· counting curvature: the numerical roots of north indian temple architecture and frank...

Counting Curvature: The Numerical Roots of North Indian Temple Architecture and FrankGehry's "Digital Curvatures"Author(s): Patrick A. GeorgeSource: RES: Anthropology and Aesthetics, No. 34 (Autumn, 1998), pp. 128-141Published by: {ucpress)Stable URL: http://www.jstor.org/stable/20140411Accessed: 23-09-2017 13:54 UTC

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128 RES 34 AUTUMN 1998

I.,

Figure 1. West facade, Temple no. 2, Bandogarh, Madhya Pradesh, India, circa eighth-ninth centuries. Photo: Patrick A. George.


Counting curvature

The numerical roots of North Indian temple architecture and Frank Gehry's "digital curvatures"

PATRICK A. GEORGE

For what it is worth, I think the younger generation ought to consider becoming the master builders again, taking over the parental role in the construction process, instead of following the current fashion in America, where general contractors are part of the power structure of the city and have virtual control of the construction process, leaving the architect to play a childlike role.'

When architect Frank 0. Gehry asks that the younger generation of architects practicing today consider becoming master builders again, he suggests that they do so by means of what may be called "digital tools," which, in a professional practice that has become abstracted and operates in a digital context, bring the primary information used to define architectural forms back into the control of architects. In order to better understand the role of the traditional master builder in the process of construction, it is instructive to examine the historical example of North Indian temple architecture, the numerical roots of which are significant for understanding the essential function of digital tools in an abstracted practice. Most significantly, many of the details of architectural

practice in India during the second half of the first millennium C.E. do not survive. Although much may be learned about the education and the role of the architect and his relation to priests, temple patrons, and craftsmen, very little is known about the architect's control and dissemination of that primary information used to define the form of the North Indian temple (fig. 1). Of particular interest in this context is the

method used to produce temple superstructures, the faceted layering of which gives the impression of curvature (fig. 2). Gehry's digital method translates the complex curved forms of his projects into constructible facets; an equivalent method that produces the constructible facets of North Indian temple superstructures is not documented and must be

interpreted from textual sources and from an analysis of the forms of the temples themselves.

The lack of knowledge about the methods of temple design and construction is not unique: in the history of the reasoning behind the production of architectural forms, much has been lost. Among the most notable losses in the history of Western architecture are those illustrations to which Vitruvius refers eleven times in order to explain the forma et ratio of architectural elements.2 Of the five separate illustrations to which he refers, which were to have been appended to the ends of chapters (subscripta erit), none survive. Three of these illustrations were intended to demonstrate methods for the production of curvature: entasis, the vertical curving taper applied to the shafts of temple columns; the horizontal curvature of the stylobates of temples (per scamillos inpares); and the volutes of Ionic capitals.3

The former two illustrations fall in the category of optical corrections to otherwise plumb and level architectural elements. These illustrations would have represented, through a process of geometric constructions, methods for the accurate calculation of optically corrected forms from which precise measurements would have been derived and applied to the construction process. Although we do not have Vitruvius's illustration of entasis, he recognized the Greek source of this concept, and thus he may have used the Greek method for its construction. It is reasonable to suppose that Vitruvius's method was derived from, or at least related to, the Greek example discovered by Lothar Haselberger on the walls of the

1. Charles Jencks, ed., Frank 0. Gehry: Individual Imagination and Cultural Conservatism (London: Academy Editions, 1995), p. 50.

2. Claudio Sgarbi, "A Newly Discovered Corpus of Vitruvian Images," RES: Anthropology and Aesthetics 23 (Spring 1 993):31-51.

3. Vitruvius, De architectura 111.3.13, 111.4.5, and 111.5.8, respectively. The remaining two illustrations were intended to demonstrate the application of musical modulations to the design of theaters (V.5.6) and the construction of a water screw (X.6.4). These illustrations may in fact have employed curvatures in their demonstrations; certainly the water screw, with its helical form, required the calculation of curvature for its construction.



Figure 2. Superstructure, Temple no. 2, Bandh6garh, Madhya Pradesh, India, circa eighth-ninth centuries. Photo: Patrick A. George.

temple of Apollo at Didyma (begun in 334 B.C.E.), "a geometric construction as simple as it was ingenious."4

Although there has been no corresponding discovery to explain the curvature of the stylobate, the method used for entasis could easily apply to this horizontal surface as well and may not be too dissimilar from the diagram proposed by William Bell Dinsmoor in 1 902.5

Whatever the method, it must have employed a technique of drawing similar to that of entasis, including geometric construction with a straightedge and compass (permitting accuracy) and the representation, wherever possible, of architectural elements at full scale (permitting precision). Joseph Rykwert suggests that the great technical achievement of these optical corrections

is "in glaring contrast to the evident lack of interest in what in the twentieth century seems the principal aim of technology: structural economy."6 To a certain extent, the primary goal of Gehry's use of digital tools is precisely this, but, as will be shown, the complexity of Gehry's forms necessitates an efficient management of structural resources in order for them to be constructible.

The remaining illustration mentioned by Vitruvius that deals with curvature is intended to demonstrate not another method for an optical correction, but rather a method for the construction of the spiral form of the volutes of Ionic capitals. Vitruvius's method was presumably to be used for the production of a paradeigma, or "sample specimen," of a column capital, from which the remaining capitals listed in the syngraphai, or "building specifications," would be copied.7 Although the forma et ratio of Vitruvius's volute

4. Lothar Haselberger, "The Construction Plans for the Temple of Apollo at Didyma," Scientific American 253 (December 1985):131.

5. William Bell Dinsmoor, The Architecture of Ancient Greece: An Account of its Historic Development, 3d ed. (1950; reprint, New York: W. W. Norton & Company, 1975), p. 168. Note also that drawings for the curvature of the stylobate may lie on stone surfaces that have been built over "and so disappeared automatically." Haselberger (see note 4), p. 132.

6. Joseph Rykwert, The Dancing Column: On Order in Architecture (Cambridge: The MIT Press, 1996), p. 226.

7. J. J. Coulton, Greek Architects at Work: Problems of Structure and Design (Ithaca: Cornell University Press, 1977), p. 54.


George: Counting curvature 131

Figure 3. Diagram of construction of a spiral of Archimedes with straight lines and sections of circles. Drawing: Patrick A. George.

is lost, his requirement for the use of a compass (ad circinum) implies a method of construction in which the constantly changing curvature of a spiral is approximated by a series of circular arcs whose centers are offset from the center of the volute in some regular pattern. Depending upon the pattern of the offsets, the curvature may approximate any number of nonmathematically definable spirals, as well as two mathematically definable ones, the spiral of Archimedes and the logarithmic spiral.

Included in the former group, represented by the series of interpretations of Vitruvius's method that began in the Renaissance, are those examples whose radii are offset in regular intervals along lines drawn from the center of the oculus of the volute.8 This group provided

the primary, if not the only, source of Ionic volutes from the Renaissance through the nineteenth century. In contrast to this group, the spiral defined and analyzed by Archimedes (circa 287-212 B.C.E.) in a book of 28 propositions on the subject is the least similar to extant Greek volutes since it is formed with radii that increase in arithmetic progression with each rotation, producing a shape similar in appearance to a coiled rope (fig. 3). The radii of a logarithmic spiral, on the other hand, increase in geometric progression with each rotation (fig. 4). The curvature of the logarithmic spiral is similar both to extant Greek volutes and to the forms of shells and horns, which are among the themes identified by Rykwert as contributing to the formation of the Ionic capital.9 As D'arcy Thompson notes in his study of growth and form, "the shell or the horn tends

8. These regular intervals may be subdivisions of a single line (Sebastiano Serl io, The Five Books of Architecture, IV. 1 0), of a square (Andrea Palladio, The Four Books of Architecture, 1.1 6), or of any other regular polyhedron, such as a hexagon, inscribed in the oculus. The subtle difference between the lengths of the intervals and the radius of the oculus produces corresponding changes in the curvature of the spiral. For an example using an inscribed square, see the illustration of

an Ionic volute according to Giuseppe Porta in Giannantonio Selva, La Voluta Jonica (Padua: 1814), reproduced by Rykwert (see note 6), p. 245.

9. Rykwert (see note 6), chap. 9, "The Mask, the Horns, and the Eyes," pp. 237-315.



Figure 4. Diagram of construction of a logarithmic spiral with straight lines and sections of circles. Drawing: Patrick A. George.

necessarily to assume the form of this mathematical figure, because in these structures growth proceeds by successive increments which are always similar in form, similarly situated, and of constant relative magnitude one to another."10

Like the spiral of Archimedes, this spiral may also be expressed mathematically. Thompson, in fact, includes tables of measurements taken from various shells, from

which he is able to derive the mathematical constants that determine the particular shape of the spiral. Once these mathematical constants are defined, the formula for the spiral may be used to accurately calculate the curvature to any desirable degree of precision. The logarithmic spiral, with its abilities to be approximated

with a geometric method of construction and to be accurately defined with a numerical, or "digital," method, provides an informative transition between the examples of the Ionic volute and the superstructures of North Indian temples. The curvatures of North Indian

temple superstructures, as will be shown, also derive from a mathematically definable process in which a series of points defines the positions of facets, the sum of which gives the impression of continuous curvature. This process thus has more in common with numerically based "digital" methods than with graphically based geometric methods.

Unlike the graphic example of Vitruvius's volute, moreover, the reasoning that generated the forms of temple superstructures was not embodied in a corpus of illustrations. At no place in the extant literature on temple architecture, during the period in which the temples under consideration here were built, is reference made to accompanying illustrations, nor is reference made to a system of graphic representation of "the appearance of the future building,"'1I as in the Greek methods of graphic representation outlined by Vitruvius, the ideae of ichnographia, orthographia, and scaenographia. There are, however, at least two extant

10. D'Arcy Wentworth Thompson, On Growth and Form (1917; reprint, Cambridge: Cambridge University Press, 1948), p. 769. 11. Vitruvius, De architectura 1.2.2.


George: Counting curvature 1 33

examples of architectural representation carved in the back face of a seat-back in the Harihara temple no. 2 at the site of Osian, one representing an orthographic projection of a Phamisana roof and the other, a diagram of a temple superstructure.12 These sketches are neither detailed nor dimensioned, and they do not seem to be demonstrations of a method for a geometric construction, but rather instructional diagrams intended to communicate the process of temple construction. In addition to four curved lines that define the right-hand face of the facade of a temple superstructure, however, the latter diagram also includes a series of marks at intervals along the central vertical axis of the superstructure and a corresponding set of marks along the outermost curved line. The sketch appears to show a division of a temple superstructure using a series of sets of two points. The single mark on the upper right appears to define both the height of the superstructure and the difference between the widths of the top and bottom. According to this interpretation, this diagram indicates that the temple superstructure is to be constructed by means of a discrete process, step by step, according to some unspecified progression. Rather than a scaled drawing of a specific temple elevation, or a diagram of a process of geometric construction, this representation appears to have been a pedagogical tool, an abbreviated explanation of a process of building that would have taken months, if not years, to complete.

Aside from rare representations such as that at Osi-an, the instructions for the production of temple forms were verbal and, traditionally at least, were transmitted orally. The efficacy of the transmission process was made possible in part by the use of an elaborate system of metrical devices in which the instructions were composed in verse, a technique common throughout Sanskrit literature. This method ensured that the instructions, or rules, fixed in meter, would be both easy to memorize and less susceptible to alteration and corruption. It also ensured the necessity of oral instruction, reinforcing the established tradition of education between teacher and student.

Some collections of rules on temple architecture survive today in the form of printed editions, typically described as "treatises on architecture," comparable to a

similar Western tradition of work of which Vitruvius is a notable example. According to the rules of one of these treatises, he who desires to become an architect must first have a religious teacher, or guru, a priest knowledgeable in the sacred texts of the vedas, agamas, and so on.'3 These sacred texts, it should be noted, are classified as sruti, or "heard," that is, from a divine source, as opposed to sastra, or "rule," for example, vastusastras, or "rules on architecture," written by men. Architects were the inheritors of this tradition and were responsible for its oral transmission, but it is clear that they were not the authors. Rather, these particular rules on architecture originated among communities of priests for the stated purpose of ensuring the ritual efficacy of the finished temples: the ability of temples to allow the accrual of merit (punya) for the architects who designed them and directed their construction, for the craftsmen

who constructed them, for the patrons who paid for them and worshipped the images of deities within, and for the priests who maintained the images and ritual processes.14 It is important to note in this context that, regardless of the analogies between the two, architects

were not priests and priests were not architects.15 In addition to a guru, the treatise mentioned above

adds that he who desires to become an architect is to have two other teachers: an architect (sthapati) knowledgeable in the treatises and skilled in making temples, and a craftsman (sthapaka) skilled in the

manual processes of building, the craft of construction. The term sthapaka itself encompasses a range of practical building tasks, typically divided into three distinct parts: sutradhara, or the "string holder," or the

12. Michael W. Meister, ed., M. A. Dhaky, coordinator, Encyclopaedia of Indian Temple Architecture, vol. 2, no. 2 (Princeton: American Institute of Indian Studies, Princeton University Press, 1 991), pls. 410, 409, respectively.

13. Srikumara, Silparatna, ed. Mahamahopadhyaya T. Ganapati Sastri, Trivandrum Sanskrit Series, no. 75, pt. 1 (Trivandrum: Superintendent, Government Press, 1 922), p. 4.

14. See Visnudharmottarapurana, Third khanda, ed. Priyabala Shah, Vol. 1, Text, Critical Notes, Etc., Gaekwad's Oriental Series, No. 1 30 (Baroda: Oriental Institute, 1958); Vol. 2, Introduction,

Appendices, Indices, Etc., Gaekwad's Oriental Series, No. 137 (Baroda: Oriental Institute, 1961).

15. The separate disciplines have often been conflated through a linguistic analogy. The patron "sacrifices" in the middle voice (yajniate), and the priests "sacrifice" in the active voice (yajnianti): the patron accrues the merit of the sacrifice itself, and the priests accrue the merit of fulfilling one of the activities (karman) of their caste (varna). This particular linguistic and corresponding semantic structure has often been used to interpret the relationship between the temple patron, who may be said to make the "sacrifice" of building a temple, and the temple architect, who officiates this "sacrifice" through the process of building.



one who measures; taksaka, or the "one who cuts," or the stonemason; and vardhaki, or the carpenter, the one responsible for scaffolding and the machinery for lifting stone blocks. The sutradhara, it should be noted, is defined in this treatise as an experienced student of the sthapati, typically one of his sons, who is an architect in-training himself and therefore not part of a separate craft tradition.

One of the most extensive compilations of architectural rules is that of the Samarahgana-sutradhara, attributed to Bhoja, the king of Malwa, who ruled during the period 1005-1054 C.E.16 Bhoja was known as a great patron of literature and as a great builder of temples, but his primary duty was that of king, as the title of the treatise implies: "architect of the battlefield." It is likely that this compilation was a collection of architectural knowledge, perhaps corresponding to a collection of architects recruited or requisitioned in the course of his conquests. Among the various sets of prescriptions compiled in this text is a chapter on the characteristics of architects (sthapatya-laksana), whose twenty-two verses begin with the following four:

Now it is told by us the steps necessary to become an architect, by which knowledge the good and bad characteristics of architects are known. [These steps are] practice, custom, knowledge, work, and instruction. He [the architect] should be a man who is grounded in the sastras, for the purpose of joining signifier and signified. The wise man should know these limbs of architectural treatises: palmistry, arithmetic, astronomy, metrics, knowledge of streams and craft practice, and the rules of machines (yantra) and production. He should recognize characteristics and be engaged in pursuing treatises.17

The purpose of the sastra is to join together a "characteristic" or the "signifying word" (laksana) and the "object to be signified" (laksya). This suggests that these treatises functioned, at least as far as the priests who composed them were concerned, primarily as assemblages of terminology that defined the technical vocabulary necessary for the production of temple architecture, producing a common ground of reference for communication between priests and architects.1 8 Had these treatises functioned as construction manuals, they would potentially have been considered substitutes for experience in the practice of architecture.19 The Samarangana-sutradhara, however, makes it clear that the architect who lacks either the knowledge of sastra or the experience in work is considered to be inadequate:

Who knows only sa>stra and is not completely skilled in work is bewildered at the time of action, appearing as if afraid in battle. Who knows only work and is not accomplished in the use of sastra is as if blind, led helpless along another's wheel ruts.20

16. P. V. Kane, History of Sanskrit Poetics (1951; reprint, Delhi: Motilal Banarsidass, 1987), pp. 257-264.

1 7. Bhoja, Samarahgana-sutradhara, ed. T. Ganapatifastri, Gaekwad's Oriental Series, nos. 25, 32 (Baroda: 1924-1925); revised by Vasudeva Saran Agrawala, Gaekwad's Oriental Series, no. 25 (Baroda: Oriental Institute, 1966), verses 44.1-4. This and all of the following Sanskrit translations are by the author.

Yantra does not mean the abstract tantric ritual "devices," but rather practical mechanical "devices." David Pingree lists the sahku, or gnomon, "employed to find the cardinal directions in the Sulbasutras," the earliest observational instrument in use in India, as a form of yantra; see David Pingree, "Mathematics," in Jyotihsastra, Astral and Mathematical Literature, A History of Indian Literature, ed. Jan Gonda, vol. 6, fasc. 4 (Wiesbaden: Otto Harrassowitz, 1981), p. 52. In the context of Indian temple construction, it is likely that yantra refers to scaffolding and to machinery for lifting stone blocks. Comparable to this example of yantra are Vitruvius's gnomice and machinatio, two of his three parts of architecture: "There are three parts of architecture

itself: building, [the making of] sundials, [and the making of] machinery." Vitruvius, De architectura 1.3.1. This and all of the following Latin translations are by the author.

18. Vitruvius emphasizes that, of all disciplines, the relationship between quod significat, "definiens," and quod significatur, "definiendum," is most important for the discipline of architecture: "As in all things, though principally in architecture, there are two things: that which is signified (quod significatur) and that which signifies (quod significat). That which is signified is the subject matter about which discourse is to be made; that which signifies is the explanation unfolded in systems of instructions. And therefore it is necessary, if one calls oneself an architect, to have experience of each." Vitruvius, De architectura 1.1.3.

1 9. In this context, it is worth contrasting the double negative given by Vitruvius, that he was unable not to praise patres familiarum eos, qui litteraturae fiducia confirmati per se aedificantes ita iudicant, or "those heads of families who fortified by a faith in [architectural] literature, are building for themselves." Vitruvius, De architectura

VI.prae.6-7. Although Vitruvius did not see his treatise as a substitute for

practical experience, it certainly had that potential and tendency. Marcus Cetius Faventinus, for instance, a Latin author writing in the third century C.E., consulted those practical examples of construction given by Vitruvius that would aid persons conducting private building projects. See Hugh Plommer, Vitruvius and Later Roman Building Manuals (Cambridge: Cambridge University Press, 1973).

20. Samarahgana-sutradhara (see note 1 7), 44.8cd-1 Oab. Similarly, Vitruvius states that: "Therefore architects, who had striven without literature [skill in writing], in order to be trained with [their] hands,

were not able to bring it about, that they had authority in proportion to [their] labors; moreover those who have relied on ratiocinatio and literature alone seem to have pursued a shadow, not the work."

Vitruvius, De architectura 1.1 .2.



Having studied the various limbs of architectural treatises, the student of architecture would have quickly learned, in the course of his practical apprenticeship, that the prescriptions defining the characteristics of temples that he had memorized through repeated recitation with his teacher were not sufficiently detailed to guide him through all aspects of temple design and construction. Perhaps the most noteworthy omission in these prescriptions, or at least the one that has most perplexed art and architectural historians since temple architecture became an object of study, is that of an adequate description of the procedures for the construction of the complex forms of temple superstructures.21 Although some prescriptions may define, for instance, the overall proportions of temple width to temple height, they do not describe the method of producing the characteristic curvature of the North Indian temple superstructure. Missing are those specific instructions, given by Indian architects to guilds of stonemasons, defining the dimensions and directing the measuring and cutting of the successive layers of stone used to construct these superstructures.

Mention of a method for producing a similar form is given, however, in a treatise on the construction of temple carts, a kind of substitution for temples in certain religious rituals. It is worth noting in this context that at least one art historian believed that the forms of temples derived from the forms of chariots.22 Historical evidence indicates, however, that temple carts were patterned after temples. Basavanna (circa 1106-11 67/8 C.E.), for instance, the founding prophet of the Lingayats, also known as VTras'aivas, developed a method of worship centered on individual devotion to Siva partly in response to the extensive material resources used in the production of temple architecture required by traditional devotional rituals. The representation of Siva as a lihgam in the inner sanctum of temples was replaced by his representation in the form of a lihgam worn around the neck of the devotee. The most important Lingayat ritual occurred during religious fairs in which the image of a

deity or saint, regarded as a real, physical manifestation, would be placed in a wheeled wooden vehicle with a towerlike superstructure and drawn past crowds of pilgrims in a religious procession.23

A maker of vehicles (rathakara), were he given a commission to build a temple cart for the purpose of carrying an image of a deity in such a procession, would have been required to know the prescriptions defining the characteristics of various types of vehicles (rathalaksana). Among other things, the rathakara would learn that the superstructure of a temple cart, like that of a temple, should be composed of levels. Unlike the prasadalaksana, however, which give no indication of the specific dimensions of each level of the superstructure, at least one source of rathalaksana specifies a relationship between the heights of the successive levels of the superstructure on a temple cart:

One should calculate this, the ground prepared for all of the levels, such that every single subsequent elevation of the levels derives from the height of the base level.24

This is the extent of the detail of the prescriptions from this compilation, and there is no further explanation of the dimensional relationship between the levels. Another source of rathalaksana, however, specifies the precise proportions of each level of the superstructure:

One should divide the height of a chariot having three storeys into forty parts.... The height of columns should be 8-1/2 parts; the entablature should be 2-1/2 parts. The columns of the upper storey should be 6 parts; the height of the entablature should be 1-1/2 parts. The height of the columns should be 5-1/2 parts and the topmost entablature should be 1-1/4 parts.25

The parts of the vehicle that correspond to the successive levels of stone in temple superstructures are

21. For a summary of the systemic origins of the form of temples, as well as an interpretation of the determination of curvature of temple superstructures, see Michael W. Meister, "On the Development of a

Morphology for a Symbolic Architecture: India," RES: Anthropology andAesthetics 12 (1986):33-50.

22. E. B. Havell, "Gupta Style of Architecture and the Origin of Sikhara," in Commemorative Essays Presented to Sir Ramkrishna Gopal Bhandarkar (Poona: Bhandarkar Oriental Research Institute, 1 91 7), pp. 443-446.

23. William McCormack, "Appendix II: On Lingayat Culture," in Speaking of Siva, trans. A. K. Ramanujan (New York: Penguin Books, 1973), p. 183.

24. Prasanna Kumar Acharya, ed., Manasara on Architecture and Sculpture: Sanskrit Text with Critical Notes, Manasara Series, vol. 3 (London: Oxford University Press, 1934; reprint, New Delhi: Oriental Books Reprint Corporation, 1980), verse 43.47. Although Acharya believed Manasara to be a North Indian composition dating to the Gupta period (circa fourth-sixth centuries C.E.), textual evidence indicates that it is a South Indian composition, dated sometime between the eleventh and fifteenth centuries C.E..

25. R. P. Kulkarni, ed., VisvakarmTya Rathalaksanam: A Study of Ancient Indian Chariots (Delhi: Kanishka Publishers, Distributors, 1994), verses 2.1 09ab-1 1 1.



the columns (pada) and entablatures (malca).26 The series of the heights of these architectural elements approximates an arithmetic progression in which the dimensions of each level are based upon the previous, diminished by an integral multiple of a module. In order to form an accurate arithmetic progression, two of the measures require modification: the height of the column of the second level should be increased by one-half of a part, from 6 to 6 1/2, and the height of the entablature of the third level should be decreased by one-quarter of a part, from 1 1/4 to 1. As a result of these modifications, the module for the diminution of the column is 1 part, so that from the first level to the second, the heights of the columns are reduced by two modules, and from the second to the third, the heights are reduced by one module, and the module for the diminution of the entablature is 1/2 part, so that, again, from the first level to the second, the heights are reduced by two modules, and from the second to the third, the heights are reduced by one module. This alteration of the heights requires a corresponding modification of the two verses translated above, a suggestion that is not unwarranted in the tradition of texts of this kind;27 the text itself suggests that dimensions may be changed under some circumstances.28

The existence of a canon of this kind for use by carpenters in the production of the superstructures of temple carts suggests the existence of a similar summary of computations for use by stonemasons in the production of temple superstructures. The nature of these specific instructions further suggests that the curvatures of temple superstructures were also the result of the application of mathematical progressions.

My analysis of the measurements and proportions of a group of extant eighth-ninth-century temples located in Bandhogarh, Madhya Pradesh, confirms that the successive levels of the superstructures at that site were constructed according to a regular arithmetic progression in which the dimensions of each layer of stone were based upon the previous, diminished by some integral multiple of a fixed module (fig. 5). In height, the layers of block decreased in units of a larger module, and in width, they decreased in units of a smaller module. This procedure of successive reduction gives the impression of continuous curvature.

It is important to note that, whatever the size of the module, the basic unit of measure was the ahgula, or "finger"; the width of four ahgula forms a musti, or "fist," abbreviated as "a" and "m," respectively, in figure 5. This unit of measure is both defined in treatises in chapters on hastalaksana, or the "characteristics of hands,"29 that is, units of measure, and confirmed by my analyses of the measurements of existing temples at a number of sites throughout north-central India.30 Although fingers provided the basic unit of measure in the carving of sculptures and construction of temples, it is not clear from the textual prescriptions whose finger provided the measure. The historical use of the measure of the sacrificer's finger to fix the dimensions of the vedic sacrificial altar, however, implies that the measure of the donor's finger was used to fix the dimensions of the temple being donated.31

The sum of the evidence given above indicates that architects would determine first the overall proportions of the temple to be built, and then the fixed width and height of the temple from which they would derive an arithmetic progression that fulfilled these dimensions. Having defined the dimensions of the successive layers of stone, the architect would then communicate these measurements to the stonemasons through the office of the sutradhara, or "string holder." The sutradhara, presumably, would monitor the dimensions of the stones

26. In the context of temple architecture, the word pada may also refer to a "wall," or more specifically a "wall frieze" containing representations of columns.

27. As the editor and translator of the Rathalaksanam notes: "From the brief description of the contents of the present manuscript, it seems that it is a collection of extracts, describing different kinds of festival cars from different chapters. The copy writer might be a carpenter who is skilled in making festival cars.... Probably he was a good car maker but was not having good knowledge of Sariiskrta." Kulkarni (see note 25), p. 65. Kulkarni neither dates nor locates the origin of this text but mentions that the manuscript is an approximately 200-year-old copy, in Devanagari script, of an original in Grantha script, located in the library of Sarasvati Mahal atTanjavur, Tamilnadu.

28. sobhasobhabalirdham ca nyunadhikyamn tu bundhiman "[There may be] an increase or decrease [in the dimensions] for the sake of strength or beauty, by one who is learned." Kulkarni (see note 25), verse 2.93.

29. Bhoja, Samarahgana-sutradhara, hastalaksana (see note 1 7), chap. 9.

30. Patrick A. George, "Construing Constructs: A Study of Temple Design and Construction in North India" (Ph.D. diss., University of Pennsylvania, 1994).

31. Fingers, symbolically and actually, had served as units of identification since the period of the vedas. See A. Seidenberg, "Geometry of the Vedic Rituals," in Agni: the Vedic Ritual of the Fire Altar, ed. F. Staal (Berkeley: Asian Humanities Press, 1983), pp. 116-117.



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and their positions, as well as the overall dimensions of the temple throughout the process of construction. The specific techniques of this construction process varied from region to region and from period to period. Each level of the superstructures of the temples at Bandhogarh, for instance, consisted of eight stone blocks: one block at each corner, probably positioned by measuring the corresponding square they formed with respect to a plumb line centered on the vertical axis of the superstructure, and four additional blocks cut to set between these, completing the sides of the superstructure (fig. 6).32

The specific methods by which architects derived arithmetic progressions and solved overall proportional requirements have not been transmitted to us, although the knowledge required to do so was well known in India during that period of temple construction. The arithmetic knowledge (ganita) mentioned above as required of architects by the Samaran6gana-sOtradhara implies an understanding of mathematical progressions, both arithmetic and geometric, as well as of algebra and geometry.33 Mathematical literature, moreover, was especially concerned with commercial and other practical applications of arithmetic,34 and the use of arithmetic progressions in the construction of temple superstructures may represent another instance of this inclination. Examples of the interest in and use of mathematical

progressions in systems of counting, in fact, recur throughout Indian culture. Knowledge of mathematical progressions was possessed in various forms and to varying degrees by all social groups in ancient India, from low-caste rajakas, or "washermen," who calculated the loss in the value of clothing through successive washing in this way, to high-caste rajas, or "kings," whose donations to priests during the period of a sacrifice decreased in an arithmetic progression daily, so that by the end of the period of sacrifice the initial allotted sum would be fully distributed. To these examples should be added the calculation of cosmological time, in which the lengths of the four ages of the world decrease in a regular progression, and the calculation of kingship, that is, the game of dicing used, among other things, in the ceremonies of royal accession. This game, unlike that in the West, did not involve chance, but rather the ability to count and calculate increasing sums of "dice," vibhTtaka nuts, thrown into the dicing circle; skill in dice was equated with skill in counting. The superstructures of North Indian temples may be interpreted as architectural manifestations of this same approach to

32. In contrast to the method of construction used at Bandhogarh, see the superstructure of the Harsa Deul Temple (circa 900 C.E.),

Jodhpur, Rajasthan, whose method of construction is visible because the veneer infill that formed the middle parts of the sides of the superstructure has fallen away, exposing the roughly cut rubble core on which they were set, and the carefully carved edges of the corner blocks, which remain intact.

33. Metrics or metrical science, moreover, includes a knowledge not only of metrical rules but also of the classification of the various meters, a procedure that involves the calculation of mathematical progressions.

34. Pingree (see note 1 7), p. 56.



Figure 6. Isometric view of superstructure under construction, Temple no. 2, B5ndh6garh, Madhya Pradesh, India, circa eighth-ninth centuries. Diagram: Patrick A. George.

counting, with all of the requisite cultural significance that this system embodies.

The textual, physical, and cultural evidence of temple design and construction indicates a division of knowledge into three distinct parts. The first division, represented by the extant treatises on architecture written by communities of priests, defined the terminology of temple architecture and specified the general characteristics of architectural education, temple forms and overall proportions, systems of measurement, and so on, necessary to communicate and, to a certain extent, control the processes of temple production. The second division of knowledge, that of temple architects themselves, is not represented by any extant textual sources but must be extrapolated from the boundaries of the domains of knowledge controlled by priests and craftsmen. The third division of knowledge, represented by the forms of the extant temples themselves and supported in part by the evidence of the carpenter's manual for the construction of temple carts, used

specific measurements and the mason's craft to produce the individual stones of temples. The second division, that of specifically architectural knowledge, included the ability to subdivide the overall proportions of temples outlined by priests into the discrete dimensions of individual stone blocks, a process that utilized the numerical methods of mathematical progressions.

The architects of North Indian temples controlled the process of construction through their control of this numerical method. In this sense, they were not master builders: the knowledge that they manipulated was not limited to a knowledge of craft, and no sum of craft experience could ever substitute for the experience required of an architect to build a new temple. Architects understood the temple form as a "type," in the sense of the concept first developed by Quatramere de Quincy, while craftsmen understood the temple as a "model." In the absence of architectural knowledge, craftsmen might be able to reproduce an existing model, but they would not be able to generate a new model from a temple type.



Analyzed in this context, Frank Gehry's suggestion that architects use "digital tools" to bring the primary information used to define architectural forms back into the control of architects may be interpreted to mean that architects need to regain control of the numerical methods used to define architectural forms. This has, in fact, at least since 1 990, been a primary concern in Gehry's architectural practice, a concern directly caused by the difficulty of defining the forms of his buildings, specifically their complex curvatures. Due to their incorporation of curvatures of this kind, Gehry's projects

were difficult to represent with traditional Vitruvian projections and therefore difficult to communicate to contractors, whose cost estimates indicated their uncertainty in interpreting the drawings and defining the specific methods of constructing the complex curves.35

Unlike the example of Vitruvius's volute, these curvatures do not derive from any process of geometric construction, nor, like the North Indian temple superstructure, do they derive from any mathematical process. Instead, they derive from Gehry's own aesthetic decisions, although they may be inspired by other requirements.36 These curvatures have been described by James Glymph, a principal in Gehry's architectural firm, as "free" in contrast to "standardized,"37 and by Coby Everdell, manager of advanced visualization for Bechtel, as "highly fluid."38 Architectural critics have described Gehry as a sculptor, and thus, critics do not generally interpret Gehry's buildings, as sculptures, beyond describing them as artistic expressions or exercises in imagination.39 One of the key projects in

the development of Gehry's current design process, for instance, the large pavilion for the 1992 Olympic Village on the Barcelona waterfront, is typically described as a "fish-shaped sculpture."40

Whatever their aesthetic sources, Gehry first develops his forms through a process of physical modeling (fig. 7). The initial model of the complex curved surfaces of the Barcelona pavilion, for instance, was created by bending wooden ribs.41 More recently, one of the models for the Guggenheim Museum in Bilbao was described as composed of "billowing sheets of paper taped together."42 In the design process as his firm currently practices it, Gehry's staff digitizes his physical models and then transfers this data into a finite-element computer-aided design program called CATIA (Computer-Aided Three-dimensional Interactive Application). The complex forms are then rationalized into mathematically definable surface patches and, finally, constructible facets (fig. 8). The database of dimensions defining these facets is further tested by constructing a model out of plastic using a CNC (Computer Numerical Control) milling machine. As Gehry summarizes:

It is an elaborate process, and a part which resembles a dentist's drill is used to digitise the shapes into the computer and analyse them. Then a robot uses this information to make a model out of plastic. This takes a

week and costs a tremendous amount, but it is like a shop drawing of the shapes completely replayed. The most exciting thing about this process is that we are able to control the costs of shapes on a big scale, and manage very tight budgets.43

Once the accuracy of the database is ensured, that is, once it sufficiently conforms to Gehry's physical models, it can be shared with contractors and manufacturers and used to control machines that mill stone and cut steel.

35. B. J. Novitski, "Gehry Forges New Computer Links: Aerospace Developed Software Translates Curved Forms into Crafted Construction," Architecture, August 1992, p. 105.

36. Acoustics, for instance, in the case of Walt Disney Concert Hall (1 988/1 992-1 997), Los Angeles, California. For additional insight into Gehry's aesthetic decisions regarding curvature, see the interview

with him and Philip Johnson regarding their collaboration in Jeffrey Kipnis, Philip Johnson Recent Work, Architectural Monographs 44 (London: Academy Editions, 1996), pp. 58-62, in which Gehry explains: "So we started doing some schemes with curves. Philip started when he brought his pear into the office and I watched him play with the computer. Although I had used double curves on Disney Hall, I hadn't made an object out of them and I thought, 'Oh my God, he's pointed to a new direction,' and then I let that concept simmer for a while."

37. Novitski (see note 35), p. 110. 38. Michael Kimmelman, "The Museum as Work of Art," The New

York Times, October 20, 1997; Aaron Betsky, "Machine Dreams," Architecture, June 1997; Herbert Muschamp, "The Miracle in Bilbao," The New York Times Magazine, September 7, 1997; Blair Kamin,

"Welcome to the Future," Chicago Tribune, October 19, 1997. The author wishes to thank Kara Heavin for assistance in this research.

39. Allan Schwartzman, "Art vs. Architecture," Architecture, December 1997, p. 59.

40. William J. Mitchell and Malcolm McCullough, Digital Design Media, 2d ed. (New York: Van Nostrand Reinhold, 1995), p. 435; Novitski (see note 35), p. 105.

41. Mitchell and McCullough (see note 40), p. 435. 42. Joseph Giovanni, "Gehry's Reign in Spain," Architecture,

December 1997, p. 66. 43. Jencks (see note 1), p. 50.



\,, . -s t 2, X A. ,* ~~~~~~~~~~~~~~~~N. X k .4 c4j 4

AA '

Figure 7. Frank Gehry & Associates, physical model of the atrium of Pariser Platz 3, Berlin, Germany, 1997. Photo: Courtesy of Frank Gehry & Associates. Cf. the photo of Gehry's model of the Peter Lewis Residence exposing the "horse's head" in Jeffrey Kipnis, Philip Johnson Recent Work, Architectural Monographs 44 (London: Academy Editions, 1996), p. 61.



Thus, although Gehry's forms do not derive from any numerical process, they cannot exist, at least within budgets, without mathematical rationalization and numerical control. Historically, computers developed as mathematical instruments, devices to rapidly and economically obtain answers to mathematical questions. Computer-aided design applications, furthermore, developed through the need for a graphic interface between the vast array of answers to

mathematical questions, that is, numerical data, and numerically controlled machines. This numerical data and CNC process are conceptually equivalent to a canon and a craft process of implementing that canon.

Viewed in this way, Gehry's current design process has some similarity to the traditional model of architectural practice exemplified by the design and construction of North Indian temple architecture in the way it controls the primary information used to define his architectural forms. This process, moreover, like that of North Indian temple architects, was not developed in order to be applied to all architectural forms, but is the specific solution to the definition of the form of an architectural type, Gehry's "free" curves, on the one hand, and the temple's manifestation of the "progressive structure" of its contemporary Indian world, on the other.

Figure 8. Frank Gehry & Associates, CATIA model of sculptural shell of conference hall in the atrium of Pariser Platz 3, Berlin, Germany, 1997. Photo: Joshua White, courtesy of Frank Gehry & Associates.


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