08 Geometry and Symmetry

Bruno Taut, Glass Pavilion. Cologne, Germany, 1914

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.

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 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.