Source: http://www.sonic.net/~tallen/palmtree/academicgeometry.htm
Timestamp: 2019-04-23 06:59:39+00:00

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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.
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.
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.
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.
There is something that can be called “the girih mode.” Girih is a term used in connection with the design of architectural decoration in the nineteenth century.
She does not say what the “highly codified mode” actually is, nor where the code is to be found, nor what a “mode of patterning” would be; the description of it as containing star-and-polygon compositions is not a proper definition. Consequently, I shall supply the following definition for use in following Necipoğlu's discussion: the phrase “the girih mode” means roughly “geometric (often star-and-polygon) designs composed upon or generated from arrays of points from which construction lines radiate and at which they intersect.” I do not understand what “broad spectrum of alternative geometric designs” is eliminated by this definition.
Master artisans were literate. The anonymous A`mâl wa ashkâl and Abu'l-Wafâ' al-Bûzjânî's A`mâl al-handasah (for which see below) “are the only works known to have directly addressed artisans, suggesting a degree of literacy at least among their masters.”6 While such addressing may have been rhetorical, and real communication may have occurred orally, the difference is not material: the point is that such communication could have occurred.
Theoretical mathematics was systematically popularized in Baghdad in the tenth century through practical manuals and through those manuals it was rapidly disseminated “to other Islamic courts” in the eleventh. “It was in this receptive setting [the popularization of geometry even among rulers] that the girih initially spread during the eleventh century when its basic repertory became elaborated in different ways in various local courts that had acquired a taste for geometry.”9 Much of the supporting material cited comes from after the period of the development of the arabesque.
It was through this new availability of classical mathematics that Muslim architects learned sophisticated geometry. “These mathematical works [listed in the Fihrist] must have had as great an impact on Islamic architectural practice as those that became available in post-twelfth-century Europe would have on Gothic design.”10 Note the implicit claim that Muslim architects did in fact learn sophisticated geometry, rather than learning how to adjust traditional craft techniques to produce more sophisticated designs.
“After the initial period of translations was over, original contributions were made in such mathematical disciplines as astronomy, optics, algebra, and trigonometry that encouraged the development of geometry in previously unknown directions[,?] promising new applications in a number of fields, including architecture and the decorative arts (e.g. arch and dome profiles, geometric patterning, and the muqarnas).”11 The assertion that new contributions to mathematics led to specific applications in the visual arts is entirely unsupported. A footnote cites M. Souissi's assertion in EI2, Supplement, s.v. “`Ilm al-Handasa,” that “the work of the sculptor in stone or in stucco was designed by the mathematician,” but that assertion is equally unsupported by anything in Souissi's article.
Mathematicians and master craftsmen collaborated in solving problems of practical geometry.
Antique emanationist philosophy was fused with Islamic notions of a unique god, providing a theoretical basis for attributing aesthetic enjoyment to the relationship between the artifact and the cosmic or heavenly thing it resembles.20 The material cited, both Antique and Islamic, is too copious to do more here than sample it. These sources claim that beauty, and in particular appropriate and proportional design, attract the viewer's soul. According to Necipoğlu, Plotinus defines the beautiful as “that which induces wonderment, longing, delight, and love because of its kinship with the soul.” The Ikhwân al-Safâ' say that that earthly music resembles the music of the spheres, which is concordant and harmonious, with well balanced melodies. This causes the human soul to rejoice. Abû Muhammad `Alî ibn Hazm, 9941064, wrote that the soul is attracted to well proportioned things like itself.
Aesthetic merit was thought to be inherent in symmetical and “well proportioned” things, including handwriting. The Ikhwân al-Safâ' wrote that the most beautiful handwriting is well proportioned.
Geometry and color were used to elevate visual design to the intellectual level of music and to allow the viewer to have a transcendental experience of heaven. The direction of the argument here is clear, although the wording is not; I have supplied the last part of the above paraphrase. Further, I have inserted wording in brackets in the following quotation to clarify what I think the intended meaning might be.
Widespread notions about the role of geometry as a bridge between the material and spiritual realms, coupled with the absolute beauty of its harmonious forms [which were thought to be] capable of purifying the mind like music, must have made geometric abstraction a particularly appealing visual idiom. The purity of a polychromatic abstract design vocabulary dominated by congruent geometric shapes approached the status of light and color, the only two properties that Ibn al-Haytham singled out as being beautiful in themselves in addition to proportionality [more properly, purity of design vocabulary corresponds to Ibn al-Haytham's “proportionality and harmony”]. This purity of sterilized forms, absolved from impure defilement, no doubt augmented the positive resonances of geometric and geometrized patterns, often accompanied by Koranic inscriptions. Moreover, the proportionally interlocking [that is, interlocking and proportionally related] colored geometric forms of girihs that [it was thought] could trigger an innate aesthetic reaction in viewers must have given medieval designers the hope of endowing their visual creations with the expressive potential and emotional immediacy of music.
Necipoğlu stops short of claiming that anyone actually obtained aesthetic enjoyment or religious experience specifically of geometric patterns on emanationist grounds, claiming rather that it was the intention of the creators of such patterns that they do so. It is interesting that she does not claim that those creators must themselves have had such experiences, else they would not have expected their creations to operate as expected.
Geometric decoration was intended to “trigger an innate aesthetic reaction” in the viewer by making it well proportioned. “To conclude, then, the so-called arabesque (whether geometric, vegetal, calligraphic, or figural) can be seen as the offshoot of a classical notion of beauty recast in abstract terms.”24 Necipoğlu here applies the argument from academic geometry to the arabesque, although much of her preceding discussion deals specifically with geometric ornament, and to a lesser extent calligraphy. Nevertheless, this is the assertion I am concerned with, and the argument from academic geometry can be dealt with on its own terms.
Literate master artisans, trained in mathematics, which was accessible to them through translations of Antique works and new Arabic mathematical literature, and conscious of the relationship between theory and practice, read manuals of practical geometry and participated in discussions with theoretical mathematicians concerning the practical problems of constructing complex geometric patterns.
The theoretical mathematicians who wrote for and discussed geometry with artisans were directly involved in developing more complex patterns, of a new type.
Those patterns were developed in order to provide the viewer with objects of contemplation that would trigger an innate aesthetic reaction, and thus a religious experience of the heavens, in accordance with neo-Platonic emanationist philosophy as modified in Islamic circles. It was believed that this triggering was possible because the patterns were well proportioned and harmonious, and because their geometric basis instantiated heavenly prototypes. This approach to design made “abstract beauty” accessible.
The predilection for geometric design was spread from tenth-century Baghdad to “other Islamic courts” in the eleventh century through the dissemination of mathematical works and “a taste for geometry”.
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.
“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].
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.
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 … .
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.
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.
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 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.
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.
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.
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.
Artisans produced geometric ornament that patrons and customers expressed pleasure in, for ordinary, not neo-Platonic, reasons. No instance of emanationist aesthetic enjoyment has been recorded in the Islamic world, and the oft-cited example of Abbot Suger (see appendix) is usually construed incorrectly. Nor is there any discussion of the virtue or pleasure derived from meditating on one design rather than another; nor, for that matter, is there any discussion in Islamic literature, so far as I know, of meditation on ornament (as opposed to wonder at great monuments) at all, while there is plenty of poetry invoking the emanationist world view.
In the continuous striving by both artisan and patron for greater visual impact that characterizes almost all Islamic art, artisans sought to increase the complexity and sophistication of their work.
In doing so they consulted theoretical geometers about designs they wished to create, who gave them practical solutions producing approximate results good enough for the task at hand.
The audience of patron and his circle appreciated the results for the same reason they appreciated earlier efforts.
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.
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.
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.
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.
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).
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.
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.
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.
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].
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).
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.
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?
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.
And of course there is the wooden mihrab of Sayyidah Nafîsah, which has a very nice example of pattern cropping on either side of the niche, and strapwork interlaced with foliage within the niche.
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).
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.
The church shines with its middle part brightened.
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.
Marvel not at the gold and the expense but at the craftsmanship of the work.
To the True Light where Christ is the true door.
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.
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.
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.
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. ….
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.
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 ….
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.
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.
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].
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.

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