This quasi-uniform packing – not unlike the packing of stones in a Gothic vault – must be reconciled with the irregular block shapes of the global urban plan. Hence the term ‘Asymptotic’, whereby the box tends towards orthogonality – without reaching it. A.03 Midterm Proposal GSD2404 LEGENDRE LIBERATORE Cara

_ranges nl0 ,1 .. essentialise and make operative the basic tensions of contrary and contested conditions in design. Continuous maps can fold, intersect with themselves, exhibit singularities; what is continuous from one point of view or notational representation may not be so from another. A blob trying to pass itself off as a box, this curious surface is produced by raising a periodic product to a (very) high exponent in order to deform an ordinary pliant surface into a ‘near box’. The medial surface is the precise surface that would induce a given set of circulation paths around and through it. In analytic terms, all mathematical parametric surfaces form and deform in direct response to numerical relations they hold in their midst. The curve skeleton, and to some extent the medial surface, are nearly self-dual: they can represent either circulation paths, or the surfaces and walls that enclose circulation paths. A logic of continuous maps is an aspiration towards that comprehensive quality

– a precise description of the local, topologically global and structurally recursive. What is more, for medial surfaces there is post-rational surface discretisation; their definition guarantees that they are rationalisable in a quad-dominated way. In the 1960s, interest in syntactic structure for the human senses produced operators that would take any shapes and automatically generate information about their fundamental spatial or topological configuration. 20 ml0 ,1 .. Instead of operating on the functional notation of the shape, it operates on the shape itself, regardless of its notational representation. IJP determined the analytic equations of the transformation and used them to invert, under Pickering’s guidance, a pair of ordinary cylinders into an intricate aggregation of (partial) cycloid surfaces. Applying the logic of the curve skeleton to a collection of curves in space produces a surface wall between each pair of curves in the set; a configuration called a ‘medial surface’ (Figure 8). Skeletal subdivision of Paris housing, late 19th century

Curve skeletons arise naturally in discretisations and packings, and as such recur in unexpected contexts. What is required is a more synthetic approach of global-local reciprocity, and an embedded logic of mathematical design. Mathematics seduces with its promise of rigorous synthesis to otherwise contradictory systems of rules. Figure 8. The promise of mathematics is that those diverse relationships and constraints can be made conceptually or notationally explicit, and their manipulation can be precise. These surfaces represent a deductive relationship between parts and whole. For example, the packing of regularly shaped apartments into irregular block shapes in Paris’s urban plan induces a curve skeleton that is legible in the plan even if the original designers did not intend it. They undulate by peaks and depressions because of a periodic internal makeup, pointing to the presence of cyclical behaviours in one of their three respective antecedents. In this sense, the office turns its back on a dazzling technological design agenda, preferring to work at an infra-technological level, where a symbolic language common to all computational design processes exists. IJP with John Pickering, F01(b), London, 2009

In this project, IJP explored the parametric deployment of a simple homothetical (scaling) transformation known as inversion, to which the Wolverhampton-based artist John Pickering has devoted several decades of work. A single room enclosed by four rigid panels of extruded acrylic, the Art Fund Pavilion may be endlessly reconfigured into a temporary support space for the neighbouring Light Box contemporary art museum for which it was commissioned. The curve skeleton appears in many contexts – as a diagram of circulation, as an aid to smooth subdivision, as an emergent property of circle and shape packings. Parametric surfaces surge upwards because of a genetic antecedent of linearity, a pattern of linear growth exhibited to some degree by one of the dependent relations they quite literally incorporate. This is evident in F01(b) (2009), a collaboration with the artist John Pickering; in the Art Fund Pavilion (2009); Yeosu 2012 Thematic Pavilion (2009); the Shenzhen Museum of Contemporary Art and Planning (2007), with architect Max Kahlen; and the Henderson Waves Bridge (2004-9) with RSP Architects, Planners and Engineers. Our contemporary opportunity is to broaden the connection of mathematics to architecture beyond intensive application of continuous surface functions to a disciplinary project that is more synthetic and spatially specific. Rather than simply consuming software, IJP produces the very material software is made of – raw equations – usually taken for granted under the hood, and hence maintains a far greater amount of control over what it designs and manufactures. Thus one may generate designed spaces of a given circulation logic that, at the same time, are also simply discretisable into flat panels. To control the logic of ruptures, architects need a semantic set of descriptors that are not merely parametric but topological, which represent,

Geodesic Parallel

Operational Definition

Figure 7. The pavilion is designed to be assembled and taken apart in 72 hours by a team of just two. From wall boundaries the skeleton describes a circulation path through them. Proposals that are drastically different in size and scope follow the same instrumental premises. There are algebraic limits to this kind of game, as infinite tangents are inadmissible. , v

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Figure 2. The promise of mathematics is that those diverse relationships and

constraints can be made conceptually or notationally explicit, and their manipulation can be precise. Curve skeletons of closed curves

The curve skeletons (below) of various closed curves (above). In post-Haussmann Paris, designers pack regularly shaped apartments into irregularly shaped city blocks. Computer vision scientists began the search for such semantic descriptors to distil shapes to their computer-readable fundamentals. Bypassing the conveniences of modelling software in favour of elemental mathematics, these projects share a common basis of analytic geometry. IJP Asymptotic Box, Parametric equations, 2004-10

The Asymptotic Box© is an implicit 3-D surface derived by analytic means. 20. Remarkably, the logic of discretisation for these surfaces follows directly from the diagram of their circulation, namely the curves here indicated in red. In short, we can broaden our vision beyond analysis and generative procedures to design. But these maps, limited as they are by the semantics of their symbolic notation, hold the seed of their own rupture, particularly when iteratively applied. IJP, The Art Fund Pavilion, Woking, Surrey, 2009

This lightweight structure is clad with a prefabricated timber lattice of highly variable, side-specific density. During the 1960s, Blum devised a construct which, given a particular shape, would generate a second encoded shape that would distil the key formal features of corners, changes in curvature and general configuration. Medial surfaces thus represent a sort of synthesis of continuity and rupture, in one simple descriptor. Architecture is the design of a felicitous relationship of parts to whole, a synthetic project of multi-objective invention. The urban form of Paris is an example of the surprising uses of the curve skeleton. In particular, architecture can begin to move beyond empty ideological distinctions that rest on notational distinctions, beyond simple dichotomies between pre-rationalisation and post-rationalisation, towards a more profound and codetermined logic of space. The skeleton metaphor is apt; the skeleton represents a sort of minimum distribution or circulation network for the interior of the shape. This shape descriptor, which is broadly applicable in design analytics, was first described by Harold Blum in his 1967 paper ‘A Transformation for Extracting New Descriptors of Shape’. They could in fact be seen as generalisations of hyperbolic paraboloids.