Following Spanish Jesuit Juan Bautista Villalpandus’s detailed reconstruction of the Temple of Jerusalem based on an interpretation of Prophet Ezekiel’s vision and published between 1596 and 1604, the large array of speculations regarding the proportions dictated by God to the builders of the temple stemmed from the belief that architectural design was ultimately an expression of demiurgic power. Actually, today’s situation is quite paradoxical insofar that under the influence of digital tools architecture has never used so many mathematical objects, from Bezier curves to algorithms, while remaining indifferent to the question of its relation to mathematics. With the recent advance of the science of chaos, for example, mathematics eclipsed by the Hilbertian heritage, such as Poincare’s qualitative methods, have

Д.03 Midterm Proposal GSD2404 LEGENDRE LIBERATORE Cara

_rangee nl := 0,1.. Hence, advocated since the 1960s by both philosophers and historians of science, the sociological dimension of mathematics eventually caught up with those thinkers so far least amenable to it. While arithmetic and geometry remained highly useful practical tools, they gradually lost their aura of cutting-edge design techniques. The founders of the journal Experimental Mathematics’4 summed up this trend unequivocally: ‘The role of computers in suggesting conjectures and enriching our understanding of abstract concepts by means of examples and visualisation is a healthy and welcome development.’5

From the 1980s onwards the increasingly pressing theme of the social dimension of mathematics gradually brought the discipline back to earth. regained an important position. Inseparably epistemological and practical, both about power and restraint, the references to mathematics, more specifically to arithmetic and geometry, were a pervasive aspect of architecture. Openmirrors. Mark Lewis, George L Legendre, Richard Liu and Kazuaki Yoneda, Parametric Seed, Option Studio Rising Masses II, Harvard Graduate School of Design, Cambridge, Massachusetts, 2010

The deployment of the arcsine function discretises and ‘straightens out’ a translated solid with a smooth, periodic profile. Another way to put it would be to say that in order to provide a truly enticing foundational model for architecture, mathematics must appear both as synonymous with power and with the refusal to abandon oneself to seduction of power. Nature has replaced God, emergence the traditional process of creation, but its power expressed in mathematical terms conveys the same exhilaration, the same risk of unchecked hubris as in prior times. This use was clearly related to the ambition to ground architecture on firm principles that seemed to possess a natural character. For this is what architecture is ultimately about: a practice, a form action that has to do both with asserting power and refusing to fully abandon oneself to it. A.03 Midterm Proposal GSD2404 LEGENDRE LIBERATORE Cara

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Until the 18th century, following the Renaissance’s preoccupations with perspective and geometry, the relations between mathematics and architecture were both intense and ambiguous. Similarly, the experimental method is no longer regarded as an antinomy to mathematics. of a basis for architecture. Envisaged in this light, proportion possessed a divine origin. Evariste Galois, Ecrits et memoires Mathematiques (Collected Writings on Mathematics), fragment from unpublished manuscript, 1832

Galois made significant contributions to the emergence of algebraic structures in the 19th century. The computer’s contribution to mathematical research was not, however, a new development; the evolutionary pattern of exploring results using the machine was already established and irreversible. It is interesting to note how the quest for restraint echoes some of our present concerns with sustainability. plms. In the 1980s, under the joint influence of technological progress and the increased awareness of the social dimension of mathematical practice, questions emerged in the community that, a few years earlier, would have been deemed totally incongruous. In this second perspective, proportion was no longer about the hubris brought by unlimited power, but about its reverse: moderation, restraint. IA ДіцілИіaЬс Fiimi Сoтім

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I ■*! Mathematics empowered the architect, but also reminded him of the limits of what he could reasonably aim at. What we might want to recover is the possibility for mathematics to be also about restraint, about stepping aside in front of the power at work in the universe. The only thing that should probably not be forgotten is that just like the use of mathematics, sustainability is necessarily dual; it is as much about power as about restraint. When they looked for foundations, 19th-century architects like Eugene-Emmanuel Viollet-le-Duc were more interested by the sciences of life than by the new calculus-based mathematics of their time. Flows on the Lorenz attractor starting from various initial values for ( x, y,z ) and with a = 10,b= 8/3,r =28. Figure from Interactive Topological Drawing (1998) by R_ G Srfiarein www. rrmhta/knolplot/

Figure 13. For the mathematical procedures architects have to deal with, from calculus to algorithms, are decidedly on the side of power. Today the image of mathematics ruled by structures and the axiomatic method is a matter of the past. Mathematics as Foundation

From the Renaissance onwards, the use of mathematical proportions was widespread among architects who claimed to follow the teachings of Roman architect and engineer Vitruvius. The 17th-century French theologian and philosopher Jacques Benigne Bossuet gave a striking expression of this conception when he declared in one of his treatises that God had created the world by establishing the principles of order and proportion.1 In this perspective, proportion was about power, about the demiurgic power of creation, and the architect appeared as the surrogate of God when he mobilised, in his turn, this power to plan his buildings. Philosopher Pierre Caye summarises this second attitude by stating that the aim of architects like Alberti was to rebuild something akin to Noah’s Ark rather than to emulate the Temple of Jerusalem.3 It is worth noting that such a conception was to reappear much later with Le Corbusier and his Modulor, which was inseparable from the attempt to conceive architecture as the core of a totally designed environment that would reconcile man within his inherent finitude.4

Exhilaration of power on the one hand, and the restraint necessary to protect man from the unforgiving power of the divine on the other: the reference to mathematics in the Vitruvian tradition balanced between these two extremes. A new type of demonstration heavily reliant on the computer, such as the four-colour theorem by K Appel and W Haken (1986), called into question the very nature of proof in mathematics: how could a human mind grasp a demonstration which filled nearly 400 pages and distinguished close to 1,500 configurations by means of long automatic procedures? As the great mathematician William Thurston wrote: ‘Mathematical knowledge and understanding [are] embedded in the minds and in the social fabric of the

community of people thinking about a particular topic

In any field, there is a strong social standard of validity and

truth ’6 Similarly, Rene Thom claimed that ‘rigour can be

no more than a local and sociological criterion’.7 When called upon to check Andrew Wiles’s proof of Fermat’s theorem in 2001, several mathematicians recognised that the social and institutional dimension of the confidence vested in them were at least as decisive a factor as the rigour of the verification they could perform. Our contemporary approach to sustainability tends to be as simplistic as our reference to mathematics, albeit in the opposite direction. Historian Joseph Rykwert has shown how influential these speculations were on the development of the architectural discipline in the 16th and 17th centuries.2 Through the use of proportions the architect experienced the exhilaration of empowerment. 360

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iThread,,,! As we will see, the strong relation between mathematics and the intuitive understanding of space was later jeopardised by the development of calculus. Figure 2. The purpose of this article is to contribute to such an understanding. Starting from computer-generated proofs, the discussion soon moved to other types of demonstrations, either unusually expansive or mobilising an extremely complex architecture of conjectures stemming from various domains – the work of Edward Witten on knot theory and string theory (after 1997) comes to mind. A better understanding of the scope and meaning that mathematics used to have for architects might very well represent a necessary step in order to overcome this indifference. Juan Bautista Villalpandus, Chart of the proportions of the entablature of the Temple of Jerusalem, from El Tratado de la Arquitectura Perfecta en La Ultima Vision del Profeta Ezequiel (Rome), 1596-1604

The architectural discipline was supposed to emulate the creative power of the Divinity by following those very rules of proportion that were constitutive of the Creation and that had been dictated by God to the builders of the Temple of Jerusalem. On the one hand the indisputable presence of the built object is synonymous with the permanence of power; on the other the same built object opposes its opacity, and a certain form of instability, at least if we are to follow Peter Eisenman’s analyses, to the sprawling domination of power. Now, one may be tempted to generalise and to transform this tension into a condition for mathematics to play the role

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