07 Math and Symmetry

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

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

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 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”