relevant passage occurs in account of Suger's decoration of the main altar with precious objects studded with gems. In Erwin Panofsky's translation, the passage runs as follows:

Often we contemplate, out of sheer affection for the church our mother, these different ornaments both new and old; and when we behold how that wonderful cross of St. Eloy—together with the smaller ones—and that incomparable ornament commonly called “the Crest” are placed upon the golden altar, then I say, sighing deeply in my heart: Every precious stone was thy covering, the sardius, the topaz, and the jasper, the chrysolite, and the onyx, and the beryl, the sapphire, and the carbuncle, and the emerald [a quotation from Ovid]. To those who know the properties of precious stones it becomes evident, to their utter astonishment, that none is absent from the number of these (with the only exception of the carbuncle), but that they abound most copiously. Thus, when—out of my delight in the beauty of the house of God—the loveliness of the many-colored gems has called me away from external cares, and worthy meditation has induced me to reflect, transferring that which is material to that which is immaterial, on the diversity of the sacred virtues: then it seems to me that I see myself dwelling, as it were, in some strange region of the universe which neither exists entirely in the slime of the earth nor entirely in the purity of Heaven; and that, by the grace of God, I can be transported from this inferior to that higher world in an anagogical manner.86

Panofsky described this passage as “a vivid picture of that trancelike state which can be induced by gazing upon such shining objects as crystal balls or precious stones [treated not as a] … psychological but as a religious experience.”87 Panofsky points out “owing to its misreading by [Arthur] Kingsley Porter, … this passage has gained currency as a characterization of the emotional impact of Gothic architecture. In reality it describes only the trancelike state induced by the intense contemplation of lustrous pearls and precious stones.”88 For Porter had translated the crucial sentence as:

When the house of God, many-colored as the radiance of precious stones[,] called me from the cares of this world, then holy meditation led my mind to thoughts of piety, exalting my soul from the material to the immaterial, and I seemed to find myself, as it were, in some strange part of the universe, which was neither wholly of the baseness of the earth, nor wholly of the serenity of Heaven, but by the Grace of God I seemed lifted in a mystic manner from this lower, toward that upper, sphere.89

That translation incorrectly has Suger transported by the building, not the gems. Panofsky cited other passages that he thought supported Suger's anagogical view of the building, but which do not involve the same intensity of personal experience. Of these passages, some, I believe, simply cannot support such an interpretation. Suger quotes the foundation inscription of the upper choir:

Once the new rear part [of the church] is joined to the part in front,
    The church shines with its middle part brightened.
For bright is that which is brightly coupled with the bright,
    And bright is the noble edifice which is pervaded by the new light;
Which stands enlarged in our time,
    I, who was Suger, being the leader while it was being accomplished.90

Even understanding that the “new light” is the light of the New Testament and taking into account that “bright” may refer to “the radiance or splendor emanating from the ‘Father of the lights,’” there is nothing in this inscription that invokes anagogical phenomenology in describing the building.

Other passages are more promising.

In another poem Suger explains the doors of the central west portal which, shining with gilded bronze reliefs, exhibited the “Passion” and the “Resurrection or Ascension” of Christ. In reality these verses amount to a condensed statement of the whole theory of “anagogical” illumination:

Whoever thou art, if thou seekest to extol the glory of these doors,
    Marvel not at the gold and the expense but at the craftsmanship of the work.
Bright is the noble work; but, being nobly bright, the work
    Should brighten the minds so that they may travel, through the true lights,
To the True Light where Christ is the true door.
    In what manner it be inherent in this world the golden door defines:
The dull mind rises to truth through that which is material
    And, in seeing this light, is resurrected from its former submersion.91

This is clearly a statement of belief in the efficacy of anagogical thought when inspired by the sight of something (“the dull mind rises to truth through that which is material”). Yet it does not say how, aside from being bright with gold and of good craftsmanship and representing Christ, the door facilitates the mind rising to truth through contemplation of it. Perhaps the reference to craftsmanship means that a poorly done image, even if gold, would not have the same effect. But the element that invokes anagogical thought cannot be mere brightness (else anything golden should lead to thoughts of Christ); it must be the iconography. Suger here is tacking anagogical phenomenology onto a straightforward iconographic theme: an image of Christ should lead the mind to contemplate not the image but Christ himself. I conclude that there is nothing here to suggest that Suger derived anagogical insight from materials or workmanship, or that his anagogical glosses indicate that he felt he had any means of designing anagogically effective works of art other than by depiction of sacred themes. Indeed, later on Suger attributes anagogical effectiveness to iconography:

Moreover, we caused to be painted, by the exquisite hands of many masters from different regions, a splendid variety of new windows, both below and above; from that first one which begins [the series] with the Tree of Jesse in the chevet of the church to that which is installed above the principal door in the church's entrance. One of these, urging us onward from the material to the immaterial, represents the Apostle Paul turning a mill, and the Prophets carrying sacks to the mill. The verses of this subject are these: [eight sets of verses invoking familiar symbolic but not anagogical interpretations of the various religious images] … Now, because [these windows] are very valuable on account of their wonderful execution and the profuse expenditure of painted glass and sapphire glass, we appointed an official master craftsman for their protection and repair, and also a skilled goldsmith for the gold and silver ornaments.92

The valuable materials and good craftsmanship are here considered solely in a practical way. Light is not even mentioned. The effectiveness of the windows lies in their imagery, supplemented by verse for the benefit of those who cannot immediately recall the esoteric meaning of such scenes as Paul grinding grain.

Panofsky insists on Suger's “neo-Platonic light metaphysics.”93 The existence of this train of thought in Suger's writing can hardly be doubted. But I can find no indication that it informed his approach to church design. Following Porter, generations of students have been taught that Gothic churches were designed to have ever bigger windows because light was intended to lead the minds of worshippers anagogically to contemplate God. Those same generations of students have struggled to understand what neo-Platonism has to do with architecture—unnecessarily. The only passage in Suger I can find that explicitly mentions design intended to admit light is entirely devoid of anagogical conceits:

It was cunningly provided that … the dimensions of the old side-aisles should be equalized with the dimensions of the new side-aisles, except for that elegant and praiseworthy extension, in [the form of] a circular string of chapels, by virtue of which the whole [church] would shine with the wonderful and uninterrupted light of most luminous windows, pervading the interior beauty.94

Suger does not even claim divine inspiration for this cunning design, nor would a metaphysics of light have led to it: only practical thinking and a desire for illumination could do that. While there is also a metaphysics of light in Islam (beginning with Qur'ân 24:35, “God is the light of the heavens and of the earth”), it did not lead to airy, light-filled buildings in Islamic architecture.95 Suger's anagogical bent seems to have been secondary to his aesthetic sensibilities. His main concern, that of all Gothic church designers, was to create an overaweing spectacle using all means available: sound, scent, poetry, rhetoric, height, precious materials, color, and light. That he laid an anagogical gloss on aspects of the spectacle from time to time does not indicate that anagogical thinking had any influence on the design of the new church.


The argument from academic geometry fails.

There is no such thing as a “girih mode” that developed in tenth-century Baghdad. Islamic geometric design continued Late Antique practice in laying out geometric designs on nets of construction lines. The designs began to grow more complex toward the end of the eleventh century in entirely evolutionary ways. There is no reason to think that master geometers created designs for artisans (as opposed to methods of construction for desired geometric figures), or that artisans or patrons created designs intended to be emanationistically effective. There is no convincing evidence that anyone except philosophers connected neo-Platonic ideas with mere geometric designs, in tenth-century Baghdad or anywhere else, and no evidence whatsoever that even philosophers connected neo-Platonic ideas with any specific geometric design or, indeed, any other object in the real world. The supposedly parallel case of Abbot Suger is built upon misconceptions and reading into his writings. There is an interesting lack of geometric patterns in the art of Umâyyad Spain, but no evidence to support the claim that geometric designs were somehow understood to be “emblematic” of the `Abbâsid caliphate and Sunnism.

The increasing complexity of geometric ornament in Islamic art and architecture is an example of the continuous striving for greater visual impact that characterizes almost all of the long span of the Islamic artistic tradition. That striving invites further investigation—into the aesthetic proclivities of Islamic civilation, 96 not Late Antique philosophy.

A. The Practicalities of Applied Geometry

To emphasize that practical concerns drive the education of architects and artisans, I present several small cases from the history of Western architecture (broadly understood), derived from my recent haphazard reading.

Railroads and Skew Vaults

The difficulty of constructing skewed vaults led to interesting solutions in Gothic architecture, as John Fitchen shows in his book The Construction of Gothic Cathedrals: A Study in Medieval Vault Erection. Similar problems arose in the nineteenth century, when “the advent of railroads demanded bridges of arched masonry many of which had to accommodate a right-of-way that crossed a stream or a roadway at an oblique angle.”

The shapes of the groin voussoirs, if properly cut to key accurately into the web stones of the two vault surfaces whose intersection they form, are very difficult to arrive at. This difficulty unquestionably accounts for the very small number of unribbed, simple groin vaults constructed by the medieval builders over naves. They could “get away with” approximate shapes ….

Today, we have nothing comparable in stone masonry to the complexity of cutting these groin voussoirs, because we no longer build structurally in stone. However, a century or more ago, before the introduction of either steel or reinforced concrete in building, the advent of railroads demanded bridges of arched masonry many of which had to accomodate a right-of-way that crossed a stream or a roadway at an oblique angle.

Laying out the complicated shapes and hewing the twisting surfaces of the voussoirs of these oblique arches created masonry problems which, for their day, were comparable in difficulty and cost to those of the determination and shaping of unribbed groin stones in medieval times. ….

The nature of the practices, the equipment, and the procedures of the masons who built them … is actually on record in a few books: surviving accounts, descriptions, drawings, and, occasionally, early photographs of an art that is practically extinct in our day. …

Not only do all these writers on oblique arches describe formulas and demonstrate the methods the draftsman uses in laying out the stone-jointing of skew arches and vaults in stone or brick, but they are also at pains to present a great deal of practical data, in considerable detail, regarding the ways by which the masons can make the many full-sized templates and moulds and “squares” they need in order to hew out the winding shapes of the web-course blocks accurately and with precision. …

As in the case of other writers, Hart (a professional mason rather than a civil engineer) frankly states (p. 30) that “It is probable that there are many bricklayers who will consider it too much trouble to observe these rules” (regarding the laying out of an oblique arch in brickwork). He continues: “I will therefore give one more in accordance with their wishes, whereby they will be able to accomplish the work, without making a drawing for any part of it.” Whereupon he proceeds to demonstrate a method of laying out the coursing directly on the formwork of the centering with the aid of straight-edges and squares.

The discrepancy between the engineering knowledge required for the design of skew bridges, and the mason's practical skill and experience in their execution, is acknowledged in Hart's preface, where he states: “My principal aim throughout has been to simplify the construction of oblique arches as much as possible …. I have been more anxious to explain them in language suited to the capacities of the men engaged in the execution of them, than to embellish the Work with scientific terms.”97

Georgian Builders

Much of Georgian London was built as a series of real estate speculations by landowners. Often, the developers of the plots within a project were master builders, usually carpenters or bricklayers, who in turn built houses on speculation, as John Summerson noted.98 Such men often played the role of architect:

Until about halfway through the eighteenth century there was no such thing as an “architectural profession” in the modern sense, and even then it was largely confined to a small number of London architects. … Architects were often recruited from the ranks of building tradesmen and surveyors ….

Speculative building and the architectural profession were two goals of success for men entering the London building trade. The roles might be, and often were, combined. But the man who was temperamentally fitted for the more academic and theoretical parts of the business usually contrived to find himself a patron, with whose help he advanced along the professional road as far as a small estate of his own and a fortune of £10,000 or £20,000. …

The ability of the craftsman to better himself by becoming an architect or quasi-architect provided a strong inducement to self-education, even to the more commercially minded man, for he could not afford to be behind in questions of taste. Self-education meant getting a hold on the artistic needs of the centres of fashion. It meant the desertion of traditional craftsmanship and the adoption of certain academic formulas. Competition made this necessary; for with the growth of capitalism in the building world the individual either had to make a place for himself or remain a journeyman all his days.

Thus we find the building industry transforming itself from a homogeous body of independent craftsmen to a body comprising, at the top the speculating master builder, at the bottom, the journeyman, and between the two, but on a pedestal of his own, raising him socially above either, the architect. The loss of status to the individual craftsman is obvious. He had to go into business as a speculator or get himself a patron and become an architect, or perish. To stand on his own feet he had to become something of an architect himself. This he was able to do through the medium of books.

Books were the most important single factor in establishing the dictatorship of Palladian taste throught the building world. The great period of book production for craftsmen began in 1715. … Within ten years … an avalanche had begun of books compiled by craftsmen and exclusively for craftsmen, with a view to instructing them in that self-improvement which would see them on the right side of the fence. …

The main output was between 1725 and 1760; it diminished with the expansion of the architectural profession and the coincident repression of the craftsman's initiative. In the latter part of the century we get a very different wave of book publishing, sponsored not by craftsmen but by architects, and designed not to instruct the workman, but to charm the potential client.99

Laying out Octagons

The practical problems outlined by Fernández-Puertas (quoted above) are still alive for craftsmen, even though they may not be making star-and-polygon designs. Practical advice on geometric construction continues to appear in special-interest books and journals. The December, 2002, issue of Fine Homebuilding carried an article by a John Carroll entitled “Laying out Octagons: Whether framing an octagonal window or laying out a bay, here are two fast ways to get the angles right.”100 The publication is pitched to today's equivalent of Georgian master builders (this item being directed to the master carpenter), and I see this piece of advice as just exactly what tenth-century Baghdadi craftsman sought from the geometers they consulted—with a decimal twist.

Carroll relates briefly a friend's difficulties in framing an octagonal window and makes reference to “the octagonal scale on his rafter square” (a photograph shows this to be a scale engraved on a fancy carpenter's square, down the middle, between the scales in inches on either side of the short arm of the square). His friend had not known how to use it, so Carroll provides two method of layout: one employing the square itself, the other based on a decimal ratio.

For a [regular] octagon, says Carroll (I have not checked the assertion), half the length of one of its sides equals 0.2071 multiplied by the length of the side of a square enclosing the octagon and coinciding with four of its sides. “Multiplying the length of a side of any square by 0.2071 equals the distance found by using the dividers on the octagonal scale or one-half the length of one of the sides of an octagon.” Thus, it is implied, one could construct an octagon by laying out its enclosing square and using a calculator to find the vertices of the sides of the octagon, a method Fernández-Puertas rightly thought beyond the reach of the mediaeval artisan, because it relies too much on understanding decimal multiplication. This method is equivalent to the use of a fraction approximating an irrational ratio, in the manner outlined by Fernández-Puertas.

The method Carroll actually illustrates relies on the use of the engraved octagonal square, another aid lacking in the premodern period, but one from the old analogue age rather than the modern digital age, which has given us the calculator (I would liken it to the scales on a slide rule, if I thought anyone still remembered what a slide rule is):

  1. To lay out, say, a 38-in. wide octagon, make a 38-in. square and mark a centerpoint (A) on each side of the square [Carroll does not say whether to do this by measurement or by geometric construction].

  2. Locate the octagonal scale on the tongue (the smaller of the two sides) of the framing square. Place one point of a pair of dividers on the 0 mark of the octagonal scale and the other point on 38, which corresponds to the width of the square in inches.

  3. After carefully removing the dividers from the framing square, place one point on the centerpoint of any side of the square. With the other point, make two additional marks (B), one toward each corner along the same side. Repeat on each of the other sides.

  4. Connect the two points (B) closest to each corner with a line.

In this procedure the use of the engraved scale substitutes for the use of some table that has been reduced to a proportional scale that need merely be consulted to set the distance between the points of the divider before proceeding with the remaining few steps of geometric construction.

These methods are long established in modern carpentry. Among my grandfather's effects I have found a copy of Smoley's Tables, subtitled Parallel Tables of Logarithms and Squares … For Engineers, Architects, and Students, first published in 1901, which facilitates solution of construction problems by computation, and a brochure for “Stanley Rafter and Framing Squares” from 1937, providing tools for solving construction problems either by computation or geometrical construction.101 In a construction industry with well differentiated trades, both methods have their audiences: those who want (or need) five centimeters of tables and a slide rule, and those who just want a roofing square.

Crop Circles

So-called “crop circles,” well established as a genre of practical joke during the late twentieth century, are constructed at night with the aid of various devices for flattening cereal crops, generally without leaving footprints. The early crop circles were simple, intended to suggest the imprints of flying saucers. With media attention, circlemakers have thrown off the restraints of hoaxing and have become more artistic. An article in Science in August, 2001, reports on and illustrates

a dazzling fractal-type design, discovered on 13 August, described as “jaw-dropping” by John Lundberg, author of the Circlemakers Bulletin. Appearing in Wiltshire in southern England, the heart of crop circle country, it comprises 409 circles [of a wide range of sizes] in a [six-armed] spiral pattern more than 450 meters across—dwarfing the 60-meter average diameter of this year's designs.

Lundberg says that the “sheer scale and complexity” of this opus has many people baffled. He estimates that for it to have been done in 4 hours of darkness, the makers had to create circles at a rate of one every 30 seconds—with no time for preliminary survey work.102

Clearly this intricate and regular design was constructed with the simplest of tools and the easiest methods of layout, although with a crew of at least six. Lundberg remarked, “I know from previous experience that after a certain length [about 60 meters] it's very difficult to hold a tape measure above the crop without it snagging.”

1. As opposed to other, less useful definitions; I have discussed varieties of arabesque in Five Essays on Islamic Art, Sebastopol, Calif., 1988.

2. Gülru Necipoğlu, The Topkapı Scroll—Geometry and Ornament in Islamic Architecture: Topkapı Palace Museum Library MS H. 1956, Santa Monica, 1995.

3. See op. cit., pp. 73–74: while it is true that so wide a definition has been used before, to use the word to mean all of Islamic decoration is to render it almost meaningless.

4. Necipoğlu, op. cit., p. 9; cf. p. 231. The value of n is not supplied but perhaps is the number of intersecting construction lines.

5. Ibid., pp. 92–93.

6. Ibid., p. 139.

7. Ibid., p. 131.

8. Loc. cit.

9. Loc. cit.; see also p. 97, right col.

10. Ibid., pp. 131–32, which stresses the presumed role of Harran; see also p. 123.

11. Ibid., p. 132.

12. Ibid., p. 132.

13. Ibid., pp. 156–60.

14. “On Interlocking Similar or Corresponding Figures and Ornamental Patterns of Cubic Equations,” Muqarnas, v. 13, 1996, pp. 191–211, with references I find easier to understand, under the title Fî tadâkhul al-ashkâl al-mutashâbiha au al-mutawâfiqa, or Interlocking Figures.

15. Ibid., pp. 167–69. Further on the tables, see Özdural, p. 194.

16. Ibid., p. 123.

17. Ibid., p. 170.

18. Ibid., p. 189.

19. Ibid., pp. 132–33.

20. Ibid., p. 187.

21. Ibid., pp. 188–89. “Unity in variety” is here used in the theological sense, although just what the last sentence means is unclear. Cf. ibid., p. 164, a use of “unity in variety” more conventional for art history. Cf. also Ernst Herzfeld, “Die Genesis der islamischen Kunst und das Mshatta-Problem,” Der Islam, v. 1, 1910, pp. 27–63 and 105–44, the relevant passage, pp. 44–45, translated in Allen, Five Essays, p. 3. For Herzfeld the principle of unity in variety as applied to Islamic art was an aesthetic, not a theological one. Incidentally, there is a very nice example of unity in variety using star-and-polygon patterns in the fourteenth-century tomb of Muhammad b. Muhammad Luqmân at Sarakhs, illustrated in Ernst Diez, Churasanische Baudenkmäler, Berlin, 1918, pl. 22, 1.

22. Ibid., pp. 189–90.

23. Ibid., pp. 193–94.

24. Loc. cit.

25. Allen, Ayyubid Architecture, ch. 11 and app. B.

26. Ahmad b. at-Tayyib al-Sarahsî, New Haven, 1943, pp. 86–94; also noted by Dmitri Gutas, Greek Thought, Arabic Culture: The Graeco-Arabic Translation Movement in Baghdad and Early `Abbâsid Society (2nd–4th/8th–10th centuries), London, 1998, p. 127.

27. Assistant to al-Muhtadî's wazir Sulaymân b. Wahb (255–256/869–870), and held lesser posts until his death in 277/890 or 273/886; see EI2 s.v. “Ibn Thawâba.”

28. Joel L. Kraemer, Humanism in the Renaissance of Islam, Leiden, 1986, p. 263.

29. EI2, s.v. “Ibn `Abbâd”.

30. As a quick sample, in the somewhat problematic list compiled by L. A. Mayer, Islamic Architects and Their Works, Geneva, 1956, among only those names beginning with the letter A there are tenth-century architects working in Jerusalem, p. 40; Tunis and Diyarbakr, p. 43; and, pp. 44–45, in the ninth century, the architect of the Nilometer, a significant work involving mathematics.

31. Greek Thought, Arabic Culture: The Graeco-Arabic Translation Movement in Baghdad and Early `Abbâsid Society (2nd–4th/8th–10th centuries), London, 1998.

32. Op. cit., pp. 47–50.

33. Op. cit., p. 29.

34. Op. cit., pp. 40–41, omitting Gutas's numbering of the successive points.

35. Loc. cit.

36. Op. cit., pp. 41, 50–51.

37. Op. cit., pp. 51–52.

38. Plans of the Hiraqlah gates can be found in Andreas Schmidt-Colinet, “Überlegungen zur Bauornamentik von Hiraqla,” Damaszener Mitteilungen, v. 11, 1999, pp. 385–89.

39. Op. cit., p. 110.

40. Loc. cit.

41. Op. cit., p. 127.

42. Tamara M. Green, The City of the Moon God: Religious Traditions of Harran, Leiden, 1992, provides a good review of relevant thought in pre-Islamic Harran, pp. 82ff.

43. This is so whether such theurgy (for which see Green, op. cit., pp. 87ff.) could be justified by recourse to Qur'ân and hadîth or not (see EI2 s.v. “Sihr” for a discussion of the varieties of magic and their theological status). Open display of designs for which some emanationist effectivity was openly claimed would have surely raised objections by various writers.

44. Mentioned above; cited by Necipoğlu, pp. 167–69; see also Özdural, op. cit., pp. 192, 195.

45. Nicely summarized by Necipoğlu, op. cit., pp. 187–88.

46. Despite “the teaching of the Pseudo-Dionysius that the visible world signifies God; thus we read in Leontius of Neapolis how God is worshipped through creation and created objects, which embrace not only ‘heaven and earth and the sea’, but also ‘wood and stone … relics and church-buildings and the cross’. Creation itself, it is argued, is a sign and mirror of God; the material objects which he has created are the signs through which he is recognized and represented.” From Averil Cameron, “The Language of Images: The Rise of Icons and Christian Representation,” in The Church and the Arts, ed. Diana Wood, Oxford, 1992, pp. 1–42, reprinted in Cameron, Changing Cultures in Byzantium, Aldershot, 1996, p. 36. In Islam, of course, it would be heresy to worship Allâh through any intermediate object.

47. For example, Fadlou Shehadi, Philosophies of Music in Medieval Islam, Leiden, 1995, p. 25, quotes al-Kindî (d. 870)

When they (the Ancient Philosophers) showed that there was nothing sensible which is not composed of the four elements and the fifth nature, I mean, fire, air, water, earth and the falak, they were driven by sagacity and guided by intelligence, and their intellect told them to devise sonorous string instruments that mediate between the self and the composition of the elements and the fifth nature (the celestial sphere). They also devised several string instruments that are in accordance with the composition of animal bodies from which (instruments) emerge sounds that are similar to the human composition. [Musawwitât, p. 71.]

Of course no musical instrument was ever developed on such a basis, and more importantly, no one suggests that any Muslim devotee of neo-Platonism, whether in tenth-century Baghdad or not, invented anything of the sort.

48. See for example the biographical sketches in Kraemer, Humanism in the Renaissance of Islam, ch. 3.

49. Necipoğlu, op. cit., p. 139.

50. Necipoğlu, op. cit., p. 93.

51. Daniel Schlumberger et al., Qasr el-Heir el Gharbi, Paris, 1986, pl. 72–81; K. A. C. Creswell, Early Muslim Architecture, 2 v., Oxford, 1932–40; v. 1 revised and published in 2 parts, 1969, pl. 59, for the Great Mosque of Damascus.

52. Op. cit., pl. 69 for Khirbat al-Minya (the design I have in mind is pl. 69 c, middle); Robert Hamilton, Khirbat Al Mafjar: An Arabian Mansion in the Jordan Valley, Oxford, 1959, pl. 76–81.

53. For the Aqsâ, see most recently Robert Hillenbrand, “Umayyad Woodwork in the Aqsâ Mosque,” Oxford Studies in Islamic Art, v. 9, pt. 2, 1999, pp. 271–310.

54. Creswell, op. cit., pl. 39 for Wâsit; Hamilton, op. cit., pl. 3, 4, 6, 46, 47, 57–63 and figures in the text. The unfinished balustrade panel is shown in pl. 61.

55. Schlumberger, op. cit., pl. 58.

56. Necipoğlu, op. cit., p. 97.

57. The aesthetic rationale for the beginnings of this brick decoration is, I think, still to be sought. Perhaps these external designs are exteriorizations of patterns found (inside) in Sâsânian stucco.

58. Loc. cit. and pl. 88. Many of these examples are also adduced by A. J. Lee, “Islamic Star Patterns,” Muqarnas, v. 4, 1987, pp. 182–97, also drawing attention to the novel radial symmetry of designs in the mosque of Barsiyân, Iran, mid-twelfth century.

59. Loc. cit. and pl. 89.

60. Op. cit., p. 99 and pl. 90, 91.

61. Such as Schlumberger, op. cit., pl. 74 a.

62. Hamilton, op. cit., p. 279.

63. A Survey of Persian Art from Prehistoric Times to the Present, ed. Arthur Upham Pope, London, 1938–39; many reprints, pl. 290; André Godard, “Isfahân,” Athâr-é Īrân, v. 2, 1937, pp. 7–176, fig. 2; Godard, “Historique du Masdjid-é Djum`a d'Isfahân,” Athâr-é Īrân, v. 1, 1936, pp. 213–82, fig. 148; Allen, Five Essays, fig. 68.

64. Antonio Fernández-Puertas, The Alhambra: I. From the Ninth Century to Yûsuf I (1354), London, 1997.

65. “Damascus—Studies in Architecture, II,” Ars Islamica, v. 10, 1943, pp. 13–70; pp. 64–66.

66. Op. cit., p. 194.

67. Op. cit., n. 16.

68. Necipoğlu, op. cit., p. 169.

69. Op. cit., pp. 194–95; n. 55 for Abu'l-Wafâ' al-Bûzjânî's own judgement that artisans were not concerned with precise accuracy.

70. Necipoğlu, op. cit., p. 97.

71. Op. cit., p. 100.

72. Op. cit., p. 101.

73. Op. cit., p. 100.

74. I have discussed these under the rubric “Transformations and Correspondences” in Five Essays.

75. Op. cit., p. 100.

76. K. A. C. Creswell, The Muslim Architecture of Egypt, 2 v., Oxford, 1952–59, pl. 7, 9.

77. Op. cit., pl. 17, right lozenge, and pl. 28 a and b.

78. Op. cit., pl. 18 b.

79. Op. cit., House V, pl. 37 e.

80. Op. cit., pl. 82 b.

81. Op. cit., pl. 102.

82. Necipoğlu, op. cit., p. 102.

83. Necipoğlu, op. cit., p. 196. Despite the existence of a book on the topic, it is clear that the Byzantines had no coherent aesthetic theory.

84. On the Mouchroutas, with argument for its having been covered with a muqarnas, see William Tronzo, The Cultures of His Kingdom: Roger II and the Cappella Palatina in Palermo, Princeton, 1997, pp. 136–37; the translation is from Cyril Mango, The Art of the Byzantine Empire 312–1453: Sources and Documents, Englewood Cliffs, 1972, pp. 228ff.

85. Necipoğlu, op. cit., p. 196. For an exposition of Suger's anagogical thinking, see Erwin Panofsky, Abbot Suger on the Abbey Church of St.-Denis and its Art Treasures, 1946, 2nd ed. by Gerda Panofsky-Soergel, Princeton, 1979, pp. 19–26, 164–65, 191, and 240.

86. Panofsky, op. cit., pp. 63–65.

87. Op. cit., p. 21. Whether a trancelike state or mere daydreaming is described may be questioned.

88. Op. cit., p. 191.

89. Medieval Architecture: its Origins and Development, New York, 1909, repr. 1966, v. 2, p. 252.

90. Panofsky, op. cit., pp. 51, 22–23.

91. Op. cit., pp. 23–24, 164–165.

92. Op. cit., pp. 73–77.

93. Op. cit., p. 21.

94. Op. cit., pp. 101, 240.

95. As pointed out by Doris Behrens-Abouseif, Beauty in Islamic Culture, Princeton, 1999, p. 119, although she seems to accept the traditional view of Suger.

96. A good example is Lisa Golombek, “The Draped Universe of Islam,” Content and Context of Visual Arts in the Islamic World, ed. Priscilla Soucek, University Park, Pa., 1988, pp. 25–38; p. 35, in which the author introduces the notion of an Islamic “love of interlace.”

97. Fitchen, The Construction of Gothic Cathedrals: A Study in Medieval Vault Erection, Oxford, 1961, repr. Chicago, 1981, app. H, Oblique or Skew Vaults of Masonry, pp. 266–68.

98. John Summerson, Georgian London, London, 1945, revised ed. New Haven, 2003, p. 55.

99. Op. cit., pp. 56–60.

100. Pp. 92–93. Carroll is identified as the author of a book entitled Measuring, Marking and Layout.

101. Smoley's Tables. Parallel Tables of Logarithms and Squares. Diagrams for Solving Right Triangles. Angles and Trigonometric Functions Corresponding to Given Bevels. Common Logarithms of Numbers. Tables of Logarithmic and Natural Trigonometric Functions and Other Tables. For Engineers, Architects, and Students, Constantine K. Smoley, Past Director, School of Civil Engineering, International Correspondence Schools, tenth ed., C. K. Smoley & Sons, Publishers, Scranton, Pa., 1941, copyright 1901, 1906, 1908, 1910, 1912, 1914, 1920, and 1941. The preface to the first edition pitches the publication to “structural draftsmen.” It is unpaginated, four to five centimeters thick, printed on bible paper, and composed of endless tables with sections of “Explanations and Examples” with diagrams. “Bevel” in the title refers to slope or pitch, for example, of a roof truss.

102. “Jaw Dropper,” Science, v. 293, no. 5534, 24 August 2001, p. 1429.