Archivos 08 Symmetry

08

ARCHIVOS DE ARQUITECTURA 08 SYMMETRY

Archivos de Arquitectura 08 Escuela de Arquitectura y Estudios Urbanos Universidad Torcuato Di Tella Symmetry The One and the Many Author David Salomon Edition Santiago Miret

This number has a prologue by Ciro Najle “Playing Dumb, or the by Julián Varas, and the essay David Salomon. The material corresponding to the seminar classes dictated by David Salomon in 2015. Each chapter Pringle Sattui that structures the organization of symmetry across case studies. The epilogues “A Manuel Mensa foster relationand direct instrumentalization of symmetry.

Archivos de Arquitectura 08 Escuela de Arquitectura y Estudios Urbanos Universidad Torcuato Di Tella Symmetry The One and the Many Author David Salomon Edition Santiago Miret Models Andrew Pringle Sattui

Universidad Torcuato Di Tella Rector: Ernesto Schargrodsky Vicerrectora: Catalina Smulovitz Escuela de Arquitectura y Estudios Urbanos Decano: Ciro Najle Carrera de Grado de Arquitectura Director: Sergio Forster Maestría en Historia y Cultura de la Arquitectura y la Ciudad Director: Julián Varas Programa en Arquitectura y Tecnología Coordinador: Francisco Cadau Programa en Arquitectura del Paisaje Coordinador: Juan Pablo Porta Programa en Preservación y Conservación del Patrimonio Coordinador: Fabio Grementieri Maestría en Economía Urbana (c/Escuela de Gobierno) Directora: Cynthia Goytia Centro de Estudios de Arquitectura Contemporánea Coordinador: Santiago Miret Archivos de Arquitectura Coordinadora: Anna Font Comité de Redacción Ciro Najle y Julián Varas

No está permitida la reproducción parcial o total del material que aquí se publica. Las opiniones contenidas en los artículos son de exclusiva responsabilidad de los autores. ISSN 2314-3029

Archivos de Arquitectura 08 Symmetry The One and the Many Autor: David Salomon Edición: Santiago Miret Modelos: Andrew Pringle Sattui Colaboraciones: Anna Font, Manuel Mensa, Santiago Miret, Valeria Ospital, Carolina Telo, Fernando Yabén Template gráfico: Departamento de Comunicaciones UTDT Material: Programa académico del seminario de posgrado Symmetry de la Maestría en Historia y Cultura de la Arquitectura y la Ciudad de la Escuela de Arquitectura y Estudios Urbanos en el año 2015 Ilustración de tapa: Torre O-14, Reiser + Umemoto RUR, Dubai, EAU, 2010. Axonometría bizantina. Dibujo de Andrew Pringle Sattui

Número de Edición: 08 Fecha de Edición: Octubre de 2018 Impresión: Ediciones Emede SA Cantidad de ejemplares: 500 Tipografía: Helvética Neue Propietario Universidad Torcuato Di Tella Campus Alcorta Avenida Figueroa Alcorta 7350 Sáenz Valiente 1010 Ciudad de Buenos Aires Argentina

Index

Playing Dumb, or the Power of Indifference

Ciro Najle

006

Theory’s Hardware

Julián Varas

010

The One and the Many

David Salomon

014

01 Architecture and Symmetry

020

02 Science and Symmetry

024

03 Biology and Symmetry

028

04 Crystals and Symmetry

032

05 Physics and Symmetry

036

06 Statistics and Symmetry

040

07 Math and Symmetry

044

08 Geometry and Symmetry

048

09 Syntax and Symmetry

052

10 Semiotics and Symmetry

056

11 Information and Symmetry

060

12 Perception and Symmetry

064

Never Even

Manuel Mensa

070

A Symmetrical Story

Santiago Miret

074

Acknowledgments

078

A system of splines is displayed between the centroids of niches and columns of the perimeter system and the central points and the different edges of the steps. The splines transversal to the axis of symmetry constitute the matrix of steps on which the flow of longitudinal splines unfolds. Michelangelo Buonarroti, Laurencian Library. Florence, Italy, 1525. Drawing by Santiago Miret

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Playing Dumb, or the Power of Indifference Ciro Najle

Yes, symmetry; because symmetry is, at present, one of the few, definitely the coolest and most gracefully effortless, and probably the most effective way to deal with unnecessary difference in a world where worthlessness has become currency, comfort zone, cliché, and commodity; because symmetry turns the inherent impoverishment of the value of difference upside down by reducing it and constructing higher levels of intelligence and beauty with its absence; because symmetry is the means to sharpen difference through sameness, distilling traits and individuations out of the general indifference resulting of the addiction to difference for the sake of it; because symmetry is a resource to the power of indifference, an aggressive cultural form that generates difference in kind by eroding from within the lowest degree of difference: that which makes no difference; because symmetry fastens up form and raises organization to the status of archetype; because symmetry dries out lagoons and secures knowledge; because symmetry is a breach opened up in the midst of the nothingness of fullness; because symmetry speaks unheard of voices and creates little cores of significance across the mute continuum of difference; because symmetry constructs speech in a speechless world without relying on discourse; because symmetry is the spoken word that both murmurs and shouts out new forms of language; because symmetry is fundamentally a culturally loaded short-circuit, an instantaneous relationship between in principle lonely fractions of matter that become a set, the most archaic and, at the same time, the ultimate technique for containment, a sudden method for enrichment, wit, and humor that reverses and reloads the magma of sheer opulence, empty extravagance, and false gravity that surrounds us; because symmetry creates apparently default conditions; because symmetry involves a tragic loss (the fatal loss of unrecoverable information) and turns the comedy of formlessness into a peculiar, at times high-pitched, often pervasive weapon; because symmetry congregates and assembles what is otherwise disintegrated and disperse; because symmetry organizes differences, forms organizations, shapes forms, and figures shapes; because symmetry transfers all of those and their knowledge across domains (material, abstract, conceptual, visual, sensible, theoretical, critical, analytical, projective), through mimicking or through rationality; because symmetry imbues appeal and permeates poignancy in anything it encounters; because symmetry is the spreading of auto-morphism both across the board of fields and between fields, until the form of knowledge becomes ubiquitous in a peculiar, distinguished manner; because symmetry is the ultimate medium for the construction of models out of systems (while “symmetry endures, symmetry changes,” as Salomon says); because symmetry is the “surprisingly supple and inclusive” medium by which architecture “simultaneously produces specific disciplinary knowledge and establishes connections with other fields;” because, far beyond the composition of parts that creates the appearance of order by means of mirrors across objects, symmetry is the slippery, kaleidoscopic choreography of those mirrors, the infusion of enigma and ingenuity where there is sheer transparency and vulgarity; because symmetry is the nesting of otherwise chaotically scattered attributes embedding one within the other, because symmetry is the incorporation of precedence where there is numbness, and the progressive configuration of timelessness by means of the propagation and contradiction of various kinds and degrees of precedence; because symmetry introduces and then blurs the distinction between the before and the after; because symmetry mingles the world and raises it above the mundane; because symmetry interlaces, stitches, sutures, and knits the world; because symmetry transforms through preservation; because symmetry is the engendering of order through the repetition and reverberation of the same; 7

because symmetry is an echo machine, a both-here-and-there, and also-here, and twistedhere, and intertwined-here, and inverted-here, and not-anymore-here, and should-be-there; because symmetry is the here-but-not-really and, therefore, the here-everywhere; because symmetry propagates resonance, structures order, and configures value where there is only nonsense; because symmetry is the nesting within, the nesting within the nesting, and the cross-nesting of sameness; because symmetry entertains (inter-possesses) diversions, flipping them, convoluting them, and turning them into a single joy machine; because symmetry cleverly enmeshes the political correctness of diversity and softly threatens the innocuousness of plurality; because symmetry involves the paradox of harm through harmony (and turns harmony into an elevated form of harm); because symmetry is the most sophisticatedly deceitful, skillfully disguised, concealed vehicle for transferring and cross-referencing across the dense sea of irrelevance; because symmetry creates knowledge out of information; because symmetry is the immersion into and out of difference, the making of sound from within the very guts of noise, the making of sound that turns noise and its pain altogether necessary; because symmetry is the deepest diagrammatic level of figures, and drafts the coordinates of its plane of immanence; because symmetry is the astringent and antiseptic formula to revitalize difference and give back life, signification, and sympathy where there is only amenity, triviality, and apathy; because difference is a sterilizer that fertilizes, the sine qua non technique to recognize, nurture, engender, consolidate, and radiate difference that makes a difference; because symmetry sinks perception down into the sleepy limbo of naturalness and raises it up to the fiction of organization; because symmetry is the most persuasive mechanism for forcing the distracted glance to look back and pay attention, the ultimate means for meaningful suspensions, the resonance box that neutralizes irrelevance, the sweet revenge of oneness in a world of multiplicities, and the fold by which indifference turns into its very opposite: singularity. Symmetry is the self-conscious act of playing dumb in a world of foolishness on steroids.

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Transversal splines are constructed by interpolating the centroids of the columns and niches of the perimeter system with the centers and edges of the steps of the central system. The splines that form the central flows emerge from the proportional subdivision of the transverse splines. Michelangelo Buonarroti, Laurencian Library. Florence, Italy, 1525. Drawing by Santiago Miret

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Degree six, plan of maximum symmetry. Six axis of symmetry organize the elements of the main plan of the Villa Stein, by Le Corbusier. Each element has originally a different degree of symmetry (from zero to six mirror conditions). A process of linear symmetrization transforms the asymmetrical into multisymmetrical, and the single into the serially singular. Le Corbusier, Villa Stein. Garches, France, 1927. Drawing by Anna Font

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Theory’s Hardware Julián Varas

For the science fiction enthusiast, the arc spanning from Steven Spielberg’s Close Encounters of the Third Kind (1977) to Denis Villeneuve’s recent Arrival (2016, based on Ted Chiang’s 1998 Story of your Life) is a clear sign of how blockbusters have contributed to upgrading the figure of the alien—and, stealthily, our own. Turning human/alien communication into a serious speculation, sci-fi no longer portrays aliens as a bunch of monstrous figures intent on subjugating or feeding on anthropos—at least not always. The narratives developed in recent films introduce an alternative figure: a non-divine, intelligent other whose mere existence seems to assert the plurality and decenteredness of consciousness, language, and culture. No longer couched as evil creatures seeking to overpower us, the notion of the communicative alien suggests a renewed philosophical realism lying at its base. The de-centering of the notion of the human wrought by science fiction is, in fact, as old as the genre. Yet, its philosophical implications only began to surface during the second half of the 20th century. The problem, as stated recently by realist philosophers such as Manuel De Landa and Graham Harman, boils down to this: human thinking has for too long been concerned with humanity and the mind-world relation as its main preoccupation and source of legitimacy. De Landa had advanced an audacious proposition to counter this trend in his book War in the Age of Intelligent Machines (1991). During the early days of the digital revolution, he had conjectured the development of the machinic phylum reaching the point where a synthetic consciousness would emerge. Not far into the future, our still fairly dumb robots could acquire an autonomy that would enable them to ponder about their existence, origins, and history. As De Landa’s parable goes, a robot-historian might have come to see itself as representing the endgame of the evolution of intelligence, not anthropos, its alleged creator. The robot might legitimately see human agency as a necessary instrument toward the emergence of synthetic intelligence—a kind of midwife that helped catalyze a more advanced species within the machinic phylum. But how can De Landa’s robot-historian and science fiction’s recasting of the alien figure be relevant to a text on architectural theory? As proven by David Salomon’s Symmetry, The One and the Many, symmetry is one of those concepts through which we have insistently attempted to outline the boundaries of our identity as a culture and species, while simultaneously appropriating its ability to bridge cultures and lines of speciation. Despite its elusiveness, symmetry appears both inherent to a variety of architectural traditions and capable of connecting them with machinic and alien ontologies. Symmetry restores architecture to a networked conception of the material world. The fact that (we) humans produce (our own) architecture as an interface for the construction of the social world, accounts for just one region of such web—and not necessarily the most significant one. The remaining relations encompass organizations studied by the natural sciences. Thus, even though science and architecture belong within culture, symmetry breaks through the boundaries of culture pointing toward realities that transcend it.

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These are the underlying premises for the ideas that David Salomon put forward at a seminar held at the Maestría en Historia y Cultura de la Arquitectura y la Ciudad de la Escuela de Arquitectura y Estudios Urbanos at Universidad Torcuato Di Tella in 2015. The development of those hypotheses has led to a tightly packed essay, in which architecture explodes into a matrix of knowledge and practices that reaches far beyond its established field of endeavor. The text drives us along the connections that root architecture within the broad purview of science, mathematics and communication. Symmetry unlocks those connections, serving at once to knit a sweeping historical narrative and to open up a field of potential for the conceptualization of contemporary design praxis. On first reading, the text appears as an essay on the history of an idea, which sprawls out into different areas of culture. Agile and concise, Salomon makes evident the resilience of symmetry in its recurrence across distant chapters in the history of architecture, even when—as in functionalist ideology—it had been discursively banished from the scene. Symmetry resonates in the broad space between architecture, society, mind, and crystals. The text indexes the migration of symmetry from architecture into mathematics, where it was redefined. From a feature of stable organizations, it was conceptualized as a process of transformation that maintains stability—a shift that was pivotal in bringing the contemporary disciplinary discourses on the generic and the parametric to fruition. However, the networks of relations that symmetry constructs throughout the book are not just rhetorical. One should avoid seeing the book as an attempt to culturalize architecture, hoping to instill new relevance into it. Salomon’s project would be misunderstood if reduced to the history of an idea. Nor is it hinting at the dissolution of architecture into a vague magma of interdisciplinarity. The parallel with science fiction and realist philosophy reveals that what Salomon is after is nothing less than an ontology of architecture. In such ordering, the human is just one idea in a network of interrelated objects, with no center to create a transcendental hierarchy that binds them. The continuities drawn up by the text are therefore more productively read as a naturalistic map of the unified configuration of the natural, social, and cultural organizations in which anthropos is involved. Even if we accept the premise that the conditions of our contact with reality (architecture) are hardwired onto the circuitry that structures our interests and sensing capacities, our conception of architecture need not be confined within those boundaries. Any sophisticated realist worldview must be able to separate what is from what can be known, and therefore concede that architecture exists prior to culture—i.e., that it is the condition of possibility of culture, its infrastructure. Symmetry is a foundational brick of that infrastructure. The tree that complexity builds is partially based on the possibilities afforded by symmetrical transformations. It gives rise to all the forms of nature, including all life forms, especially those that develop self-awareness, language, tool making, and radically artificial environments. David Salomon’s text is part of a project to illuminate how those relations escalate, cutting through eons of evolutionary development. It seeks to turn architectural theory again into a grand narrative—a new account of the cosmic, this time without irony or finalism.

1 In the 19th century, science fiction became a privileged experimental genre for the expansion of the conception of the human. Alongside Darwin’s Theory of Evolution and Marx’s Historical Materialism, science fiction began to hint at the crisis of the modern subject and its pretense to liberty, integrity and sense of finality. 2 De Landa picks up the notion of the machinic phylum from Gilles Deleuze and Felix Guattari’s A Thousand Plateaus (1980), and elaborates an extensive and systematic theorization thereof. The machinic phylum refers to a flow of matter that cuts across conventionally established divides such as culturenature or organic-technological. “The idea of a ‘machinic phylum’ would then be that, beyond biological lineages, we are also related to non-living creatures (winds and flames, lava and rocks) through common ‘body-plans’ involving similar self-organizing and combinatorial processes. As if one and the same material ‘phylum’ could be ‘folded and stretched’ to yield all the different structures that inhabit our universe.” See: Manuel De Landa, “The Machinic Phylum,” in Joke Brouwer et al. (eds.) Technomorphica (Amsterdam: De Balie, 1997).

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Degrees one, two, three, four, and five, plans of progressive symmetrization. Six axis of symmetry organize the elements of the main plan of the Villa Stein, by Le Corbusier. Each element has originally a different degree of symmetry (from zero to six mirror conditions). A process of linear symmetrization transforms the asymmetrical into multisymmetrical, and the single into the serially singular. Le Corbusier, Villa Stein. Garches, France, 1927. Drawing by Anna Font

13

An axis of vertical symmetry determines the proportion of the column by an aureus rectangle, whose side is the maximum width of the column. Two equilateral triangles are defined from the rectangle: the lower horizontal symmetry axis is determined by the proportions of the lower triangle. Estudio CEPRA and Clorindo Testa, Banco de Londres. Buenos Aires, Argentina, 1966. Drawing by Fernando Yabén

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The One and the Many David Salomon

Architectural symmetry, we have been told, is obsolete. It is dangerously essentialist and authoritarian. It is superficially applied. Its rigid rules are an indication of indifference—to site, to structure, to use, to climate.1 Symmetry is a problem to overcome, not something to explore. Outside of architecture, however, the opposite is true. At the same time as its architectural relevance was being challenged—at the turn of the 20th century—its importance in math and science increased. Physics, chemistry, and biology recognized the multiple forms and uses of symmetry, and effectively employed it in a variety of contexts.2 A closer look at modern architecture reveals that it never truly abandoned symmetry, it only discursively deserted it. Given the continued presence, even today, of symmetry, this book examines how architecture, math, and science have persistently used it to simultaneously produce specific disciplinary knowledge and establish connections with other fields. But, why revisit symmetry, why now? Because the need for linking diverse ideas and objectives is increasingly crucial today. In a world where the side effects of disciplinary specialization and technological optimization can no longer be solved by more isolation, synthetic modes of inquiry are necessary. Despite its reputation as ideal and autonomous, symmetry has proven to be surprisingly supple and inclusive, and it is worth reexamining it today to see what it can—and what it might already effectively—integrate. In short, given its long history, its covert presence in modernism, and its continued use in contemporary architecture, one must ask if today symmetry still functions as an expression of power and permanence, or if it can be used as an effective device to compare and combine seemingly unlike things. Defining Symmetry Before addressing these issues we need to be clear about what symmetry is. For many, something is symmetrical if it is composed of two equal and identical parts facing each other across an axis. This is typically called mirror or reflective symmetry. The human face is a prime example. However, this is neither the original meaning of symmetry nor its contemporary mathematical use. This phenomenon had not been codified as “symmetry” until the 17th century by the French architect Claude Perrault, and, in the late 18thth, century by the French mathematician Adrien-Marie Legendre.3 An earlier definition of symmetry, which goes back to Vitruvius, defines it as the regular and pleasing harmony that is created by the correct use of proportion. Today, following the findings of group theory in math, symmetry is the quality of anything that remains invariant despite it being transformed. In all of these definitions, symmetry is plural. For something to be symmetrical there must be more than one element present. But, these elements must be distributed in a particular way. It is a particular kind of pattern, a pattern being defined as an array of elements distributed according to a specific logic or structure.4 Symmetry is a structure that demands a specific kind of equivalence between the parts—the parts of a body, of a formula, of a building, or of anything else.

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Formally speaking, an image or object is symmetrical if you can pick it up, move it, then place it back down in a new position without an observer being able to tell it had been moved. Mirroring, or reflection, is one operation that can produce these results. There are three more: translation, rotation, and glide reflection. Mathematicians—building on work done by crystallographers—have proven that there are only seven ways in which these four operations can be used to create one-dimensional patterns (often referred to as frieze patterns). They will also tell you that they can only produce seventeen two-dimensional figures (known as wallpaper patterns) and 230 three-dimensional configurations. Symmetry is by no means infinite, but there is definitely more than one. The Use of Symmetry While the number of actual permutations possible using the four operations was established mathematically in the 19th century, they had been understood and used for millennia in architecture and design. One could say that symmetry was ornamental before it was mathematical. In other words, traditional symmetrical ornaments are not illustrations of mathematical concepts: they are the engines that produced them. Such knowledge was widespread. In his 1856 book The Grammar of Ornament, Owen Jones attempted to capture how this information was used in every historical era and geographic region in a single volume. At roughly the same time crystallographers and chemists were establishing the symmetrical structures that underlie the appearance of matter. Like Jones, they sought to capture, once and for all, the consistent source of inanimate form. Thomas Beeby has documented how the foundational work of Frank Lloyd Wright, Le Corbusier, and Mies van der Rohe was directly influenced by the symmetries illustrated by Owen Jones and subsequently described by crystallography.5 All three used symmetry operations to organize and distribute spaces, structural elements, and forms. Symmetry was used as an active organizational and form-making device. It was not just a descriptive idea, but a heuristic, productive device. Meanwhile, in physics and math symmetry became ever more useful in the 20th century; not because of its empirical presence but because of its theoretical value. If in modern architecture symmetry was found in physical objects but was absent from theory, in the natural sciences the opposite was true. Symmetry was a powerful idea, even if it was rarely encountered in its pure state. If symmetry was the rule, then many other phenomena made sense.6 The fact that symmetry is both a property of objects and an underlying concept is also significant. Objects and ideas: for a long time the latter have held sway as the primary source of knowledge production. However, the story of symmetry suggests that this sequence can work in reverse, that aesthetic operations and artifacts can precede and even help produce ideas. This book outlines symmetry’s productive trafficking, especially in the 19th and 20th centuries, between these two ways of knowing. This combination is itself a decidedly architectural move. Architects are alchemists who combine seemingly unrelated parts into complex wholes. Why revisit symmetry now? To examine how this seemingly inevitable architectural trope has and might still effectively integrate a number of architectural ideas and elements with one another. 1 Phillip Tabor, “Fearful Symmetry,” Architectural Review no. 1023 (May, 1982). 2 Giora Hon and Bernard Goldstein, From Summetria to Symmetry: The Making of a Revolutionary Scientific Concept (London: Springer, 2008). 3 Giora Hon and Bernard Goldstein, “Unpacking ‘For Reasons of Symmetry’: Two Categories of Symmetry Arguments,” Philosophy of Science no. 73 (October, 2006), 419-439. 4 Paul Andersen and David Salomon, The Architecture of Patterns (New York: Norton, 2008). 5 Thomas H. Beeby, “The Grammar of Ornament/Ornament as Grammar,” VIA III (1977), 10-29. 6 Giora Hon and Bernard Goldstein, “How Einstein Made Asymmetry Disappear: Symmetry and Relativity in 1905,” Archive for History of Exact Sciences 59, no. 5 (2005), 437-544.

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An axis of vertical symmetry determines the geometry of the column by an aureus rectangle, whose side is the maximum width of the column. Estudio CEPRA and Clorindo Testa, Banco de Londres. Buenos Aires, Argentina, 1966. Drawing by Fernando Yabén

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Matrix of symmetry analysis. Plans and axonometries. From top to bottom, one project per row: Maison Carrée, Pazzi Chapel, Santa Maria Novella, Villa Capra, Barrière de la Villete, Altes Museum, Darwin D. Martin House, Glass Pavilion, Villa Savoye, Crown Hall, Teatro del Mondo, O-14. From left to right, one type of symmetry per column: mirror reflections, translations, rotations and glide reflections, and the eight possible combinations between them

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Maison Carrée, Nimes, France. 16 A.C.

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01 Architecture and Symmetry

Symmetry endures. Symmetry changes. This paradox is evident in the varying definitions and fluctuating influence that symmetry has had on architectural design and discourse for over 2,000 years.1 This chapter outlines symmetry’s consistent presence in Western architectural thought and reveals how it has often been used to establish an objective, quantitative means for creating and evaluating work, while still allowing for subjective innovations within its well-defined limits. In other words, it will examine how symmetry’s disciplinary stability is a function of its flexibility. Classical Symmetry The conventional understanding of symmetry is technically defined as reflective symmetry, that is, the relationship between two equal and opposite figures on either side of an axis.2 While the use of this figural trope is indeed ancient, it was not the original definition of symmetry. For the Roman architectural theorist Vitruvius, symmetry described the good and beautiful use of proportions to create a composition, one where the parts were harmoniously related to one another and to the larger whole. Symmetry was not a simple technique; it was a complex method for creating harmony out of a variety of different parts. The Latin term that Vitruvius used to describe the mathematical relationship between parts and wholes was respondere. The numerical ratios used to create this effect were embodied in the different columnar orders (the Doric, the Ionic, and the Corinthian), and were typically defined by whole-number-based ratios found in the relationship between the width and the height of a column. This ratio, and its sub-ratios, established every dimension found in the design of a truly symmetrical building. While this often resulted in reflective plans and elevations, creating a bilateral image was never an explicit goal and is never mentioned in Vitruvius’ treatise, The Ten Books of Architecture.3 Medieval Symmetry Although often defined by its irregularities, medieval architecture also contains multiple symmetries. This is particularly true in the composition of Gothic cathedrals. Symmetry was a requirement for these buildings because of the historical association between, stability, god, and symmetry: “(…) the universe owes its stability to the perfect balance of its elements as instituted by the Creator, a stability that will be denied to any building that does not possess symmetry.”4 Here too, symmetry was a mathematical technique for producing a consistent and underlying objective standard for architecture to conform to. Symmetry was not limited to the ground plans and elevations of Gothic cathedrals. It helped define the overall geometric relationships between part-to-part and part-to-whole. It also defined the local relationships in structural bays, arches and vaults, and guided the designs of intricately patterned surfaces. Nor was symmetry limited to the bilateral variety. The geometric order found in Gothic architecture contains examples of what 19th century mathematicians would later categorize as the four basic symmetrical operations: reflection, rotation, translation, and glide reflection.5

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Renaissance Symmetry

Neo-Classical Symmetry

Despite his direct debt to Vitruvius, the term symmetry is absent from Leon Battista Alberti’s 1452 treatise, On the Art of Building. What Alberti does include, which Vitruvius did not, is a description of elements with equal dimensions being identically distributed on either side of a central element. The word Alberti uses to describe this arrangement is respondere. This is the same term Vitruvius used to describe the proportional relationships between parts and wholes within a composition. Alberti does not abandon the idea that beauty requires the use of proportioning systems. He referres to this relationship as concinnitas, and it too is a prerequisite for beauty. In transferring the meaning of respondere, Alberti gives equal value to bilateral symmetry (without using that term) and proportion. While what we now call reflective symmetry was possible in Vitruvius’ theory, it is a requirement of Alberti’s.6

The placement of identical elements on either side of a reflection line would come to define the arrangement and configuration of many monumental buildings in the 18th and 19th century. The organization of circulation and programs astride symmetrical axes became the academic and disciplinary norm in the teachings of J.N.L. Durand at the École Polytechnique and the lecture halls and ateliers of the École des Beaux-Arts, both in Paris. Whether located in buildings with pure geometric envelopes or irregular urban lots, symmetry was used to orient and guide people through a building.10

Enlightenment Symmetry The search for a simple, consistent, underlying logic that explains a large set of phenomena is a hallmark of modern science. In late 17th century France, Claude Perrault attempted to apply this ethos to architecture. After reviewing a variety of architectural treatises and studying the actual measurements of buildings, he came to the conclusion that there was no empirical or theoretical evidence that allowed one to objectively establish one set of proportions as being any better or more beautiful than any other. These empirically established variations led him to the conclusion that proportions were an “arbitrary” or “conventional” standard for producing and evaluating architecture and architectural beauty.7 In contrast, Perrault argued that the size of buildings, the quality of their materials and craftsmanship, and the presence of symmetry were not arbitrary but “positive” and “convincing” sources of architectural beauty. He maintained that these were convincing traits because their “presence in works is bound to please everyone, so easily apprehended are their value and quality.”8 In other words, they are objective. Symmetry could be understood as such because Perrault redefined it as referring exclusively to the reflection of identical parts on either side of an axis. While this trope was not new, Perrault was the first architectural theorist to define symmetry in this way.9 With Perrault, mirror symmetry is established as an objective prerequisite for architectural beauty. It is at this moment that symmetry leaves the realm of opinion and enters the world of reason (something is either symmetrical or it is not). It would take more than a century before the mathematicians Legendre and Galois would establish a similarly central and objective place for symmetry in science and mathematics. 22

The Beaux-Arts emphasis on bilateral symmetry was codified by the École’s professor of architectural theory, Julien Gaudet, in his 1901 treatise Eléménts et théorie de l’architecture. These conventions were carried on in 20th century books on architectural composition.11 It was only with the acceptance of the Bauhaus model of basic design, and with Modernism’s emphasis on letting program, structure, and materials directly influence architectural form, that the spell of symmetry was broken. Modern Symmetry In 1977, the midst of the postmodern era, architect Thomas Beeby showed how three modernist masters– Frank Lloyd Wright, Le Corbusier and Mies van der Rohe– were all well-versed in the use of symmetry. This skill came not from the study of architecture proper, but from their experience with ornament, especially the intensely symmetrical plates found in Owen Jones’ 1856 book, The Grammar of Ornament. Rather than letting their form follow function, Beeby shows how they distributed structure and space using the four symmetrical operations of reflection, rotation, translation and glide reflection.12 In the 1960s a few architects familiar with the scientific understanding of symmetry attempted to incorporate the new mathematics of symmetry–most notably Alfred Neumann and Anne Tyng–into their work.13 However, the modernist stigma against symmetry, its association with applied ornament, and the radical nature of their work–full of rotations, translations and reflections–kept symmetry out of the spotlight. Postmodern Symmetry In his 1982 essay entitled “Fearful Symmetry,” Philip Tabor outlined the historical arguments made for and against architectural symmetry.14 The “fear” in Tabor’s title refers to a poem by William Blake, but it also echoes modernists’ anxiety around the reappearance of bilateral symmetry in the work of postmodern architects like Aldo Rossi in Italy and Michael Graves in the United States.15

Tellingly, Tabor only discusses mirror symmetry. He does not address the other symmetrical operations or the changing uses and definitions of the term in math and science. His emphasis is on what mirror symmetry represents and means. Looking at the history of Western architecture he notes that the arguments made regarding symmetry relate to: “1) bodily experience, 2) the notion of redundancy, and 3) the notion of hierarchy.” The first is related to the empathy humans feel for symmetrical things. We recognize ourselves in these forms. The second category emphasizes the economy of means and ends present in the production and reception of symmetrically composed buildings. They are easier to design and build because they have fewer unique parts. They are easier to understand and remember because they are predictable. Finally, the imposed consistency required with symmetry is often used by and associated with the conformity and authority found in political and religious institutions. In short, symmetry is human, symmetry is quantifiable and easy, and symmetry is powerful. Contemporary Symmetry The influence of postmodernism and its use of traditional architectural tropes was challenged in the 1990s by the rise of digital design tools, methods, and forms. Despite the avant-garde stance towards form, the direct rejection of symmetry and the ease with which these tools create asymmetrical forms and patterns, symmetry (especially reflections, rotations, and translations) can still be regularly spotted in the diverse practices of Greg Lynn, FOA, and Norman Foster.16 Even a decade into the 21st century, the four symmetrical operations are still easily found in work done by “postdigital” era architects who are either interested in revisiting postmodernism, or who are importing ideas from the philosophy of Object Oriented Ontology into architectural design and discourse. Despite the oppositional formal and ideological differences of these two trends, in both cases symmetry provides a clear perceptual limit that enables one to grasp what would otherwise be disorienting effects. It is useful but not universal. To conclude, the constant presence of symmetry should not be mistaken as an architectural essence or prerequisite. Rather, its eternal return reveals that despite the many fluctuations in its status and definition, symmetry endures as a stable reference point for change. In other words, architectural symmetry is itself symmetrical, as its presence has remained consistent despite its definition having been transformed.

1 Giora Hon and Bernard Goldstein, From Summetria to Symmetry: The Making of a Revolutionary Scientific Concept (London: Springer, 2008). 2 Giora Hon and Bernard Goldstein, “From Proportion to Balance: The Background to Symmetry in Science,” Stud. Hist. Phil. Sci. No. 36 (2005): 1-21; Phillip Tabor, “Fearful Symmetry,” Architectural Review No. 1023 (May 1982). 3 Hon and Goldstein, “From Proportion to Balance”, 4-6. 4 Nelly Shafik Ramzy, “The Dual Language of Geometry in Gothic Architecture: The Symbolic Message of Euclidian Geometry versus the Visual Dialogue of Fractal Geometry,” Perigrinations: The Journal of Medieval Art & Architecture 5, No. 2 (2015), 140. 5 Ibid. 6 Hon and Goldstein, From Summetria to Symmetry. 7 Claude Perrault, Ordonnance for the Five Kinds of Columns after the Method of the Ancients (Santa Monica: Getty Center for the History of Art and the Humanities, 1993). Hon and Goldstein, “From Proportion to Balance”,: 1-21. Perrault’s solution to this problem was not to abandon proportion, but to take the average of the ratios used in the “best” ancient and contemporary buildings, rounding the numbers to the nearest whole number so that they would be easier to remember and use, and then establishing them as a disciplinary standard. 8 Perrault, Ordonnance, 50-52. 9 Hon and Goldstein, “From Proportion to Balance.” 10 Michael Dennis, Court & Garden: From the French Hôtel to the City of Modern Architecture (Cambridge: MIT Press, 1986). 11 Howard Robertson, The Principles of Architectural Composition (London: The Architectural Press, 1942). John F. Harbeson, The Study of Architectural Design, with Special Reference to the Program of the BeauxArts Institute of Design (New York: The Pencil Points Press, 1926). 12 Thomas H. Beeby, “The Grammar of Ornament/Ornament as Grammar,” VIA III (1977), 10-29. 13 George Teyssot, “Toward a Cyborg Architecture,” in A Topology of Everyday Constellations (Cambridge MA: MIT Press, 2013). Ann Tyng. “Geometric Extensions of Consciousness,” Zodiac 19 (1969). Alfred Neumann. “Architecture as Ornament.” Zodiac 19 (1969),: 90-94. 14 Tabor, “Fearful Symmetry.” Tabor’s was one of a few texts at the time that looked at symmetry during the postmodern era, a time when traditional architectural tropes, such as symmetry and ornament, were being reintroduced to the discipline. 15 “The Art of Symmetry,” entire issue. Daidalos 15 (1985). 16 Greg Lynn, “The Renewed Novelty of Symmetry,” Assemblage 26 (1995), 8-37.

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Filippo Brunelleschi, Pazzi Chapel. Firenze, Italy, 1429-1443

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02 Science and Symmetry

Historians Giora Hon and Bernard Goldstein have traced symmetry’s presence in mathematical and aesthetic thought back to ancient Greece.1 The mathematical trajectory begins with it being used to describe proportional relationships. In the aesthetic realm, symmetrical elements served “the function of harmonizing the different elements into a unitary whole.”2 The two were intertwined, as it was the task of the artist to use proportions to create harmonious and pleasing work. However, from these beginnings the understanding of symmetry has changed dramatically. The following brief history of symmetry in math and science shows how its meaning moved from commensurable, to correspondence, to equivalence, to invariance, revealing how these shifts increasingly broadened its predictive and projective powers.3 Ancient Proportions The mathematical concept of symmetry described in Euclid’s Elements, circa 300 BC, was defined in terms of commensurability. This describes a situation where two objects, magnitudes, or numbers can be described in terms of a third. The numbers 9 and 12 are symmetrical because they can be divided by 3. Or, two pieces of lumber are symmetrical when one has a length of 2 units and a width of 3 units and another has length of 8 and a width of 12. In classical aesthetics, the combination of these differently sized but similarly proportioned elements into a unified whole was a prerequisite for beauty. This was the same definition that Vitruvius gave to symmetry, and it was the one that held sway in architectural theory until the 17th century, and in mathematics until the 19th century. Copernicus, Galileo, Kepler, and Leibniz, all defined symmetry in this classical framing. Copernicus and Kepler even cited Vitruvius’ combination of commensurability and harmony as the source of their astronomical theories of the heavens.4 Symmetrical Operations Hon and Goldstein credit the mathematician Adrien-Marie Legendre with making symmetry a modern scientific concept. However, his use of the term has its origins in the aesthetic practice of architecture. Legendre was aware of how fellow Frenchman Claude Perrault had redefined architectural symmetry in the 18th century as the correspondence of identical elements equally placed on either side of a central feature in a building’s plan or elevation.5 In other words, he limited its definition to what we now call reflective symmetry. The metaphor Perrault used to describe this situation was a balance. Legendre replaced it with a mirror, and in doing so expanded the concept’s influence. Legendre was studying the equivalence of solid angles in polyhedra when he used the term symmetry to describe a situation in which two concave solids faced each other. He wrote: “two equal solid angles, which are formed (by the same plane angles) but in the inverse order, will be called (…) symmetrical angles.” The key phrase here is “inverse order.” Before Legendre, this reflected configuration would not have been understood as equivalent or symmetrical. Only objects that were automorphic themselves (i.e. that could be mapped onto themselves without detection), like a rectangle or an equilateral triangle, or the orb of an eye could be. Asymmetrical objects, like human hands, were not.6 However, in Legendre’s new definition, two hands reflected across a line of reflection were symmetrical. The important change is the inclusion of an operation. The hands needed to be understood as having been repeated. After Legendre, symmetry was the result of a process, a very limited process. Only the four rigid movements of reflection, rotation, translation, and glide reflection could produce the symmetrical relationship that he described. These limitations gave symmetry the power of being a clearly defined, objective, measurable, and repeatable quality. 25

Symmetry Groups While Legendre’s concept would prove immediately productive in crystallography, the power of symmetry as a set of objectively equivalent phenomena would be made mathematical in the work of Évariste Galois. In his investigations into the algebraic logic of polynomial equations he found that the set (or group) of answers that solved them had a symmetrical structure. That is, they could replace one another without producing a change. Galois discovered this through the use of permutations. For example, according to the rules of multiplication, there are six ways to arrange the variables a, b, and c, with each permutation producing the same result. That is, a*b*c=b*c*a=c*a*b=x. This equivalence is an algebraic automorphism, or symmetry. In both algebraic and geometrical terms, symmetry could thus be defined as a transformation that preserves structure, or in which its structure is invariant. In the above equation, the structure that is preserved is the transitive property, while the transformation is the order of the variables themselves. In the geometric example of Legendre, the location of the solid angle is changed, but the orientation and angles themselves are preserved.7 By providing an abstract logic to symmetry, Galois made symmetry a generalizable condition. Symmetry could now be understood as any permutation-based process that produces equivalent results. The new mathematics of group theory establishes the structure and notational system for expressing these equivalencies. In other words, it makes them not only analytical but also projective. Galois’ symmetry groups make it possible to predict when symmetry will or will not be present. If Legendre had turned symmetry into an objective geometric property, Galois transformed symmetry from a descriptive to a heuristic process, one that would ultimately be used in a wide variety of scientific fields. Crystalline Symmetry The development of the link between symmetrical forms and structures was led by crystallographers in the 19th century. They discovered that the seemingly random forms of crystals were governed 26

by a limited set of underlying symmetries. In 1815, René-Just Haüy was the first to apply symmetry to physical objects. Building on Haüy’s work, and on the mathematical systematization of symmetry, in 1830 Johann Hessel established that there were only thirty-two different combinations of symmetry elements in crystals. In 1848 Auguste Bravais found that there were only fourteen different orientations or lattices in which the symmetrical distribution of molecules within a crystal could be arranged. Finally, in 1891, using these restrictions as a starting point, Evgraf Fedorov and Arthur Schoenflies found that there are 230 ways in which the molecules of a crystal could be symmetrically distributed.8 That same year, 1891, Fedorov established that there were only seventeen possible permutations that will symmetrically distribute elements on a two-dimensional plane and only seven unique ways to symmetrically arrange them in a onedimensional line. The fact that these are commonly referred to as the seventeen wallpaper groups and the seven frieze groups not only recognizes their direct relationship to group theory, but also shows that while it was not until the end of the 19th century that these symmetries were established mathematically, reflections, rotations, translations, and glide reflections had been used in architectural ornament for millennia before that.9 Symmetry and Physics The notion proposed by crystallographers that there was an underlying symmetrical structure to the physical world was well-established by the time Einstein published his ground-breaking paper on relativity.10 Among the leaps that Einstein took in his essay was making symmetry axiomatic. As Nobel laureate David Green put it, Einstein’s great advance in 1905 was to “put symmetry first,” that is, to regard the symmetry principle as the primary feature of nature that constrains the allowable dynamical laws.11 Thus, the transformation properties of the electromagnetic field were not to be derived from Maxwell’s equations, as Lorenz did, but rather were consequences of relativistic invariance. By assuming that symmetry not only has a descriptive power, but a predictive one as well,

physicist and mathematician Hermann Weyl extended Einstein’s intuition when he pioneered (with others) the use of symmetry and group theory to understand quantum mechanics. But he took symmetry even further. He held that symmetry, as expressed in group theory, was foundational to the creation and the understanding of a multiplicity of phenomena. In his 1952 book Symmetry, Weyl provides the now common definition of symmetry as any situation in which invariance persists despite a transformation having taken place.12 This is true for physical forces and particle fields, in the positioning of molecules in elements, and in architectural ornaments. While Weyl argued that symmetry provided the link between a variety of natural and cultural phenomena, it was the mathematics of groups that was at the core of those connections. Central to establishing this relationship is the importance of how multiple scenarios or permutations could produce the same result. That is, far from limiting output to a single outcome, the symmetry of groups allows for multiple (but not an infinite number of) results. For Weyl, symmetry is both an analytic and a heuristic mechanism, a tool that describes the existing world but can also create new ideas and new worlds. It was expansive not only because it generated form, but also because it established a connection between two ways of knowing— science and art—that were thought to be separate; both having the capacity to reveal and make new (symmetrical) patterns. As in the classical definition, symmetry is a concept in which a unit of different and equal elements are combined. However, because these forms are grounded in the logic of group theory, they were no longer associated with harmony or beauty. Rather, they are presented as an objective, mathematical quality. Contemporary Symmetry Symmetry continues to play a central role in physics, particularly in the search for a Grand Unified Theory that will account for all the forces present in the universe. David Green notes that “in the latter half of the 20th century, symmetry has been the most dominant concept in the exploration and formulation of the fundamental laws of

physics.”13 Following Weyl, Green emphasizes the conceptual and expansive power of symmetry. Symmetry is an objective, invariant cause (produced by permutations), not an effect.14 It is the source of forms, not their shape. Despite these changed definitions, the usefulness of symmetry persists. And, no matter its definition, symmetry still represents “a unity.” However, “the way in which this unity is realized (…) and how the equal and different elements are chosen” has changed over time.15 One could again say that symmetry is itself symmetrical, as it retains its invariant presence in art and science, despite the many transformations that have occurred to it as a concept.

1 Giora Hon and Bernard Goldstein, From Summetria to Symmetry: The Making of a Revolutionary Scientific Concept (London: Springer, 2008); A.V. Shubnikov and V.A. Koptsik, Symmetry in Science and Art, translated rrom Russian by G.D. Archard, edited by David Harker (New York: Plenum Press, 1974); Ian Stewart, Why Beauty Is Truth: A History of Symmetry (London: Basic Books, 2007). 2 Katherine Brading and Elena Castellani, “Symmetry and Symmetry Breaking,” The Stanford Encyclopedia of Philosophy (2013). https://plato. stanford.edu/entries/symmetry-breaking/ 3 Ibid; Bas C. Van Fraassen, Laws and Symmetry (Oxford: Clarendon, 1989); Hermann Weyl, Symmetry (Princeton: Princeton University Press, 1952); Shubnikov and Koptsik, Symmetry in Science and Art; Stewart, Why Beauty Is Truth. 4 Hon and Goldstein, From Summetria to Symmetry, 157-177. 5 Claude Perrault. Ordonnance for the Five Kinds of Columns after the Method of the Ancients (Santa Monica: Getty Center for the History of Art and the Humanities, 1993). 6 Hon and Goldstein. From Summetria to Symmetry, 203-219. 7 Stewart, Why Beauty Is Truth, 118-123. 8 Shubnikov and Koptsik, Symmetry in Science and Art. 9 Donald Crowe, “Introduction to the Plane Symmetries,” in Symmetry Comes of Age: The Role of Pattern in Culture, edited by Dorothy K. Washburn and Donald W. Crowe (Seattle: University of Washington Press, 2004), 3-17. 10 Giora Hon and Bernard Goldstein, “Unpacking ‘For Reasons of Symmetry’: Two Categories of Symmetry Arguments,” Philosophy of Science No. 73 (2006), 419-439. 11 Ibid. 425. 12 Weyl, Symmetry. 13 Hon and Goldstein, “Unpacking ‘For Reasons of Symmetry’”, 426. 14 Van Fraassen. Laws and Symmetry; Weyl, Symmetry; Hon and Goldstein, “Unpacking ‘For Reasons of Symmetry’”. 15 Brading and Castellani, “Symmetry and Symmetry Breaking.”

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Leon Batista Alberti, Santa Maria Novella. Firenze, Italy, 1456-1460

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03 Biology and Symmetry

At the end of the 18th century, scientists sought empirical evidence and a unified theory to account for the common elements and processes that lay beyond the appearance of the natural world. Biologists and crystallographers were particularly interested in the potentially shared morphology of living and non-living things.1 Architects and architectural historians followed these scientific pursuits, often incorporating and translating their findings into their own ideas and work.2 For both scientists and architects, symmetry was integral to these pursuits. Haeckel and Symmetry Ernst Haeckel, a German biologist/morphologist and a devout disciple of Darwin, has often been accused of finding symmetry where it does not exist; doing so not only in the depiction of the thousands of species he studied, but also in the famous plates that illustrated his influential treatises General Morphology (1866) and Art Forms in Nature (1899).3 In these plates he arranged (and changed the scale of) specimens according to the reflective symmetry of the page—rather than deploying a more linear or numeric classification system. A close look at the plates shows that while the underlying structure is symmetrical, the actual organization only approximates this state. This condition of near or broken symmetry was a fundamental technique of Haeckel and an index of his understanding of natural forms.4 When drawing any one of the thousands of species of single-celled radiolarian he discovered while aboard the ship Challenger, Haeckel consciously chose to emphasize their shared translational, rotational, and reflective symmetries. While he was aware of the uniqueness of each specimen, these composite images were purposefully meant to show what they had in common. This was not an oversight. Haeckel was aware of contemporary crystallography’s classification of matter according to its underlying symmetrical structures.5 In other words, symmetry was not just an artistic or subjective decision to make things easier for the illustrator: it was the illustration of a scientific idea. For Haeckel symmetry was not an ideal state, it was a generic one. It was the quality that was passed on from generation to generation; an invariance the persisted despite individual transformations that occurred in space and time. Symmetry was the law, and asymmetry was the rule. Symmetry is what linked the past with the present, and one species to another. Within this mindset, the underlying symmetry he observed in the silica skeletons of single-celled protozoa and the symmetry crystallographers found in minerals was evidence that evolution (and life itself) had emerged out of inorganic substances. In the larger context of evolutionary development, symmetry was neither essential nor obsolete. Although present at the beginning of the forming processes, it survived because it proved useful. The persistence of symmetry in diverse contexts and environments was a sign of its evolutionary fitness. If it were not useful, it would have disappeared. Symmetry was not superficial: it was pragmatic.

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Art Nouveau and Symmetry

Beaux-Arts architecture the highest form of organization.9

Generally speaking, buildings do not move. However, they exist in order for living and non-living entities (people, air, water, heat) to travel. In other words, the integrating of inorganic and organic elements with one another is an older problem for architecture than it is for science. If the symmetry of 19th century Beaux-Arts architecture emphasized the inert aspects of architecture, Art Nouveau architects sought to add the latter, or the image of the latter, into it. Many of them used Haeckel’s ideas and artwork for inspiration. The biologist’s influence on late 19th century avant-garde architecture is well known and documented.6 Architects such as Victor Horta, Hector Guimard, Antoni Gaudí, and Louis Sullivan were quite familiar with his work. And, like him, their designs used symmetry as a backdrop for their asymmetrical elements.

Louis Sullivan was the product of a partial Beaux Arts education. He was also a believer in epigenetic processes, and in adaptation and evolution. Sullivan was an avid reader of science and popular science literature, including Haeckel. He too sought to synthesize the geological with the biological, the scientific with the artistic, the romantic with the rational.10 For him, this meant that the hegemony of Beaux-Arts symmetry and style needed to be challenged—but not erased.

However, the symmetries one finds in the plans and elevations of Horta, Guimard and Gaudí are different from those in 19th century neo-classical and 19th century Beaux-Arts architecture. Where in the latter they were compositional devices, symmetry is used by the Art Nouveau to accommodate pragmatic needs. The symmetrical elements one finds in Horta’s Tassel House, Gaudí’s Casa Batlló and Sagrada Familia are used to carry structural loads or contain conventional programs. In these examples, symmetry lingers because it efficiently handles generic architectural issues that had not gone away: structure and program. Where there is change—in the city, in the economy, in new materials, in sensibility— symmetry is broken, and asymmetric, vegetal, and crystalline forms, appear. This relationship between stasis and change is what Louis Sullivan was referring to when he famously wrote “form ever follows function,” that is, only new demands require new kinds of shapes, spaces, and surfaces.7 Sullivan and Symmetry At the turn of the 19th century, biologist William Bateson noticed that certain body deformations in animals resulted in extra symmetries. For example, when an extra leg in a bug or a finger in a human hand appeared, their relationship to the normal configuration was symmetrical.8 Bateson concluded that these deformations were due to a lack of genetic information during the embryological process. If we take Bateson’s understanding of symmetry as axiomatic—i.e. that symmetry is an index of missing information—what is missing in a symmetrical building is information that comes from a building’s physical, social and historical context. Information from the site, the climate or the program was not to interfere with the symmetrical arrangement of forms and spaces it demanded. Thus, what is an initial or a simple state in biology was in 30

Sullivan’s interest in the transition from symmetrical figures to asymmetrical ones is clearly articulated in his System of Architectural Ornament. Yet, he consistently returned to symmetrical forms by reflecting, translating and rotating these asymmetrical forms. In the first instance, form is the result of a series of deformations of simple geometric figures. In the latter, symmetrical figures arise out of the aggregation of identical pieces. In both cases form is the result of a set of operations applied to standard elements. Both tactics are in evidence in the terracotta tiled facades of his tall buildings, where symmetrical figures made from asymmetrical elements abound. Form is the result of a limited set of operations on a given set of figures. And, as with Haeckel’s plates and species, symmetry is the law but does not govern the individual results. In relation to Bateson’s definition of symmetry, Sullivan used the same ornamental motif until a new source of information or influence made him augment or abandon it. For example, he used the same sized tile and motif over and over again across a façade until the function of what he was cladding changed. Columns were treated one way, capitals another, door jambs another, and window frames yet another. The same was true for every floor level. Each motif was symmetrical, but each was unique to its specific location and function. This same logic was applied to differentiate the overall form of the building, which can be seen in his design of the Guarantee Building in Buffalo, New York. From a distance, the Guarantee Building seems straightforward. It is a simple box on a corner site. Its two public facades display reflective symmetry, while its plan reveals a slight break from symmetry to allow for a light well on the western edge of the site. The overall shape and size are determined by what Sullivan recognized as the dominant “social conditions” of his day; conditions such as the economic value of the land, the invention of the steel frame structure, and the size of the standard office space. These are understood as the initial undifferentiated starting points: they are not things the architect can change. The architect’s task is to inform this generic condition and make it specific to its time and context. For Sullivan this meant manipulating the building

envelope to transform its inert mass into a “proud and soaring thing.” Sullivan does this by articulating on the facade the way in which the different functions are distributed in the building. He follows his creed that “where function does not change, form does not change.” In other words, different functions demand different forms and surfaces. The entry portals that provide access from the street have unique shapes. The space and size of the windows on the first and second floors are different from one another because of the different types of commercial spaces they house. The office floors above the two-story base are treated identically—despite climatic differences—because their function is the same. The attic story and cornice serve unique functions and are given different forms and ornament. Social conditions—rather than urban or climatic ones—inform the overall texture of the project and make the building-as-organism change its form. The separate areas are then given their own ornamental treatment. Consistent areas are treated similarly and with symmetrical motifs. For example, the office floors are clad in repetitive bands of tiles with geometric motifs, while moments of transition, like the capitals and the cornice are wilder and more vegetal. Places of literal transition or movement, such as the stair balustrades and the elevator cab and grate, are given the most intricate treatment. And yet, all differences are bound within reflectively symmetrical facades, which can themselves be described as being comprised of symmetrically translated pieces. Every surface is made up of reflections, translations and rotations. Within these rigid operations are motifs that are at once sinuous and faceted, vegetal and crystalline, symmetrical and asymmetrical. In Sullivan’s hands symmetry’s role is not an image or an idea to imitate, nor a superficial application. Rather, it is the starting point—and the structure—that links diverse things with one other: the past to the future, the organic with the inorganic, the generic and the specific. It is a different symmetry than Beaux-Arts’: not an ideal to conform to but a logic and a set of operations put to use. This is the function that symmetry would continue to play throughout the 20th century in science. In contrast, modern architecture would abandon symmetry and instead seek out design devices that focused on the optimization and articulation of individual elements rather than on their integration. This was, at best, a shortsighted shift, given that the combination of unlike things—matter and culture, present and the past, structure and ornament—is an inevitable architectural task. Symmetry has proven itself as a technique for creating such connections, and a useful one at that.

1 Caroline Van Eck, Organicism in 19th Century Architecture (Amsterdam: Architectura & Natura Press, 1994). 2 Amy Kulper, “Of Stylized Species and Specious Styles,” The Journal of Architecture No. 11 (2006) 391-406; Barry Bergdoll, “Of Crystals, Cells, and Strata: Natural History and Debates on the Form of a New Architecture in the 19th Century,” Architectural History No. 50 (2007), 1-29. 3 Robert J. Richards, “Haeckel’s Embryos: Fraud Not Proven,” Biology and Philosophy No. 24 (2009), 147-154. 4 Robert J. Richards, The Tragic Sense of Life: Ernst Haeckel and the Struggle Over Evolutionary Thought (Chicago: University of Chicago Press, 2008). 5 Sypros Papapetros, “On the Biology of the Inorganic: Crystallography and Discourses of Latent Life in the Art and Architectural Historiography of the Early Twentieth Century,” in Oliver A. I. Botar and Isabel Wunshe, eds., Biocentrism and Modernism (Surrey: Ashgate, 2011), 77-106. 6 Kulper, “Of Stylized Species and Specious Styles” 391-406; Bergdoll, “Of Crystals, Cells, and Strata” 1-29; Robert Proctor, “Architecture from the Cell-Soul: Rene Binet and Ernst Haeckel,” The Journal of Architecture No. 11 (2006) 391-406; David Brody, “Ernst Haeckel and the Microbial Baroque,” Cabinet No. 7 (2002). 7 Louis Sullivan, “The Tall Building Artistically Considered,” Progressive Architecture No. 38 (June, 1957) [1896], 204-206. 8 William Bateson, Materials for the Study of Variation: Treated with Especial Regard to Discontinuity in the Origin of Species (London: Macmillan, 1894). 9 Greg Lynn, “The New Novelty of Symmetry,” Assemblage No. 26 (1995), 11-25. 10 Bergdoll, “Of Crystals, Cells, and Strata”, 1-29.

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Andrea Palladio, Villa Capra. Vicenza, Italy, 1566

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04 Crystals and Symmetry

Where does form come from? What are the underlying elements and processes that govern the formation and shapes of organic and inorganic objects? What role does symmetry play in creating and understanding them? Addressing these questions—all of which revolve around the relationship between invisible interior structures and exterior shapes—is a cultural task that is often assigned to scientists and philosophers. However, the same issues have long been of interest to architects and architectural historians. During the 19th century these questions and these disciplines often overlapped with one another. Symmetry was particularly important in establishing the relationship between the internal organization and the external appearance of crystals. And crystals and symmetry held an important place in architectural history and design. Crystal Structures Plato believed that the basic units of the cosmos were the five-regular polyhedron: the tetrahedron, the cube, the octahedron, the icosahedron, and the dodecahedron. These elegant elements, each composed of identical planes symmetrically arranged at equal angles, were thought to be the building blocks from which the universe was constructed. The importance of these symmetrical figures held sway in the scientific theories of Johannes Kepler and Robert Hooke in the 17th century. Their respective investigations of macro and microcosms relied on the heuristic power of their predictable geometry. In Hooke’s 1665 Micrographia, he theorized that the internal organization of crystals conformed to the angles present in Plato’s polyhedra (30, 60, 90, 120, 180). He held that the specific forms that crystals took adhered to the geometry produced by the periodic packing of spheres. Four years later the Danish scientist Nicolas Steno would reveal the limits of the Platonic geometries. Based on accurate measurements of literal crystals, he agreed with Hooke that the angle between their faces was always consistent. However, the angle that described that relationship did not always adhere to the five found in the Platonic solids. Steno’s advances were aided by applying the descriptive techniques of measuring and making polyhedra using templates of tessellated planes developed by the German artist Albrecht Dürer in the 16th century.1 Soon after the turn of the 18th century, and after Legendre’s redefined notion of symmetry was published in 1794, crystallographer René Just Haüy “was able to formulate the law relating the measured angles of crystal forms to an internal repetition of identical molecules” inside of them. From this hypothesis he was able to begin to establish the symmetrical internal molecular structure of all natural elements.2 Empirical analysis showed, and in 1840 Johann Hessel proved, that there were only thirty-two symmetrical combinations of molecules in crystals. In 1848 Auguste Bravais established that there were only fourteen different orientations or symmetrical lattices in which these molecules could be arranged. Finally, in 1891 Evgraf Fedorov and Arthur Schoenflies found that there are only 230 symmetrical configurations in which the internal molecular structure of a crystal could be arranged.3 Thus, despite the myriad of external forms, the internal logic of minerals was discovered to remain always symmetrical. In other words, symmetry was always present in the world, even when our senses could not detect it. If it was the job of scientists to establish these facts in the 19th century, then it was the task of artists to translate this information into recognizable cultural codes. 33

Reflections As both a model and a metaphor, crystals were central figures in 19th and early 20th century art and architectural history. As in crystallography, the goal of many artists and historians was to establish a clear relationship between the external appearances and the invisible structures that governed the creation of material and cultural artifacts. One artistic genre in which the affinity between natural and artistic forms was found especially productive was architectural ornament.4 Prominent art historians, such as Alois Riegel, Wilhelm Worringer, and Aby Warburg, saw in the symmetries found in traditional ornament a link between crystal formation and artistic form.5 In architecture, literal references to crystals were also used to produce new forms. As a process with a consistent internal structure that could produce asymmetric forms, it was an ideal analogy for those wishing form to be both guided by rules and free from their prescribed results. In the early 20th century crystals were associated with mystical and utopian projects, such as Bruno Taut’s Alpine Architecture.6 In the latter half of the century they were used in a more literal fashion by architects like Zvi Hecker and Alfred Neumann.7 Perhaps the subtlest and most unexpected translation of the inner symmetries of crystals into architectural form is found in the work of Frank Lloyd Wright. Throughout his career local and global symmetries lurk underneath the dynamic massing and spatial arrangements of the body. These hidden symmetries had their source in both crystallography and in traditional ornaments. It is well documented that, as a child, Wright played with the so-called Froebel “gifts.”8 Less well-known is the fact that, before he became a childhood education theorist, Friedrich Froebel was a crystallographer in the early 1800s. The fact that the components of Froebel’s toys were all symmetrical, as were the operations he prescribed for manipulating them, had its origin in the structure of crystals. His goal was that, by physically rotating, reflecting, and translating these materials, a child would learn the underlying logic of the physical world. However, this knowledge did not come from controlled experiments within language or mathematics, but via direct experience with the body.9 When Wright claimed that he could “still feel the blocks in his fingers” eighty years after playing with Froebel’s blocks, he was not just recalling his childhood, but unknowingly mimicking the underlying symmetries found in the material world. Wright’s work does not look crystalline though. But a review of it, especially of his plans, reveals the multiple 34

reflections, rotations, and glide-reflections at work. They are the same operations prescribed in Froebel’s system, and the same ones found in the thirty-two nets and fourteen lattices that describe crystalline structures. They are also the operations found in traditional architectural ornament. Wright learned the logic of ornament in a similar way by which he learned about three-dimensional form from Froebel: not abstractly, but physically. He told the story of his tracing every line in his uncle’s copy of Owen Jones’ book The Grammar of Ornament, a book that documents symmetrical ornamental motifs from around the globe. The site plan of the Darwin D. Martin House in Buffalo (1905) reveals a series of local rotational symmetries across the property, while the plan of the main house reveals reflective symmetries in each of the main social spaces, as well as a global reflection that organizes the library, the dining room, the living room and the porch. However, as with crystals, its asymmetrical external appearance and its internal spatial configuration mask these underlying operations. Likewise, his early suburban planning schemes, such as the Quadruple Block Plan (1901), subject an asymmetrical house to a series of reflections to produce a symmetrical effect. Similar results are seen in his unofficial entry to the Chicago City Club Housing Competition (1916) for an urban subdivision. In fact, the larger the scale that Wright worked at—like the Crystal City outside of Washington, D.C. (1940)—the more symmetries one finds. His plan for the Unity Temple in Oak Park, IL (1908) shows a clear globally reflective organization. However, one rarely sees or experiences the project in this way. In the interior, the visitor is always taken off the central axis, and the still symmetrical translational and rotational effects are made more visually prominent. The tension between the two tall, centrally organized masses and the overall horizontal and asymmetrical effect reinforces the co-presence of symmetry and asymmetry: while symmetry is persistently present at the level of organization, it is not at the level of perception. A common motif in the Froebel system is the rotationally symmetric pin-wheel. This can be found in many of Wright’s skyscraper schemes, such as St. Mark’s in the Bouwerie (1931) and the Price Tower (1956). The Guggenheim Museum (1959) also conforms to a rotational logic. The spiral is a common crystalline structure, achieved by a consistent rotational movement in the x-axis and a translation in the y-axis. Across Wright’s diverse sensibilities and appearances—the long and low Prairie houses, the pin-wheeled towers, the massive Unity Temple and Guggenheim—what remains consistent is the underlying symmetrical operations— just as one finds in crystals.

Translating Symmetry It is significant that Frank Lloyd Wright’s symmetries mostly occur in plan. The plan is to architecture what the diagram is to science, or what the equation is to math. It is not something one encounters in the world. Rather, it is the thing that encapsulates ideas while abstractly organizing spaces, activities, movements, and perceptions. Like the symmetrical nets and laws governing the form of crystals, the plan invisibly manages forms and behaviors. In chemistry, knowing how matter was organized enabled it to be manipulated in novel ways. The results of these operations are what we call chemicals. In architecture, Wright’s mastery of symmetry (and steel and concrete) allowed him to create a new architectural idiom, what he called “nature patterns” or “integral ornament,” both dependent on the understanding of the “nature of materials.” “And when I say Nature, I mean inherent structure seen always by the architect as a matter of complete design. It is in itself, always, nature-pattern.”10 “(…) Integral ornament is the developed sense of the building as a whole, of the manifest abstract pattern of structure itself (...) Integral ornament is simply structurepattern made visibly articulate and seen in the building (…) It is the expression of inner rhythm of Form.”11 Where does form come from? In the case of Wright, it comes from a variety of cultural contexts, but also from the inner workings of crystals, from architectural ornament, and from their common symmetrical structures.

1 Cecil J. Schneer, “The Renaissance Background to Crystallography: The Search for Harmonious Proportions and Perfect Shapes in the Natural World Opened the Way to the Science of Crystals,” American Scientist vol. 71, no 3 (1983) 254-263. 2 Ibid. 262-263. 3 A.V. Shubnikov and V.A. Koptsik, Symmetry in Science and Art, edited by David Harker (New York: Plenum Press, 1974). 4 Barry Bergdoll, “Of Crystals, Cells, and Strata: Natural History and Debates on the Form of a New Architecture in the Nineteenth Century,” Architectural History 50 (2007), 1-29. 5 Spyros Papapetros, “On the Biology of the Inorganic: Crystallography and Discourses on Latent Life in the Art and Architectural Historiography of the Early Twentieth Century,” Biocentrism and Modernism, edited by Oliver Arpad Istvan Botar, Isabel Wünsche (London: Routledge, 2011) 77-106. Spyros Papapetros. “On the Afterlife of Crystals,” On the Animation of the Inorganic: Art, Architecture, and the Extension of Life (Chicago: University of Chicago Press, 2012) 113-160. Amy Kulper, “Architecture’s Lapidarium: The Life of Ten Geological Specimens in Architecture,” The Anthropocene, edited by Etienne Turpin (Michigan: Open Humanities Press, 2013), 87-110. 6 Rosemarie Haag-Bletter, “The Interpretation of the Glass DreamExpressionist Architecture and the History of the Crystal Metaphor,” Journal of the Society of Architectural Historians vol. 40, no. 1 (1981) 20-43. Gyorgy Kepes, “Thing, Structure, Pattern, Process” and “Transformation, Physical, Perceptual, Symbolic,” in The New Landscape in Art and Science (Chicago: Paul Thebold, 1956). 7 Georges Teyssot, “Toward a Cyborg Architecture,” A Topology of Everyday Constellations (Cambridge: MIT Press, 2013) 183-218. Alfred Neumann, “Architecture as Ornament,” Zodiac 19 (January 1969) 90-98. Ann Tyng, “Geometric Extensions of Consciousness,” Zodiac 19 (January 1969), 130-162. 8 Frank Lloyd Wright, An Autobiography (New York: Duell, Sloan and Pearce, 1943) 13-14. Frank Lloyd Wright, A Testament (New York: Horizon, 1957) 19-21, 63, 100, 206-207, 220, 300. Edgar Jr. Kaufmann, “Form Became Feeling: A New View of Froebel and Wright,” Journal of the Society of Architectural Historians 40 (1981), 130-137. Richard C. MacCormac, “The Anatomy of Wright’s Aesthetic,” Architectural Review 113 (February 1968) 143-146. Richard C. MacCormac, “Froebel’s Kindergarten Gifts and the Early Work of Frank Lloyd Wright,” Environment and Planning B, no. 1 (1974) 29-50. Stuart Wilson, “The ‘Gifts’ of Friedrich Froebel,” Journal of the Society of Architectural Historians no. 26 (December 1967) 238-241. Edgar Jr. Kaufmann, “Centrality and Symmetry in Wright’s architecture,” Architects’ Yearbook 9 (1960), 120-131. 9 Jeanne Spielman Rubin, “The Froebel-Wright Kindergarten Connection: A New Perspective,” Journal of the Society of Architectural Historians vol. 48 no. 1 (March 1989), 24-37. 10 Frank Lloyd Wright. “In the Nature of Materials,” in Architecture Culture 1943-1968, edited by Joan Ockman (New York: Rizzoli, 1993), 37. 11 Ibid. 29.

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Claude Nicolas Ledoux, Barrière de la Villete. Paris, France, 1784-1788

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05 Physics and Symmetry

Before 1905 time was time, and space was space. Albert Einstein’s paper “On the Electrodynamics of Moving Bodies,” changed that. Einstein was troubled by the fact that James Maxwell’s equations regarding electric and magnetic forces “when applied to moving bodies, leads to asymmetries that do not seem to adhere to the phenomena.” Instead of changing the equations Einstein changed the theory explaining them. Maxwell assumed that space and time were distinct entities. Einstein did not. He argued that if one accepts that time and space, as well as the electric and magnetic force, are fully integrated with one another, then these asymmetries disappear.1 In the wake of his paper, symmetry—especially the transformation and permutation-based symmetries described by the mathematics of group theory—took on ever-greater importance in physics.2 What does this dramatic change in our conception of how the physical world works tell us about where form comes from and how (or if) it is dependent on symmetry? And, how were these ideas—particularly regarding symmetry—translated into architectural discourse and form? In regard to the first question, Einstein’s special relativity helps explain the external forces—or charged fields—that determine the shape of matter.3 In terms of its relationship to symmetry, while symmetry is less defined in geometric terms and is instead described in terms of the automorphisms and equivalencies between mathematical groups, the function of symmetry remains unchanged: it provides an objective and invariable boundary within which differences occur and are measured. Regarding the relationship between new ideas in physics and architecture, in the early 20th century one also finds architects challenging the distinctions between a variety of disciplinary binaries—inside and outside, structure and ornament, etc. However, one equally finds the continued presence of the traditional, reflective, form of symmetry. And yet, its presence can be understood as fulfilling a similar function, establishing an unchanging frame within which to observe and account for change. New Theory=New Symmetry Instead of two neutral and independent categories, special relativity understands the relationship between time and space as a dynamic field, or “inertial frame,” whose limits are defined by the constant speed of light.4 It also holds that the set of coordinates and events described in any one inertial frame can be transformed into any other. While each frame is specific, it can be mapped onto any other. In other words, they are isotropic or symmetrical.5 This symmetry is mathematically confirmed by group theory, specifically by the Lorentz group of transformations.6 As Hermann Weyl put it: “What Einstein did was this: without bias he collected all the physical evidence we have about the real structure of the four-dimensional space-time continuum and thus derived its true group of automorphisms (the Lorentz group).”7

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Weyl’s reference to “physical evidence” is significant, as it distinguishes Einstein’s achievement from the theoretical and mathematical work on the subject (by Hilbert, Poincaré and Minkowski). It also speaks to the (four dimensional) form that space-time takes, not just the laws governing it.8 In his later paper on general relativity (1915) Einstein more directly addresses form when he incorporates the curved nature and effect of mass and gravity in space-time. The automorphisms that Weyl refers to are also significant, as it is this formal consistency that establishes the fact that the (new) laws of physics remain invariant in all situations. In spatial terms this means that forces behave in the same way no matter their orientation, i.e. up and down, left or right, past or future. These equivalences are captured in Minkowski’s space-time diagram—with two cones whose apexes expand out at 45 degree angles from the (0,0,0) point on an x, y, and z axes—diagram that itself exhibits rotational and reflective symmetry. As with previous conceptions of symmetry, both group theory and Minkowski’s manifold provide a limit condition within which a variety of events can happen; the limit in space-time being the constant speed of light. For Weyl, this diagram showed how “the world has an objective causal structure described by these light cones issuing from every world point.”9 The key terms are “cones,” “plural,” and “every.” They indicate that there can be many objectively established world points, yet each one is symmetrical, or automorphic, with every other one. No two points are identical with one another, but they are part of a related set of similarly limited permutations or possibilities, inscribed by a specific mathematical group.10 The difference between any two points, or inertial fields, is a relative difference, not an absolute one, and it is what enables the asymmetries to disappear. Symmetry Reappears If space and time are inextricably interlocked, what are the implications for cultural production? How can, or should, this new fact be translated into intentionally aesthetic disciplines? Must it be subconsciously smuggled in, as Frank Lloyd Wright did with the laws of crystallography, or can it be directly imported? What is the role of symmetry in establishing this relationship? Many critics and historians have noted that among the tropes found in Modern architecture in the years before and after Einstein’s 1905 paper is the use of new materials to create new shapes, spaces and surfaces.11 In turn, the increased presence of steel, glass, and concrete created objects and spaces that allowed one to more easily penetrate them with one’s eyes and body. This 38

visual transparency and spatial plasticity literally opened up the formal and perceptual limits of what an architectural object and experience could be. It also broke down the strict divisions and autonomous status of structure, space, ornament, form, and function. These were now functions of one another, rather than discrete, additive entities. These losses of distinction allowed Sigfried Giedion, in his influential book Space, Time, and Architecture, to draw attention to their affinity to the new physics. However, as Giedion himself noted, these architectural effects are not analogous to the kind of space-time events described by Einstein. His comparison of architecture and physics can be thus better understood as a metaphor. This is not a problem, especially in terms of symmetry. The logic of metaphors demands that relationships remain stable even when the elements themselves (architectural, algebraic, electromagnetic or other) change dramatically. In other words, metaphors have the same structure as symmetry. However, the elements being compared in a metaphor are by definition not identical to one another: they are only interchangeable. This quality enables metaphors to be flexible and surprising, despite their rigid underlying structure. Despite the dramatic reframing of the physical world by theoretical physics, and the remaking of architecture by new materials and sensibilities in the early 20th century, certain things remained the same. The list of canonical modern buildings that make use of new structural, spatial, and atmospheric effects but also have reflectively symmetrical plans is surprisingly large. It includes: Labrouste’s Sainte-Geneviève Library (1850), Paxton’s Crystal Palace (1851), the Eiffel Tower (1889), Perrault’s 25 rue Franklin (1903), Wright’s Larkin Building (1904) and Unity Temple (1908), Behrens’ AEG Turbine Factory (1909), Sant’Elia’s Citta Nouva (1914), Taut’s Glass Pavilion (1914), Gropius’ Werkbund Pavillion (1914), Mendelsohn’s Einstein Tower (1921), Mies’ Friedrichstrasse Skyscraper (1921), Le Corbusier’s Ville Contemporaine and Cruciform Towers (1922), Taut and Wagner’s Berlin Britz Housing (1925), Mies’ Crown Hall (1956) and Le Corbusier’s Carpenter Center (1963). There are a myriad of sensibilities, spatial effects and ideologies in this group. There are also countless counterexamples that exhibit hardly any symmetry, reflective or otherwise. Nevertheless, the size of this list begs the question: why the persistent presence of reflective symmetry in modern architecture? The presence of bilateral symmetry is especially surprising in the work of the Italian Futurist Antonio Sant’Elia. The Futurists rejected all previous cultural forms as obsolete. They celebrated industry, science, technology, speed, violence, and war.

Sant’Elia’s work has been identified as coming closest to embodying the logic of the physics of electro-magnetic fields. His drawings for his Cittá Nuova show infrastructure becoming architecture, structure morphing into monuments, and stable bodies pierced by animated objects. As Sanford Kwinter has noted, instead of forming an isolated and unified composition, the buildings in Cittá Nuova appear to plug into one another to create an “indeterminate whole.” As such, their form is not an expression of an interior (crystalline) logic but of “an exterior syntax of combination and connection (…) [where] individual units are mere operators or commutation devices within a much larger assembly whose greater intensity they modulate and control.”13 Kwinter’s reading is accurate, but it is also incomplete. By stressing the dynamic and “dissymmetrical” nature of Sant’Elia’s forms, it ignores the effects of their consistently symmetrical organization. What is the function of all that symmetry? Is it a remnant or a homage to tradition? Futurism banned such sentiments. It is not a placeholder for movements yet to come either. Rather, it can be best described as a symmetry group. A symmetry group is a subgroup of all the possible transformations that an object can make, while leaving its structure unchanged.14 In the Cittá Nuova, bilateral symmetry is the structure that stays constant despite the changes in the scale, shape, function, location, or material of the objects found within it. In Weyl’s terms, reflective symmetry is the “objective causal structure” that links each of its “world points” with one another. It provides the limit condition within which otherwise fluid forms emerge, and a reference point for comparing otherwise unlike objects. In other words, symmetry functions as the stable structure that organizes a series of visual metaphors.

And yet, these should not be understood as oppositional states. Rather, they represent the maximum and minimum condition on an ever-changing continuum of possible formal effects—a continuum in which symmetry is an inevitable, rather than an ideal state. In other words, symmetry should not be understood as the negation of asymmetry but as its partner. When combined they produce a synthesis in which the stable and the fluctuating parts, the familiar and the new are codependent rather than contradictory. As Einstein recognized, symmetry preserves relationships and thus needs to be recognized. And, as in the detection and measurement of moving bodies, dynamic architectural forms require a stable position to understand where things are and where they are going. In this architecture, as in physics, the clear distinction between one thing (or category) and another has been replaced by a continuum of ever-changing yet still symmetrical effects.

Dynamic Symmetry Space-time does not eliminate space or time, nor does it blend them into a homogenous solution. Instead, each category is preserved and co-present in the resultant fourdimensional world described by Einstein. Likewise, in Sant’Elia’s city, symmetry is neither lost nor is form completely dynamic. Symmetrical and asymmetrical forms are combined to produce a consistent yet unique set of effects. Infrastructure and architecture, movement and stasis, and orthogonal, diagonal, and curved forms are similarly merged yet articulated. A comparison of Sant’Elia’s sketch with his finished delineated drawing for the Cittá Nuova’s Central Station illustrates this productive tension between fluidity and concreteness. The sketch exhibits the active forces within and around its forms. The measured drawing shows the rigidity of materials and the clarity of geometry. In the sketch it is unclear where the building ends and where the movement of machines or the sky begins. In the finished drawing these are more stable.

1 Giora Hon and Bernard Goldstein, “Making Asymmetry Disappear: Symmetry and Relativity in 1905,” Archive for History of Exact Science vol. 59 no. 5 (2005), 437-544. Ian Stewart, Why Beauty is Truth: A History of Symmetry (New York: Basic Books, 2007), 185-196. 2 Hermann Weyl, Symmetry (Princeton: Princeton University Press, 1952). Giora Hon and Bernard Goldstein, “Unpacking ´For Reasons of Symmetry’: Two Categories of Symmetry Arguments,” Philosophy of Science vol. 73 no. 4 (October 2006), 419-439. 3 Sanford Kwinter, “La Cittá Nuova: Modernity and Continuity,” in Architecture Theory Since 1968, edited by K. Michael Hays (Cambridge: MIT Press), 586-613. 4 Ian Stewart, Why Beauty is Truth: A History of Symmetry (New York: Basic Books, 2007), 192-193. 5 Weyl, Symmetry, 131-132. Stewart, Why Beauty is Truth, 193-196. 6 Weyl, Symmetry, 131-132. Stewart, Why Beauty is Truth, 97-123. 7 Weyl, Symmetry, 131. 8 Hon and Goldstein, “Making Asymmetry Disappear.” 9 Weyl, Symmetry, 132. 10 Ibid. 11 Sigfried Giedion, Space, Time, and Architecture, 5th ed (Cambridge: Harvard University Press, 1967) [1941]. Reyner Banham, Theory and Design in the First Machine Age (New York: Praeger, 1962), Manfredo Tafuri and Francesco Dal Co, Modern Architecture 1 & 2 (New York: Rizzoli, 1986). 12 Gregory Bateson. “Style, Grace, and Information in Primitive Art,” in Steps Toward an Ecology of Mind (New York: Ballantine, 1972), 138-142. 13 Kwinter, “La Cittá Nuova”, 599. 14 Stewart, Why Beauty is Truth: 160. Weyl, Symmetry, Preface.

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Karl Friedrich Schinkel, Altes Museum. Berlin, Germany, 1830

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06 Statistics and Symmetry

Whether created for the empirical analysis of an existing urban area or for the proposal of a radically new city, a surprising number of urban diagrams and designs from the first half of the 20th century share a symmetrical organization. Despite their differences in political and physical form, they each use symmetry to structure the distribution of their elements. A closer look at a number of canonical projects reveals symmetry’s presence in their street, buildings, and units plans.1 A plan has many definitions. It is a strategy for controlling behavior, a drawing showing the separation and connection of spaces, and an indication of social relationships and priorities.2 As with symmetry, a plan functions to create a predictable relationship between the present and the future.3 In addition to their geometric similarities, a large number of modernist proposals rely on statistics to guide their decision-making. Statistics are employed to predict future behaviors based on past actions.4 In the language of (group) symmetry, both plans and statistics are deployed to create automorphic conditions, that is, to secure invariance in the face of transformations. The combination of symmetrical plans and statistical analysis is most obvious in the architectural discourse around the existenzminimum dwelling in Europe in the 1920s, and in the search for efficient unit and site plans in America in the 1930s.5 Long before big data, symmetry and statistics were the most efficient tools for addressing issues associated with the creation of radical new cities and of various forms of housing. Rotation Only a dozen or so of the one hundred and fifty pages in Ebenezer Howard’s Garden Cities of Tomorrow (1902) are dedicated to its physical design. These include a few diagrams that illustrate its organizational details. The bulk of the book focuses on its political and economic organization. There is much numerical data outlining its revenues and expenses, as well as the analysis for how its infrastructural needs will be met. Howard proposed a cooperative scheme where land would be held in common to keep it free from the forces of financial speculation. In the book’s first edition he included a drawing titled Slumless and Smokeless Cities. It depicts a constellation of six circular, 32,000 person settlements orbiting a 58,000 person “Central City.” Although each of the cities has a slightly different internal configuration, each is also rotationally and reflectively symmetrical. Howard makes it clear that these are diagrams and not the plan for his new city, but the emphasis on the circle, and the resultant rotational and reflective symmetry of the drawing are not random. The early build-outs of Howard’s vision at Letchworth and Welwyn did not take this exact shape. Their overall outline, and the profile of individual blocks and buildings are irregular. There is, however, much symmetry to be found in Parker and Unwin’s street plans and buildings in Letchworth, reinforcing the collective nature of the scheme. As with all symmetry groups, there are no a priori hierarchies between each entity or any orientation. All are equivalent. Symmetry is not a neutral actor but is an ideological entity. It is presented as a clear icon of the egalitarian society that the design is meant to produce.

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Reflection Through his decorative arts training Le Corbusier had an intimate relationship with symmetrical ornament and traditional ornamental techniques. Later he would indicate his commitment to a mathematical basis for architectural composition.6 These dual influences are best on display, not in his buildings, but in his radical urban proposals from the decades of the1920s and 1930s.7 Both the Ville Contemporaine (1922) and the Ville Radieuse (1933) are marked by a strict adherence to reflective symmetry. Moreover, unlike Howard’s diagrams, they were intended to be built that way. While there is a tension between symmetry and asymmetry in many of his buildings from this era, his urban schemes and the buildings represented in them are dominated by mirror reflections. The redent or “setback” apartment blocks are bilaterally symmetric, while the immeuble villas and the cruciform towers are biaxially symmetric. Within the overall plan of the city these three building types are isolated from one another via the symmetrical subdivision of the scheme by orthogonal and diagonal roadways. It is only at the level of the individual dwelling unit where the strict symmetry begins to break down. Le Corbusier was also dedicated to industrial methods of building and thinking.8 This included a strong interest in statistics. He devoted an entire chapter in his Urbanisme (1925) praising them for their honesty and for the empirical limits that they established. These facts were to be used as the “jumping off point for poetry:”9 poetry with a decidedly symmetric structure. Bruno Taut’s work is similarly paradoxical. Although celebrated for his expressionist use of form, color, and material, his work—such as the Glass Pavilion and Alpine Architecture—is almost always symmetrical. His design (with planner Martin Wagner) for the Britz or Horseshoe Siedlung (1927-33) is organized via a series of local and global symmetries and near symmetries. Reflective and translational symmetries can also be found in the individual buildings. The project’s cooperative financial structure, its size, its location on the outskirts of the city, and the diversity of its building types were all grounded on the statistical analysis of economic and infrastructural data done in Berlin’s city planning office, headed by Wagner.10 The combination of rigorous numerical analysis and symmetrical plans would soon make its way to the United States. By the 1930s, the US government was publishing pamphlets showing ideal housing unit plans based on statistical analysis.11 It also produced guidelines for site plans for low-cost housing that were almost exclusively arranged symmetrically. Subsequently, most 42

limited profit, non-profit, and even some public housing projects built from the 1930s through the 1960s followed their lead.12 Examples such as Radburn in New Jersey, Stuyvesant Town and the Red Hook Houses in New York City, and Baldwin Hills Village and Park La Brea in Los Angeles share both alternative financing structures and a symmetrical shape. Translation Reflection was not the only variety of symmetry used by modernist planners. Translational symmetry was also prevalent, especially in German Zeilenbau schemes composed of parallel and equally spaced buildings. A translation is a movement of an entity in a specific direction and distance without a rotation or a reflection. In other words, it is the simple repetition of an element at a consistent interval. Translational patterns are often referred to as frieze patterns and are commonly used in the horizontal band of ornament found in the entablatures of classical architecture. They also describe the modular repetition of standardized elements found in much industrial era architecture. Many of Walter Gropius’s designs for housing estates from the decade of 1920 are defined by translational symmetry. His deployment of parallel rows of townhouses and apartment blocks were arranged to maximize the amount of light, air and views for each unit.13 Based on geometric analyses, it was concluded that the higher the building the better access one had to these environmental amenities, as well as allowing for increased density on the site. In short, the symmetry was informed by data. However, these were not the only considerations. Gropius also relied on sociological statistics to argue that different demographic groups could best be respectively served by low, mid, or highrise building types.14 These qualities are on display in the plan for the Dammerstock Complex in Karlsruhe. The repetition of the linear buildings gives the project a prototypical modern sensibility. However, a closer look reveals that in addition to translations, there are multiple reflections and even rotations in the overall plan and in the plans of the buildings themselves. Ludwig Hilberseimer also studied the formal and spatial effect that maximizing sunlight and ventilation had on building and block types. He was committed to symmetry. His perspectives and plans in Großstadt Architektur (1927) show a series of urban blocks distributed in a relentless application of translations across the city. The blocks are made up of buildings that accept rotations and reflections. His proposals for the American landscape are similarly symmetrical.15 The basic “settlement unit” he proposed for this context is distributed on the land via translations. The unit is internally organized around a reflection, and the individual

houses on each street are translations of one another. A relatively small version of this regional vision was built at the Lafayette Park development in Detroit, designed in conjunction with Ludwig Mies van der Rohe and Alfred Caldwell.16 Equivalence Does the presence of symmetry in these canonical and often-radical modernist projects undermine their modernity? Or, was symmetry also updated, as it was in physics and math? Certainly, there is no attempt, neither discursively nor formally, to adhere to the classical definition of symmetry as the harmonizing of unlike parts (via the use of proportion) to create objectively beautiful objects. Rather, in these urban schemes one finds an emphasis on consistency and predictability. What is emphasized is the equivalency of the parts. Things are not proportional to one another, they are identical with one another. This shift from harmony to uniformity is precisely the change symmetry made in math and science. Harmony is the integration of multiple elements—sounds, people, or colors—into a complex whole. In contrast, equivalence is the repetition of identical elements to make a simple one. This is the logic of mass production. This is a significant change. To live uniformly is very different from living harmoniously. If traditionally symmetry was a guarantor of beauty, it is now a guarantor of uniformity. Symmetry is still symbolic, but instead of representing an aesthetic hierarchy it expresses a set of social and statistical equivalences.

1 Catherine Bauer, Modern Housing (Boston: Houghton Mifflin, 1934). Richard Pommer, “The Architecture of Urban Housing in the United States during the Early 1930s,” Journal of the Society of Architectural Historians vol. 37 no. 4 (December 1978), 235-264. Roger Sherwood, Modern Housing Prototypes (Cambridge: Harvard University Press, 1979). Manfredo Tafuri and Francesco Dal Co, Modern Architecture 1 & 2 (New York: Abrams, 1979). Peter Rowe, Modernity and Housing (Cambridge: MIT Press, 1993). 2 Robin Evans, “Figures, Doors and Passages,” in Translations from Drawing to Building and Other Essays (London: Architectural Association, 1997), 55-91. 3 Manfredo Tafuri, Architecture and Utopia: Design and Capitalist Development (Cambridge: MIT Press, 1979), 125. 4 Ian Hacking, The Taming of Chance (New York: Cambridge University Press, 1990). 5 On existenzminimum see Eric Mumford, “CIAM Urbanism after the Athens Charter,” Planning Perspectives vol. 7 no. 4 (1992) and The CIAM Discourse on Urbanism 1928-60 (Cambridge: MIT Press, 2000). Martin Steinmann (ed.), CIAM: Dokumente 1928–1939 (Stuttgart: Birkhäuser, 1979). On the emphasis on efficiency and statistics in the United States, see Richard Plunz, A History of Housing in New York City (New York: Columbia University Press, 1990), 207-246. US Federal Emergency Administration of Public Works, Unit Plans (Washington D.C., 1935). “Housing Number,” Architectural Record 77 (March, 1935), 148-189. 6 Paul V. Turner, The Education of Le Corbusier, a Study of the Development of Le Corbusier’s Thought 1900 1920 (New York: Garland Publishing, 1977). Reyner Banham, “Conclusion: Functionalism and Technology,” Theory and Design in the First Machine Age (New York, Praegar, 1962), 328. 7 Thomas H. Beeby,“The Grammar of Ornament/Ornament as Grammar,” VIA III (1977), 10-29. 8 Mary McLeod, “Architecture or Revolution: Taylorism, Technocracy, and Social Change,” Art Journal 43 (Summer 1983), 132-147. 9 Le Corbusier, “Statistics,” The City of Tomorrow and Its Planning (New York: Dover, 1987) [1929], 105-126. 10 Esra Akcan, Architecture in Translation: Germany, Turkey, and the Modern House (Durham: Duke University Press, 2012), 152-177. Barbara Miller Lane, Architecture and Politics in Germany 1918-1945 (Cambridge: Harvard University Press, 1985) [1968]. 11 US Federal Emergency Administration of Public Works, Unit Plans (Washington D.C., 1935). “Housing Number,” Architectural Record 77 (March, 1935), 148-189. 12 For their presence in New York City, see Richard Plunz, op. cit. 13 Susan R. Henderson, Rationalization Takes Command: Zeilenbau and the Politics of CIAM in Building Culture: Ernst May and the New Frankfurt Initiative 1926-1931 (Bern, Frankfurt, London, New York: Peter Lang, 2013). 14 Walter Gropius, Sociological Premises for the Minimum Dwelling of Urban Industrial Populations in Scope of Total Architecture (New York: Collier, 1955) [1929], 91-102. 15 Ludwig Hilberseimer. Metropolisarchitecture and Selected Essays, translated by Richard Anderson (New York: Columbia University Press, 2014) [1927]. 16 Caroline Constant, “Hilberseimer and Caldwell: Merging Ideologies in the Lafayette Park Landscape” in Charles Waldheim (ed.) CASE Hilberseimer/Mies van der Rohe, Lafayette Park Detroit (Munich: Prestel, 2004), 95-111.

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Frank Lloyd Wright, Darwin D. Martin House. New York, United States, 1905

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07 Math and Symmetry

In the wake of World War II, the status of science and technology was surprisingly ambivalent. At once responsible for the conflict’s greatest triumphs, they were also guilty of its greatest horrors. In the years immediately following the War’s end there was much talk within architectural discourse for the need for a balanced or more “harmonious” relationship between humans and our inventions.1 Such was the subtext surrounding architecture’s reevaluation of the relevance of proportion in the late 1940s and early 1950s. Following the publication of a number of texts on the topic, including Rudolph Wittkower’s book Architectural Principles in the Age of Humanism (1949), architects actively debated the merits of the harmonic proportions favored by Palladio and Pythagoras, the relevance of the Vitruvian Man, and the efficacy of the Golden Section.2 While these had the authority of age value, because they emphasized the use of standardized, number-based systems, they were seen as being compatible with the repetitive methods of industrial production. The focus on proportion and harmony echoed Vitruvius’ definition of symmetry as the skilled use of ratios to create beautiful effects.3 This is precisely how Le Corbusier defined symmetry. In his 1950 treatise outlining his eclectic proportioning system, the Modulor, he writes that symmetry is a term that “(…) comes from the very essence of civilization, a word which can contain our desire: symmetry expressing a limitless relationship between two terms, each raised above all vulgar acceptance, both placed, one in relation to the other, in positions that are unforeseeable, unexpected, astonishing, stupefying, enchanting: poetry.”4 Dismissing the “increasingly scientific mathematical truths” that symmetry was helping to produce, Le Corbusier took it back to its ancient roots.5 Rejecting the “false meaning of equality” underlying the modern mathematical definition of symmetry, he put it “back in its proper place, on the plane of equilibrium: the very essence of proportion.” But, because proportion was also “concretely linked to questions of measures, dimensioning, strictly objective relationships,” he preferred the concept of harmony, as it better addressed the poetic goal of turning unlike parts into a beautiful whole.6 Le Corbusier was mostly likely made aware of the contemporary notion of symmetry via his association with Andreas Speiser. Speiser was the brother-in-law of one of his early clients, Raoul La Roche. And, through La Roche, he was acquainted with the avant-garde art scene in 1910s-1920s Paris.7 Speiser was a prominent mathematician who worked on group theory and mathematical symmetry.8 He and Hermann Weyl had completed their dissertations on the topic under David Hilbert in Gottingen, Germany. It was Speiser who informed Weyl about the connection of symmetry to ancient aesthetic practices, and it was Weyl who kept Speiser abreast of its usefulness in physics.9 While Weyl would help establish symmetry as a vital tool in theoretical physics, and was an important player in the Manhattan Project, Speiser focused on the presence of the same symmetries in pre-modern contexts.10 For Speiser and Le Corbusier the reemergence of symmetry in the 20th century was not a break with the past but a continuation of it, whereas for Weyl it was a step towards the future.11

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Weyl and Symmetry Weyl did share Speiser’s interest in the relationship between mathematical facts and cultural forms, and Speiser believed, as Weyl did, in math’s universal influence. In a letter to Le Corbusier, published in Modulor, Speiser wrote: “It can be said that our duty on Earth and during the whole of our life consists precisely in this projection of forms issued forth from numbers, and that you, the artists, fulfill that moral law to the highest degree.”12 Weyl also saw symmetry as a device that connected math with physical phenomena found in the world. However, based on his experience in physics, he came to the opposite conclusion that Le Corbusier did. Instead of seeking a return to harmony, in his 1952 book Symmetry, he argued how symmetry’s value lied in its progression away from a loose definition of harmony and towards the objective status of a mathematical law.13 Weyl argued that underlying a variety of cultural, physical, and geometric forms were the permutations and sets found in group theory. A woman’s hat, a Byzantine ornament, a bee hive, the location of electrons in an atom, the placement of molecules in a crystal, the organization of petals on a flower, the composition of a renaissance painting, architectural objects, and ornamentation—all of which are literal examples in Weyl’s book on symmetry—could be compared, because they all shared the same underlying logic. By 1950 the group theory definition of symmetry had proven useful in physics for both observing and predicting phenomena. Symmetry provided the literal connection between the objective mathematical logic and the conceptual and empirical phenomenon. For Weyl, the relationships established by symmetry were anything but false. Rather, they were nothing less than a universal law that permeated every layer of reality, from the most obvious to the least visible. Those layers were organized in a nested hierarchy, from the cultural to the mathematical, and were represented in the structure of his book, progressing from looking at the “somewhat vague notion” of symmetry found in art, to geometry and crystallography, to finally the mathematical description of symmetry as an “invariance of a configuration of elements under a group of automorphic transformations.”14 The relationship between Weyl’s disparate phenomena is not harmonious, and is not one that could be synthesized at the level of perceived form. Their correspondence can be found at the invisible level of an abstract structure.15 Placing the logic of mathematics, specifically group theory, at this foundational level, challenges any categorical distinction or hierarchy between natural and cultural artifacts, and provides 46

a unified theory for where (symmetrical) form comes from. This is possible because groups, symmetrical and otherwise, are by definition plural. Because they are premised on operations and their permutation, they allow for multiple conditions to satisfy the same condition without conforming to a static model, such as the Golden Mean, the Classical Orders, or Le Corbusier’s Modulor. As such, they are not positioned as a transcendental truth to be illustrated, but as a dynamic and evolutionary condition that can be made and remade over and over again. As shown by Weyl it is an empirical truth not an a priori one. Group theory—which broadly accounts for all kinds of operations, and not just symmetrical ones—has codified and provided the nomenclature for documenting the plural but limited possibilities for any situation— spatial, temporal, physical—in which such invariance is present despite something else being transformed. Such relationships can be found inside of molecules, in crystals, as well as in architectural ornaments and objects. It also describes the logic of non-Euclidian objects and phenomena, such as the quantum movement inside the atom, the metrics of n-dimensional spaces, as well as the curved geometry of spheres and space. Many of these things look very different from one another. Some of them are impossible to see, with human or even cyborg eyes. But symmetry is not limited to experience.16 In addition to making symmetry plural, Weyl’s and others’ contribution was to recognize that symmetry is less a quality than it is an operation. Le Corbusier and (Modern) Symmetry In 1951 the First International Congress on Proportion met in Milan. The title of the event was “Divine Proportions.” It was attended by leading scholars, artists, and architects, including Wittkower, Le Corbusier, Sigfried Giedion, Ernesto Rogers, Matila Ghyka, and Max Bill. Andreas Speiser gave the only paper on group theory. The dominant focus was on historical and contemporary use of proportion in aesthetic discourse and practice.17 Two years later, at a subsequent meeting attended by both Speiser and Le Corbusier, the group agreed to abandon the term “proportion,” because it emphasized the study of the past rather than the design of the future. They renamed themselves Groupe Symétrie.18 Despite this title, in his writings Le Corbusier remained committed to the ancient definition of symmetry. And yet, when one looks at his work from around this period one finds it to adhere to the modern logic of permutations and groups rather than to the ancient idea of harmony. This is particularly evident in the facades and section of his Unite d’Habitation in Marseilles (1947-52). In cross section, the two interlocking L-shaped living units

that make up the dominant module of the scheme are rotationally symmetrical. The short, southern elevation shows how pairs of these modules are reflected onto one another, with this grouping then translated vertically three times up the facade, with a slight interruption for two single-height floors. The long east and west elevations are less complex and are dominated by translations. These facades are broken horizontally by a column of units arranged along a series of local rotations, and vertically by the different fenestration pattern used for the shopping streets on the seventh and eighth levels. These breaks give the elevations an overall asymmetrical effect and undermine any reading of them as monotonous or globally symmetric.19 The final result is a patchwork of local symmetries. Despite the fact that all its dimensions conform to the Modulor, the effect is less harmonious than episodic. All local relationships are defined by symmetry, but they are juxtaposed rather than blended with one another. Conclusion The overall asymmetry of the Unite d’Habitation can obscure the importance of the simple, local symmetrical operations present in them. In these projects symmetry is not an a priori goal governing the global composition. It is not a prohibition. Instead, it establishes a limited set of operations to be enacted on a limited set of elements, or modules. When these elements are combined they produce a variety of different, yet still symmetrical effects. The final form is the outcome of these permutations rather than the illustration of a fixed proportion. The result is symmetrical, as the underlying logic is invariant despite the multiple transformations undertaken by the elements. And, it is also harmonious simply because it is an “orderly or pleasing combination of elements in a whole,” rather than because it adheres to a mathematical idea. In other words, it is simultaneously ancient and modern.

1 William Graebner, The Age of Doubt: American Thought and Culture in the 1940’s (Boston: Twayne, 1991). Sarah Ksiazek and Mitchell Schwarzer, “Modern Architectural Ideology in Cold War America,” in The Education of the Architect: Historiography, Urbanism, and the Growth of Architectural Knowledge, edited by Martha Pollak (Cambridge: MIT Press, 1993). 2 Rudolf Wittkower, Architectural Principles in the Age of Humanism (New York: Norton, 1949). Rudolf Wittkower, “The Changing Concept of Proportion,” Daedalus 89 (Winter, 1960), 199-215. Henry A. Millon, “Rudolf Wittkower, Architectural Principles in the Age of Humanism: Its Influence on the Development and Interpretation of Modern Architecture,” Journal of the Society of Architectural Historians vol. 31 no. 2 (May, 1972), 83-91. Alina A. Payne, “Rudolf Wittkower and Architectural Principles in the Age of Modernism.” Journal of the Society of Architectural Historian vol. 53 no. 3 (September 1994), 322-342. Christopher Hight, “A Mid-Century Renaissance,” Architectural Principles in the Age of Cybernetics (New York: Routledge, 2008), 71-89. 3 Vitruvius. The Ten Books on Architecture (New York: Dover, 1960), 3-17. 4 Le Corbusier, Modulor I and II (Cambridge: Harvard University Press, 1980) [1952], 149. 5 Anna Chiara Cimoli and Fulvio Irace, “Triennial 1951: Post-War Reconstruction and ‘Divine Proportion’” Nexus Network Journal 15 (April, 2013), 3-14. 6 Ibid.,154-155. 7 Lynn Gamwell, Mathematics and Art: A Cultural History (Princeton: Princeton University Press, 2015), 265-266. 8 Andreas Speiser, Die Theorie der Gruppen endlicher Ordnung, 2nd edition (Berlin: Springer, 1927). 9 Gamwell, Mathematics and Art, 265-266. 10 Andreas Speiser, Die Mathematische Denkweise (Berlin: Rascher, 1932). Andreas Speiser, “Symmetry in Science and Art,” Daedalus 89 (Winter, 1960), 191-198. 11 Gamwell, Mathematics and Art, 259-264. 12 Le Corbusier. Modulor I & II, 77. 13 Hermann Weyl, Symmetry (Princeton: Princeton University Press, 1952). 14 Ibid., “Preface.” 15 Anthony Zee, “Symmetry and the Search for Beauty in Modern Physics,” New Literary History 23. (Autumn, 1992), 815-838. 16 Ibid. 17 Cimoli and Irace, “Triennial 1951”, 3-14. 18 Cimoli and Irace, “Triennial 1951”, 12. 19 Julián Varas, In the Name of the User. Social Housing and the Agenda of Architectural Heterogeneity. Unpublished PhD dissertation. Santiago, Chile; Pontificia Universidad Católica de Chile, 2016. See: Chapter 2, Section 2: “Steps Toward a Socio-Plastics: Type, Program and Expression in Le Corbusier’s Unite d’Habitation”

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Bruno Taut, Glass Pavilion. Cologne, Germany, 1914

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08 Geometry and Symmetry

In his 1971 essay “Structure and Patterns in Science and Art,“ Arthur Loeb brutally—if not conventionally—divided the world in half when he claimed that scientists discover patterns while artists create them. He quickly made it clear that, despite this separation, the goal of his text was to show how these two ways of knowing and working overlapped, when stated that the “Interaction between art and science may occur in two ways: (a) when a scientist studies the relations occurring in the patterns created by an artist and (b) when an artist uses relations discovered by a scientist.”1 The patterns he looked at were symmetrical tessellations of two-dimensional planes. Their ornamental and geometric associations were well established and well known to Loeb. He was a chemist, a crystallographer, a mathematician, a musician, a dancer, and a teacher in the Visual and Environmental Studies program at Harvard University. He had written a mathematical treatise on symmetry and color,2 had contributed an essay to Gyorgy Kepes’ book Module Proportion Symmetry Rhythm,3 was a founding member (along with material scientist Cyril Stanley Smith and evolutionary biologist Stephen Jay Gould) of the Philomorph Group, and had helped found the International Society for the Interdisciplinary Study of Symmetry. What connected these pursuits were questions such as: Where does form come from? What is the relationship between natural and cultural form-making processes? What role does symmetry play in them? While many others were interested in these questions, what made Loeb’s inquiry particularly fruitful were his own experiences as a practitioner and a teacher of both science and art.4 Loeb defined patterns as an array of elements arranged according to a structure. A structure could be symmetrical or mathematical, though it need not be. A symmetrical pattern was thought to be one in which structure was limited to the four automorphic operations of reflection, rotation, translation, and glide reflection. Though he understood Weyl’s group theory derived definition of symmetry, he did not refer to it in his aforementioned text. Instead he provided a more limited theorem for all symmetrical, two-dimensional patterns: “(…) the coexistence in a plane of a k-fold and an I-fold rotocenter implies the existence in that plane of an m-fold rotocenter, where 1/k+1/l+1/m=1.”5 To use an architectural metaphor, this simple equation is the structure, the strong, stable system that holds an entity or edifice together and combats the forces that would otherwise undermine it. However, unlike its use in architecture, here structure is an abstract, immaterial set of relationships. No matter the shapes used, or the forms and effect they produced, any symmetrical, tessellated surface pattern shares this structure. Loeb illustrates this point with a triptych painting that he created, entitled Disciplined Freedom, in which three identically symmetrical geometric patterns are shaded in such a way that they each appear unique. In this example the physical and the conceptual structure of symmetry is a mechanism for exerting control and spatially predicting what comes next, while the addition of color shows that this consistency does not limit the possibility for a variety of affects. Symmetry is only the shared starting point for individual transformations, providing the structure to compare and relate different formal iterations with one another. 49

Kepes and Transformation Hungarian-born and Bauhaus-trained, Gyorgy Kepes shared Loeb’s desire to stitch science and art together, and positioned patterns and symmetry as two means for doing so. As an artist, curator, writer, editor, and teacher of the visual arts at MIT, Kepes was long dedicated to the cause of bridging these “Two Cultures.”6 By the late 1950s, when he published The New Landscape in Art and Science, any uncertainty surrounding the status of science and technology had vanished.7 The Space Race and numerous hot and cold wars had made their development a priority. Kepes did not challenge this emphasis. What concerned him was that human beings’ perceptual apparatus was not keeping up with the changes they were creating. Echoing Marshall McLuhan, Kepes believed that artists were uniquely qualified to help human kind to adjust to the new sensorial and mental reality being produced by a variety of technological tools. The task for the artist was no longer to illustrate or hold a mirror (or a microscope, or a telescope) up to the natural world. Citing Hemholz and Mondrian, Kepes argued that these new facts had to literally and figuratively be exaggerated and “transformed in order to evoke aesthetic sensations.”8 Like Loeb, this meant working on, thinking with, and manipulating patterns. And, like Hermann Weyl, he held that the key phenomenon was transformation. The new scientific knowledge, from atomic transfiguration to artificial mutation of genes, made transformation no longer a philosophical issue, but a vital experience.9 For Kepes a static visual pattern was always a snapshot of a spatial-material-temporal process; it captured movement or was the result of a sequential set of operations. Visual patterns were the temporary markers of a transformation. He argued that the world of patterns gave us a new way to recognize these processes. Perception based on the isolation of individual units gave way to the recognition of relationships.10 As in Loeb’s examples, symmetrical patterns were particularly good at revealing what stayed the same while other qualities changed during processes of physical transformation. For example, they showed the affinity between the similar branching patterns that were found in organic contexts, such as trees, and in inorganic ones, such as rivers. That being said, Kepes noted that nature’s processes and our pictures of them are not identical. He argued that a new vocabulary of visual thinking was required to focus our attention on the fundamental significance of transformation.11 What art provided were the means for creating this new language. Kepes maintained that while science made these patterns available to our senses, it would be artists who needed to translate them so that they could be made sense of. Art had a different cultural 50

and social function. In Weyl’s terms, it operated in a different inertial frame, and thus needed different forms to communicate and fulfill its task. Such patterns would be dramatically different from their natural sources, as they needed to act directly on the body, itself understood in a culturally and historically specific way. In the Vision & Values series of books that Kepes edited in the mid-1960s, he brought together texts by scientists, artists, designers, architects, and historians (including Loeb) who addressed these issues from within their respective disciplines. Issues of proportion, modularity, structure, and symmetry where presented as the means of creating and showing the shared and unchanging relationships that existed between genres, eras, media, and disciplines.12 In other words, in order to understand what is transformed, one also needs to understand and show what is invariant: one needs to see symmetry. Tyng and Kahn: Raw and Refined Geometry Architects were also interested in symmetry during the 1960s. A 1969 edition of the Italian journal Zodiac featured the work of a number of architects—including Anne Tyng, Alfred Neumann, Zvi Hecker, Moshe Safdie, and Buckminster Fuller—who were experimenting with architectural forms influenced by both the geometric and group theory definitions of symmetry. Space frames, which mimicked crystal structures and platonic solids, scaled to room size and repeated using the four basic symmetrical operations, were common tropes found throughout the issue. Tyng literally was a student of geometry and proportion. She earned her PhD at the University of Pennsylvania— under the guidance of Buckminster Fuller and Robert Le Ricolais—with a dissertation entitled “Simultaneous Randomness and Order: The Fibonacci-Divine Proportion as a Universal Forming Principle.” Her academic work focused on the archetypal presence of the Golden Section in architectural form and its mathematical connection to the Platonic solids. In her text for Zodiac she made the mathematical case for the fundamental evolutionary importance—for humans and architecture— of these devices. However, looking back at her and her peer’s work published in Zodiac, Tyng conceded that, in the work shown, there was less “integrated architecture” than there was “raw geometry.”13 In Kepes’ terms, the symmetrical logic was only scaled up: it had not been transformed into something that produced a new mode of perception. The kind of transformation that she recognized was achieved more fully in the work that she did in collaboration with Louis Kahn. Kahn’s use of symmetry is often associated with the Beaux-Arts training that he received at Penn from Paul Cret.14 While this helps account for his regular use of an

axial organization strategy, the symmetrical repetition of geometrical structural units is more indebted to Tyng. In projects such as the Trenton Bath House, the Erdman Hall at Bryn Mawr College, and the proposal for the City Hall Tower in Philadelphia, she recalled that she would be responsible for the initial mathematical “order” of the project, while he would subsequently “randomize it.”15 This geometric logic often became the physical structure of the project, and was turned architectural by making it “charming with all these little forms.”16 For example, her original scheme for Erdman Hall was dubbed the “molecular plan” due to its field or wallpaper like organization of rooms. Kahn revised it to be a linear sequence of three-diamond shaped modules, joined by a central axis that also connected a series of shared communal spaces. Globally, the scheme contains clear reflective and translational symmetries, while each of the three large modules is also rotationally symmetric. In making his modifications, Kahn exaggerates both the presence of the module and their relationship. Like Tyng, Kahn often spoke in transcendental terms about architecture. He too was in search of an inherent “Order”—both mathematical and existential—to ground architectural production. The task of the architect was to use “Design” to give form to that “Structure.”17 Perhaps, the best example of an underlying symmetrical order transformed through a “Structure” of “charming little forms,” is the National Assembly Building in Dhaka. Although simpler geometries of squares and hexagons are present in the project, the overall scheme in plan is comprised of eight segments flanking a sixteen-sided, rotationally symmetrical central space. There are four identical segments on the perimeter and four unique ones. Each of these eight pieces, as well as the entire composition, is divided into two reflected halves (save for the mosque on the southern end, which is skewed to face Mecca). These symmetries are immediately evident on the exterior, as the massive elements are punctuated with large rectangular and triangular openings that establish the sections as having been actively reflected. A closer look at the individual elements in the plan and the interior elevations reveals a few broken or near symmetries. These asymmetries are easier to find when the view of the project is limited and seen obliquely. However, every asymmetrical element has a twin that brings it into a symmetrical relationship, even if one cannot see it. In short, the seemingly static figure that Kahn presents is the result of a number of symmetrical operations. In this and other buildings, though symmetry’s presence is sometimes independent of the immediate image at hand, Kahn consistently exaggerates symmetry’s physical presence, transforming the literal and conceptual structure into a series of sensorial experiences. As Kepes requested, the experience of any

independent unit reveals a larger set of relationships. In other words, more than an image, an object is the embodiment of the process that formed it—a process guided by both mathematical and artistic symmetries.

1 Arthur Loeb, “Structure and Patterns in Science and Art,” Leonardo 4 (Autumn, 1971), 339. 2 Arthur Loeb, Color and Symmetry (New York: John Wiley & Sons Krieger, 1978) [1971]. 3 Arthur Loeb, “The Architecture of Crystals,” in Module Proportion Symmetry Rhythm, edited by Gyorgy Kepes (New York: George Braziller, 1966). 4 Arthur Loeb, “Symmetry in Court and Country Dance,” in Symmetry: Unifying: Human Understanding, edited by Istvan Hargittay (Oxford: Pergamon Press, 1987), 29-640. 5 Loeb, “Structure and Patterns in Science and Art”, 344. 6 C.P. Snow, Two Cultures and the Scientific Revolution (New York: Cambridge University Press, 1961). 7 Gyorgy Kepes, The New Landscape in Art and Science (Chicago: Paul Theobald, 1956). 8 Piet Mondrian, quoted in Gyorgy Kepes, The New Landscape in Art and Science (Chicago: Paul Theobald, 1956), 229. 9 Kepes The New Landscape in Art and Science, 229. 10 Ibid., 226. 11 Ibid., 279. 12 Gyorgy Kepes (ed.), Structure in Art and in Science (New York: George Braziller, 1966). 13 Anne Tyng, “Number is Form and Form is Number, Interview by Robert Kirkbride,” Nexus Network Journal 7 (2005), 134. 14 Kenneth Frampton, “Louis Kahn and the French Connection,” Oppositions 22 (September 1980), 20-53. 15 Anne Tyng and Antonio Juarez, “Aleatoriedad y orden en la arquitectura de Louis I. Kahn = Randomness and order in Louis I. Kahn’s Architecture,” Via Arquitectura (March 1998), 91-97. 16 Ibid., 91. 17 Louis Kahn, “Order is,” Perspecta 3 (1955), 59.

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Le Corbusier, Villa Savoye. Poissy, France, 1929

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09 Syntax and Symmetry

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.

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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 54

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.

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Ludwig Mies van der Rohe, Crown Hall. Chicago, United States, 1956

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10 Semiotics and Symmetry

By the 1960s there were many complaints about the state of modern architecture. These came from inside and from outside the discipline. While some lamented the sterile, minimal living conditions it produced, others focused on the dullness of its forms. Its inability to produce positive meanings and associations was also decried. It was not just ugly or uncomfortable: it was indecipherable. One response to architecture’s inability to communicate was an interest among architects in semantics and semiotics. Semantics looks at how meanings are produced. Semiotics explains how linguistic signs operate and communicate meaning. Instead of looking at the internal structure (or syntax) of form, these communication-oriented architects were interested in how form produced cultural meaning.1 Toward this end, they looked at contemporary cultural contexts that were successfully speaking with the public—namely, the mass media.2 It also spurred a renewed interest in how past architectural styles continued to produce meaning.3 In this postmodern context the answer to the question “where does architectural form come from” was: traditional architectural tropes. Among these motifs was symmetry, in particular, reflective symmetry. Meaning The production of meaning can be understood as a type of symmetry making and breaking. As the polymath Gregory Bateson noted, the goal of any communicative act is “the creation of redundancy, meaning, pattern, predictability, information, and/or the reduction of the random by ‘restraint.’”4 In other words, the goal is an invariance of information despite its transformation from the sender to the receiver. Among the things that can disrupt this symmetrical condition are poorly formulated messages, the use of the wrong medium, the presence of noise in the communication channel, or the unequal background and skills of those wishing to communicate with one another. Bateson recognized that any system that relied on the exchange and feedback of information existed within a larger, more complex, but similarly structured context. This was particularly true of the messages sent by the less rule-bound practice of art: “It is, I believe, of prime importance to have a conceptual system which will force us to see the message (i.e., the art object) as both itself internally patterned and itself a part of a larger patterned universe—the culture or some part of it.”5 The internal pattern is the form (i.e. words, colors, textures, shapes) of the message proper, while the larger external pattern is “derived from, or determined by, other characteristics of cultural and psychological systems.”6 Of course, not all messages get through, and not all meanings are the intended ones. Bateson understood that most of the time the goal was to avoid this asymmetric condition. However, he also recognized that moments of miscommunication could play a valuable cultural role: “All that is not information, not redundancy, not form and not restraints—is noise, the only possible source of new patterns.”7 57

Noise, randomness, and misinterpretations were not always inefficient or unproductive: sometimes they were the source of cultural evolution. However, these new patterns would only survive if the existing context could support them. The mutation that was modern architecture was patterned in such a way as to be indecipherable by the context it was born into. Instead of attempting to bend the “larger patterned universe” to match the messages, modern architecture was sending, postmodern architecture used traditional tropes in the hopes of reestablishing a rapport with a larger audience. In addition to including classical features like pediments and pilasters at the level of organization, bilateral symmetry was a frequent feature of “Postmodern Buildings.” Grassi: From Transformation to Tradition Giorgio Grassi belonged to the group of architects who looked to historical architectural tropes as a means to reconnect architecture with its various publics. His neorationalist work interestingly spans the gap between architecture as an autonomous practice and as a social art, combining traditional building types and materials with a minimal modern sensibility. It also frequently makes frequent use of symmetry. Grassi was a member of the Tendenza group that formed in Italy in the 1960s. As with the original rationalists of the 1920s and 1930s, they sought to strip architecture to its core.8 They were not seeking out architecture’s deep structure, but rather looking for its most basic and most legible units. Their designs emphasized the serial repetition of windows, walls, doors and columns. They used traditional building elements, such as pediments, domes and colonnades. And, they consistently deployed conventional building typologies, including courtyards, stoas, and slabs, in symmetrical arrangements. The emphasis on these well-known architectural components was not a move towards autonomy, but a prerequisite for engaging the social sphere. Through his writing in 1980, Grassi proposed that architecture must first come to terms with itself, that is, with its specific characteristics, and at the same time with its particular social responsibility. For this reason, he argued that the language of architecture is—or should be—accessible.8 In Grassi’s case accessibility meant rejecting any avant-garde experiment. The ad hoc expression of materials, function, and structure was replaced by the compositionally consistent characteristic of reflective symmetry. Grassi’s Housing Project in Ticino (1972), the Palace of Administration in Trieste (1974), the Student Housing project in Chieti (1976), the Roman Theater at Sagunto (1994) and the University Library in Valencia (1998), among other projects, combine 58

symmetrical plans with simple, spare, and serially repeated punched windows and doors in masonry walls.9 On the one hand, these spartan buildings are modern. They are large, functional, minimal, out-of-scale, and even threatening. On the other, they are formal and familiar. They appear as uncanny combinations of Ledoux’s neo-classicism and HIlberseimer’s modernism. Formal types and details appear as updated versions of ancient and modernist models. They have been transformed yet remain invariant.10 What is difficult to decipher is their function. Echoing the projects shown in the 19th century textbooks of French theorist J.N.L. Durand, the same plan type, the same window treatment, the same site strategy can be used for a housing complex, office building, prison, library, or hospital. Not only can the same forms accommodate multiple functions, but they can also accommodate multiple, often contradictory, meanings. As Philip Tabor noted, this flexibility is an important feature of buildings with axial symmetry. Symmetry has been associated with both democracy and fascism. It has been interpreted as beautiful and boring. It has been judged as being efficient and wasteful, timeless and obsolete.11 Yet, the key term here is “and.” Grassi’s work shows that both of the terms bound up in these binaries can coexist in one project. There is enough “noise” coming from the use of conventional architectural tropes (including symmetry) to upend the modernist myth of aesthetic progress. But, there is enough newness to undermine the interpretation of tradition as an uninterrupted and timeless quality. In short, their physical qualities - symmetry, serial repetition, spare surfaces, and materials—embody the message that architecture’s specific characteristics can remain invariant and adapt themselves to meet the transformations caused by historical disruptions. Graves: From Type to Expression Another postmodern architect who incorporated traditional details with symmetrical plans was Michael Graves. However, the contexts and messages of his buildings were dramatically different from those of Giorgio Grassi. His forms were as loud as Grassi’s were quiet. Where the Italian neo-rationalists were minimalists, American postmodernists like Graves, Robert Venturi and Charles Moore were maximalists. Not surprisingly, their work is exuberant where Grassi’s is somber. If Grassi tended to abstract and simplify traditional forms and details, Graves exaggerated them. Projects like the Portland Building (1982), the Humana Tower (1985), and the Swan and Dolphin Resort at Disney World (1990) are all out-of-scale. Their forms are inflated and their colorful surfaces flattened. Traditional ornamental details are

distorted. Every elephantine element (and its symmetrical twin) announces, if not shouts, its presence. I am a keystone! I am a capital! I am a pilaster! On the one hand, the shapes of these details send a message that they are associated with the past. On the other, they are as big, bold, and colorful as billboards. The accessible language that Graves deployed was taken as much from marketing as from mannerism.12 Ambiguous and Exaggerated Conventions Despite their differences, Graves and Grassi shared a reliance on reflective symmetry to organize the plans, facades, and shapes of their buildings. What message does the common presence of this staid form of symmetry send? Is it just a neutral technique, indifferent to stylistic and ideological differences? Does it signify a point of connection between the two seemingly polar positions? Or, does it highlight their differences? The axial symmetry and classical iconography used by both architects indicates a desire to connect their projects with old architectural traditions. Both architects attempted to update and bring the past into the present, and symmetry was a marker of that tradition par excellence. The eerie silence of Grassi’s forms also reveals his stated belief that architecture needs to communicate to—and to be able to communicate in— the future.13 As such, they cannot be identified as belonging to a specific time period. The unchanging nature of symmetry is not necessarily timeless, but it also does not belong to any one era either. In contrast, Graves’ use of symmetry appears, like his exaggerated ornaments, as a historically specific scare-quote from the past, a quote that signifies that symmetry is being used to mark a historically specific moment rather than connect the present with the future. Symmetry can be a very slippery signifier. Grassi and Graves’ were also popular, if only for a short time and with a limited (architectural) audience. Today, even though the “larger patterned universe” has dramatically changed, their easily recognizable, internal (symmetric) patterns allow them to still convey meaning—even if those meanings have changed. No longer radical, Graves’ buildings are now closely associated with the specific time and place in which they were made. Grassi’s imposing structures are harder to pin down. They do not seem out of date, nor do they appear contemporary. They may not be indecipherable, but they are ambiguous. The blunt presence of symmetry in plan and elevation helps produce this out-of-this-time effect.

1 Charles Jencks and George Baird, Meaning in Architecture (London: Barrie & Rockliff, 1969). Geoffrey Broadbent, “A Plain Man’s Guide to the Theory of Signs in Architecture,” Architectural Design vol 47 no. 7, (1977), 474-482. 2 Robert Venturi, Denise Scott Brown and Steven Izenour, Learning from Las Vegas (Cambridge, MA: MIT Press, 1977). Charles Jencks, The Language of Post-Modern Architecture (London: Academy, 1977). 3 Robert Venturi, Complexity and Contradiction in Architecture (New York: Museum of Modern Art, 1966). Aldo Rossi, The Architecture of the City (Cambridge: 1982) [1966]. 4. Gregory Bateson, “Style, Grace, and Information in Primitive Art,” Steps Toward an Ecology of Mind (New York: Ballantine, 1972), 130. 5 Ibid. 6 Gregory Bateson. “Cybernetic Information,” Steps Toward an Ecology of Mind (New York: Ballantine, 1972), 416. Gregory Bateson, Mind and Nature (New York: Dutton, 1979), 42-45. 7 Marco De Michelis, “Aldo Rossi and Autonomous Architecture,” in The Changing of the Avant-garde: Visionary Architectural Drawings from the Howard Gilman Collection, edited by Terence Riley (New York: Museum of Modern Art, 2002), 89-98. Giorgio Grassi, “Interview with Giorgio Grassi,” Architectural Design vol. 77 no. 5 (September-October 2007), 26-29. 8 Giorgio Grassi, “Avant-Garde and Continuity,” Oppositions 21. (Summer 1980), 32. 9 “Casa dello Studente, Chieti, Italy, 1976-1980,” Architectural Design vol. 52 no. 5 (1982), 118-124. Giorgio Grassi, “Scena fissa: progetto per il teatro romano di Sagunto = Fixed stage: project for the Roman Theatre of Sagunto,” Lotus International 46 (1985), 7-21. Kenneth Frampton, “Modern Drama: Roman Theater of Sagunto, Sagunto, Spain, Giorgio Grassi, Architect,” Architecture vol. 83 no. 11 (November, 1994), 98-105. “Biblioteca del Nou Campus, Valencia = New Campus Library, Valencia,” AV Monografías = AV Monographs no. 75-76, (January-April, 1999), 60-65. 10 Giorgio Grassi, “On the Question of Decoration,” Architectural Design vol. 54 no. 5-6, (1984), 10-13, 32-33. 11 Philip Tabor, “Fearful Symmetry,” Architectural Review vol. 173 no. 1023, (May, 1982), 18-25. 12 Martin Filler, “Michael Graves: Before and After,” Art in America vol. 68 no. 7, (September 1980), 99-105. David L. Gilbert, “The Portland Building,” in The Critical Edge: Controversy in Recent American Architecture, edited by Tod A. Marder (Cambridge: MIT Press, 1985), 163-173. Sylvia Lavin, “Michael Graves: Humana Building,” Domus no. 667, (December, 1985), 1-5. Mark Aiden Branch, “Story Time,” Progressive Architecture vol. 71 no. 3 (March, 1990), 76-83. 13 Grassi, “Avant-Garde and Continuity.”

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Aldo Rossi, Teatro del Mondo. Venice, Italy, 1979-1980

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11 Information and Symmetry

Architecture’s two-decade long emphasis on language, linguistics, and meaning was challenged in the early 1990s. The penetration of information technologies and computational capacity into the design process made it easier to represent and build dramatically different forms—forms that looked unfamiliar and performed in unfamiliar ways. Computer aided design and manufacturing methods made it almost as easy to make one-of-a-kind asymmetrical surfaces, spaces and structures as regular, repeated, and symmetrical ones. The answer to the question “where does form come from?” was increasingly: from “computer software, hardware, and algorithms.” However, this did not mean that symmetry was abandoned. As the title of Greg Lynn’s 1995 essay “The New Novelty of Symmetry” suggests, symmetry was still present, but in a different way.1 Drawing on lessons from biology and genetics, for Lynn the novelty was in positioning symmetry as an initial condition to be overcome rather than an essence to aspire to. Similarly, Preston Scott Cohen’s Contested Symmetries uses symmetry as a foil against which to argue for more complex geometries.2 These were not attempts to discard this disciplinary constant once and for all. Rather, they were operations that re-appropriated and rebooted it as an important beginning rather than an ideal end. For these and other authors and architects, instead of being positioned as a sign of stability and beauty, symmetry is understood as a primitive state to be evolved away from. As in embryology, symmetry can be understood as a homogenous state that will become differentiated as energy and information are added to it. The analogies of embryology and evolution are important ones, for along with their cultural equivalents of repetition and learning, they provide insight into the understanding of the relationship of symmetrical and asymmetrical architectural form in the information age. Difference and Information Among the traditional arguments for symmetry is the allegedly empathetic relationship it produces between our bodies and buildings. Symmetry allows us to recognize ourselves in the things we make for ourselves. It has also been understood as the ideal version of a body.3 But a building is not a body, or not a biological body at least. Its form does not come into being in the same way that biological forms do. Nor does symmetry function in the same way in architecture as it does in the natural world. Philip Ball has studied where biological form comes from, how it is related to symmetry and patterning, and the underlying processes that govern them. One question he asks is how do undifferentiated elements, such as a fertilized egg or a seed, transform themselves into a highly specific one? How does formal and functional specificity emerge out of homogeneity? In particular, how does this epigenetic process consistently and automatically occur?4 One answer is that stable formal patterns emerge from generic ones only in conditions that are far from equilibrium—conditions in which there is a steady supply of energy that fuels the change and/or a predictable exchange of information that guides the transformation process.5 In these situations symmetry is a starting condition that is broken, reformed, and broken again. Symmetry is the necessary starting point of the forming process. Where one finds symmetry as an end state, for example, in the skeletons of mammals or the distribution of leaves on a plant, it is not because their symmetrical distribution of elements mimics an a priori ideal, but because this configuration provides an adaptive advantage in the natural selection process. 61

Gregory Bateson observed how this natural process is categorically distinct from how humans create form, architectural or otherwise. He argues that there are two parts to the forming process: the epigenetic and the evolutionary. In the former, there are forces and impacts. In the latter, there is information and differences. He notes that “the essence of epigenesis is predictable repetition, the essence of learning and evolution is exploration and change.”6 In epigenetic growth, formal effects occur automatically and are the results of “concrete conditions or events—impacts, forces, (...) and energy exchange.”7 In evolution and learning—“the world of communication and organization”—information has to be consciously preserved, taught, and relearned.8 In this realm, formal change is produced by informational “differences” rather than force.9 The passing on of cultural knowledge is a hybrid process, “it must attempt to use the phenomena of learning for the purpose of repetition.”10 In other words, it must produce the effects of epigenesis, i.e. the consistent passing on of useful information, via a distinct method. If information, or form, is not useful, it will disappear. This sequence is the same in natural and cultural systems, where new forms and ideas are accepted or rejected by the context in which they emerge from. Symmetry is one idea that has survived. Having entered into natural and cultural realms, it has proven itself useful at preserving and passing on information from one generation to the next. It does this precisely because of its ability to remain invariant in the face of multiple transformations. Embryological Form Following Bateson, we can say that invariance is an indication of the absence of difference in a system. In biological bodies symmetry can also be a sign that information is missing. Gregory Bateson’s father, the biologist William Bateson observed that mutations occurring during the embryological process often produced symmetrical forms where, ordinarily, they would have been asymmetrical ones. He concluded that this was due to a lack of genetic information. For example, the presence of an extra leg in a beetle or an extra thumb on the human hand will always create a symmetrical relationship with its neighboring appendages. This led him to conclude that the basic biological default, or uniformed, state was symmetrical. This means that, instead of being a marker of higher orders, symmetry is a sign of missing information. Broken symmetry illustrates when a system is functioning properly, and asymmetrical forms are more complex than symmetrical ones. That the former required more information than the latter would prove to be paradigmatic in an age where information and 62

information technology would be the source on new architectural forms. When Greg Lynn referenced William Bateson’s work in “The New Novelty of Symmetry,” the latter’s findings were almost a century old. What had changed was the computational capacity to measure and manipulate this law of nature. The order present in asymmetry, long hidden, could now be revealed, as the increased speed of information processing made it possible to understand and manipulate this more intricate order. Lynn’s work investigated how these insights and tools can be used to create architectural form using the same symmetry-breaking processes. In his aforementioned text he illustrates techniques he used to design his competition entry for the Cardiff Bay Opera House. Generic diagrams—made with computer software— illustrating the symmetry-breaking and branching processes found in natural systems are juxtaposed with those showing a similar process used to create the project. Both diagrams and design drawings use an oval as a “primitive” geometric shape. As the oval is divided, repeated, and scaled up or down, a globally asymmetrical configuration, made up of symmetrical elements, emerges. Symmetry is not abandoned. It still outlines the initial condition from which asymmetry will emerge, as well as governing local relationships. The latter occurs when there is no new information put in the system, for example, when two spaces have the same program and size their relationship to one another is symmetrical. Symmetric Matter Whereas Greg Lynn’s point of reference for producing new architectural forms was biology, Reiser + Umemoto’s starting point was geological. In the book, Atlas of Novel Tectonics, their interest is positioned as a move away from geometry. Instead, their focus is on the organizational logics found in the literal matter from which things are made.11 Such a position is not dependent upon or driven by digital tools. However, those tools do make it easier to unpack the complex informational structures that govern the formation of matter, and they make it easier to translate their logic into architectural space and form. Despite their declared de-emphasis of geometry, there is a consistent presence in their work of two and threedimensional diagonal grids. Jesse Reiser has attributed his interest in this configuration to his childhood fascination with the Wellington Bomber. The diagrid also has an architectural pedigree. Buckminster Fuller’s interest in the structural potential of this uniform system is well known and highly influential. Reiser + Umemoto deployed diagrids in a number of proposals in the 1990s and the early 2000s, including their competition entries

for the Yokohama Port Terminal, the Alishan Railway Route, and the Westside Convergence Center. In each case, a symmetrical diamond pattern was transformed into an idiosyncratic one that accommodates different functional and structural tasks. The dynamic between symmetry and asymmetry, between a uniform diagrid and a variety of specific architectural performances is also present in their O14 Tower in Dubai, completed in 2010. The base diagrid has uniform openings with rounded corners. In order to create different lighting and thermal conditions, as well accommodating different views, they needed differently sized apertures. This, in turn, produced specific structural demands, which required smaller openings. They ultimately settled on four sizes of diamond-shaped apertures: smaller ones to handle structural and shading needs, and larger ones for views, ventilation, and daylight. The R+U team developed and ran computer algorithms to create designs. Each integrated and satisfied the multiple demands placed on it. Multiple iterations satisfied the goals that the code was asked to resolve. In parametric design processes such as this, any solution that does not satisfy the requirements would be eliminated. This should not be mistaken as an example of learning as described by Bateson. The criteria for success were given in advance, with the code used to uncover and replicate all the good answers. This process is less successful when it comes to making the evolutionary act of selection. None of the solutions offered by the algorithm were acceptable to Reiser + Umemoto. The ultimate selection required a more random input—namely, the personal sensibility (or aesthetic preferences) of the designers themselves. A few subjective adjustments were made to the digital outputs (“by hand”) to the otherwise objectively equivalent solutions.12 This allowed for a solution that was more aesthetically appealing (and only slightly less optimal at a functional level) to the designers—a requirement that was not a part of the original algorithm. It would be a mistake to conclude that only idiosyncratic, aesthetic decisions qualify as creative and/ or random. The process consistently combines epigenetic and exploratory processes, which are less contradictory than they are complementary, as they combine transformations with invariances. This combination is registered in the final design of the O14 Tower. Despite the irregular distribution of the different opening types, the original symmetrical diagrid is clearly present. The undifferentiated/symmetrical portions of the diagrid serve as the initial and limiting condition from which the specific form emerged. Symmetry may no longer be an end goal but, as a starting point, its power lies in its ability to combine stability and change, to accommodate information without being completely overwhelmed by it.

More Information In an age of almost instant communication, it has been easy to replicate the underlying processes and sensibility found in Reiser + Umemoto’s digital diagrid projects. It remains to be seen, however, if the cultural and historical context into which these techniques and forms have been placed will ultimately accept or reject them. Will they force their environment to adapt to their presence or will their randomness be rejected? Such evolutionary selections cannot be predicted in advance, they can only be tested by more repetitions and explorations, actions that will in turn produce additional transformations and invariances. In other words, they will continue to be defined in symmetrical terms.

1 Greg Lynn, “The Renewed Novelty of Symmetry,” Assemblage 26 (April, 1995), 8-37. 2 Preston Scott Cohen, Contested Symmetries and Other Predicaments in Architecture (New York: Princeton Architectural Press, 2001). 3 Philip Tabor, “Fearful Symmetry,” Architectural Review vol. 173 no. 1023, (May, 1982), 18-25. 4 Philip Ball, Bodies in The Self-Made Tapestry: Pattern Formation in Nature (New York: Oxford University Press, 1999), 77-109. 5 Ibid. 6 Gregory Bateson, Mind and Nature (New York: Dutton, 1979), 44. 7 Gregory Bateson, “Form, Substance, Difference,” Steps to an Ecology of Mind (New York: Ballantine, 1972), 459. 8 Bateson, Mind and Nature, 44. 9 Gregory Bateson, “Form, Substance, Difference,” Steps to an Ecology of Mind (New York: Ballantine, 1972), 459. 10 Bateson, Mind and Nature, 44. 11 Jesse Reiser and Nanako Umemoto, “Matter,” Atlas of Novel Tectonics (New York: Princeton Architectural Press, 2006), 71-162. 12 Jesse Reiser, Public panel discussion for the exhibition The Synthetic Intelligence of Patterns, Graduate School of Design, Harvard University, (February, 2009).

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Reiser + Umemoto, O-14. Dubai, United Arab Emirates, 2010

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12 Perception and Symmetry

Invariant, equivalent, identical, uniform, easy, efficient: these are a few of the qualities associated with symmetry. What unites them are the themes of consistency and utility. Symmetry is stable. Symmetry is useful. Thus, it should come as no surprise that these qualities and the symmetrical figures and operations that embody them are present— sometimes overtly and sometimes covertly—in contemporary architecture. Disappearing As the proceeding chapters have documented, while its definition and reputation has waxed and waned, symmetry has had a constant presence in architectural thought and design. Sometimes symmetry is so pervasive that it seems to disappear. The ability of symmetry to hide in plain sight has also been recognized by scholars who study the physiology and psychology of vision.1 They have found that it is so predictable and so easily perceived that symmetrical figures require less effort to detect than asymmetrical ones. As art historian Ernst Gombrich noticed, once recognized symmetry provides no new information for the eye’s “break-spotter” to engage, and so it fades into the perceptual background.2 As with William and Gregory Bateson’s biological analysis, optic symmetry is an indication that something is missing. The mathematician and psychologist Michael Layton has studied how people respond to the objects they encounter in their perceptual environment. Using empirical methods to observe people’s vision and behavior, and group theory to study the logic of regular and irregular form, he has concluded that symmetry is a device for erasing memory.3 He maintains that while asymmetries are perceived as being transformed over time from an originally symmetrical state, the processing of symmetrical form does not. This is because symmetry is understood as being in a permanently fixed state. In other words, it has no history, and as such, there can be no memory of where it came from.4 In contrast, asymmetric figures do have a history, and our perceptual apparatus retraces the steps that they took from their original symmetrical configuration. For example, when we see a rotated parallelogram, our mind subconsciously rotates it, straightens it into a rectangle, and then shrinks it into a square. In Layton’s words, “the mind extracts the past history that produced a shape—i.e., the sequence of causal forces that produced the shape.”5 Once we get to the square the process stops. We don´t (subconsciously) inquire as to where the square came from. According to Layton’s research, we accept it as having always being that way. Following Layton and Gombrich, symmetrical form has the paradoxical quality of being easily recognizable yet easily forgettable. It is so efficient at communicating its unchanging nature that it does not demand our full attention to understand it. It is precisely the capacity to be both recognizable and unchanging that may help explain the intentional re-emergence of symmetry in contemporary architecture. In an age where differences, complexity, and variation can so easily be produced and reproduced, symmetry’s stability and simplicity appears as a rare and useful commodity. 65

There are at least two ways in which symmetry is deployed today. One uses it as a figural limit that contains and contrasts itself with otherwise loosely organized shapes and spaces. The other sees symmetry less as a mode of organization than as a self-conscious sign that provides projects with an immediately recognizable, yet still elusive identity. While both modes can be found in a variety of practices, the former is more often found in those whose intricate forms are made using digital design software, while the latter strategy is more graphic and can be seen in what could be called the postdigital or neo-postmodern work. Tension The competition drawings for Foreign Office Architects’s Yokohama Port Terminal (1995-2002) show a series of smooth, undifferentiated interior spaces and exterior forms. Along with the diagram depicting multiple circulation loops, they are all asymmetrical. Renderings and models of the exterior, however, reveal a bilaterally symmetric condition along the building’s long axis. Looking at the project’s plans and sections, this symmetry is even more prominent, albeit with a series of breaks within it. The cross sections also reveal an underlying reflective symmetry, with slight changes in the center point from one section to the next. In other words, the line of symmetry moves or changes to accommodate specific spatial and programmatic needs.6 If one were to look at each section individually, one might conclude that the building is quite static. However, if one looks at them as a series, the history of the project emerges. In short, while one experiences a sequence of locally asymmetrical spaces, they are made possible by an underlying symmetrical structure. The axial symmetry that one finds in these sections comes out of a solution to a construction problem, or rather to a spatial-construction problem.7 Making the cross sections symmetrical, but with different profiles, allowed the otherwise fluid forms to be easily built up from a kit of (relatively) simple, symmetrical elements. Here symmetry provides the stable armature in which fluid and complex spaces and experiences emerge. In other words, while symmetry is present in the physical and organizational structure, experientially it disappears. The office of Young & Ayata also juxtaposes symmetry and asymmetry, but they use symmetry in a more direct way. In both their elegant conceptual drawings and their designs for buildings a symmetrical figure is often used to create a fixed border, in which clarity is not subsequently eroded or transformed despite it being filled with an intricate assortment of forms, spaces, and programs. Projects like the Dalseong Gymnasium (2014), the Busan Opera House (2011), and the Bauhaus Museum (2015) establish a clear relationship 66

between symmetry and asymmetry. The drawings for the Dalseong Gymnasium show a pronounced contrast between the reflective symmetry of the perimeter and main spaces and the formless impression on the landscape. In plan, the reflectively symmetric entry portals and the almost symmetric circulation paths are juxtaposed with the seemingly formless outdoor spaces and the reflected ceiling plan. An observation Young & Ayata make about their drawings applies equally to their architecture: “Symmetry is used here to fix and stabilize the figural representation (…) The figures bend and layer, sometimes twitch and wiggle under their symmetrical pinning (…) although the global gestalt is one of pure symmetry, as one looks at the images longer it becomes clear that there is no element that is purely symmetrical.”8 Symmetry serves to frame and control, not eliminate, the random. However, unlike FOA’s Yokohama Terminal, where the symmetry is hidden in the structure, Young & Ayata forcibly foreground the visual and tactile encounter with it. The tension between parts that are easily recognizable—and thus quickly fade to the background—and the more active ones that demand one’s attention illustrates once more Layton’s and Gombrich’s analysis of the inherently dialectic relationship between symmetric and asymmetric form. Recognition Another trend in contemporary architecture is the use of immediately identifiable architectural tropes, including symmetry, which, combined in unexpected ways, produce uncanny effects. MOS has designed a number of houses that aggregate symmetrical parts to create figural effects at once graphic and ambiguous. For example, the gable-ended modules used in the design of their Element House (2014) in New Mexico appear as bloated versions of a child’s drawing of a house.9 The combination of this well-known profile with the unexpectedly puffy sensibility—created by the thick structurally insulated panel construction system—makes for an oddly cartoonish effect. The modules are triangular in plan and, when aggregated, produce a combination of rotationally symmetrical patterns at the center of the organization and irregularly shaped appendages at its edges. Given its modularity and its clear iconographic agenda, the effect is surprisingly chaotic and disorienting. Despite the local symmetries, there is no clear hierarchy and no one place to focus one’s attention. And, in Layton’s terms, although the modules are always in a symmetrical relationship with their neighbors, the overall configuration allows one to easily image the sequence of events that produced the overall asymmetry.

One also finds multiple symmetries present in their design for a House 10: House with a Courtyard (2017). Pairs of reflectively symmetrical wings flank three sides of a square courtyard. The single leg on the fourth side appears poised to have its missing limb added at any moment. As a group, the linear organized legs have been slightly rotated relative to the courtyard, creating an unexpectedly irregular relationship between the two major compositional components. There is a tension between symmetry and asymmetry in the elevations as well. The gable-roofed wings terminate in a symmetric, iconic shape that says “I am a house.” But, there are five of these, each with a different orientation, so that no two can be seen frontally at any time, undermining any simple axial reading and making it surprisingly difficult to find a focal point on the three facades. While no two sides of the house are the same, there is seemingly no hierarchy between them. Again, even though multiple symmetries are present, their relative arrangement to one another creates a global asymmetric condition that allows one to easily imagine the “history” of the project’s formation process. The symmetry in the Cut/Fill housing proposal (2016) by Central Standard Office Design is more easily detected. The design diagrams reveal the sequence of operations that turned five typical Chicago housing lots into a 4 x 4 grid that are subsequently sheared in plan and section to produce a scheme that is rotationally symmetric in two and three dimensions.10 Each of the quadrants is symmetrically subdivided into four more zones. Each zone accommodates four L-shaped buildings. Each of the sixteen “houses” has a pitched roof on one side and a vaulted roof on the other, with an optically ambiguous roof connecting the two. The combination of easily recognizable references—“this is a gable;” “this is a vault;” “this is a house;” or “this is symmetrical”—is combined with a massing strategy and color scheme that blurs the line between sold and void, and between one unit and another. There is enough symmetry to make it appear stable, but enough asymmetry to allow one to see the design’s history. As in any good symmetrical figure, identity and invariance are equally and ambiguously present. Conclusion The presence of symmetry in the work of these diverse contemporary practices presented illustrates the lasting and elastic qualities of symmetry, and is evidence of its continued relevance, or more accurately, of its usefulness. Whether helping to integrate a structural idea with a spatial one, providing a stable background for exuberant compositions, or giving graphic cover for otherwise complex objects, symmetry is still an effective architectural device. As documented, symmetry may not have the status or the job that it once had—in math, in

science, or in architecture—but it does continue to do work. The question is how does it remain relevant. Is it because it is both an abstract idea and a literal organizer of matter? Is it because it can connect mental with physical processes? Is it because it is easy to use and recognize? Yes, yes, and yes. Symmetry may be sometimes simple, but it is also surprisingly plural. As such, in architecture, and elsewhere, it will continue to transform while remaining invariant. This is the symmetrical paradox: symmetry changes and symmetry endures.

1 Ernst Gombrich, The Sense of Order: A Study in the Psychology of Decorative Art (London: Phaidon, 1979). Michael Leyton, Symmetry, Causality, Mind (Cambridge: MIT Press, 1992). Michael Leyton, A Generative Theory of Shape (Berlin: Springer-Verlag, 2001). Michael Leyton, Process Grammar: The Basis of Morphology (Berlin: Springer, 2012). J. J. Gibson, The Perception of the Visual World (Boston: Houghton Mifflin, 1950). Rudolf Arnheim, Art and Visual Perception: A Psychology of the Creative Eye (Berkeley: University of California Press, 1974) [1954]. Rudolf Arnheim, The Dynamics of Architectural Form (Berkeley: University of California Press, 1977). 2 Gombrich, The Sense of Order, 121. 3 Michael Leyton, “Group Theory and Architecture,” Nexus Network Journal vol. 3 no. 2, (2001), 43-45. 4 Ibid. 5 Ibid., 44. 6 Foreign Office Architects, “National Glass Center, Sunderland/ Redevelopment of Cartuja Island, Seville/Yokohama Port Terminal,” AA Files no. 29 (summer, 1995), 7-21, in Albert Ferrer (ed.), The Yokohama Project: Foreign Office Architects (Barcelona: Actar, 2002). 7 Alejandro Zaera Polo, “Roller Coaster Construction,” Verb: Processing no. 1 (2001), 12-18. 8 Young & Ayata, “Symmetry Series” (2013). Accessed at http://www. young-ayata.com/funnyhairysymmetry 9 Mos Architects, “Element House,” Log 29 (Fall, 2013), 66-75. 10 Kelly Bair, “Cut/Fill,” MAS Content 29 (Spring, 2016). Accessed at http://www.mascontext.com/issues/29-bold-spring-16/cutfill/

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Matrix of symmetry analysis. Plans of projects with mirror reflections, translations, rotations and glide reflections. From left to right and top to bottom: Maison Carrée, Pazzi Chapel, Santa Maria Novella, Villa Capra, Barrière de la Villete, Altes Museum, Darwin D. Martin House, Glass Pavilion, Villa Savoye, Crown Hall, Teatro del Mondo, O-14

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The plan emerges from the construction of an ellipse by two pairs of circles in six iterations, of the following systems: lantern dome, lantern apse, main dome, main apse, minor apse, and façade. The subdivision of circles distribute columns, chapels, altars and alcoves in each iteration. Francesco Borromini, San Carlo Alle Quattro Fontane. Roma, Italia, 1599-1667. Drawing by Carolina Telo

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Never Even Manuel Mensa

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Two pairs of symmetrical circles, one to the access, the other to the apse of the church, establish scalar variations by the connection of its iterative subdivision. Francesco Borromini, San Carlo Alle Quattro Fontane. Roma, Italia, 1599-1667. Drawing by Carolina Telo

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Longitudinal plan lines of structural and circulatory components. The angle of inclination in respect to the longitudinal central axis is lower than 45°. Transversal structural lines. Segments connecting the intersections of longitudinal and transversal lines and the longitudinal central axis. Interpolation of these segments. Foreign Office Architects FOA, Yokohama Port Authority Terminal. Yokohama, Japan, 2002. Drawing by Valeria Ospital

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A Symmetrical Story Santiago Miret

Since it is not contemplated from strictly operational notions, symmetry implies broader conceptual constructions that define positions regarding the discipline of architecture as a whole. Symmetry organizes the essential attributes of objects. Its presence involves the generic, the undifferentiated, the primitive, and the synthetic. Given that symmetry is a form of order, control, and unity, its appearance requires the consolidation of an object. The ability to visualize symmetry in an organization allows us to understand it as a whole. It reassures us. It gives us sternness by letting us understand the whole. However, the symmetry that David Salomon presents us is not reassuring. In fact, it is the opposite: disturbing, omnipresent, and elusive. As soon as we think that we have brought it under control, it sneaks back and filters through the slits of control that it produces, generating uneasiness, uncertainty, and perplexity because it is not symmetry as a practical tool that Salomon is presenting us, but as a complex construct that includes the technic and overcome it. This research is not intended to develop a historicist notion of the idea of symmetry as a technique, but to present it as a specter, which operates architecturally at many levels, not only in the generative process of the project but also in its form, its idiosyncrasy, its reason of being, and its inspiration. Symmetry is an unassailable presence that has determined the historical development of architecture. In this pursuit, though, Salomon does not restrict the idea of symmetry to the domain of architecture. He starts from the hypothesis that architecture is informed by other disciplines, expanding its register. In this context, symmetry is not restrained to the known geometric operations of reflection, translation, rotation, and glide reflections, for there is also a conceptual manifold of scientific, biological, physical, sociological, mathematical, syntactic, semiotic, informational and psychological notions and approaches to its concept, all of them deploying a novel understanding of its reaches. That is one of the virtues of the book: its multiplicity, as its title indicates, since these notions are presented as stratifications that need to be fully deciphered. Throughout the book, the thesis is constructed that symmetry has had a recurrent presence in the architectural project. The digital turn that began in the 1990s and the contemporary post-digital turn seemed to neglect it, but the complex configurations developed by digital architects cannot escape repetition and geometric reflection techniques. Modernism, in all its versions and protagonists, is marked by the use of symmetry both as an explicit technique of geometric generation and as an implicit manifesto in search for order and control. Renaissance is perhaps the period in which symmetry was most widely experimented, expanding its register and improving its scope with respect to its effects and modalities. It was during the classic period that symmetry was strengthened as a canon and, since then, it was consolidated as a legitimating beacon, inevitable and permanently present. The thesis presented in this book is that an architecture of symmetry can allow complexity to coalesce without being simplistic or dull. Expanding the register of this notion allows us to approach the concept with seriousness, and ask ourselves about its pertinence and contemporaneity without falling into contingent trends widely spread around the idea. 75

Salomon argues that symmetry is not order, but something beyond, uneasy to portray. Symmetry is the structure that operates among things, but, unlike topology, its rules cannot be determined from form, but from operations of differentiation. If symmetry implied order to the classics, today it is the infrastructure that allows us to think new orders: orders that go beyond theory, methodology, or sheer technique, orders that are not canonical but that can override the canon, dynamic, unstable orders, appropriable by whomever is attentive to its their intangible presence. In The Renewed Novelty of Symmetry, Greg Lynn states that symmetry has a direct relation with information: the more information, the less symmetry, and vice versa. However, what we here learn from Salomon is that more information does not necessarily imply less symmetry, but a transformation of symmetry. Information is the factor of complexation of the substrate that provides a symmetrical structure. Symmetry in architecture, unlike in the organic entities realm, does not necessarily represent the zero state of a primitive structure. Rather, it compels us to see beyond and operate from within a substrate that identifies variations, which can be manipulated. Here, the relation with homogeneity and heterogeneity becomes key. Like Lynn, Salomon understands symmetry as an opportunity for the generation of novelty, and proposes that the idea of novelty, as naturally linked to the heterogeneous, is outdated. Rather, when novelty operates between gradients of differentiation, distorting values here and intensifying operations there, it can become more than just a tool (flattening its power of transformation) but as a modality of organization that has repercussions on new notions regarding control, related to contingent yet systematic protocols of action. To understand symmetry as an agent of command proliferation rather than a global strategic rule from above, is to open the possibilities to active and continuous transformation of organizational form. Going back to the title of the book and its reference to multiplicity, that is, to the presence of symmetry everywhere, it is important to account for symmetry not as something immeasurable, out of our possibility of understanding. Symmetry is a modality of organization of the form that operates materially, consolidating its internal consistency through the scientific disciplines. Lynn’s argument implies a retrocession to magical thought, where the primitive conception of the virgin world is absolutely symmetrical and where only the artificial intervention of the accident is capable of transforming it. The present book insists on constructing the argument by which symmetry does not imply an untouchable universal truth of mystical characteristics that escape reason, but that is a material construction with specific not fixed organizational aims. Finally, the transdisciplinarity ubiquitous in the present investigation, does not seek to find foundations for symmetry alien to architecture, but humbly learn the arguments, reasons and values that other disciplines have built towards it. This perspective seeks to expand the material corpus of the conceptualization of symmetry as a modality so that its conceptualization can overcome the idea of geometric control, towards a notion of symmetry as a source of disciplinary novelty.

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Longitudinal lines interpolating the longitudinal plan lines and the longitudinal central axis. Foreign Office Architects FOA, Yokohama Port Authority Terminal. Yokohama, Japan, 2002. Drawing by Valeria Ospital

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Acknowledgments

Symmetry, The One and the Many, is the result of a graduate seminar conducted by David Salomon at the Maestría en Historia y Cultura de la Arquitectura y la Ciudad of the Escuela de Arquitectura y Estudios Urbanos of the Universidad Torcuato Di Tella. The seminar took place during the spring semester of 2015, and counted with the participation of the following students: Luciana María García Campos, Mariano Galíndez, Agustina María González Morales, Agustín Llobera, Florencia Mischelejis, Rosario Pastore, Santiago Peña, Sofía Perazzolo, Lucía Rush, Carlos Sala, Nicolás Vaz, Inés Verna, Magdalena Viegener, Lucía Paula Viviani. We would like to thank the students for their dedication and commitment, and extend our gratitude to Andrew Pringle Sattui for the production of the symmetry models, to Anna Font, Manuel Mensa, Santiago Miret, Valeria Ospital, Carolina Telo, and Fernando Yabén for their drawings and essays.

Universidad Torcuato Di Tella Escuela de Arquitectura y Estudios Urbanos Campus Alcorta Avenida Figueroa Alcorta 7350 Sáenz Valiente 1010 Ciudad de Buenos Aires Argentina

La Universidad Torcuato Di Tella de lucro fundada en 1991, heredera del espíritu innovador industrial de la fábrica SIAM Di Tella (1910) y de la visión artística y social de vanguardia del Instituto Torcuato Di Tella (1958). Su misión es la formación de las nuevas generaciones empresariales, políticas, académicas, sociales y artísticas de nuestro país y la producción de conocimiento básico y aplicado, en el marco de la excelencia académica, el pluralismo de ideas y la igualdad de oportunidades.

currency, comfort zone, cliché, and commodity. Symmetry

Rector: Ernesto Schargrodsky Ph. D. in Economics Harvard University Vicerrectora: Catalina Smulovitz Ph.D. in Political Science The Pennsylvania State University

of the nothingness of fullness. Symmetry speaks unheard

means for meaningful suspensions, the resonance box that

opposite: singularity. Universidad Torcuato Di Tella Escuela de Arquitectura y Estudios Urbanos Campus Alcorta Avenida Figueroa Alcorta 7350 Sáenz Valiente 1010 Ciudad de Buenos Aires Argentina

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