# e-book Mathematics of Space (Architectural Design, July-August 2011)

Until that time, arithmetic and geometry had been a constant reference for the architect, as foundational knowledge as well as practical tools, as empowerment and as incentive for restraint, as a means of control and standardisation as well as a guideline for surprising invention. Galileo Galilei, Animal bones, from Discorsi e dimostrazioni matematiche intorno a due nuove scienze, The example is used by Galileo to illustrate how strength is not proportional to size, contrary to what proportion-based theories assumed.

In all these roles, mathematics had a strong link with spatial intuition. Arithmetic and geometry were in accordance with the understanding of space. This connivance was brought to an end with the development of calculus and its application to domains like strength of materials. First, calculus revealed the existence of a world that was definitely not following the rules of proportionality that architects had dwelt upon for centuries.

Galileo, for sure, had already pointed out the discrepancy between the sphere of arithmetic and geometry and domains like strength of materials in his Discorsi e dimostrazioni matematiche intorno a due nuove scienze Discourses and Mathematical Proofs Regarding Two New Sciences published in But such discrepancy became conspicuous to architects and engineers only at the end of the 18th century.

The fact that some of the operations involved in calculus had no intuitive meaning was even more problematic. It meant that the new mathematics were like machines that possessed a certain degree of autonomy from intuitive experience. A century later, nascent phenomenology would return to this gap and explore its possible philosophical signification. Among the reasons that explain such a gap, the most fundamental lies in the fact that calculus has generally to do with the consideration of time instead of dealing with purely spatial dimensions.

What was the most puzzling, like the so-called infinitely small, were actually elementary dynamic processes rather than static beings. Calculuss most striking results were by the same token related to dynamic phenomena like hydraulics and the study of flows. Later on, calculus would also be instrumental in the development of economic theory by providing the means to study the circulation of goods and capitals. Another reason explaining the growing gap between architecture and mathematics was the new relation between theory and practice involved in the transition from arithmetic and geometry to calculus.

In the past, mathematical formulae were seen as approximate expressions of a higher reality deprived, as rough estimates, of absolute prescriptive power. One could always play with proportions for they pointed towards an average ratio. The art of the designer was all about tampering with them in order to achieve a better result. As indicators of a higher reality, formulae were an Computer simulation is instrumental in the exploration of such complex organisational patterns.

Michael Weinstock, Drosophilia Wing Development, The emergence of small complex anatomical organisations makes possible the emergence of ever larger and even more complex organisations. Complexity builds over time by a sequence of modifications to existing forms. Michael Weinstock, Architectural and urban forms in Mesopotamia The organic property of emergence is supposed to apply to both the natural and the human realms. In the new world of calculus applied to domains like strength of materials or hydraulics, mathematics no longer provided averages but firm boundaries that could not be tampered with.

From that moment onwards, mathematics was about setting limits to phenomena like elasticity, then modelling them with laws of behaviour. Design was no longer involved. Theory set limits to design regardless of its fundamental intuitions.

Theorists like Violletle-Duc or Gottfried Semper are typical of this reorientation. Despite claims to the contrary made by architects like Le Corbusier, this indifference to mathematics was to remain globally true of modern architecture. The Ambiguities of the Present Today the computer has reconciled architecture with calculus. For the first time, architects can really play as much with time as with space. They can generate geometric flows in ways that transform architectural forms into sections or freezes of these flows. But this has not led so far to a new mathematical imaginary.

To put it in the historical perspective adopted here, mathematics appear neither as foundational nor as tools. Various reasons may account for this situation. First, the mathematical principles are very often hidden behind their effects on the screen. In many cases the computer veils the presence of mathematics. This is a real issue that should be overcome in the perspective of truly mastering what is at stake in computer-aided design.

Second, one has the disturbing nature of the underlying mathematical principles mobilised by design. In computer-aided design, one no longer deals with objects but with theoretically. One deals also with relations. This is what parametric design is about: considering relations that can be far more abstract than what the design of objects usually entails. Scripting and algorithmics reinforce this trend. With algorithmics, one sees the return of the old question of the lack of intuitive content of some operations.

This might result from the fact that the polarities evoked earlier have not been reconstituted. In architecture, todays mathematics is about power and invention. Restraint and control through the establishment of standards have been lost so far. The reconstitution of this polarity might enable something like the restoration of an essential vibration, something akin to music. Architects need mathematics to embrace the contradictory longing for power and for restraint, for standardisation and for invention.

To achieve that goal, one could perhaps follow a couple of paths. One has to do with parametricism, but parametricism understood as restraint and not only as power, and also parametricism as having to do with the quest for standards and not only of invention. Another path worth exploring is simulation. Simulation goes with the new importance given to scenarios and events.

In this perspective, architecture becomes something that happens, a production comparable to a form of action, an evolution that lies at the core of todays performalist orientation. Notes 1. A medial surface discretised by parallelepipeds The medial surface represents a class of surfaces that synthesise global topology with local discretisation. There is currently a disjoint between the enthusiasm that is expressed for geometry in architecture and the disparate manner in which it is applied spatially.

Dennis R Shelden and Andrew J Witt of Gehry Technologies here seek to address this by reconnecting theory and practice with developments in modern mathematics. It can be argued that architectures contemporary embrace of the geometries of modern mathematics has occurred derivative of, but largely removed from, the corresponding evolution in the foundational basis of space and shape that these advances propose.

As algorithmic design has emerged through application of a collection of discrete geometric techniques, the contemporary language of form has become a disparate archipelago of geometries with unique topological signatures, collectively instantiated into space but otherwise disconnected from any unifying framework. The project featured here is twofold. First, it seeks to reconnect the theory and discourse of contemporary architectural form to its origins in the development of modern mathematics, and in doing so bring to light the radical implications these theoretical developments offer to the epistemology of form.

From the basis of this emergent theoretical foundation, a framework for the examination of form is proposed that reveals the distinct topologies of contemporary architectural form as aspects of a synthetic and unifying problematic. As a central example, the framework is applied to the oppositions of continuous and discrete topologies, and demonstrates that these apparently contrary signatures can be seen as duals, co-emerging from a common origin.

Space and Shape From antiquity until the present, architecture has been founded on the principles, constructs and, to no small extent, the ontologies of the Euclidean and Cartesian systems. Often used interchangeably, these systems both individually and in concert make specific assertions on the nature of geometry and its relationship to space.

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Euclids Elements1 establish geometry through assertions on constructions of shapes the lines and arcs, their measurements, angles and intersections without directly referencing a spatial medium. They establish shape as construction. The Cartesian system declares shape as an algebraic function on points in real numbered coordinate space Rn.

The fact that the Euclidian constructions hold when described. However, no less remarkably, the Euclidean axioms do not presume or require the presence of any space, real numbered or other, to be complete. The nature of space, in which the constructions occur and the axioms hold, has been debated throughout the history of spatial ontology,2 and specifically whether space is absolute, discrete from geometries it contains, or relational, sufficiently defined by relationships between spatial phenomena.

This containing space is presumed Euclidean: linear, continuous, absolute and singular; there is only one such space in which all shape occurs. Shape grammar theory, as established by George Stiny,4 provides an important counterpoint to this largely pervasive view of design shape as occupancy of point set topology. This work re-prioritises shape over space, and re-establishes an axiomatic system of shape as an algebraic topology of shapes and their parts. As with the Euclidean elements, shape grammars form a complete system of shape description whose closure is independent of any containment space.

While much of the application of this system has been concerned with developing substitution grammars of Euclidean transformations implicitly deployed in the context of a Euclidean spatial medium, the shape grammar system has demonstrated applicability to problems involving non-Euclidean elements and their transformations as well. In the last half-decade, the available descriptions of architectural form have radically expanded beyond the Euclidean to include new geometries: the non-Euclidean, the fractal, the procedural and the parametric.

The existence of such geometries has been supposed over the past three centuries, but prior to digital computation they could be treated only in their most general forms and through their most simplistic examples, Gehry Partners, Walt Disney Concert Hall, Los Angeles, The enclosure detailing connects features from two distinct mappings between 2-D and 3-D space: the ruling lines of the global developable surface, and a patterning of lines in the unfolded surface space, injected back into 3-D space as geodesic curves.

The tangent developable surface The tangent developable surface is a locally continuous surface that has a global singularity at the edge of regression. These geometries are no longer seen as monstrous or pathological, devised to challenge the limits of the Euclidean, but rather as generalisations of the classic geometries, formalisms of utility and applicability to architecture, and indeed of everyday experience.

Collectively they can be seen as positing a view of shape as space; moreover as connections via mappings among a disparate network of Euclidean, non-Euclidean and more general topologies. These geometries occur as R2 x R3 mappings between two or more distinct topological spaces: an intrinsic two-dimensional parametric space, and the containing three-dimensional space outside the surface.

The extrinsic space contains the shape as points of occupancy, while the intrinsic space the space of the surface is the basis by which its shape is described, measured and traversed, and the perspective from which its continuity emerges. Its signature as a continuous surface emerges from both its real numbered Cartesian intrinsic and extrinsic structures, and by the specific coordinate relationships defined across its mapping.

By extension to alternative dimensions, the two- and three-dimensional Euclidean and non-Euclidean shapes including points, lines, curves, surfaces and volumes are described. This mapping is itself a space the product space of the intrinsic and extrinsic and an instance in the space of the family of all similarly structured mappings. This framework extends directly to the parametric, wherein shapes are instances of geometric functions driven from spaces of discrete parametric values. The spaces are not atomic, but in turn disaggregate into subspaces of individual parameters, subshapes and their products.

As shapes aggregate through their combinatorics, so do their individual connected spaces connect into larger networks. Shape exists in and as this network of spatial connection. In the purely digital realm, both intrinsic and embedding spaces are by necessity Euclidian real numbered coordinate. Mathematical models from Institut Henri Poincar, Paris, France, late 19th century Models illustrating three possible cubic ruled surfaces. These surfaces each demonstrate remarkable singularities in the form of self-intersections. The geometry and formal structure of the self-intersections may not be immediately obvious from the standard analytic representation of these surfaces.

However, this manifold structuring applies in a formally rigorous manner when extended to a much wider spectrum of embedding topologies. Manifolds can be embedded into any topological space where locally continuous measurement by real numbered coordinates holds. This broad class of admissible embedding topologies includes the affine, vector and tensor spaces among many others. We can in fact rigorously consider manifolds that bridge from the digital into the worldly topologies and transformations of physical space.

The measurements and mappings, historically the realm of craft, are now conducted through increasingly sophisticated machines providing direct and continuous transformation between numerical coordinates and physical location. What emerges is a view of space and shape that is a radical expansion of the Cartesian system. Space and shape are no longer distinct, but synonymous. Shapes emerge from, within and as a system of spatial networks of heterogeneous dimension and signature, no longer inert but active and dynamic, continuously created, connected and destroyed by design.

Within this system, Euclidean geometries and spaces take a natural place as the restricted class of linear transformations in the more general class of differentiable mappings. The Euclidean re-emerges locally within the network as a regularising structure wherever Cartesian product spaces of independent real variables and their linear transformations occur. In the context of such an expanding constellation of interconnected space, the notion of a singular and privileged containing space loses hold as a necessary or even relevant construct.

Special relativity dictates that the container view of space cannot hold, but we do not need to recourse to the very large, very small or very fast to witness the efficacy of the relational spacetime view. At the scale of human experience, we may arbitrarily select a containing Euclidean space worldview of specific dimensionality, measured by a specific coordinate system, etc, and normalise shape by its embedding into this arbitrarily privileged frame.

But this reductionism erases the. This evolution of the spatial fabric presupposed by contemporary geometry, of shape as space, and of space as relational, localised and connected, is arguably the central ontological advance of contemporary form-making and associated architectural description. Most significant for design is the migration of forms locus, which emerges not simply as the occupancy of any specific Cartesian space, nor its numerical description, but resides in and as the connection between spatial frames the intrinsic and the extrinsic, the Euclidean and non-Euclidean, the continuous and the discrete, the digital and the physical.

Continuous Maps and Their Epistemic Limits The new geometries are uneasily classified as either continuous or discrete, a dichotomy whose simple and axiomatic distinction in the classical view between the real and the integer no longer so simply holds. The non-Euclidean shapes are intrinsically continuous, but can demonstrate folds and singularities in the embedding space. The procedural shapes of subdivision surfaces may be extrinsically continuous but arise out of discrete intrinsic operations.

Parametric shapes may be both continuous in state space and their extrinsic instantiation, but may exhibit singularities in either, and no longer maintain any topological similarity between intrinsic and extrinsic views. The impulse to equate continuous maps to complete definitions of architectural elements is compelling because it has proven so germane to problems of constructability, rationalisation and parametric control Figure 2. If one understands the surface as a purely functional space, problems of design rationalisation become more precise and tractable.

Unfortunately the lure of such problems has kept recent applications of mathematical approaches within architecture focused on technical problems of surface resolution and Amiens Cathedral, Amiens, France, 13th century left and opposite top: As systems of one logic are sequentially propagated along surfaces of a divergent logic, ruptures inevitably occur. Such ruptures are common in Gothic design, for example here in the vaults of Amiens Cathedral. Fascinating as issues of surface differential geometry are, the more fundamental formal issues play out at the scale of the global surface that is, how surfaces enclose and partition spaces, how one circulates among them, and resolution of spatial connectedness and separation.

How can we bridge the gap between the mathematics of continuous discretisation and the syntax of architectural spaces? To answer, we must consider why the functional definition of surface geometry has become so distinct from the topological one. It follows from the axiomatisation of mathematics in the 19th century.

During this time the mathematician Felix Klein was preoccupied with the question of unifying a multiplicity of theoretical geometries. Kleins ambition was to classify the varieties of surfaces through the sets of maps or functions that left these surfaces invariant. Kleins proposal for a unified classification of surfaces through their nested invariant functional meta-behaviour is known as the Erlangen programme. While the Erlangen programme opened profound new understandings in the mathematics of geometric group theory, it also effectively divorced geometry from spatial intuition: since facts about commutative algebra became facts about the surface itself, spatial visualisation became superfluous to the mathematical study of surfaces.

By distancing geometry from visualisation, Kleins Erlangen programme lay the seeds of the divorce between geometry and design. This split is fundamental because continuous mathematical functions, algebraically expressed, often obscure rather than reveal spatial or topological facts. The local, closed, analytic representation of the surface does not communicate its key spatial properties the moments of self-intersection, self-tangency, the way it partitions space.

Consider, for example, the functional expression of the tangent developable Thus for designers, a purely functional approach obscures as much as it illuminates. What designers need are descriptors of shape and space that encompass, but move beyond, notions of functional continuity to include singularities, ruptures and exceptional conditions. What we need are richer descriptions of topology that embed also implicit logics of construction and concurrent local discretisation that emerge organically from the global topology itself.

Of course the tools we have are new, but this synthetic ambition for deductive relationships of local parts to global whole is a fundamental tension within the project of design itself Figure 4. Continuous Maps and the Topological Exceptions of the Gothic The dialectic between global continuity and local discontinuity forms a clear thread within design history, emerging from material laws of aggregation and deformation.

Certain designers strive for perfect and unobstructed continuity, and others for punctuated discontinuity. The tension is illustrated in the topological exceptions of Gothic vaults moments where continuity is frustrated by ruptures in the logic of module propagation itself. Gothic builders attempted to build complex vault surfaces with modules bricks with no explicit mathematical relationship to the vault geometry.

The divergence of each successive row of aggregation from the ideal design surface accumulated to the point of system rupture the necessary introduction of a distinct material and module. As the logic of discrete material confronts the desire of global continuous expression, the need for more integrated descriptors of shape emerges Figure 5.

Semantic Descriptors of Global Ruptures Mathematics seduces with its promise of rigorous synthesis to otherwise contradictory systems of rules. To control the logic of ruptures, architects need a semantic set of descriptors that are not merely parametric but topological, which represent,.

Skeletal subdivision of Paris housing, late 19th century Curve skeletons arise naturally in discretisations and packings, and as such recur in unexpected contexts. For example, the packing of regularly shaped apartments into irregular block shapes in Pariss urban plan induces a curve skeleton that is legible in the plan even if the original designers did not intend it. Curve skeletons of closed curves The curve skeletons below of various closed curves above. The skeleton metaphor is apt; the skeleton represents a sort of minimum distribution or circulation network for the interior of the shape.

It also represents the locus of singularities as a boundary is continuously and uniformly offset. This shape descriptor, which is broadly applicable in design analytics, was first described by Harold Blum in his paper A Transformation for Extracting New Descriptors of Shape.

Architecture is the design of a felicitous relationship of parts to whole, a synthetic project of multi-objective invention. The promise of mathematics is that those diverse relationships and constraints can be made conceptually or notationally explicit, and their manipulation can be precise. Medial surfaces generated from sets of curves The surface is generated so that there is always a wall between two distinct curves. Remarkably, the logic of discretisation for these surfaces follows directly from the diagram of their circulation, namely the curves here indicated in red.

They could in fact be seen as generalisations of hyperbolic paraboloids. Thus one may generate designed spaces of a given circulation logic that, at the same time, are also simply discretisable into flat panels. In addition, the discretisation has a direct connection to the implicit logic of self-intersection of the surface.

In this context we mention two constructs the curve skeleton and the medial surface that suggest generative topological tools and semantic descriptors of shape. Computer vision scientists began the search for such semantic descriptors to distil shapes to their computer-readable fundamentals. In the s, 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.

During the s, 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. This second, encoded shape often indirectly revealed features of the first shape that were difficult to detect directly. Blum called his shape descriptor the curve skeleton Figure 6. Instead of operating on the functional notation of the shape, it operates on the shape itself, regardless of its notational representation.

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. The urban form of Paris is an example of the surprising uses of the curve skeleton. In post-Haussmann Paris, designers pack regularly shaped apartments into irregularly shaped city blocks. 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.

The solution is remarkable: the curve skeleton, this essential diagram of shape, appears not as a planned structure but as an emergent trace, an inevitable consequence of uniform packing within a non-uniform boundary Figure 7. Curve skeletons can be generalised from planimetric to surfacial constructs. 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.

The medial surface is the precise surface that would induce a given set of circulation paths around and through it. 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.

From wall boundaries the skeleton describes a circulation path through them. Conversely, from a circulation path the skeleton will describe walls that induce that circulation. The curve skeleton thus represents something fundamental about space and circulation, a reciprocity between the singularities that structure both. 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 fact, there is an elegant connection between the global and local forms of these medial surfaces since the joint lines between different surfacial domains extend continuously from one to the next. Medial surfaces thus represent a sort of synthesis of continuity and rupture, in one simple descriptor. With these curve skeleton diagrams and medial surfaces, the topological properties of space connectivity, passage, edge and rupture follow directly from the connective paths of the designed promenade.

These surfaces represent a deductive relationship between parts and whole. 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. In short, we can broaden our vision beyond analysis and generative procedures to design. Conclusion Architecture is the design of a felicitous relationship of parts to whole, a synthetic project of multi-objective invention.

The promise of mathematics is that those diverse relationships and.

## Epub Mathematics Of Space Architectural Design July August 2011

A logic of continuous maps is an aspiration towards that comprehensive quality a precise description of the local, topologically global and structurally recursive. 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.

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. The identification of these ruptures, both at the local scale and at the global, topological scale, becomes key to understanding space itself and the descriptive, analytic and designed project of architecture the description of the thing itself, beyond its multiple notational representations. 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.

What is required is a more synthetic approach of globallocal reciprocity, and an embedded logic of mathematical design. GV Leroy, Die philos. Probleme in dem Briefwechsel Leibniz und Clarke, Giessen, Eschewing packaged software, IJP develops its own equations, while its spatial model is underwritten by mathematical surfaces. Legendre describes the pervasive influence of the discipline across the offices output.

IJP with John Pickering, F01 b , London, 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. IJP determined the analytic equations of the transformation and used them to invert, under Pickerings guidance, a pair of ordinary cylinders into an intricate aggregation of partial cycloid surfaces. 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.

There are algebraic limits to this kind of game, as infinite tangents are inadmissible. Hence the term Asymptotic, whereby the box tends towards orthogonality without reaching it. Since its inception in , the London-based practice IJP Corporation has been using a mathematical knowledge model as a blueprint for the design of novel structures. Proposals that are drastically different in size and scope follow the same instrumental premises. Bypassing the conveniences of modelling software in favour of elemental mathematics, these projects share a common basis of analytic geometry.

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. Character Limit The Privacy of the Self. The Jewish : a ranking of the most influential Jews of all time. Le PC : Internet et les emails. The Geeky Chef cookbook : unofficial recipes from Doctor Who, game of Thrones, Harry Potter, and more, real-life recipes for your favorite fantasy foo.

Description Additional Information Reviews 0. Additional Information. Add a review. Message Character Limit I think this helps my understanding of architecture because it sharpens my ability to think of multiple ways in which something can be built, and also increases my ability to quickly analyze different combinations of materials in order to produce the most beautiful and yet efficient solution to a design problem. Any and all of the aforementioned off-the-clock activities gives me a chance to appreciate my surroundings. And some of those surroundings are architectural.

## Architectural Form as Space-time Cell

Working in architecture informs my leisure pursuits more than the other way around; leisure pursuits are just thatâ€”leisure. The past can provide us with knowledge on improving how the future is built. By understanding and observing architecture in other geographies, we can better create a more relevant and purposeful built environment, regardless of destination. Travel provides a means to understand them. I love exploring new cultures with unique contexts and perspectives. As architects, we need to consider and question numerous factors that inform every project.

I look to my travel experiences to help elevate myself as a designer using the many examples of aesthetic and functional inspiration I have encountered. Reading presents a more frequent and accessible means to explore many of these other perspectives from the couch with my feet up.

Cooking and playing piano are my stress relievers. In the kitchen everything else fades away. Piano provides a brief moment of solace and meditation from an otherwise busy day.

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I believe good design must address inequity and is best served by a diversity of thought and experience from the designer s. My ceramic works consist of homewares and architectural features, including handmade decorative tile. My performance work is another way that I practice critical cultural engagement and place making. In my opinion photography and other forms of architectural visualization are vital in conveying the ideas of architect to a larger audience.

Christine joined the team in She is a graduate of University of Houston and Yale University.

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So, whenever I visit a place that I want to remember, I draw it. The labor of looking at a space over and over again to get it down on paper engrains it into my memory for future study. Zach joined DKA in People are asking more from architects than ever before. We should be more than just designers; We should be part system builder, and scenario planner.

We should be more than just draftsman; we should be part photographer, cinematographer, and digital artists. We should be more than architects; we should be part scientist, anthropologist, historian, and psychologist. Katherine joined the team in Elizabeth began working at the office in She is a graduate of the Art Institute of Houston.