09 Syntax and Symmetry

Le Corbusier, Villa Savoye. Poissy, France, 1929

Transformation is intimately connected with symmetry. Hermann Weyl’s now normative definition that symmetry is an “invariance despite a transformation” makes the relationship between the two axiomatic. Transformation is foundational to the mathematical topics of groups, topology, and patterns. It is also essential to certain linguistic theories. In short, it has been essential to the study of form, specifically, the relationship between the outward appearance and the inner structure of mathematical, physical and cultural artifacts. In linguistics, transformational grammar describes how the syntactical structure of language is preserved even as it is transformed into a limitless number of unique combinations. This too can be understood as symmetry, as it is alternatively defined in science as any structurepreserving operation.

Operations are the link between transformations and symmetry. Only certain operations produce symmetrical transformations. In Euclidian geometry, there are the four rigid operations: translation, reflection, rotation, and glide reflection. In topology, it is the “continuous homeomorphic function” that provides the consistency between one shape and another.1 In linguistics, “move,” “merge,” and “labeling” operations enable basic grammatical entities to be combined into more complex groups without undermining the basic syntactical rules.2 In each of these contexts, form is defined in terms of the relationship between an unchanging, inner organization and a set of limited operations that preserve structure and produce specific configurations. Architecture of Syntax

In the 1960s, linguistics, in particular Noam Chomsky’s transformational grammar, became an influential model for understanding the relationship between shape and structure in a variety of fields, including architecture. The syntactical logic that Chomsky defined involved different levels of rules, but the underlying and abstract “deep structure” remained unchanged, even after it had been subjected to a set of syntactical operations, technically called transformational rules, producing unique “surface structures,” i.e. sentences.3 As with all other symmetries, transformational grammar describes a situation where a limited set of operations generate an extremely large number of forms.4

Chomsky did not address the specific meanings and associations that these surface structures generated. His theory only dealt with how the system manipulated the placement of words to generate sentences, which could then produce meaning. He focused on how forms came into being, not on the effect that they produced. He maintained that the ability to transform the deep structure into specific surface structures was universal and innate in human beings. Specific languages, like the specific shapes of crystals, were the result of the interaction between an unchanging, deep structure and historical circumstances.

Linguistic Architecture

At least four lines of architectural inquiry with a connection to Chomsky’s ideas emerged in the 1960s and early 1970s: the design methods begun by Christopher Alexander at Berkeley, the design research emphasis started by Leslie Martin at the University of Cambridge—expanded by Lionel March and Philip Steadman, the shape grammars developed by Georgy Stiny at UCLA and James Gips at Stanford, and Peter Eisenman’s “Conceptual Architecture.”5 These four positions had different agendas and ideologies, sometimes radically so. Yet, what they had in common was a common interest in Chomsky’s theory, which involved the desire to uncover a common source from the diversity of architectural form and an intuition that architectural form could be more clearly and rationally understood, even if architectural design could not. They also shared an almost uncanny—and unstated—reliance on symmetrical forms and operations to make and illustrate their claims.6 Chomsky’s theories at the time did not emphasize symmetry, though his later work would. Yet, in drawing after drawing, and diagram after diagram, symmetrical figures and patterns arising from a series of symmetrical operations are ever-present in texts of these architectural researchers, theorists, and designers.

Eisenman’s Symmetry

While Peter Eisenman expressed an interest in a more “logical” basis for design, he was not interested in using the “new math” that appealed to March and Steadman to do so.7 Nor was he looking for the science of shapes as Stiny, Gips, and Terry Knight were. He was not trying to find the range of architectural forms that would best fit a specific climate, site, or program either, as Steadman’s and Alexander’s work did. He was decidedly not interested in creating “good environments or better environments.”8 Rather, his was a heuristic approach, inspired by the methods of conceptual artists like Sol LeWitt. He sought to move architecture away from an empirical and functionalist approach, and was equally skeptical of the phenomenological understanding of architecture. Instead of focusing on the use and sensorial experience of architectural form, he was interested in exposing and expressing the underlying—and rational— concepts that were unique to architecture.9

In both historical architectural examples and in Chomsky’s syntax, Eisenman looked for the transformational grammar and operations that transformed the deep structure of architecture into evernew surface structures.10 In his essay “Cardboard Architecture” Eisenman describes his work as being guided by the rules of a “process of transformation,” which would take form-making away from the “imprecise” and “metaphorical,” and make design “more logical and rational.”11 This required a change of emphasis, from form to formal structures. As he put it, he attempted “(…) to bring [formal relationships and qualities] into some sort of theoretical construct by focusing on the aspects of formal structure, (…) [this theory being] concerned with the relationship between objects, rather than the object itself. This framework has two aspects; dual deep structural and transformational operations (…)”12

The translation from linguistics required establishing the architectural equivalents of words and sentences. Eisenman concluded that “within the conditions of deep [architectural] structures it is possible to identify (…) two irreducible sets of formal integers. The first is solid and void; the second, centroidal and linear. These conditions do not exist without each other; they are interdependent.”13 These concepts are distinct from literal lines, planes and volumes, and are rather understood as relationships within and between them. As such, they cannot be directly perceived: they need to be transformed in order to be recognized. Such formal transformation was the task of Conceptual Architecture.

Following the lead of his teacher Colin Rowe, one of the operations that Eisenman used to transform these deep structures into new surface structures was phenomenal transparency, or the “ambiguity of layered planar space and volumetric space.” In his analysis of Guiseppe Terragni’s Casa del Fascio he concludes that from this “one transformational device (…) an infinite range of specific forms can be conceived, (…) based on a limited series of formal universals.”14

However, there is another transformational device found in Eisenman’s work from that period—namely, the four symmetrical operations. The diagrams that explain the generation of his early houses, especially House II and House IV, reveal their consistent presence.15 In House II the boundary of the overall shape is established at the beginning as a cube. Its four vertical surfaces are translated equally to create a nine-square grid of space inside the cube. The diagrams show how the first six steps preserve the cube’s rotational and reflective symmetry. Even the one asymmetrical move, of sequentially shortening the planes and volumes into one, two and three bay lengths in the seventh and eighth diagrams, have a symmetrical relationship with one another.16 By combining these diagrams with one another, and repeating the operations in section, the result is a highly complex object produced via the aggregation of a very limited set of symmetrical operations.17

In House IV there is little to no asymmetry to be found in the diagrams that produced it. Every plane, point and volume has its twin and every symmetrical operation,

including glide reflection, is deployed in plan and section. Phenomenally transparent, layers are again present, and they too have a symmetrical relationship with one another. As with House II, it is the density of these operations that transform a simple cube into a dizzying, spatial object. Despite its surface complexity, the rules and elements of its deep structure are limited to symmetrical movements and to a few points, lines, planes, and volumes.18 While perhaps irrational from an experiential or habitation standpoint, its operation follows a strict (symmetrical) logic. In doing so, it helps make Eisenman’s point that architectural form is independent from human use or perception.

Operations over Communication

Like Chomsky, Eisenman was not driven by the desire to create specific affects or meanings. Rather, he was interested in how these new forms could “accept a new and greater range of iconographic and metaphoric meanings.”19 Such meanings would be the unexpected result of this process, but not its goal. Symmetry—as a set of operations rather than a quality—provided Eisenman with a “found” technique for producing new formal configurations that contained both invariance and transformation. There are multiple locally automorphic elements in these early projects, but their combination produces a formal and spatial whole that is not. As with its use in science and linguistics, symmetry is not an ideal to aspire or adhere to, it is a tool to help understand the underlying structure of reality, and in turn generate new versions of that reality. In experimenting with symmetry’s operations, Eisenman helped to transform architectural discourse, pedagogy, and practice from a strictly professional to a more intellectual pursuit. In other words, his symmetrical operations produced a variety of asymmetrical effects.

1 Theodore W. Gamelin and Robert Everist Green, Introduction to Topology (New York: Dover, 1999). 2 Barbara Citko, Symmetry in Syntax: Merge, Move and Labels (New York: Cambridge University Press, 2011), 210. 3 “(…) Transformational Generative Grammar (…) [has] as its principal objective the formulation of a finite set of basic and transformational rules that explain how the native speaker of a language can generate and comprehend all its possible grammatical sentences (...).” Beatriz Rodríguez-Arrizabalaga, “Modification,” Encyclopedia of Linguistics (New York: Fitzroy Dearborn, 2005), 697. 4 Noam Chomsky, Syntactic Structures (The Hague: Mouton & Co, 1965), see also: Aspects of the Theory of Syntax (Cambridge: MIT Press, 1965). 5 Stephen Grabow, Christopher Alexander, The Search for a New Paradigm in Architecture (Boston: Oriel Press, 1983), 48-49. Philip Steadman, “Research in Architecture and Urban Studies at Cambridge in the 1960s and 1970s: What Really Happened,” The Journal of Architecture vol. 21 no. 2, (2016), 291-306. Lionel March, “Architecture and Mathematics Since 1960,” Nexus IV: Architecture and Mathematics, Kim Williams and José Francisco Rodrigues eds. (Florence: Kim Williams Books, 2002), 9–33. James Gips and George Stiny, “An Investigation of Algorithmic Aesthetics,” Leonardo vol. 8; no. 3, (summer, 1975), 213220. George Stiny, Shape (Cambridge: MIT Press, 2006). Lionel March, “Forty Years of Shape Grammars, 1971-2011,” Nexus Network Journal vol. 13 no. 1, (winter, 2010), 5-13. Peter Eisenman, “Notes on Conceptual Architecture: Towards a Definition,” Casabella vol. 35 no. 359 (1971), 4858. Peter Eisenman, “Cardboard Architecture,” Casabella vol. 37 no. 374, (February, 1973), 17-31. 6 Noam Chomsky, Aspects of the Theory of Syntax (Cambridge: MIT Press, 1965). Noam Chomsky, Syntactic Structures (The Hague: Mouton & Co, 1965). Noam Chomsky, Cartesian Linguistics (New York & London: Harper and Row, 1966). 7 Eisenman, “Cardboard Architecture,” 17-31. Eisenman, “Notes on Conceptual Architecture”, 48-58. 8 Eisenman, “Cardboard Architecture,” 24. 9 Eisenman, “From Object to Relationship II: Casa Giuliani Frigerio: Giuseppe Terragni Casa del Fascio,” Perspecta 13/14, (1971), 36-65. 10 Eisenman, “Cardboard Architecture,” 23. 11 Ibid., 22. 12 Ibid. 13 These quotes appear in Mario Gandelsonas, “Linguistics in Architecture,” Casabella vol. 36 no. 374, (February, 1973), 17-31. Gandelsonas attributes them to “Notes on Conceptual Architecture II” but gives no source information. Eisenman would later publish “Notes on Conceptual Architecture (II): Double Deep Structure,” A+U 3, (March 1974). (Japanese version only), “Notes on Conceptual Architecture II,” in Environmental Design Research, proceedings of the fourth international Environmental Design Research Association conference vol. 2. Wolfgang F. E. Preiser ed. (Stroudsburg: Dowden, Hutchinson & Ross, 1973 1974). 14 Eisenman, “From Object to Relationship II,” 61. 15 Eisenman, “Cardboard Architecture.” 16 Ibid., 20. 17 Ibid., 21. 18 Ibid., 30-31. 19 Ibid., 24.