05 Physics and Symmetry

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

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

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 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.

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. 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.

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.