Omar Khayyam, Mathematicians, and "Conversazioni" with ArtisansAuthor(s): Alpay ÖzduralSource: Journal of the Society of Architectural Historians, Vol. 54, No. 1 (Mar., 1995), pp. 54-71Published by: University of California Press on behalf of the Society of ArchitecturalHistoriansStable URL: http://www.jstor.org/stable/991025Accessed: 09/06/2010 09:19

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Omar Khayyam, Mathematicians, and Conversazioni with Artisans

ALPAY OZDURAL, King Faisal University

T he intricate geometric patterns that decorate the monu- ments of the Islamic world have always intrigued contem-

porary architectural historians. These patterns, cleverly inter-

locking with each other to create infinite compositions on wall

surfaces and highly sophisticated configurations of muqarnas

(stalactites), are believed to have been created by architects or

artisans who were not only masters in their own crafts but also

competent in geometry. The general expectation is that these

architect-artisans, like all practical people, were not in the habit

of producing any sort of written material; therefore, their

exquisite works are regarded as the only evidence of their skill

in geometry. The remarks of a young Ottoman geometer, however, cast a

shadow of doubt on this assumption. While reading from a

book on geometry and explaining it to the mother-of-pearl workers (specialized carpenters) at their workshop in the

gardens of Topkapi Palace, he said:

Regarding that which is called the science of geometry [handasa], in this

age [1570], if the science of geometry is discussed among architects

[mi'mar] and learned men ['dlim], each one will answer, "Yes, we have

heard of it, but in essence we have not heard how the science of geometry

works and what it deals with." Now this noble book fully describes that

fine science. As long as a person does not understand this rare and

agreeable science, he is not capable of the finest working in mother-of-

pearl, nor can he be expert and skilled in the art of architecture.1

Considering the fact that the young geometer made this

remark while Ottoman architecture was enjoying its golden

period under the leadership of the great architect Sinan, the

implication is outrageous. It can, of course, be dismissed

conveniently as a gross exaggeration. Even the underlying

reason for such an exaggeration, one has to admit though, is

enough to contradict the general assumption of attributing a

genius for geometry to architect-artisans.

The famous mathematician-astronomer Abu '1-Wafa' al-

Bfizajamni (940-98), who wrote a book on geometry specifically

for artisans, Kitdbfimd yahtaju ilayhi al-sani' mmin amal al-handasa

(The book on what the artisan requires of geometric constructions),

hereafter Geometric Constructions, was sharing the same concern

for the inadequacy of artisans of his time in geometry.2 He

wrote:

I know that artisans [sunna] construct figures in round forms unmethodi-

cally.... In order to create fine works, the artisan has to quit working by

the eye-measure. Instead, he must determine the dimensions of sides of

the pentagon, hexagon, decagon, or other figures as we explain in this

book.3

To make his point clearer, he expanded on the same issue:

What an artisan illustrates is an approximation of the geometric

construction, which he perceives to be correct through his senses and

observations. He is not concerned with diagrammatic proofs. [On the

other hand], when a geometer [muhandis] establishes the proof of a

problem by deduction, he never questions whether the correctness of the

construction is observable. However, it is not justified to distrust every-

thing that an artisan sees as correct, since these are usually taken from

constructions which were previously proven by geometers. The artisan

and the surveyor [mdsah] take only the end product of a problem but pay

no attention to how the correctness is determined; therefore they may

commit fallacies and errors. The geometer believes in the correctness

through required proofs if he can derive the meaning of the construction

of the artisan and the surveyor.4

More than six centuries had passed between these comments of

the two mathematicians. During this period in the Islamic

world, the field of mathematics enjoyed great advances, and

numerous monuments were produced in the field of architec-

ture with increasingly exquisite applications of geometry. The

words of both mathematicians indicate a close collaboration

between geometers and architect-artisans, and, in so doing,

suggest a relation between the developments in these two

fields. One can not help but wonder why, therefore, the

Ottoman geometer was still claiming architect-artisans had no

notion of geometry. How could the achievements in mathemat-

ics, which were apparently reflected on architectural monu-

ments, have had no effect on the improvement of artisans'

knowledge? Another common point between these two sources seems to

provide us with a plausible answer to this question. The

54 JSAH / 54:1, MARCH 1995

a

FIG. I: a) Abu 'l-Wafa's figure proving the

Pythagorean theorem; b) The ornamental pat-

tern of Abu 'I-Wafa's proof, general; c) The

ornamental pattern of Abu 'I-Wafa' 's proof, the

ratio of 1:2.

Ottoman geometer mentioned that the science of geometry was discussed among architects and learned men. It seems logical to infer that, in sixteenth-century Istanbul, architects and mathematicians were in the habit of coming together at special gatherings in order to discuss the application of geom- etry to architecture. Abu 'l-Wafa' was quite precise about this sort of meeting (hereafter referred to as a conversazione).5 He stated:

I was present at some conversazioni [majdlis] held among a group of

artisans and geometers.6

He then gave the account of a particular conversazione at which they discussed the problem of "composing a square from three squares," in other words, the construction of a square the side of which is equal to /i3. Evidently, in tenth-century Bagdad, it was customary for artisans to meet with mathematicians to seek advice on certain problems concerning the application of geometry to architecture and related arts.

If conversazioni of this sort were common enough in two

major cities with different cultural and political settings, sepa-

rated from each other by six centuries, it would be reasonable to think that this sort of gathering represented a widespread phenomenon in the Islamic world. It can thus be expected that when architectural and scientific activities were being concen- trated in urban centers, there was a constant dialogue between architect-artisans and mathematicians in the form of conversa- zioni that served as the vehicle to exchange knowledge between the two groups. Practitioners, who were deficient in theoretical knowledge, had convenient access to advanced mathematics by way of these meetings; scholars, who were not experienced in practical applications, there found the opportunity to be involved in architecture, the visual results of which were pleas- antly rewarding. Though solving the immediate problems by way of a dialogue was evidently a convenient means for artisans to learn, this probably explains why there was no real improvement in their knowledge of geometry during those six centuries, as they would tend to implement only capsulized solutions.

Unlike Greek mathematicians who developed mathematical sciences for the sake of rigorous thinking and the disciplinary

OZDURAL: OMAR KHAYYAM AND THE ARTISANS 55

value of the subject, Muslim mathematicians were always more

concerned with the practical and immediate consequences rather than the theoretical qualities of their work.7 It can be

easily be imagined that, throughout the centuries during which

mathematical sciences have flourished, some of the great

mathematicians, such as Abu 'l-Wafa' enjoyed being involved in

architecture and related arts through conversazioni. Thus, some

of the aesthetic, structural, or spatial innovations that we

observe in the products of the major architectural centers of the

Islamic world may be explained as the contributions of certain

mathematicians.8

The proposition that the conversazione acted as the mode of

exchange for architectural and geometrical knowledge in ma-

jor urban centers of the Islamic world can be supported by

further references to this phenomenon in several other sources.

For instance, Ghiyath al-Din Jamshid al-Kashi (d. 1429), an

outstanding mathematician-astronomer, related in the letter to

his father a debate between him and the master mason, and

other mathematicians who sided with the mason, about the

leveling instrument used at the site of the Samarqand Observa-

tory.9 What Al-Kashi described there can be considered as a

conversazione at the construction site. This sort of conversazione

seems to have been a common phenomenon, particularly in

fifteenth-century Khurasan. In the literary sources that give the

accounts of the laying out of buildings and the start of building

operations, geometers (muhandisdn) were always mentioned as

being present together with architects, masons, and/or other

sorts of artisans. 1 In the gatherings that were held to start the

construction works, apparently, the expertise of the geometers

was considered essential.

Another document of significance is an untitled treatise

about a geometric problem written by the celebrated poet-

philosopher-mathematician-astronomer Omar Khayyam

(1048-131).11 The solution of this very same problem later

occured in the form of an ornamental pattern in an anonymous

Persian treatise written exclusively for artisans, Fi taddkhul

al-ashkdl al-mutashdbiha 'aw mutawdfiqa (On interlocking similar or

correspondingfigures), hereafter Interlocking Figures. 12 At the end

of the untitled treatise, Omar Khayyam commented on what

gave him the incentive to write it:

If it were not for the highness of this meeting... and for the obligation to

the proposer of the question ... I would have been far away from this

wilderness.13

The present article expands on Omar Khayyam's untitled

treatise so as to establish that the meeting he attended was

indeed a conversazione with artisans, the proposer of the ques-

tion was most probably an architect-artisan, and, therefore,

"wilderness" refers to the field of architecture. To prove its

point, the present study follows the story of the special ornamen-

tal pattern that originated from a proof in Abu 'l-Wafai's

Geometric Constructions, brought into realization by Omar

Khayyam in this treatise, and several practical constructions of which are illustrated in Interlocking Figures. By studying this

unique example we also gain insight into how mathematicians and artisans collaborated and into the actual results of this collaboration. Our search starts with Geometric Constructions, in which Abu '1-Wafa' gave the account of the conversazione that he

attended in Bagdad.

Abu 'l-Wafd's book Geometric Constructions is a unique combination of practical geometry and theoretical geometry. As a work on applied

geometry, it is comprehensive and highly didactic; as a work on

pure geometry, it is the best one ever written by a Muslim

mathematician.14 It contains almost everything that an artisan would require of geometry and is organized in a very systematic manner. Abu 'l-Wafa"s objective in writing this book was to lead

artisans along the methodical path of theoretical geometry so

that they would achieve excellence in their works. At the

beginning of the chapter in which he discussed the conversazi-

one, he mentioned that dissecting the geometric figures was a

technique widely used by artisans and adds:

In this chapter we set the rules, all that are to be used by artisans,

because, according to the principles, they commit gross mistakes in

dissecting and composing [squares]. 15

In a book that was necessarily intended for artisans, Abu 'l-Wafa'

singled out this chapter and devoted it exclusively to artisans;

indeed, he treated it in a distinctive manner. When a geometer

proposed an algebraic solution to the problem of "composing a square from three squares," none of the artisans were

satisfied. They wanted to see the three squares being dissected

into parts and then reassembled into a single square. Realizing that the only way for artisans to accept a proof was to see it in

tangible shapes, Abu 'l-Wafa' thus offered an ingenious solution

to that effect. Evidently, the mission he assigned himself was

not to teach artisans the most advanced mathematical tech-

niques of his time, but to initiate a sound understanding of

geometry. During the same conversazione, Abu 'I-Wafa' provided the

artisans with a novel proof of the Pythagorean theorem. Here, he neither multiplied the lines, as Al-Khwarizmi did in his

proof, nor described the squares on lines as Euclid had. Those

notions would have been too abstract for artisans to come to

terms with. Instead, he dissected the given square into four

congruent right triangles rotating around a central square in

such a way that these parts could be reassembled into two

squares that corresponded to the sides of the triangles; so that if

one of the sides was known, the other could be found [Figure

la]. What Abu 'l-Wafa' displayed here reflects only his own

resourcefulness in finding a way to demonstrate an abstract

56 JSAH / 54:1, MARCH 1995

theorem to practical-minded artisans, not the general level of Muslim mathematics of his time.16

This proof, which was offered in a conversazione, was the starting point of the creation process of the pattern under consideration; the realization of it was later achieved by the efforts of Omar Khayyam, after another conversazione-the point that this study aims to demonstrate. In order to bridge the gap, the process in between is reconstructed hypothetically, supported by some physical evidence and mathematical inter- pretations, in the remaining part of this section.

The inherently ornamental quality of Abu 'l-Wafa"s figure, which was originally conceived for didactic purposes, appar- ently caught the attention of artisans. The revolving symmetry of the figure, they realized, could easily be translated into a dynamic decorative pattern. By joining four more of the congruent triangles along their hypotenuses to the original square, a larger square composed of four rhomboids could be obtained [Figure lb]. After this transformation, Abu '1-Wafa"s figure became one of the popular ornamental motifs that we now observe on a number of architectural monuments, such as the west iwan of the Masjid-iJami of Isfahan [Figure 2].

The rhomboid is one of the common components of the ornamental geometric patterns throughout the Muslim world, and it plays an important role in the subsequent discussion. It thus needs to be explained first. This kite-shaped figure was, and still is, generally known to artisans under the names in different languages that correspond to almond (hence, hereaf- ter it is referred to as such).17 When two non-isosceles congru- ent triangles are joined together along their longest sides (the

03

&' ' 5 60. [|L^UcO [|

r,^ -

I1^ ^

FIG. 2: Two variations of the motif of Abu 'I-Waf" 's proof: the west iwan of

Masjid-i Jami' of Isfahan (from Rassad, "Masjed-e Jame"' [see n. 43], 3).

hypotenuse, if it is a right triangle), the combined form is either a parallelogram [Figures 3a, 3b] or an almond [Figures 3c, 3d]. General properties of the parallelogram were well-known and widely discussed, particularly in Greek geometry; but the almond had not previously been a very popular topic in mathematical literature.18 On the other hand, artisans were familiar enough with the basic properties of the almond to make extensive use of them in their designs. These properties can be summarized as follows: the line on which the congruent triangles are joined becomes the primary diagonal and the symmetry axis of the almond; the two diagonals intersect each other perpendicularly, and the bisecting lines of the opposite angles meet each other on the axis; if the length of the shorter sides of the almond are marked on the longer sides, the intersection point of the bisecting lines is equidistant from the marked points and the comer of the shorter sides; the almond can thus be subdivided into three smaller almonds, two of them being congruent, by drawing lines from the intersection point to the three equidistant points [Figures 3e, 3f]. Such subdivision of the almond was one of the common tools of artisans to generate a variety of interlocking patterns of almonds, poly- gons, and polygonal stars.

Abu 'l-Wafai"s figure represents a general theorem that can be applied to any ratio between the two unequal sides of the almond. This ratio, at the same time, corresponds to the tangent of the angle of rotation. By definition, the following condition always exists: if the longer side of the almond is called x and the shorter one y, then the side of the central square is equal to x - y, and the side of the outer square is equal to x + y [Figure lb]. The neatness of this property apparently encour- aged artisans to explore the possible variations of Abu 'l-Wafa "s figure. It could simply be used as a single motif by choosing any ratio that they liked. The property also allowed artisans to generate compositions of various interlocking patterns, if the appropriate ratios were selected. Note that the main consider- ation for Muslim artisans in selecting a proportion was the flexibility it offered and the constraints of the geometric properties of the pattern, not the reputed superiority of certain systems.19

So far as can be observed in the published illustrations, most of the existing examples of the pattern have a ratio of 1:2 between the sides of the almond; for example, see Figure 2. These are all single motifs, but the reason why artisans favored this ratio was most likely the following consideration: when the sides of the almonds are related to each other by a ratio of 1:2, the side of the central square corresponds to the full length of shorter side, y, and the side of the outer square becomes 3y [Figure lc]; in this arrangement the subdivision of the almonds allowed artisans to generate a composition of interrelated squares and almonds by repeating the main square unit in either direction of the rotation angle [Figure 4].

OZDURAL: OMAR KHAYYAM AND THE ARTISANS 57

FIG. 3: a, b) General properties of the parallelo-

gram; c, d) General properties of the almond;

e, f) Subdivision of the almond into smaller

almonds.

What the ratio of 1:2 rendered was fine; but for artisans, no

achievement was final. Once a pattern was found, its variations

were exploited by generations of artisans, but, as far as we

know, the same composition was never repeated. Artisans were

always in need of fresh patterns to add to their repertoire. Abu

'l-Wafa"s proofwas a rich source; they could explore it in depth to discover new versions. The majority of the examples that we

can observe on existing buildings are from Persia and Khurasan

and it appears that the pattern of Abu 'l-Wafa"s proof was

particularly popular among the artisans of that area. Indeed,

most of the extant copies of Abu 'l-Wafa" s Geometric Construc-

tions are Persian translations.20

Hypothetically, we can imagine a clever artisan in Isfahan

envisaging a potential special ratio between the sides of the

almond, which makes it possible to generate a more elaborate

composition [Figure 5]. This in fact is another version of Abu

'l-Wafa-"s figure. The visualization is not very difficult; it re-

quires little to imagine that the subdivision of the primary almonds is performed by drawing a perpendicular through the

axis so that the longer sides of the secondary almonds are equal to the shorter sides of the primary ones, y, and the shorter sides

of the former to the side of the central square, x - y. The

realization of the pattern, one supposes though, is beyond the

reach of the artisan. To cope with that sort of a problem, he has

to be equipped with an advanced knowledge of mathematics.

Given his limited knowledge in practical geometry, the only

option open to him is to ask advice from mathematicians. Let

us suppose that he does so in a conversazione, and Omar

Khayyam responds in the following way [Figure 6]:

[What the artisan wants is to construct a right triangle, ERT, with the

necessary condition that, if it is dissected into a right triangle and an

almond, the following relations would exist: RI = BI; ER = EB = TI. Let

us suppose that the required triangle ERT is constructed. Erect the

58 JSAH / 54:1, MARCH 1995

perpendicular HR on the diagonal ET, then the triangle REH is

congruent to the triangle TIB (since corresponding angles are equal and

TI = ER). Hence RH = TB and EH = IB. But the triangle REK is similar

to the triangle RIL (since corresponding angles are equal). Then

ER: EK = RI: RL. But EK = HR and RL = HB and RI = EH. Then

ER: HR = EH: HB. Let the circle ADCBR be drawn having the point E

as its center and AE as its radius. Consequently, the specific question of

FIG. 4 (Top): The composition generated by the ornamental pattern of Abu '1-Waf" 's proof subdivided in accordance with the ratio of 1:2.

FIG. 5 (Bottom): The composition generated by the ornamental pattern of Abu

'I-Waf" 's proof subdivided in accordance with the special ratio.

the artisan can be formulated as a general problem of ratios:] to divide

the one-fourth AB of the circle ABCD by a point R into two parts such

that if RH is drawn perpendicular to the diameter BD, the ratio of AE to

RH is the same as EH to HB.21

With the final formulation of the problem, which Omar

Khayyam posed in the untitled treatise, the hypothetical recon- struction ends. After attending the meeting, apparently, he worked out the solution and wrote a treatise about it. In the

following section, so as to supply credence to the foregoing reconstruction, the treatise is analyzed with the aim of demon-

strating that he addressed it to artisans as well as to his learned

colleagues.

Omar Khayyam's treatise The untitled treatise by Omar Khayyam, which was discov- ered around 1960, is available in various languages and so far has only attracted the attention of historians of math- ematics.22 The treatise is about a problem for which the author offered a variety of solutions: a cubic equation, two

geometric constructions by means of conic sections, and a numerical interpolation in trigonometric tables. This prob- lem appeared to be of particular interest to Omar Khayyam as he wrote a whole essay on it. It is of critical importance to the history of mathematics as it motivated Omar Khayyam to make major contributions to algebra and number

theory.23 It is also of great significance to the history of Islamic art and architecture as it becomes the evidence of Omar Khayyam's involvement in the ornamental arts, and

FIG. 6: Hypothetical reconstruction of the artisan's problem concerning the

special ratio.

OZDURAL: OMAR KHAYYAM AND THE ARTISANS 59

j_ -11- > ) X el -- ^1 -> A

Ir-- l "I -> ) I < -)< J 1-1 V e-

4 r . < - v 1-1

thus indicates his familiarity with the problems related to architecture.

According to the historian Daoud S. Kasir, "Omar Khayyam followed the tradition of Muslim writers by pursuing mathemati- cal investigations only so far as they were needed to express and

interpret problems arising from investigations in such sciences as astronomy and surveying and from commercial transactions and inheritance law."24 This recently-discovered treatise allows us to add architecture and related arts to the fields of investiga- tion that Omar Khayyam was involved in.

The story of the pattern under consideration proceeds with the passage towards the end of the treatise in which Omar

Khayyam remarked on that particular meeting:

This is what passed through my mind considering my dissipated

thought, disturbed mind, and being occupied with facts which prevent

me from paying attention to these simple ideas. If it were not for the

highness of this meeting whose highness be lost forever and for the

obligation to the proposer of the question whom God will bless, I would

have been far away from this wilderness. My efforts are solely concen-

trated upon facts which are to me more important than those simple

ideas. My efforts are spent on those.25

Omar Khayyam's words give us some clues about what sort of a

meeting it was. His praises of the meeting may suggest that

somebody from the court was present.26 The impression we get is that the proposer of the question was a respected person; but, to our disappointment, Omar Khayyam did not mention his rank or his profession. He did say, however, that his feeling of

obligation towards that person prompted him to write this work. It would be difficult to imagine that such a technical

question leading to the problem under consideration could have been raised by royalty, unless, as seen in very rare cases, he

himself was a scientist or belonged to a craft.27 Omar Khayyam noted with some regret that, prior to the meeting, he was

occupied only with "facts" and paid no attention to "simple ideas." For a philosopher-scientist like him, "facts" were the

truths reached through philosophical or theoretical studies, and "simple ideas" were the mere practicalities of ordinary works. We can infer therefore that at this meeting Omar

Khayyam was introduced to a practical field that he felt

enthusiastic about. As the word "wilderness" suggests, he

considered that what has been opened to him is a new field to

explore. He seemed to enjoy acquainting himself with those

"simple ideas," but again to our disappointment, he did not

specifically mention that these were the practicalities of architec-

ture. As we examine the remarks he inserted along the course

of his analysis, however, it becomes increasingly clear that

Omar Khayyam's main concern in writing the treatise was to

make it useful for artisans. Omar Khayyam started the treatise by posing the problem

in the form of the aforementioned definition. After supposing

the desired division is performed, that is, AE: RH = EH : HB,

he stated that the construction is possible if a hyperbola is

drawn so that it passes through the points E and L and its

asymptotes are the lines TM and TK [Figure 7a]. But the

positions of the point L and the asymptote TK are not known.

He realized that completion of the construction would be

difficult because it "needs a few introductions from Conic

Sections." Instead of carrying out the process to the end, he cut it

short: "Those who know conics can, if they wish, finish it

later."28 Apparently, Omar Khayyam was not much interested

in accommodating mathematicians who already knew about

conic sections.

After the first unsuccessful attempt, Omar Khayyam tried an alternative approach to the problem. As he

introduced it, he made a point of being more concerned

with practical people:

This method also needs some elements of conic sections but in many

ways is much easier than the first method, and its ideas are more useful.29

Here he tried to find a method that involves as few conic

sections as possible, so that it would be less difficult for people other than mathematicians to grasp. "Useful" seems to be the

operative word. As long as both methods produce the correct

result, under which condition is "its ideas.. .more useful"? If

"ideas" are interpreted as "practical properties," then being

"useful" applies to a practical field.

Omar Khayyam started the analysis of this alternative method

by supposing again that the desired division has been made,

that is, AE: RH = EH: HB [Figure 7b]. He then proved ET =

ER + RH, or in his own words:

This analysis leads to a right triangle with the condition that the

hypotenuse is equal to the sum of one of the sides of the right angle and

the perpendicular to the hypotenuse.30

His intentions become more evident in the following passage:

This idea, that is, a triangle with mentioned properties, is very useful in

problems similar to this one. This triangle has other properties. We shall

mention some of them so that whoever studies this paper can benefit

from it in similar problems... .Another property of such a triangle is that

of the two sides of the right angle the longer one is equal to the sum

of the shorter one and the segment that the perpendicular separates

from the hypotenuse is toward the shorter side [that is, RT = ER +

EH].31

The solution of a geometrical problem can be "useful" only to

"similar cases of applications," not to "similar [geometrical]

problems." What Omar Khayyam meant, therefore, was "cases"

similar to the one he was dwelling upon. We can hence infer

that Omar Khayyam was dealing with this triangle with the aim

of making it "useful" to a field in which potential "similar cases"

were waiting to be applied. Architecture, with its related arts,

60 JSAH / 54:1, MARCH 1995

FIG. 7: a) Omar Khayyam's attempted solution

of the problem by means of conic sections

(adapted from Amir-Moez, "A Paper" [see n.

II], 324, fig. 2); b) Omar Khayyam's alternative

approach to the problem by using a right triangle

(adapted from Amir-Moez, "A Paper," 325,

fig. 3).

suggests itself as the field which would make the most use of such a variety of geometric properties.

Omar Khayyam started the algebraic analysis of the prob- lem with a note of apology, apparently directed to his nonmath- ematician readers:

As the intelligent mathematicians of the past have used notations of

algebraists in order to simplify the intuitive solutions, we shall also follow

them. But the notations of algebraists are not necessary. We can do just as well without them. However, with these notations, multiplications and

divisions will become easier.32

In his major work on algebra, which he wrote exclusively for his learned colleagues, he did not explain himself for using algebraic notations.

Omar Khayyam drew the triangle ABC and supposed that AC = AB + BD as the basis for the algebraic analysis of the problem [Figure 8].33 He assigned the "unknown," x, for BD and a rational length, 10, for AD and thus reduced the problem to the solution of a cubic equation:

x3 + 200x = 20x2 + 2,000.34 After achieving the equation of the triangle (hereafter referred to as Omar Khayyam's triangle), Omar Khayyam commented on the general issues of algebra. He defined and explained

algebraic terms, brought forward an outlined classification of various types of algebraic equations, gave a brief account on the works of previous mathematicians concerning cubic equations, and added:

But whenever cubes (x3) come in,...we need solid geometry, and

especially conics and conic sections because a cube is a solid... .For

people who do not know conics, certain instruments are used.35

The last remark concerning the use of certain instruments for executing conic sections is of crucial significance. In the context of his previous remarks it leaves no room for doubt that the practical people whom Omar Khayyam was addressing were actually artisans. This remark serves as the link between what Abu 'l-Wafa' proposed for constructions that involve conic sections and the instrument used for the construction of Omar Khayyam's triangle in Interlocking Figures.

In Geometric Constructions, Abu 'l-Wafa' offered mechanical solutions for the special problems, "the duplication of the cube" and "the trisection of the angle."36 Those were the problems that occupied quite a number of Greek mathematicians and, as the solutions required cubic equations, motivated the discovery of conic sections. The solutions they proposed either made use of conic sections or were reduced to mechanical procedures

OZDURAL: OMAR KHAYYAM AND THE ARTISANS 61

FIG. 8: The right triangle that serves as the basis

for the algebraic equation of Omar Khayyam

(adapted from Amir-Moez, "A Paper" [see n.

II1], 327, fig. 4).

that they called neusis (verging). In a verging procedure, a given segment-or two equal segments-is inserted between two

given straight or circular lines in such a way that the segment verges to a point. In a few cases certain devices were used, but

mostly it was performed by trial and error using rulers. Verging procedures were also known to Muslim mathematicians under the name "moving geometry." Abu 'l-Wafa"s treatment of

special problems was very suggestive. While borrowing from Greek verging solutions, he was careful in selecting the ones that were most convenient for artisans.37 The message he

apparently wanted to convey was that in dealing with cubic

equations, verging procedures are the most suitable means for artisans because they are both accurate and easy to operate. It can thus be inferred from Omar Khayyam's remark that he shared Abu 'l-Wafa"s opinion and was advising artisans to use

verging instruments. From Interlocking Figures, which was writ- ten for artisans, we also learn that the cubic equation of Omar

Khayyam's triangle was actually solved by the aid of a moving instrument called the "ruler-triangle." Considering Omar

Khayyam's previous remarks as well, it now seems safe to conclude that the "people" who he is concerned with and "who do not know conics" were in fact artisans. As he wrote the treatise in response to a question raised in the aforementioned

meeting, it would be logical to assume that the question was

asked by one of the artisans; since he was a respected person, most probably he was an architect. The meeting, therefore, can

be defined as a conversazione.

Thereafter, Omar Khayyam worked out the solution of the

cubic equation by means of conic sections [Figure 9a]: intersec-

tion of the hyperbola NDK, which has AC and EC as the

asymptotes, and the semicircle DKB at the point K. Conic

sections were used by mathematicians to reach theoretical

proofs, but never to obtain actual measurements. He promptly advised artisans to that effect:

By saying it is known in value I do not mean that its magnitude is known

because these two ideas are different. By being known in value I mean

what Euclid meant in the book of Constructions . That is, we can construct

a magnitude equal to it.38

To facilitate a practical construction of the problem, Omar

Khayyam offered an interpolated solution of the cubic

equation. He apparently assumed some artisans, like sur-

veyors, were familiar with trigonometric tables and angular measurements:

Whoever wants to know this in arithmetic, if he looks carefully, he will not

find a way to it because whatever is obtained by conic sections cannot be

62 JSAH / 54:1, MARCH 1995

_? ???

obtained by arithmetic. If the seeker is satisfied with an estimate, it is up

to him to look into the table of chords of Almagest, or the table of sines

and versed sines [that is, 1 - cosine] of Motamed Observatory . He should

find an arc in the table that is the ratio of sixty, which is supposed to be

half of the diameter of the circle, to the sine of that arc is the same as its

cosine to its versed sine. We shall find this arc about fifty-seven degrees,

of which the circle is three hundred and sixty, its sine about fifty pieces, its

versed sine about twenty-seven pieces and one-third piece, and its cosine

about thirty[-two] pieces and two-thirds piece. It is possible to compute

more carefully to the extent that the error would not be felt.39

Normally, Omar Khayyam should have ended the treatise

with the aforementioned closing remark; but afterwards he

offered an alternative solution by means of conic sections. He

apparently realized that the cubic equation he previously

proposed was rather a clumsy one, and offered a neater version

[Figure 9b]. Here the solution was more direct and reached by

intersecting the hyperbola AR with the semicircle EARC at the

point R. In this construction all the coefficients of the supposed

third-degree equation, which he did not cite, are taken as the

unit, 1. It corresponds to:

x3 +x2 + x-140 In light of the present study, it can be maintained that a

problem asked probably by an architect-artisan in a conversazi-

T E C

M

B L D

one prompted Omar Khayyam to write a treatise, which moti-

vated him to make a major contribution to the science of

algebra.41 He outlined his subsequent work, The Algebra, so

accurately that it appears to have been almost ready in his

mind; therefore, it should not have taken too long to write. The

Algebra is dated to ca. 1074, and he was invited to Isfahan by

Saljukid Sultan Malikshah in 1073 and was put in charge of the

new observatory.42 Thanks to the construction work taking

place in Masjid-i Jami'-possibly in the south dome-during that period, Isfahan was the hub of architectural activity and

certainly an ideal place for conversazioni. It is only natural to

expect that Omar Khayyam, the brilliant mathematician who had recently come to Isfahan, was invited to attend one of these

meetings. It seems likely therefore that he wrote the untitled

treatise shortly after 1073, that is, ca. 1074.43 It is thus not

farfetched to suppose that the question was asked by the architect who was at the time in charge of the construction-

possibly, the architect of the south dome, Abu 'l-Fath the son of Muhammad the treasurer.

After publishing the untitled treatise, it can be presumed, Omar Khayyam explained his findings to artisans in another

conversazione and there he offered the practical solution of the cubic equation by means of verging procedures. This appar- ently is the solution reported by the anonymous author of

FIG. 9: a) Omar Khayyam's solution of the cubic

equation by means of conic sections (adapted

from Amir-Moez, "A Paper," 332, [see n. II],

fig. 7); b) The alternative solution of the problem

by means of conic sections (adapted from Amir-

Moez, "A Paper," 336, fig. 10).

A

a

b

OZDURAL: OMAR KHAYYAM AND THE ARTISANS 63

l x

Interlocking Figures; but he recorded four more constructions

concerning the same problem. Those are all approximate solutions with varying degrees of accuracy yielding all but the

pattern under consideration. As the correct solution was avail-

able, why did artisans still produce the incorrect ones? The

following analysis seeks to answer this question.

The anonymous Persian treatise

Interlocking Figures, as a work on geometry, has been totally

ignored by historians of mathematics. For instance, Woepcke,

who published a comprehensive analysis of the Persian transla-

tion of Geometric Constructions based on the Paris manuscript,

made no mention of Interlocking Figures which followed Abu

'l1-Wafa"s work in the same manuscript.44 Some historians of

architecture, on the other hand, attribute a high value to its

discussions of ornamental geometry.45 The treatise can be

described as a collection of geometric constructions concerning

various procedures and ornamental patterns and a few instru-

ments used by artisans. A critical assessment of its mathematical

content, particularly when compared to Geometric Constructions,

indicates that it was the work of an artisan who had but a shallow

acquaintance with geometry, rather than of a geometer who

had been occupied with the ornamental arts.46 Its anonymous

author, to put matters succinctly, was deficient in the essential

knowledge and skills of geometry; his presentation lacked the

necessary organization, coherence, and clarity that are the

qualities normally expected from a geometer. In spite of all

these, his work, being the only known surviving authentic

written material on the subject of artisans' practice in geometry,

is a very valuable source of information for the history of Islamic

art and architecture.47

The recordings in Interlocking Figures illustrate with certainty

that the triangle discovered by Omar Khayyam was actually

used by artisans as an ornamental pattern. The most direct

reference to this triangle, albeit mistaken in authorship, is

found in the following passage:

Correlations involved in this drawing concern conic [sections]. The

objective of it consists in constructing a right triangle in such a way that

the sum of the perpendicular and the shorter side is equal to the

hypotenuse. Ibn Haytham wrote a treatise on the construction of such a

triangle, and there he described the conic sections, which turned out to

be a hyperbola and a parabola.48

As we have seen earlier, it was "a hyperbola and a circle," not "a

hyperbola and a parabola," the intersection of which solved the

cubic equation; it was Omar Khayyam, not Ibn al-Haytham

(965-1040), who wrote a treatise on the problem that the

anonymous author defined.49 These misquotations would al-

most certainly not have occurred had the anonymous author

read Omar Khayyam's treatise. His source of information was

apparently an oral one. He should not have mistaken

"Khayyam" for "Haytham" had he seen it in writing; but these

names do sound alike. It can thus be inferred that he probably heard about this problem during a conversazione but recalled it

mistakenly.50 While Omar Khayyam's treatise was discussed

during this gathering, it would be reasonable to assume that he

was a mathematician who explained the solution of the prob-

lem and its construction.

The anonymous author gave more information, in his

disorderly and unknowing way, about the construction of the

pattern based on Omar Khayyam's triangle:

Here, the objective can be achieved by the aid of a "ruler-triangle."

As mentioned above, the objective of our drawing is four conical

figures [that is, almonds, which he calls turunj (orange) at other

places] with two right angles that surround an equilateral right-

angled quadrilateral [that is, a square]. Such conical quadrilaterals

[that is, almonds] AIHK, CHMN, DMLX, and BLKO describe the

four-cornered [that is, square] KHML [Figure 10]. Since the corner

H of the quadrilateral [that is, square] consists of two perpendicular

lines, then KH and [H]D are necessarily straight, but the triangle

AKC is right-angled and equal to the triangle CHD. This triangle is

right-angled too, because it is inscribed in the semicircle. Therefore,

the point H should be found on the arc [C]E. If the corner F on our

ruler is perpendicular, then the side AB is both perpendicular and

corresponds to the side AB [that is, AC, a mistake of carelessness] of

the square. And Allah knows best.51

What he set as the objective was to construct a pattern

composed of four of Omar Khayyam's triangles that sur-

round a central square, that is, the pattern under consider-

ation. What he described, however, amounted only to an

incomplete proof of its properties. Apparently, his recollec-

tion failed him in achieving the objective. When his quasi-

proof was compared with the proofs given by Omar

Khayyam, it becomes painfully transparent that our author

was only pretending to be a mathematician.52 Omar

Khayyam followed a logical sequence of theorems accord-

ing to a predetermined plan in order to reach the required

conclusion; whereas the anonymous author did not seem to

have a plan to follow or, for that matter, a conclusion to

reach. By using complicated terms instead of the ordinary

ones, he only exhibited his vanity. It is not difficult to understand why the anonymous author

was so confused. Conic sections was too advanced a topic for

artisans of the time to fully comprehend. Despite the ambigu-

ity, however, his description provides us with sufficient informa-

tion to reconstruct what he failed to achieve. "If the comer F on

our ruler is perpendicular," he said, then, the ruler must be the

scale on the perpendicular leg of the triangle that he is

referring to [Figure 10]. When the triangle is slid along the side

AB, it cuts the semicircle at a point, H, and FH + HG = AC =

AB = CD. Since this condition is true for every position of H,

64 JSAH / 54:1, MARCH 1995

FIG. 10: The construction of the pattern of

Omar Khayyam's triangle by means of the

verging procedure in Interlocking Figures (adapted

from Bulatov, Geometricheskaia garmonizatsiia

[seen. 12], 342, fig. 36).

the anonymous author was unable to realize the construction of the pattern. He apparently failed to remember the following step: in order to fix the position of the point H on the semicircle, FH is required to be equal to HC; only then the condition of the problem would be satisfied, that is, HC + HG = CD. The problem can thus be reduced to a verging procedure so as to determine the position of the point H at which HC = HF.

Omar Khayyam said, "For people who do not know conics, certain instruments are used." Indeed, the "ruler-triangle" turns out to be precisely the verging instrument to perform the construction of the pattern of Omar Khayyam's triangle. The following is the reconstruction of the verging procedure that was presumably proposed by Omar Khayyam in Isfahan and transferred from one conversazione to another until it reached the one that the anonymous author attended, probably in thirteenth-century Diyarbakir:

[Draw a semicircle on the given line AB and place a straightedge along

the line. Slide a right-angled triangle, which has a scale on its perpendicu-

lar leg, along the straightedge so that it always intersects the semicircle at

a point H. With an additional ruler measure and simultaneously

compare the distances HF and HC. Repeat this process until you reach a

position at which HF = HC. Mark this position of the point H and

complete the pattern.]53

The anonymous author was apparently unfamiliar with

verging procedures which, as Abu '1-Wafa' indicated earlier, were the most convenient means for artisans in coping with

problems that involved conic sections. However, it can be said

that his ignorance of this matter was not shared by all artisans.

The fact that Omar Khayyam's solution occurred in Interlocking Figures, probably around 150 years later, points to the existence of at least a small number of scrupulous artisans who were

willing to follow the advice of mathematicians.

The anonymous author described four more methods to construct the pattern in question. Three of these were pre- sented on the same drawing, but he did not mention elsewhere that these drawings, and Omar Khayyam's solution as well, were meant to produce the very same pattern. His silence

strongly suggests that he was not the author of any of these methods but was simply recording the solutions worked out by other artisans.54

He started the first construction by assigning an arbitrary length, AD, to the diagonal of a square [Figure 11].55 He marked the point C on a horizontal line at a distance of 2AD and extended the line CD until it met the perpendicular at the

point E. He then marked the point H on the perpendicular at a distance of 2AC from the point E and drew the line CH. From the given point K, he drew a line parallel to CH to determine the position of the point L, which concluded the construction.

OZDURAL: OMAR KHAYYAM AND THE ARTISANS 65

What he failed to mention is that all the steps of this construc- tion are in fact relevant for the execution of the pattern. The

angle of rotation corresponds to CG, to which the diagonal LR of the primary almond is parallel; half of the angle of rotation

corresponds to CE, to which the diagonal LP of the secondary almond is parallel. This construction turns out to be quite a successful approximation of Omar Khayyam's triangle. The

angle of rotation it yields deviates only 0.2 per cent from the theoretical value.

One of the methods in the second drawing was virtually identical to the construction described above. Here, he started the construction by assigning an arbitrary length, BC, to the

side of a square [Figure 12a]; the rest of the process depends precisely on the same geometrical procedure as the previous one [Figure 11].56 Such a slight variation was enough for our author to be confused and regard these two as different methods of construction.

In the third method, the position of the point Q was

determined as the arithmetic mean between LP, half of the side

of the square, and LZ, half of the diagonal of the square [Figure 12b]. The deviation between the result of this and the real value, 1.4 per cent, is rather large for a geometer to tolerate. The

fourth method produced yet a larger deviation, 2.9 per cent,

FIG. I 1: One of the approximate methods of

constructing the pattern of Omar Khayyam's

triangle in Interlocking Figures (adapted from

Bulatov, Geometricheskaia garmonizatsiia [see n.

12], 340, fig. 33).

which is not acceptable even for an artisan if he is meticulous

[Figure 12c]. Here, the length LX was found by intersecting the

quartercircle AMK and the semicircle KML at the point M,

erecting the perpendicular MN to the side KL, and making LX = 2LN. The whole process amounts to 3LX = 2KX. Whoever was the author of this last method seems to have had a

unique talent in complicating simple matters. Had this method been used, Omar Khayyam's triangle would hardly be recogniz- able, and to generate a composition out of it would have been an impossible task.

The existence of these approximate methods suggests that

the verging construction of the pattern in question was either unknown to, or not accepted by, most of the artisans. It might be that they did not have access to Omar Khayyam's solution, had not comprehended it fully as indicated by the anonymous author, or were showing their preference for the traditional methods instead of trying an unusual technique, however

simple and correct it was. Moving geometry was considered

inadmissible by the majority of Muslim mathematicians, let

alone the conservative-minded artisans. Whatever the reason

might be, it can be inferred that a conversazione, as a verbal

mode of transmitting knowledge, was a convenient means for

artisans but it was not always effective and productive in

/

!

66 JSAH / 54:1, MARCH 1995

C

b

I I

- -

\ \ \ \

a

FIG. 12: Three of the approximate methods of

constructing the pattern of Omar Khayyam's

triangle in Interlocking Figures (adapted from

Bulatov, Geometricheskaia garmonizatsiia [see n.

12], 338, fig. 28).

promoting new ideas. Not all artisans were receptive to advice

coming from mathematicians (say, intellectuals of today; appar- ently not much has changed since then). This may also explain why the young Ottoman geometer seemed so displeased with the artisans of his time.

Conclusion

The history and interpretation of Islamic architecture is full of

geometrical issues, and some of these are still unresolved. The

present study illustrates a case that proves to be informative to both the history of architecture and the history of mathematics. As the number of studies in depth on mathematical works of a similar nature increases, one hopes, more of these issues will be resolved.57

Primarily, this study aims at providing evidence for the existence of conversazioni, at which mathematicians and artisans collaborated to find solutions to the problems concerning the application of geometry to architecture and its related arts. The untitled treatise of Omar Khayyam proves to be a convincing document in this respect. Other

points that the study draws attention to can be summarized as follows:

a) Some mathematicians, such as Abu 'l-Wafa' and Omar

Khayyam, were more than willing to offer their expertise to

artisans; in the particular case of cubic equations, their advice to artisans was to utilize verging procedures;

b) Interlocking Figures, an anonymous work on ornamental

geometry that was presumably written by an artisan, does not, as a technical work, deserve the high praise it receives from some scholars;

c) The information provided by the anonymous author

suggests that a verbal mode of transmitting knowledge through conversazioni prevailed among artisans, but they were not always informed about, or receptive to, mathematicians' advice.

Appendix A list of publications and commentaries on Abu 'l1-Wafa"s book, On What the Artisan Requires of Geometric Constructions:

1. Mashhad, Rida 37 (Persian), end of the tenth or the early eleventh century. The earliest extant manuscript that was

prepared at the request of Abu Mansuir Baha' al-Dawla, the

Bujid ruler of Persia from 998 to 1013, by a translator whose name would perhaps have been ascertainable if the manuscript had not been defective at the end; C. A. Storey, Persian

Literature, 2 vols. (London, 1972), 2:2-3. As the date can be fixed to 998-1013, this manuscript is possibly one of the two original Persian translations of Abu 'l1-Wafa"s work, either prepared by Najmaddin Mahmfid Shah or Abu

OZDURAL: OMAR KHAYYAM AND THE ARTISANS 67

Ishaq ibn Abdallah Kfitibani; see the Paris manuscript, appen- dix no. 6.

2. Tahran, Danishgah 2876 (Persian), the eleventh or twelfth

century; Fuad Sezgin, Geschichte des arabischen Schrifttums, 7 vols.

(Leiden, 1974), 5:324.

3. Mashhad, Rida 5357/139 (Arabic commentary), by Ka-

maladdin Mfisa ibn Muhammad ibn Man'a (1175-1242), the

early thirteenth century. According to Ibn Khallikan, Ibn Man'a

highly extolled Abu 'I-Wafa' and possessed a number of his

books; Ibn Khallikan, Biographical Dictionary, trans. Mac Guckin

de Slane, 4 vols. (London, 1868), 3:320.

4. Cairo, Dar, riyada 260 (Arabic), the thirteenth century;

Sezgin, Geschichte (Leiden, 1978), 7:408.

5. Istanbul, Ayasofya 2753 (Arabic), the early fifteenth cen-

tury (Siileymaniye Kutiiphanesi, microfilm archive no. 375).

This manuscript was presented to Ulugh Beg in Samarqand. As

mentioned on the tide page, Abu 'l1-Wafa"s book was dedicated

to Mawlana al-Malik Shahanshah al-Ajall al-Man-suir Baha'

al-Dawla. He could have assumed the tide of Shdahadnshah only after Persia fell under his reign in 998. Abu '1-Wafa' died either

in 997 according to Ibn Khallikan, or in June 998 according to

Ibn al-Qifti; Ibn Khallikan, Biographical Dictionary, 321; Ibn

al-Qifti, Ta'nih al-hukamd', ed. J. Lippert (Lepzig, 1903), 288.

Thus, it seems more likely that the manuscript was not copied from the original work but translated back into Arabic from one

of the earlier Persian translations, possibly the manuscript

Mashhad, Rida 37 (appendix no. 1).

Ayasofya 2753 includes annotations apparently added by the

copyist himselfwho, in all probability, was one of the mathema-

ticians gathered in Samarqand during Ulugh Beg's reign.

Those annotations, which particularly deal with theoretical

proofs of Abu 'l1-Wafa"s constructions, give us a better insight

into the practical approach of Abu '1-Wafa'. As a plausible

possibility, it can be suggested that 'Ali Kushchu, who was the

last one in charge of the observatory, took the manuscript with

him to Istanbul after the death of his friend Ulugh Beg and

placed it in the library of Ayasofya Madrasa, where he served as

professor for the rest of his life.

6. Paris, Bibliotheque Nationale, ancien fonds persan 169

(Persian), the early seventeenth century; Storey, Persian Litera-

ture, 2:2. According to the information given by the translator,

Abu Ishaq ibn 'Abdallah Kftibamni, he made use of an earlier

translation prepared by his contemporary Najmaddin Mahmud,

a talented mathematician. Bulatov dates this manuscript to the

early eleventh century by arguing that Najmaddin Mahmfid

had lived during that time; Midhat Bulatov, Geometricheskaia

garmonizatsiia arkhitekture SredneiAzii IX-XVvv (Moscow, 1978),

51-52. Bulatov's argument is very convincing; but it applies to

the original translation, possibly the manuscript Mashhad,

Rida 37, certainly not to this late copy. 7. Mashhad, Rida 144 (Persian commentary), by Muhammad

Baqir Zain al-'Abidin (active in 1637-38), the seventeenth

century. 8. Milan, Biblioteca Ambrosiana, arab. 68 (Arabic). 9. Uppsala, Tomberg 324 (Arabic), 933-34 [?]. This manu-

script, although virtually identical to Abu 'l1-Wafa 's work, has a

very unusual title, Kitdb al-hiyal al-rahdniya wa-l-asrdr al-tabfa 'yafi

daqdiq al-ashkdl al-handasiya (The book of pneumatics and the

natural mystery in subtleties of the geometric figures ), and is wrongly attributed to Al-Farabi (d. 950). The mistake is corrected by Toomer and by Hogendijk; G. J. Toomer, Diocles on Burning Mirrors (Berlin, Heidelberg, and New York, 1976), 23; J. P.

Hogendijk, Ibn al-Haytham's 'Completion of Conics' (New York,

Berlin, Heidelberg, and Tokyo, 1985), 62.

10. Francois Woepcke, "Analyse et extrait d'un recueil de

constructions geometriques parAboul Wafa',"JournalAsiatique 5,

no. 5 (1855): 218-56,309-59; the Paris manuscript. Mainly due

to the fact that Abu 'l1-Wafa 's name and title, professor, was cited

several times in the text of the Paris manuscript, Woepcke assumes that Geometric Constructions was actually composed by a

student of Abu '1-Wafa'. Since the Paris manuscript was copied from an earlier Persian translation, this fact can be traced back

to the earliest one, possibly the manuscript Mashhad, Rida 37

(appendix no. 1), and Najmaddin Mahmfid might indeed be a

student of Abu 'l-Wafa'. While translating the book, it seems, he

felt entitled to insert the name and title of Abu 'l-Wafa' in the

text.

11. Heinrich Suter, "Das Buch der geometrischen Konstruk-

tionen des Abu '1 Wefa'," Abhandlungen zur Geschichte der Natur-

wissenschaften und Medizin 4 (1922): 94-100; the Milan manu-

script. 12. S. A. Krasnova, "Abu-l-Vafa al-Buzjani, 'Kniga o tom chto

neobkhodimo remeslenniku iz geometricheskikh postroeniy'," Fiziko-matematicheskie nauki v stranakh Vostoka 1 (1966): 42-140;

the Istanbul manuscript. 13. S. A. Krasnova and A. Kubesov, Al-Farabi, matematichekie

traktati (Alma-Ata, 1972); the Uppsala manuscript. The authors

acknowledge the close resemblance between the Uppsala manu-

script and Abu 'l-Wafa"s work but, instead of questioning the

authorship of the manuscript, they suggest the latter was

perhaps based on the former.

Notes For the transliteration of the Arabic, Turkish, and Russian words that appear

in the present text, the system adopted by The Encyclopedia of Islam (2d ed.) is

followed. For the sake of convenience, subscript bars are omitted and "j" is

substituted for "dj" and "z," "q" for "k," and "ch" for "c." I would like to express my gratitude to Taner Avci for his assistance in the translations of the Arabic

passages. I CaTer Efendi, Risdle-i Mimariyye, an Early-Seventeenth-Century Ottoman Trea-

tise on Architecture, trans. Howard Crane (Leiden, New York, Copenhagen, and

Cologne, 1987), 28; see also Orhan SenfGorkyay, "Risale-i Mimariyye-Mimar Mehmed Aga-Eserleri" (Treatise on architecture, the architect Mehmed Agha and his works), in Ismail Hakkz Uzuncarszlzya Armagan (Ankara, 1976), 113-215.

68 JSAH / 54:1, MARCH 1995

The only manuscript of Risale-i Mi 'mdriyye exists in the Topkapi Sarayi Museum Library, YY339.

2 Al-Nadim, who gives full information about Abu 'l1-Wafa 's life and all his works, does not cite Geometric Constructions in Fihrist, which was completed in 988-89; Al-Nadim, The Fihrist, trans. Bayard Dodge, 2 vols. (New York and London, 1970), 1 :xxi. It can thus be concluded that Geometric Constructions was written sometime between 988-89 and 997 or 998, the two alternative dates given for Abu 'l-Wafa 's death. The original work in Arabic, which was composed of thirteen chapters, is not extant today. All the existing copies are in one way or another incomplete. The fact that most of the copies are Persian translations indicates that it was popular in Persia. For the list of all the available publications and commentaries, see appendix. In the present text, the references are made to the Istanbul manuscript, Ayasofya 2753, which includes fifteenth-century annotations; see appendix, no. 5.

3 " 'A'lamu 'inna al-sunna' ya' maluina al-'ashkali fil mudawwaraat wa 'alayha bil qismah ... fa' inna al-sina' al-jayida 'indahum an yabtadi al-sana' bidarb min al-nadhir al-qarniya yasilu ila miqdar dil' al-mukhammas 'aw al-musaddas 'aw al-mu'ashshar 'aw ghayruha min al-'ashkal kama bayannah6 fi hadha al-kitab ..." Ayasofya 2753, 21.

4 "Fainna al-sana' 'aradahf ma yuqrab 'alayhi al-'amalf wa yazhar lahu sihhat ma narahu fi al-his wa '1-mushahada wala yubali bil barahin al-khututiya wa 'l1-muhandis idha qama lahu al-burhan 'ala al-shay bil tawassum lam yas' al sihhat dhalika bil mushahada 'aw lam yassuhhu 'ala anna lashakka inna jami' ma yarahiu al-sana' innama huwa makhfdh mimma ya'malahu al-muhandis 'awwala qama' al-burhan ala sihhatahu fainna al-sana' wa 'l-masah innama ya'khidu min al shay zibtadahu wala yufakkiru fi al-wujfuh allati tathbitu sihhat dhalika bihi waliajil dhalika qat yaqa' al-ghalat wa 'l-khata fa' amma al-muhandis faqad 'ulima sihhat ma nuridhu bil barahin idha kana huwa al-mustakhrij lilma'ani al-lati 'amala 'alayhi al-sana' wa 'l-masah . .." Ayasofya 2753, 52-53.

3"Conversazione: a meeting for conversation, esp. about art, literature, science, etc." Longman Dictionary of Contemporary English (Bath, 1978), 241.

6"Laqad hadartu fi ba'ad al-majalis wafihi jama'a min al-sunna' wa '1 muhandisin..." Ayasofya 2753, 53.

7 This point is discussed in more detail by Daoud S. Kasir in The Algebra of Omar Khayyam (New York, 1931), 18-19.

8 Renata Holod argues that traditional modes of transmitting architectural knowledge through example and through verbal or visual notation do not allow for the explanation of those innovations, and suggests that a more thorough awareness of the history of engineering may be necessary; Renata Holod, "Text, Plan and Building: On the Transmission of Architectural Knowledge," in Theories and Principles of Design in the Architecture of Islamic Societies, ed. M. B. Sevcenko (Cambridge, Mass., 1988), 1-2, 11, n. 4. She cites the works of Banui Musa ibn Shakir as the examples to be studied. Although one of the three brothers was a technologist as well, all were prominent mathematicians, and, like all others, they were referred to as muhandis (geometer; see nn. 4, 6). The word muhandis, which originally meant "one who does geometry," in the modern usage corresponds to engineer with a narrower meaning, "one who applies geometry." It seems more reasonable to think of muhandis in its original meaning rather than engineer, unless it is used in late sources. Therefore, Holod's suggestion and my proposition actually amount to the same point, that is, a thorough awareness of the history of mathematics may prove to be useful in explaining some of the innovations that we observe in major monuments of the Islamic world. For a probable illustration of this point, see n. 43.

9 Aydin Sayili, Giydth al-Din al-Kdshf's Letter on Ulugh Beg and the Scientific Activity Patronized by Him (Ankara, 1960), 101-2; see also E. S. Kennedy, "A Letter ofJamshid al-Kashi to his Father," Orientalia 29 (1960): 191-213. The debate took place in the presence of Timurid Prince Ulugh Beg, the founder of the observatory, and other high-ranking people, but they did not take part in it. Various aspects of this highly interesting debate is discussed in Alpay Ozdural, "Giyaseddin Jemshid el-Kashi and Stalactites," Middle East Technical University,

Journal of the Faculty ofArchitecture 10(1990): 34-35. 0 Bernard O'Kane, Timurid Architecture in Khurasan (Costa Mesa, 1987),

37-38. O'Kane expresses difficulty in interpreting the prominence given to muhandis in these accounts (he gathers a list of twelve citations), and suggests that it would more closely approximate surveyor rather than architect. When Al-Kashi referred to mathematicians, he used the title muhandis. There appears no reason why the very same word should assume a different meaning when it occurred in other contemporary sources that include the accounts of building operations. It

thus seems logical to take muhandis in its original usage, geometer, in this context too; see n. 8.

" Ali R. Amir-Moez, "A paper of Omar Khayyam," Scripta Mathematica 26 (1963): 323-37. See also G. H. Musahib, Hakim Omare Khayyam as an Algebraist, Persian translation, Arabic edition, facsimile of the manuscript (Teheran, 1960); S. A. Krasnova and B. A. Rosenfeld, "Omar Khayyam pervy algebraicheskiy traktat," Istoriko-matematicheskie issledovaniya (1963), fascicle 15.

12 The only manuscript of this treatise exists in Paris, Bibliotheque Nationale, ancien fonds persan manuscript 169, following the Persian translation of Abu 'l1-Wafaf 's Geometric Constructions (see appendix, no. 6). It is published in Midhat S. Bulatov, Geometricheskaia garmonizatsiia v arkhitekture Srednei Azii IX-XV vv (Moscow, 1978), appendix 2, 325-54. The second word of the title, which Bulatov reads as maddkhil (introduction), can also be read as taddkhul (interlock- ing). I adopted the latter since it answers the content of the treatise better; hereafter the treatise is cited as Interlocking Figures, and references concerning the text and figures are made to Bulatov's publication.

13 Amir-Moez, "A Paper," 336. 4 M. Soussi, "'Ilm al-handasa," Encyclopedia of Islam 3 (1982): 414. 5 "Wa naj'al laha qawanin narja' ilayha fa'anna jami ma yasta'miluhu al-

sunna' fi hadha al-bab bima 'usil ya'mal 'alayhi wa li'ajil dhalika taq'a al-ghalat al-kathir fama yuqassimfinahuwa yurrattibunahu.. ." Ayasofya 2753 (see appen- dix, no. 5), 47.

16 Some scholars are of the opinion that Abu 'l-Wafa' borrowed this figure from Indian mathematics; Moritz Cantor, Vorlesungen uber Geschichte derMathema- tik, 4 vols. (reprint, New York and Stuttgart, 1965), 1:744-45; Adolf P. Youschkevitch, Les mathematiques arabes, trans. M. Cazenave and K. Jauishe (Paris, 1976), 110. It seems more reasonable to consider it as a prompt response to artisans' requirements. He might, however, have been influenced by Thabit ibn Qurra's (836-901) method of dissection; see Thomas L. Heath, Euclid's Elements, 3 vols. (New York, 1956), 1:365.

17 According to the information given by Al-Kashi in the chapter on architec- ture in his book, this figure was called ladwza (almond) in fifteenth-century Persia and Khurasan; Ghiyath al-Din Jamshid al-Kashi, Miftdh al-hisab (Key for arith- metic), ed. Nabulsi Nader (Damascus, 1977), 220, 382. As Al-Jazari informs us, the same Arabic name was used by Artukid artisans in the thirteenth century; Ibn al-Razzaz al-Jazari, The Book of Knowledge of Ingenious Mechanical Devices, trans. Donald R. Hill (Dordrecht and Boston, 1974), passim. Its current Turkish name among stonemasons of Anatolia, based on my own observations, is badem (almond). In another part of the Muslim world, Morocco, traditional artisans of today continue to call it lawza; Andre Paccard, Le Maroc et lartisanant tradionnel islamique dans l'architecture (Annecy, 1983), passim. It has to be noted, however, that the anonymous author of Interlocking Figures used two different names for this figure, turunj (orange) and conical quadrilateral; Bulatov, Geometricheskaia garmonizatsiia (see n. 12), 339, 341-42, 344.

18AI-Kashi briefly explained how to calculate the area of the almond and treated it separately as one of the key elements of muqarnas; Al-Kashi, Miftdh al- hisdb, 222-25, 381-90. For the English translation of Al-Kashi's section on muqarnas, see Ozdural, "El-Kashi and Stalactites" (see n. 9), 37-43.

19 For further discussion of this point, see Lisa Golombek and Donald Wilber, The Timurid Architecture of Iran and Turan (Princeton, 1988), 137-38.

20 See n. 2 and appendix. 21 Amir-Moez, "A Paper" (see n. 11), 323. 22 Seen. 11. 23 Amir-Moez, "A Paper," 323. 24 Kasir, The Algebra (see n. 7), 2. 25 Amir-Moez, "A Paper," 336. 26 As the debate that Al-Kashi related indicates, it was not unusual for royalty

to show interest in such meetings, particularly when the discussions concerned architectural problems of the buildings they sponsored; see n. 9. It would not be surprising, therefore, if Saljukid Sultan Malikshah or his vizier, Nizam al-Mulk (who, according to an unconfirmed story, was an old friend of Omar Khayyam), was present at that meeting given that they were the sponsors of the south dome of the Masjid-iJami' of Isfahan.

27 For instance, Ulugh Beg was a well-known astronomer, and Ottoman Sultan Abdulhamid II was an accomplished carpenter. We do not have any such information, however, concerning contemporary Seljukid royalty.

28 Amir-Moez, "A Paper" (see n. 11), 325. 29 Amir-Moez, "A Paper," 325.

OZDURAL: OMAR KHAYYAM AND THE ARTISANS 69

30 According to Euclid's Elements 3.16, the tangent RT is drawn at the point R,

and the line EB is extended until it intersects the tangent at the point T. Draw

the line RE. Since the angle ERT is a right angle, the line RH is perpendicular to

the hypotenuse ET and, according to Elements 6.8, EH: HR = HR: HT. Hence, HR2 = EH. HT. Similarly, HR2 = DH . HB. Therefore, DH . HB = EH. HT.

According to Elements 6.16, DH: EH = HT: HB. By decomposition of ratios, ED: EH = BT: BH. But, AE: RH = EH: HB is the given condition. By two

exchanges, AE: EH = RH: HB. But, AE = DE. Thus, RH: HB = BT: HB.

Therefore, according to Elements 5.9, RH = BT. But, RE = EB. Consequently, ER + RH = ET; Amir-Moez, "A Paper" (see n. 11), 325-26.

31 Since ED: EH = BT: BH, by composition of ratios DH: EH = TH: HB.

By exchange of ratios, DH: HT = EH: HB. But, EH: HB = ER: RH. Since the

triangles ERH and RHT are similar, RE: RH = RT: HT. Thus, RT: HT = DH:

HT. Therefore, RT = HD. But, HD = ER + EH. Consequently, RT = ER +

EH; Amir-Moez, "A Paper," 326. 32 Amir-Mo6z, "A Paper," 327. 33 Dotted lines are added to facilitate the comparision with Fig. 6. 34 If BD = x and AD = 10, then AB2 = x 2 + 100, according to Euclid, Elements

1.47. Since the triangles ABC and ABD are similar, AC : AB = AB : AD and

AB2 = AC AD. Then, AC = AB2/AD = 10 + x2/10. ButAC = AB + BD. Then, AB + BD = 10 + x2/ 0. If BD is subtracted, then AB = 10 + x2/10 - x. If this

equation is multiplied by itself, then 100 + 3x2 + x4/ 100 - 20x + x3/5 = 100 +

x2. When algebraic procedures are applied, the equation becomes 2x2 +

x4/100 = 20x + x3/5. When every term is divided by x, it becomes x3/100 + 2x =

x2/5 + 20. When it is multiplied by 100, we obtain the equation x3 + 200x =

20x 2 + 2,000; Amir-Moez, "A Paper," 327-28. 35 Amir-Moez, "A Paper," 329. Several mathematicians attempted to devise

an instrument, which they called a perfect compass, to perform drawings of conic

sections; see F. Woepcke, 'Trois trait6s arabes sur le compas perfait," Notices et

extraits 22 (1874): 1-176. Those instruments, which hardly had any practical value, were actually intented for mathematicians, not "for people who do not

know conics"; therefore, what Omar Khayyam was referring to could not be a

perfect compass. Omar Khayyam concluded his comments on algebra by providing a brief

outline of his subsequent work, The Algebra: If the opportunity arises and I can succeed, I shall bring all of these fourteen

forms with all branches and cases, and how to distinguish whatever is possible or impossible so that a paper, containing elements which are greatly useful in

this art will be prepared; Amir-Mo6z, "A Paper," 331. 36 Ayasofya 2753 (see appendix, no. 5), 13-14, 2.19, 20, and 22. 37 The Greek solutions that Abu '1-Wafa' borrowed are: Heron of Alexandria,

Mechanics, 2.11; Archimedes, Book of Lemmata, 18; Pappus, Collections, 4.36-42.

For more information on verging solutions in Greek mathematics, see T. L.

Heath, A History of Greek Mathematics, 2 vols. (Oxford, 1965), 1:235-70; T. L.

Heath, The Works of Archimedes (New York, 1921), c-cxxii. 38 Amir-Moez, "A Paper" (see n. 11), 333. 39 Amir-Moez, "A Paper," 336. In accordance with Omar Khayyam's advice,

the angle BAC (Fig. 8) can be determined by more precise calculation as:

57? 03i 53ii 34iii.

The trigonometric relation that he describes corresponds to:

1: sin BAC = cos BAC: 1 - cos BAC. 40 If the shorter side of the triangle, AB, is taken again as 1 but the

perpendicular BD is designated as x, then the equation becomes:

x3 + 2x2 = 2. 41 See n. 35. 42 Youschkevitch, Mathimatiques arabes (see n. 16), 94. 43 Oleg Grabar, following the late Eric Schroeder, suggests that possibly

Omar Khayyam was behind the conceptual thinking that created the north dome of the Masjid-i Jami' of Isfahan (1088, the first known example of

geometric decorations on dome surfaces); Oleg Grabar, The Great Mosque of Isfahan (New York and London, 1990), 85, n. 5. Since the application of the

pentagram on the surface of the dome required the knowledge of spherical trigonometry in order to transform the straight lines of the two-dimensional schema into curves, it is plausible to think that Omar Khayyam, who was well

versed in mathematics and astronomy, and who had a delightful imagination, was possibly the one who conceived of such an idea. It can reasonably be

assumed that, between ca. 1074 and 1088, Omar Khayyam had attended several conversazioni and had acquired a familiarity with architecture. To claim his

authorship of the design of the north dome appears now more likely than when Grabar suggested it as an "unverifiable but attractive" possibility. It can also be noted with interest that the proportion of the north dome is almost precisely equal to the ratio of Omar Khayyam's triangle: the diameter at the springing level/the height from the springing level to the apex = 1028 cm/666 cm = tan 57? 03i 45ii; the measurements are obtained from the photogrammetric survey of the monument in Rassad Survey Company, "Masjed-e Jame' Esfahan," published paper presented to Symposium on the Photogrammetric Survey of Ancient

Monuments (Athens, 1974), 13. With its superbly arranged proportions, finely articulated hierarchy of structure, and richly outlined muqarnas, the space underneath the astounding north dome is certainly one of the greatest achievements of the architecture of the Islamic world. If the foregoing argument is indeed true, then what we observe here is the inspiring contribution of Omar Khayyam, a very talented mathematician and a brilliant poet, to architecture.

44 See appendix, no. 10. 43 For example Bulatov regards it as a textbook that explains the basic

principles of geometric decorations and which reflects the progress of applied geometry of its time; Midhad S. Bulatov, "U istikov arkhitekturnoy nauki srednegovostoka,"NarodyAziiiAfriki 1 (1973): 99, 252; Bulatov, Geometricheskaia

garmonizatsiia (see n. 12), 52. Chorbachi goes even further and claims, "in contrast to the simple shapes and polygons of Abu 'l1-Wafa' 's manuscript, the complex geometric shapes in Interlocking... Figures indicate a much higher and later stage of development ..."; 1 Wasma'a K. Chorbachi, "In the Tower of Babel: Beyond Symmetry in Islamic Design," An International Journal: Computers and Mathematics with Applications 17 (1989): 755.

46 Most significantly, the anonymous author exhibits a total ignorance of Geometric Constructions. This point clearly indicates that Interlocking Figures was

not intended as the appendix to Abu 'l-Wafa-"s work, as Bulatov wrongly assumes; Bulatov, "U istikov," 99; Bulatov, Geometricheskaia garmonizatsiia, 51-52, 325. He dates the Persian translation of Geometric Constructions (the Paris

manuscript) to the early eleventh century and applies this date automatically to Interlocking Figures; see appendix, no. 6. According to internal evidence, however, it seems likely that the anonymous treatise was written sometime in the

early thirteenth century, after 1206, probably in Diyarbakir (a major city in southeastern Anatolia); Alpay Ozdural, "An Ornamental Pattern of Cubic

Equations," Muqarnas (publication pending). Generally the constructions in Interlocking Figures are of a stenographic

nature. It should, therefore, be considered as a sample of the repertoire of the artisans working probably in the thirteenth-century Diyarbakir area. It would be

misleading, however, to assume that all the patterns and methods in this treatise were the peculiarity of this period and area. As architectural knowledge was transmitted mainly through examples, it is expected that some of these were introduced earlier, and some continued to be employed by artisans of later

generations in several other locations. 47 Interlocking Figures appears to be not the only work of this sort. Cafer

Efendi, the author of Risale-i Mi'mariyye, informs us that he had published a treatise on geometry prior to Risdle-i Miumdriyye (see n. 1); but this can not be traced:

Because we have been connected with him [Mehmed Aga] for many years until the present time [1614], for the most part closely, when certain subjects concerning the science of geometry were being discussed, this humble servant took and wrote down everything. In accordance with this, he set down and

composed a treatise concerning the science of geometry. However, previous to this, books of deeds were written and composed about some of the chief-architects. As books of deeds were written down for them, it was

necessary for us to write, in addition to that treatise on geometry, a book of deeds about our generous Aga. Efendi, Risale-i Mi'mdriyye, 22-23. 48 "Sootnosheniya etogo cherteja zaklyuchayutsya v konicheskikh [secheni-

yakh]. Tsel sostoit v tom, chtobi postroit takoy pryamougolniy treugolnik, visota i

samaya korotkaya storona kotorogo vmeste bili bi ravni gipotenuze. Ibn

Khaysam napisal traktat o postroenii takogo treugolnika, i tam govoritsya o

konicheskikh secheniyakh, yavlyayushchikhsya giperbolami i parabolami." Bula-

tov, Geometricheskaia garmonizatsiia (see n. 12), 341. 49 Ibn al-Haytham, a prominent physicist-mathematician, wrote more than

180 treatises. The titles of all are listed and around seventy are extant today. Although he made extensive use of conic sections, none of his works concerns

this particular problem. One of the treatises is about the perpendiculars of the

triangles, but there he investigates the sum of the perpendicular distances from a

70 JSAH / 54:1, MARCH 1995

point inside a triangle to the sides. For more information on Ibn al-Haytham's works, see Fuad Sezgin, Geschichte des arabischen Schrifttums, 7 vols. (Leiden, 1974), 5:365-74; H. M. Said, ed., Ibn al-Haytham, Proceedings of the Celebrations of the IOOOth Anniversary (Hamdard, 1971); Jan P. Hogendijk, Ibn al-Haytham's 'Completion of the Conics' (New York, Berlin, Heidelberg and Tokyo, 1985); R. Rashed, "La construction de l'heptagon regulier par Ibn al-Haytham,"Journal for History of Arabic Science 3 (1979): 309-87; R. Rashed, "Ibn al-Haytham et le theorme de Wilson," Archive for History of Exact Sciences 22 (1980): 305-21; R. Rashed, "Ibn al-Haytham et la mesure du paraboloide," Journal for History of Arabic Science 5 (1981): 191-262; A. I. Sabra, "Ibn al-Haytham's Lemmas for Solving 'Alhazen's Problem'," Archive for History of Exact Sciences 26 (1982): 299-324.

5 The anonymous author's inaccuracies suggest that a verbal mode of transmitting knowledge through conversazioni was prevalent among artisans. They seemed to have the tendency of obtaining information through dialogue, not by reading, even if they were literate. Holod argues that the existence of Abu 'l-Wafa 's book indicates a modicum of literacy among artisans; Holod, 'Trans- mission of Architectural Knowledge" (see n. 8), 3. The anonymous author himself is the most convincing case for her point; but even then, he did not manifest any knowledge of Abu 'l-Wafai's work; see n. 46. As Bloom rightly points out, learning something by reading a book is a distinctly modem practice, and it is not likely to imagine artisans borrowing Abu 'l-Wafa 's book from a library to read his instructions; Jonathan M. Bloom, "On the Transmission of Designs in Early Islamic Architecture," Muqarnas 10 (1993): 21. According to the aforementioned account of the Ottoman geometer, it appears that the same type of learning also prevailed among the artisans of the imperial court in sixteenth-century Istanbul. As the geometer was reading from a book on geometry, he was narrating and explaining each section to the artisans; Efendi, Risdle-i Mimdriyye (see n. 1), 28 (according to the references given in the text, the book that he was reading from must have been written sometime between 1540 and 1570). Listening to oral explanations can not be the most effective way of acquiring a real familiarity with geometry. If the Ottoman geometer's account and Interlocking Figures are considered as sufficient evidence for artisans' verbal

mode of learning, then Abu 'l-Wafa" 's book, in spite of all his efforts in making it comprehensible to artisans, was apparently a futile exercise. Perhaps Abu 'l-Wafa' concieved of his book as an effective replacement for conversazioni, but if so, then it failed to compete with the convenience that that sort of meeting provided for artisans. The same fate, it seems, was shared by Omar Khayyam's untitled treatise. He addressed it to artisans as well as his learned colleagues, but it had not been read even by the anonymous author, who was certainly one of the highly literate artisans.

51 "Zdes tsel mojno dostignut 'lineykoy-treugolnikom'. Kak bilo skazano vishe, tsel nashego cherteja-chetire konichesie figuri s dvumya pryamimi uglami, okrujayushchie ravnostoronniy pryamougol'niy chetirekhugolnik. Ta- kovi konicheskie chetirekhugolniki AIHK, CHMN, PMLX i BLKO, opisivayush- chie chetirekhugolnik KHML. Poskolku ugol H chetirekhugolnika obrazavan dvumya perpendikulyarnimi liniyami, to neizbejno KH i [H]D-pryamie, a treugol'nik AKC-pryamougolniy raven treugolniku CHD. A etot treugolnik pryamougolniy poskolku vpisan v polovinu okrujnosti. Itak, nujno nayti tochky H na duge [C]E. Esli na nashey lineyko ugol F-pryamoy, to storona AB- pryamaya i sovpadaet so storonoy AB kvadrata. AAllakh znaet lusche." Bulatov, Geometricheskaiagarmonizatsiia (see n. 12), 341-42, fig. 36. Fig. 10 of the present text is adapted from the figure given by Bulatov, except for the ruler, which I added in order to make the verging construction possible.

52 For comparision, see nn. 30 and 31. 53 For the date and place of Interlocking Figures, see n. 46. 4 See n. 46.

55 Bulatov, Geometricheskaia garmonizatsiia, 339-41, fig. 33. 36 Bulatov, Geometricheskaia garmonizatsiia, 337-38, fig. 28. In both cases, the

tangent of the angle of rotation is determined as 4/(4 - V2). 57 For instance, Al-Kashi's Key forArithmetic (see n. 17) provides us with some

crucial information to resolve the seeming complexity of the geometry of muqarnas; see Ozdural, "El-Kashi and Stalactites" (see n. 9), 31-47; Alpay Ozdural, "Analysis of the Geometry of Stalactites: Buruciye Medrese in Sivas," Middle East Technical University, Journal of the Faculty of Architecture 11 (1991): 57-71.

OZDURAL: OMAR KHAYYAM AND THE ARTISANS 71