Figure 9. To write equations, on the other hand, is to work, if not at the bottom of the pyramid, at least pretty down low, where most of the room lies but little if anything is predefined. uj D*un point donncA hors Ic ccrclc BD mener la li* gnc A В qui touche 1c ccrclc. %—–

a£і^=->/ьіі^>. To explore the terms of a fair and mutual rapport, this collection of essays departs from the habitual emphasis on computational morphogenetic design that has dominated theoretical discourse in the last one and a half decades. This direct mathematical approach to practice has already inspired a new generation of outstanding young architects, including Max Kahlen (see his Rising Masses project on pp 108-11) and Ana Marfa Flor Ortiz and Rodia Valladares Sanchez (The Hinging Tower, pp 11217), who studied under myself at the Architectural Association in London and Harvard University’s Graduate School of Design (GSD), and whose recent final theses are in part reproduced in this issue. The relationship between computation and architectural geometry looms large in the collective argument exhumed in this issue. The mi

nimum value

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□nge (1.. The pattern-oriented design strategies devised by Daniel Bosia’s Advanced Geometry Unit (AGU) at Arup (pp 58-65) provide a strong survey of the incredible creativity and pragmatic application, in different contexts and at different scales, of non-linearity, branching, recursion, complex proportional relationships and mathematical parametricism in practice. Ema Bonifacic, Suk-Kyu Hong and Jung Kim, Figure 11. Ema Bonifacic, Suk-Kyu Hong and Jung Kim, Degenerate Weave, Diploma Unit 5 (Engineering the Immaterial), Architectural Association, London, 2003

The iterative summation of a complex periodic function causes a weave of indicial threads to veer into a hyperactive, disorderly pattern. What distinguishes these segments, qualitatively speaking, is less what they do, than how they do it; and in this sense, the key difference between contemporary architectural geometry and, say, that of Andrea Palladio (1508-80), is not only that we no longer believe in the ideal figure of the circle, but that when we do use it we choose to construct it with Cartesian or polar (parametric) coordinates, rather than with a ruler and a compass. And Adams Kara Taylor (AKT) research associates Panagiotis Michalatos and Sawako Kaijima have devised an application of topology optimisation theory to facilitate an intuitive yet rigorous approach to structural scheme design in the early parts of the design process (pp 66-9). When it comes to solving problems and creating new things, working with mathematical concepts and equations, rather than with the standard modelling software disseminated by the industry, implies a direct recourse to generative symbols and marks. « mm* wmra»*L*s,

Вaм U ргміaa mri<« par I* ftaUr, w CXI Лі» иі. Parametric periodic curves subjected to a ‘thickening’ function that reproduces the inflections of a calligrapher’s brushstroke. Oh divifera deux dcs angles du triangle, comme ACB, ВAС en deux cgalcment par les ligncs GG, lij

Figure 4. ; We:

_rangee nl0,1 .. As Fabian Scheurer and Hanno Stehling demonstrate in

it U Gtomttrit. иaмс ІШ рміaaa aиНиaaiaaat la dmftw и» ж m/}-j mixJj— ffMj. Threads that used to be parallel are now secant (the intersections are flagged in red). com

Amplitude

Minima

Minima

created by phased periodic movement differentiate surface texture. Similar concerns about spatial and organisational patterns, networks and scaling animate Michael Weinstock’s masterful study of territorial growth and self-organisation (pp 102-7), reconciling the latest heuristic paradigms of flow and network topologies with a time-honoured progressivist model of city growth, where the mathematics of space operate at a large scale.3

Another line of argument unfolds in Scheurer and Stehling’s practice, designtoproduction, where such issues are taken at the other end of the spectrum to the micro-level of component fabrication, resulting in ground-breaking structures in collaboration with architect Shigeru Ban. The essential issues of representation, abstraction and reduction of data are still very much there, and must be disentangled through a careful interplay of mathematical and computational resources, driven to a large extent by the unforgiving bottlenecks of machinic performance and physical materiality. Recalling Felix Klein’s and David Hilbert’s famous theoretical consolidations of geometry in the 19th century (also mentioned by Amy Dahan-Dalmedico), Shelden and Witt suggest that the recent breakthroughs of computing may demand a similar approach; that innovation may bring about a further generalisation of historical precedent, that the feedback loop between the development of technology and the history of mathematics is still up and running, and hence that the relationship between them is alive and well, despite the more cautious prognoses spelled out by historian of science Dahan-Dalmedico at the close of her article. In March 2010, Bernard Cache, Amy Dahan-Dalmedico, Antoine Picon, Dennis Shelden and myself participated in a conference on a related subject that I convened at Harvard, during which many ideas presented in this issue were initially discussed. Similar concerns abound in our own work at the London – based IJP (see the articles pp 44-53, 100-1 and 118-21). Structural

requires

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Continuous Broken Primary Threads

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рнккмaІріІМa an шaіax, і»j cl Aгa І. Mathematics or Computation? Among other things, these school projects for a new type of global high-rise building demonstrate the application of periodic equations and Fourier summations to the production of building diagrams while pointing to many new future directions. Line Thickness (2 independent variables, 2 ranges)

f(x)

Non-Parametric Function (1 dependent, 1 independent) Parametric Function (2 dependent variables)

f(x) = Sin((x~2) + x) + и

y(x) = cos((x~2) – x) – x

Thickness to the line can be achieved by adding

□ noth it. Omar Al Omari, Superficial Thickness II,

Diploma Unit 5 (Learning Japanese), Architectural Association, London, 2004

Periodic pleating. In all these projects the material considerations and an intimate knowledge of physical behaviour go hand in hand with a rigorous mathematical formalisation, abetted by the latest computational facilities. V/

Openmirrors. pour faire suite aux developpements de geometrie (Application of the Latest Developments in Geometry and Mechanics to Marine Engineering and all Manners of Infrastructure), Bachelier (Paris), 1822. From

LEcole des arpenteurs ou l’on enseigne toutes les pratiques de geometrie qui sont necessaires a un arpenteur (The Survey School where all Manners of Geometry Needed by Surveyors Are Taught), Thomas Moette (Paris), 1692. Omar Al Omari, Superficial Thickness II, Diploma Unit 5 (Learning Japanese), Architectural Association, London, 2004

Periodic Pleating. 50

G5D2404 I LIBERATORS Cara

Praxis

This issue would not be relevant without an applied survey of what mathematics can actually do for practice. In the 18th century, a further seismic shift towards calculus, detailed by Antoine Picon in his account of the turbulent relationship between geometry and architecture, alienated our profession’s narrowly intuitive dimensional sensibility, leading to a protracted estrangement from mathematics, only weathered today, thanks in part to the advent of computation, which has imbued the relationship with a new lease of life. er range

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on only. As the consolidation of symbolic algebra and its emancipation from the figures of geometry began around the middle of the 16th century, the Renaissance architect (born in 1508) whose villas embody the ultimate expression of classical geometry lies

on the wrong side of modernity by a couple of decades. The simplistic notions that computation constitutes an ‘automation’ of mathematics (a probable side effect of the introduction and popularity of early CAD systems), or conversely that mathematics is only a slower, static expression of computational activity, must be dispensed with. Line studies — Indexical Model

Figure 6. 1 The other half is removed, and the remainder laminated into a self-supporting structure that shares the morphological characteristics of half and whole. Writing forms and processes in this manner requires an authorial mindset. (n) n

unges between 1

and 2. Analytic geometry at work

The applied mathematics of space in the 19th century is no longer concerned with drawn figures: all steps are now written (and calculated). Hence to design with mathematics in 2011 is not to design free of software, a futile if not wholly impossible claim in an age where software is the only idiom available. From Charles

Dupin, Applications de geometrie et de mecanique a la marine, aux ponts et chaussees etc. Mathematics is a broad topic that we must necessarily restrict, for the purpose of brevity and coherence, to the subject of geometry. y(x) =

= yM). Sadly no trained architect since Desargues in the 17th century has managed to contribute disciplinary knowledge to mathematics, but the movement across the (increasingly wide) boundary between geometry and architecture has nonetheless been continuous. Regardless of emphasis (algorithmic or otherwise) and despite the occasional claims to the contrary, the overwhelming geometric implications of any structured design process will not go away – so we may as well discuss them seriously. Modelling software being generally built by ‘chunking’, or consolidating lower-level steps into higher-level ones – like

In all these projects the material considerations and an intimate knowledge of physical behaviour go hand in hand with a rigorous mathematical formalisation, abetted by the latest computational facilities. The difference is not only technical. Ema Bonifacic, Suk-Kyu Hong and Jung Kim, Degenerate Weave, Diploma Unit 5 (Engineering the Immaterial), Architectural Association, London, 2003 Thanks to the multiple intersections, half of the form is becoming potentially self-structural. Or, ki Rjiilmi 1* la aanaala <aal

f «*— a—

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bac la traa Mffar* dwM Mat i

к. Figure 12. Over the past six years it has developed a unique body of knowledge about periodic equations, variously consistent with the constraints of numerical fabrication machinery (some better suited to sheet cutting, others to lamination). Hence the architectural, engineering and computational proposals illustrating our theme (as well as most geometry produced and consumed today in the world at large) can be said to be calculated rather than figured, and written rather than drawn. The algebraists of the 16th and 17th centuries set out the future innovations of analytic and differential geometry, this new geometry of symbols and operators rather than lines and figures, which subtends our contemporary understanding of mathematics in space and, of course, enables the recent innovations of computation. Omar Al Omari, Superficial Thickness I, Diploma Unit 5 (Learning Japanese), Architectural Association, London, 2004

Analytic geometry at work. In any с

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nge (2. Thickness, achieving thickness by adding another range and by derivation based on material behaviour and structural limits.