Concrete curves: architectural curvilinearity, Descartes’ Géométrie, Leibniz's calculus and Eero Saarinen's TWA terminal

ABSTRACT

Since the 1990s, analyses of architectural curvilinearity have drawn upon Gilles Deleuze's The Fold, a philosophical consideration of the Baroque in terms of Gottfried Wilhelm von Leibniz's calculus and theory of matter. Deleuze's analogy between a curvilinear aesthetic and Leibniz's conception of curvature and matter as forces has influenced investigations of both Baroque and contemporary architecture. I argue that architectural curvilinearity can also be understood in terms of another mathematical conception of curves – namely, René Descartes’ geometric representation of curvature. Like Leibniz, Descartes’ understanding of mathematical curvature resembles his theory of matter. I examine how Eero Saarinen's TWA terminal, a Baroque-inspired building, can be considered in terms of Leibniz's notion of curves and matter as forces as well as Descartes’ conception of curves and matter as extensions in space. This paper offers a new understanding of how Deleuze's ideas in The Fold are applicable to investigations of architecture.

KEYWORDS:

• Deleuze
• Leibniz
• Descartes
• TWA terminal
• curvature
• architecture
• Baroque
• The Fold

AMS SUBJECT CLASSIFICATION:

• 00A67
• 01A45

Acknowledgements

I would like to thank Rob Stone, Rhodri Windsor Liscombe, Charlotte Townsend-Gault, Sherry McKay and the peer reviewers for their suggestions on various drafts of this article.

Disclosure statement

The research in this article is not tied to any financial interest or benefit for the author.

Notes

1. For a cursory overview of Baroque art and architecture, see [1].

2. For examples of architecture inspired by The Fold, see [16,29,31,46]. See also the Architectures non-standard exhibition catalogue (Centre Pompidou, 10 December 2003 to 1 March 2004) which considers CAD-designed architectural form in terms of Abraham Robinson's non-standard analysis as well as Deleuze's writing on Leibniz [8].

3. See Michael Ostwald's analysis of the ‘New Baroque’ and his critique of the assessments that associate CAD-designed architecture of the late twentieth and early twenty-first century with Baroque architecture [36].

4. Greg Lynn introduced Deleuze's ideas in The Fold and its pertinence to architectural theory in the March–April 1993 issue of Architectural Design which he guest edited. In his essay, ‘Architectural Curvilinearity: The Folded, the Pliant and the Supple’, he draws upon Deleuze's interpretation of Leibnizian calculus and philosophy to consider the newly emerging CAD design process being used by architects to generate forms [27].In his 2004 retrospective account of this seminal issue of Architectural Design, he explains: ‘For me, it is calculus that was the subject of the issue and it is the discovery and implementation of calculus by architects that continues to drive the field in terms of formal and constructed complexity. The loss of the module in favour of the infinitesimal component and the displacement of the fragmentary collage by the intensive whole are the legacy of the introduction of calculus’ [32, p. 11]. Lynn also argues that the CAD design process is defined by a type of vitality akin to a Leibnizian notion of matter [32].

5. Descartes' and Leibniz's differing notions of curves illustrate how a mathematical problem can be understood using a number of different methods. Emily R. Grosholz posits that these two conceptions of curved lines complicate the notion that there is only one universal language of mathematics [19,20]. Furthermore, Descartes' and Leibniz's notions of curves illustrate how different modes of representing a problem – namely, whether geometry or calculus is used to represent a curved line – bring a greater level of intelligibility to our understanding of the particular problem.

6. None of these authors address the differences between Descartes' and Leibniz's notions of curvature and why this pertains to the history of mathematics and their respective conceptions of matter.

7. Since classical antiquity architects have used regular compasses to draw circles. Like Descartes, Baroque architects in the latter half of the seventeenth century started using different types of compasses to draw curves that were not circles or segments of circles [18,22]. François Blondel, a French architect and civil engineer, designed a column's entasis using a trammel compass that drew Nicomedes' first conchoid. This compass was similar to those examined in Descartes' Géométrie. He also used pantographs and other special compasses to draw ellipses that are mathematically accurate, unlike those drawn by earlier Baroque architects such as Francesco Borromini. Furthermore, Blondel investigated how arches designed with parabolic and hyperbolic curves could be raised, lowered, widened or narrowed and allowed for more practical applications than semi-circular arches drawn with a regular compass.

8. Lachterman posits that Descartes' method differs from Euclid's since it is defined by invention and is intended to exhibit how the mind discovers the solution to a problem. He notes: ‘While the latter [a Euclidean theorem-proving method] is bent on inculcating the appropriate virtues in the learner qua learner, the Cartesian ethos… concentrates on exhibiting the virtuosity of the artisan qua inventor’ [25, p. 151].

9. Some of Frank Gehry's architectural forms are inspired by folded fabric, and he notes that he was particularly influenced by Gian Lorenzo Bernini's rendering of folds on his marble sculpture of Saint Theresa [9]. This quintessential Baroque artwork was completed in 1652 for the Cornaro Chapel in the Santa Maria della Vittoria in Rome.

10. Even though Leibniz's philosophy differs from Descartes', many of his concepts derive from Descartes' ideas and philosophical method [12].

11. Leibniz introduced the Latin word functio as a mathematical term when he described the tangent, chord, abscissa and ordinate as ‘functions’ of a curve, and the word's modern meaning is implicit in his work [43]. The definition of a mathematical function was refined after Leibniz's death. This includes Leonhard Euler's definition in 1748 which describes how a function of a variable can generate straight or curved lines.

12. The TWA terminal's stark contrast with the predominantly rectilinear architecture designed at the time resembles the marked difference between Baroque architecture and the styles upon which it was based. Francesco Borromini's San Carlo alle Quattro Fontane in Rome (1638–1677) is perhaps the most iconic example of Baroque architecture. Like many other churches built in the region during the previous few centuries, its façade consists of elements derived from Ancient Greek and Roman architecture such as columns with capitals and entablatures with architraves, friezes and cornices. However, unlike these other church designs, the San Carlo alle Quattro Fontane does not have a rectilinear façade but one that is defined by an undulating concave and convex curve.

13. See also Félix Candela's experimentation with concrete shell construction techniques [17].

14. Schiphol's architect Jan Benthem notes: ‘What makes an airport different is that it is not a finished building. It's always being built’ [10, pp. 232].

Additional information

Funding

This work was supported by the Social Sciences and Humanities Research Council of Canada [grant number 752-2012-2578].