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Original Research Article

Topological Calculation in Architecture

A Historical and Conceptual Investigation of a Cultural Technique and Its Vectors of Variation

Working paper

Published online: 15 March 2015

Abstract

This paper proposes an analysis of software as an object *in-the-world*. By conceptualising topological calculation and its computation as *cultural techniques*, it is argued that techniques can and should be critically analysed in their own right as *mediators* with the capacity to introduce new and specific modes of ordering space and time-structures by reconfiguring or modelling their relationships differently. Building on previous insights by Luciana Parisi on algorithms in architectural and interaction design (2012, 2013), two lines of enquiry are developed to demonstrate what such an approach might look like: first, topological computation is situated within a larger conceptual space or genealogy of concepts and methods that developed across a variety of fields and disciplines. Second, I examine how these particular concepts and methods are mobilised or converge in specific instances of topological computation as technique or practical rationale, tied in with practices of contemporary computational architectural design and engineering that give it meaning and purpose. Rather than taking an *object* as the starting point of enquiry, the approximation to defining its contours of demarcation as an object in-the-world is taken as the main *objective*. This crosscutting approach to software and culture through techniques invites us to be skeptical of bifurcations – between the arts and sciences, the theoretical and empirical, the abstract or formal and concrete, the digital and physical, the structural and aesthetic, the skeleton and skin – and forces us to rethink concepts and methods based on geometrical assumptions.

Keywords

cultural techniques, topological calculation, invariance, control, digital design, computational architecture

The knowledge that divined, at *random*, signs that were absolute and older than itself has been replaced by a network of signs built up step by step in accordance with a knowledge of what is probable. Hume has become possible.

— Michel Foucault ([1966] 1970, 60, emphasis in original)

The traditional problematic of space, form, and order is informed, reformed, and transformed by the new possibilities and strategies open through technological cross-polination, particularly digital technologies and a new computational relationship to space and time. The theoretical possibilities cut across disciplines and are affected by concepts and mechanisms that have no precedence.

— Kostas Terzidis (2003, 2)

1. Introduction

Rather than an ontological category, software exists as an object *in-the-world*;^{1} it takes form in the world through multiple means and is something that can, and should, be critically analysed accordingly. It is not just a set of machine-readable instructions that direct a computer's processor to perform specific operations; computation not just calculation; and an algorithm not just “a finite set of rules which gives a sequence of operations for solving a specific type of problem” (Knuth 1968, 4). Neither their mathematical underpinnings (i.e., mathematical formalism), nor their physical underpinnings (i.e., empiricism) can provide an exhaustive or sufficient explanation by themselves of what or where these concepts and methods are today, what or where they have been in the past, or what they could potentially become in the future. In order to analyse software as an object in-the-world, a crosscutting approach is proposed which engages with them as “cultural techniques” (Siegert 2013, 2014; Macho 2003, 2008) for ordering what Rob Kitchin and Martin Dodge have dubbed “code/space”, which is neither completely material (i.e., code), nor completely abstract (i.e., space), bur rather mutually constituted (2011, 16). In so doing it becomes a Cartesian exercise in the suspension of judgement concerning the question of what these objects are and thereby “shift the analytic gaze from ontological distinctions to the ontic operations that gave rise to the former in the first place” (Siegert 2013, 48). More specifically, topological calculation is analysed as operative procedures of control in the domain of architectural design and engineering, a field undergoing fundamental change in terms of techniques, rationalities, and practices as a consequence of these new mechanisms. This mode of control works not to prevent, but rather to prescribe. Writing on topological thinking as a new method of quantification of uncertain states, Luciana Parisi argues, “[t]he spatial architecture of points and lines, of discrete and finite states, has been superseded by topological methods of measuring infinitesimal quantities and establishing neighbourhood proximity through the function of the constant invariant” (2012, 172). Furthermore, her observation that topology – “the mathematical study of properties that are preserved through continuous deformations, twistings, and stretchings of objects” (Weisstein, n.d.) – has come to invest all fields of aesthetics and culture (165) is indicative of its profoundness. As a mathematical technique, topology enables one to capture or measure things that would not be discernible physically, thereby expanding vision, and to *reason* with these (i.e., manipulate, work with). But the focus ultimately is neither on topology, nor on computation *per se*; but instead on the computation of topological properties characterised by invariance, and its value or purpose within a particular domain of application. More broadly, this paper asks how we can understand techniques of topological calculation and their computation as particular kinds of mediators. How does software introduce new and specific modes of ordering space and time-structures? What concepts and methods are mobilised to achieve such order? And how do techniques distinguish themselves according to their domains of application? In short, what do topological calculation and its computation bring to the table – how to understand their involvement?

The proposed style of analysis is indebted in particular to Michel Foucault's writing of a “history of the present” ([1975] 1995, 31), Ian Hacking's adoption of some of these concepts and methods in *The Emergence of Probability* (1975), as well as Bernhard Rieder's more recent analyses of Google's *PageRank* algorithm (2012) and Bayesian information filtering (2013). Namely, my analysis is an attempt to trace early ideas, concepts and methods, in terms of their resonance in contemporary applications in the domain of architectural design and engineering. The result is an approach to writing history as fragmented bits and pieces that are mobilised in new ways in the present. As in Hacking's book, the point of doing this is not to recite the history of these ideas, but as a way of understanding the particular space of possible meanings these ideas have today when they enter the domain of application in the form of specific techniques, rationalities, practices. In other words, the “invention” of topological computation^{2} as new procedures of control – the cultural technique in question – should not be mistaken for a sudden discovery, but is rather a multiplicity of “often minor processes, of different origin and scattered location, which overlap, repeat, or imitate one another, support one another, distinguish themselves from one another according to their domain of application, converge and gradually produce the blueprint of a general method” (Foucault 1995, 138). However, although each of these authors favours an analysis from the ground up, going down to the “little dramas” (Hacking 2002, 75), none of the approaches is identical with the one proposed here. For instance, Foucault's and Hacking's analyses are less technologically informed, and none of these authors could have argued for studying software as cultural techniques. This approach to software as an object in-the-world more generally proceeds from a place conditioned by *Actor–Network Theory* (ANT) for instance in asking first “Where should we start?”, to which Bruno Latour has responded “[a]s always, it is best to begin in the middle of things, *in medias res*” (Latour [2005] 2007, 27, emphasis in original). Thus, what matters is the way one chooses to approach the task of tracing the network of connections, however these are defined. While for Latour actor–networks are articulated as traces left behind by actors and their associations, this investigation attempts to trace precisely those connections that do not necessarily show themselves in positively discernible traces. Conceived as such, the objective of the enquiry is to excavate the historical *a priori* of the object; a diachronic approximation to defining its contours of demarcation (i.e., the demarcating criteria) that define the possibility space of this puzzling object.

The changing cultures or ideas of order that emanate from the particular techniques analysed, the *leitmotiv* of this paper, can be conceived as highly formal in themselves (i.e., in mathematics) as well as through their relations or tensions with their concretisation in the domain of application from where they can also be distinguished from one another. Digital or computational architecture, if they are indeed proper umbrella terms for a range of adjunct architectural and engineering trends, practices, rationales and techniques, offer one possible approach to the question of what topological calculation – and its computation by extension – has brought to (the design of) architecture and urban space. The domain of application denoted by these umbrella terms include a wide array of variations, including for instance “responsive architecture” (Negroponte 1975), “responsive environments” (Krueger 1977), “relational architecture” (Lozano-Hemmer 1994), “algorithmic architecture” (Javier Fernández Herrero 2002; Terzidis 2006, 2004), “digitally-driven architecture” (Bier and Knight 2010), kinetic architecture and “robotic environments” (Bier 2011), “parametric architecture” (Schumacher 2009, 2011), “interactive architecture” (Oosterhuis et al. 2012), “permutation design” (Terzidis 2014) and many others still.^{3} Constituting one specific intersection from where changing ideas of order may be identified, this particular field enunciates these multiplicities of disparate processes, mobilising concepts and methods that have been excavated in the first part of the paper to particular ends. The intersections of these trajectories, each bringing their own histories, result in new tensions, shifting boundaries between abstract or highly formal concepts, ideas, or techniques on one side, and concrete instances of those concepts, ideas, or techniques in practices on the other. Architecture – traditionally understood as both the process and the product of planning, designing, and constructing physical structures – is transformed not just significantly, but in fact also fundamentally as it takes up such techniques, rationalities and practices as computer modelling, programming, simulation and imaging to *design*^{4} both virtual forms and physical structures. Moreover, as Tony Kotnik observed, digital architectural design transforms into an exploration of computatable functions (2010). In other words, it engages in the production of new types of (architectural) knowledge through digital design. Thus, architectural engineering is interesting for the way it commits, and has committed itself historically, to a wide range of dichotomies and distinctions that interesect with the question of ordering spaces, which involves both symbolic and material work. Through identifying the operationalisation of distinctions in computational architectural engineering, we start to see contours of a new “epistemic culture” (Knorr-Cetina 1999) coming into being around these new techniques. It is here that the larger *archive* of ideas, concepts and methods that made these new techniques possible (as Hume became possible for Foucault) defines the conditions of emergence for different ideas of order to come into being.

The analysis will proceed on these two levels: on the one hand it constitutes an investigation of topological calculation and its computation as cultural techniques distinguishing themselves according to the domains of computational architectural design and engineering. At the same time, however, it is also proposes a reflexive attitude towards the question of taking software as an object of study. In doing so this paper contributes to two ways: first, to the emerging field of software studies (Berry 2011; Fuller 2008) it proposes an approach to study software as an object in-the-world. Second, to cultural analysis it contributes with a historically-informed account of the emergence and adoption of a cultural technique, enriching the way we think about the question of how culture itself develops or evolves, how concepts and methods are mobilised and made to sustain, disseminate, internalise, endure in the face of other parallel changes. In particular, it is a contribution to a recent stream of cultural theory inspired by the concepts and ideas of topology and topological thinking sometimes referred to as “cultural topology” (Lury 2013; Marres 2012; Shields 2012; Barreneche 2012; Lury, Parisi, and Terranova 2012; Fuller and Goffey 2012; Parisi 2012; Hendrix 2012; Sampson 2007; Flusser 2005; Mol and Law 1994). Finally, to those studying spatial architectural problems, it provides historical awareness of how specific computational techniques, rationales, and practices may gradually come into being and stabilise into more general methods – or what Bruno Latour has called an “immutable mobile” (1986) ready to be applied in other situations.^{5}

2. Topological Calculation as Cultural Technique

A proper “history of the present” as Foucault himself practiced it starts with a diagnosis of the current situation; a question posed in the present from which an appropriate specification of the problem may be derived (Garland 2014, 376–379). Luciana Parisi's *Contagious Architecture* (2013) offers such an excellent diagnosis within the field of digital architecture. As she starts the preface, the logic of computation – a mathematical calculation – increasingly ingresses into culture (ix). At the same time, she states that the last twenty years or so have seen the digital mapping of space become increasingly intersected by a new tendency in digital design that has more fully embraced the power of computation to generate new architectural forms or smooth surfaces (xi). These new architectural forms or smooth surfaces rely on a “new centrality of generative algorithms (but also cellular automata, L-systems, and parametricism) in digital design has led to the construction of various topological geometries and curvilinear shapes that have come to be known as blob architectures” (xi). Moreover, “the new function of algorithms within the programming of spatiotemporal forms and relations reveals how the degree of *prehension* proper to algorithms has come to characterise computational culture. Algorithms are no longer seen as tools to accomplish a task: in digital architecture, they are the constructive material or abstract ‘stuff’ that enables the automated design of buildings, infrastructures, and objects” (xii, emphasis in original). Parisi's philosophical enquiry takes *uncertainty* and *incomputability* as intrinsic characteristics to computation, bringing her “to challenge the metacomputational approach of understanding algorithms according to a set of simple rules whose combination produces complex results” (Ikoniadou 2014), and into the realms of speculative realism and object-oriented philosophy. This in turn leads her to “[leave] behind the topological schema that presumes algorithms to be subjected to continuous, sequential order results and evolving in time, . . . [turning] instead to parametricism and the mereotopological order of events.” (Ikoniadou 2014). Thus, Parisi's contribution is interesting for the way it explores a philosophical paradox constituted by, on the one hand, the (algorithmic) calculation of *invariance* (a continuous function) as an aesthetic ideal as encountered for instance in the design of contemporary digital architecture, and on the other hand a fundamental discreteness of computation, mathematic calculation, and algorithmic procedures. The book then speculates about the possibility of a continuous concept of computation and of topology. But although Parisi's diagnosis of the role of topological calculation in contemporary digital architecture is certainly useful for the purpose of this paper, the question of software or topological calculation as cultural techniques is simply not within the scope of her project. Put differently, the aim of this analysis is not to explore uncertainties or the uncomputable quantities – following Gregory Chaitin's work on algorithmic information theory – nor is it to close the gap between the discrete and infinity with a continuous “mereotopological order of events” – after Alfred North Whitehead's philosophical writings. Rather, in conceiving of software or topological calculation as cultural techniques, the same diagnosis may serve to examine their conditions of emergence, and the way they subsequently distinguish themselves in operationalising (ontological) distinctions according to their domains of application, thus “presenting a critique of our own time based upon retrospective analyses” (Simon 1971, 192).

Operationalising this relationship between the present and the past is not without its problems, for the approaches taken by Foucault and Hacking are not commensurable *per se* with those of media archeologists focusing on cultural techniques like Thomas Macho or Bernhard Siegert.^{6} To investigate software in general, and topological calculation more specifically, as cultural techniques is to work against a background of understanding wherein “*media* are scrutinised with a view towards their technicity, technology is scrutinised with a view towards its instrumental and anthropological determination, and culture with a view towards its boundaries, its other and its idealised notion of bourgeois *Bildung*” (Siegert 2013, 58, emphasis in original). Following Bernhard Siegert's interpretation of the concept – that is as a reorientation of German media theory (54–55) – five further features are listed that characterise the theoretical profile of cultural techniques, which in turn helps inform the demarcation of the conceptual search space for the purpose of this paper. First, extending Thomas Macho's often-cited definition of cultural techniques as operative chains that are always older than the concepts they generate (2003, 179), Siegert points out that in speaking of cultural techniques, “we envisage a more or less complex actor network that comprises technological objects as well as the operative chains they are part of and that configure or constitute them” (58). Second, objects tend to be tied in with practices to which these objects lend their specific meaning. In effect, cultural techniques presuppose a plural notion of culture (58–59). Third, cultural techniques are differentiated from other technologies because they involve symbolic work (59; cf. Macho 2008, 99). At the same time, this differentiation also cannot be attributed to any static concept of technology or of symbolic work, necessitating a *processual*, rather than ontological, definition of first-order and second-order techniques (Siegert 2013, 60). As a result, focus shifts to the (recursive) operative chains that constitute such distinctions. Fourth, it is presumed that “[e]very culture begins with the introduction of distinctions” (61). Consequently, to analyse cultural techniques is to observe and describe techniques involved in operationalising the ontological distinctions that we encounter as discernible unities of distinctions (62). Fifth and finally, cultural techniques not just “sustain, disseminate, internalise and institutionalise sign systems, they also destabilise cultural codes, erase signs and deterritorialise sounds and images” (62). In other words, if we accept that operative chains may reinforce or introduce new distinctions, then it is also reasonable to accept they may unmake present ones, or remake previous ones. Cultural techniques as a particular kind of media then are crucial in this process of negotiation, as they constitute the position of a third, appearing as “code-generating or code-destroying interfaces between cultural orders and a real that cannot be symbolised” (62).

Aligning the concept of cultural techniques with the methodology as outlined so far has implications for the style of analysis pursued here. For the historian of ideas, it means to expand its primary focus on empirical historical objects with a present-day critical technologically informed view focusing primarily on techniques rather than concepts, and for the media archeologist it means to enrich its primary focus on digging up (material) discourse networks of technologies in the past with a conceptual view towards writing a “history of the present” – conceived as a particular method of critically examining the relationship between past and present “as a means to re-value the value of contemporary phenomena” (Garland 2014, 365). The structuring of the remainder of this paper follows from these insights, diverging into two stages of enquiry. In the next or third section, mobilised concepts and methods originating from different and scattered locations will be located within a larger archive of ideas so as to trace the historical *a priori* of the cultural technique in question. As Hacking demonstrated in his study of the emergence of ideas of probability, the preconditions for the emergence of these ideas also determine the space of possible theories and applications based on these ideas. The focus, however, is limited to a handful of significant moments of reconceptualisation or realignment of particular concepts and methods. In the fourth section, the focus is on the way these concepts and methods – which, as Foucault notes, “overlap, repeat, or imitate one another, support one another” (1995, 138) – distinguish themselves from one another according to their domain of application. Thus, the point is to examine where and how the topological calculation and its computation are tied in with techniques, rationales, and practices of present-day architectural design and engineering that give meaning (and purpose) to it. These cultural techniques play a key role in constituting the symbolic as well as the material embodiment of digital or computational architecture, introducing new distinctions and, by extension, different cultures that emanate from them (cf. Siegert 2013, 60). Finally, the concluding section presents a synthesis of both levels of the analysis and reflects on implications for changing cultures or ideas of order implied by the investigated cultural technique.

3. Topological Computation in the Archive

Guiding this search for topological computation in the archive are three main threads each with their own distinct lines of conceptual descent and patterns of distribution: changing notions of prehending space, time-structures, and their expressive power and associated modes of control. If, as Macho (2008) and Siegert (2013) have suggested, cultural techniques involve both symbolic and material work, then conceiving of topology in purely mathematical terms (e.g., as axioms, formal language, notations) is not sufficient. Instead, like Kitchin and Dodge's elaboration of “code/space” as something mutually constituted (2011, 16), the concept of cultural techniques helps think in such symbolic–material pairs as captured in a compound term like topological computation. Following Macho's proposition to think of cultural techniques as “second-order techniques” (2008, 100), topological calculation and its automated computation – being cultural techniques for the prehension of space simultaneously on a symbolic and material level – should then be conceived in terms of two aspects. On the one hand, the concepts and methods of topology enable a particular expressive power or *expressiveness* through which space can be formalised. That is, it provides new modes of “calculating, ordering, classifying, predicting, profiling, constraining, and rescripting data” (Parisi 2013, 266).^{7} On the other hand, as a second-order technique (i.e., enabled by these modes of prehending space), topological computation also offers new modes of control that rely on these underlying topological matrixes (i.e., digital design and modelling). In the following paragraphs, multiple patterns of scattered concepts and methods mobilised by topological computation are traced on these two levels (Table 1), and are argued to be a consequence not of mathematics alone, but of the drive towards a particular mode of control and *reasoning* (i.e., manipulating, working with) simultaneously enabled and constrained by its expressive power. Interestingly, the first moment in this narrative already provides us with the concept of topological properties and invariance, yet it took over two centuries to find meaningful technological innovations to apply them to architectural spatial design problems.

**Table 1.** Overview of retrieved patterns of concepts and methods per section grouped by category.

Section | Period/year | Prehension of space | Time-structures | Expressiveness and Control |
---|---|---|---|---|

3.1. | 1736; 1895 | Topology; relative positions; graph theory; topological geometry. | Discrete steps; spatialisation of time; Eulerian path and Eulerian cycle. | Euler characteristic; topological properties; invariance; algebraic topology; essential relative properties match geometrical shapes; geometry defines topology. |

3.2. | 1948; 1970s | Non-linear situations; degrees of freedom of an environment. | Bergsonian time, or time as duration; time series; storage of messages as memory system; historical data for predicting; morphology; autopoiesis. | Closed feedback systems; acting upon empirical data; sensory-motor coupling; homeostasis and heterostasis; self-regulating systems; negative feedback to correct performance; control as regulation; model-based control over degrees of freedom of an environment; equilibrium state; organism-plus-environment as a single circuit. |

3.3. | Late 1960s; 1975 | Responsive environment; matrix of sensors; environments to a participant's behaviour. | Responsivity; personal action sequences; patterns of living. | Model–form pair; entanglement of control and composition; aesthetics–engineering pair; computer-aided participation; conditions of interactivity; shape and structure directly connected to environment; structures for human activity. |

3.4. | 1982; early 2000s | Cellular automata; composition of elementary particles; state spaces; permutations. | Time as generation or steps; natural selection and fitness functions; coupling time-structure with spatial axis; morphogenesis. | Control points, weights, and knots; behavioural modelling; simple instructions to produce complex forms; rule-based systems; generative algorithms; self-organising sytems. |

3.1. The Seven Bridges of Königsberg and *Analysis Situs* (1736, 1895)

It is notable that the emergence of the mathematical subject of topology occurred simultaneously with the representational concept of graph theory. This idea was established through a discussion of the famous problem known as the Seven Bridges of Königsberg (Kaliningrad), a problem set by the inhabitants that had been solved by Leonhard Euler in 1736, providing the basis for a more general method to solving problems relating to the “geometry of position”^{8} (Euler [1741] 1752; Shields 2012). The problem asks whether it is possible to traverse all seven bridges in a single stroll without crossing any of the bridges twice (a “Eulerian path”), and with the additional requirement that the trip ends in the same place it began (a “Eulerian cycle”). In his solution to the problem, Euler reconceptualised and abstracted a Euclidean space of discontinous points separated by distances into a non-Euclidean field of relations, thereby positioning the field of topology in relation to geometry by demonstrating that all information necessary to solve this problem is essentially relative. Thus, although topology came into being in relation to a problem about land masses and bridges with physical properties like volume, angles and corners, weight, textures, densities, measurements, and so forth, its solution relies only on a representational model that matches only its essential relative properties; those that are *topologically invariant* or remain unchanged under continuous deformation.^{9} These insights have resulted in a two distinguishable families of ideas: those relating to geometry as a branch of mathematics concerning properties of space and of figures in space (e.g., Platonic solids, polygons, convex and non-convex polyhedra), and those relating to topology as a branch concerned with only those properties of geometrical figures that are left unchanged under continuous deformation like stretching and bending, but not cutting or gluing together (e.g., Riemann's idea of manifolds, Möbius strips, knots). Furthermore, geometry and topology do not just concern different kinds of properties of space and of figures in space, but also require a different vocabulary to express those different aspects. For example, while measurements in geometry may be expressed with the standardised metric system or other units of measurement, topology and graph theory had to invent a different vocabulary to express concepts of continuity, connectivity, and boundaries (i.e., consisting of vertices and edges, surfaces, dimensions, vectors and fields, directionality, and so forth). With regards to this problem, Henri Poincaré is considered to be the founder of the field of algebraic topology – concerned with its symbols and rules of manipulation – and of the theory of analytic function with the publishing of Analysis Situs in 1895, which is also one of the earliest systematic theorizations of topology. Although the specifics of the notation are not of concern here, it is important to note that the topological design or modelling of digital architecture requires both vocabularies to be translatable. That is, the topological and geometrical, and the symbolic and material, have to maintain some sort of stable relation, a method or system of symbolic representation able to express the rules of transformation from the topological into the geometrical or vice-versa.

3.2. *Cybernetics* and Neocybernetics (1948, 1970s)

Jumping roughly two centuries ahead to the publication of Norbert Wiener's *Cybernetics: Or Control and Communication in the Animal and the Machine* in 1948 and republished in extended form in 1961, we find the origin of another family of ideas, concepts and methods that have left their mark in contemporary digital architecture and engineering. Having not much to do with Euler's original ideas, these are primarily relevant for providing a particular notion of control as regulation enabled by Claude Shannon's mathematical concept of information as something that may be measured (i.e., assigned a quantity; 1948), and recursive mechanisms of negative feedback systems to correct or otherwise act on these measurements towards some state or equilibrium defined by its function (as encountered for instance in servomechanisms). Wiener's contribution is to the problem of how to gain or maintain control over non-linear situations (e.g., the prediction of tomorrow's weather), where the initial conditions are largely unknown but outcomes are defined in a narrow statistical range (indeed a “miracle”). It is interesting then that his first chapter compares Newton's philosophical system to Bergson's with regards to their treatments of concepts of time and space, and their measurements. While Newtonian mechanics consists in *time-reversible processes*, presuming a separation between absolute properties of time and space – independent of our perceptions and measurements – and those that are relative, Henri Bergson posited time as duration, implying incapacity to measure time itself. The latter grapples with *time-irreversible processes* according to the Second Law of Thermodynamics. On the basis of this distinction, he then continues, “the modern automaton exists in the same sort of Bergsonian time as the living organism; and hence there is no reason in Bergson's considerations why the essential mode of functioning of the living organism should not be the same as that of the automaton of this type” (1961, 44).^{10} There is, in other words, no effective distinction for Wiener between animals and machines, for both systems have behaviours subject to control and, more precisely, *regulation*. Further, besides the notion of “time series” – where the “message” is “a discrete or continuous sequences of measurable events distributed in time” (8) – the concept of time also appears as the physical storage of information in relation to machine learning and memory (i.e., nervous system); in the need to store the algorithms for processing data and the challenges involved in implementing a suitable memory system. The storage of such “messages” is crucial to the prediction problem for time series, which typically lends its effectiveness to the “rational, consistent treatment of many problems associated with non-linear prediction, non-linear filtering, the evaluation of the transmission of information in non-linear situations, and the theory of the dense gas and turbulence” (80). These are problems of communication to which, in Wiener's view, computer programmes would offer the best hope.

In the 1970s, Wiener's classical cybernetics was contrasted to “second-order cybernetics” (or “neocybernetics”) from the realisation that concepts from control theory could offer the conceptual tools for understanding behaviour in complex systems and nature itself. Gregory Bateson and Margaret Mead had initially described this second order concept in 1976 in a conversation with Stewart Brand as “essentially your ecosystem, your organism-plus-environment, is to be considered as a single circuit. . . . but . . . you are part of the bigger circuit . . . the engineer is outside the box, and Wiener is inside the box; I'm inside the box” (40). Heinz von Foerster (1981; Abramovitz and Foerster 1974), in contrast, attributes the origin of second order concept to the attempts of classical cyberneticians to construct a model of the mind (i.e., of “understanding”). In his view,

What is new is the profound insight that a brain is required to write a theory of a brain. From this follows that a theory of the brain, that has any aspirations for completeness, has to account for the writing of this theory. And even more fascinating, the writer of this theory has to account for her or himself. Translated into the domain of cybernetics; the cybernetician, by entering his own domain, has to account for his or her own activity. Cybernetics then becomes cybernetics of cybernetics, or *second-order cybernetics*.” (2003, 289, emphasis in original)

It is in this sense that cybernetic systems could be conceived of as emergent model-based control of the degrees of freedom of an environment. In both cases, new ways of thinking about the relations between, and boundaries of, systems were catalysed, including conversation and interaction theories (Pask 1996; Frazer 1993; ). As Bruce Clarke and Mark Hanson (2009) have observed, this implies a shift from *homeostasis* as a mode of autonomous self-regulation in mechanical and informatic systems to concepts of self-organisation and *autopoiesis* in embodied and metabiotic systems. Moreover, the second-order concept of cybernetics assumes that any observer is always-already part of the system or model itself (and of its performance), rather than being outside of it, thereby replacing the subject–object problem with a concept of self-referentiality (autopoiesis) that enables a system or model to produce its own rules of transformation or variation (morphology), or modify its own instructions (i.e., algorithms operating on algorithms as on any other types of data). This is why Parisi (2013) also makes a connection, central to her argument, between digital architecture and neocybernetics to explain the capacity of computational algorithms to generate instructions that change the model itself based on some kind of sensory-motor response or interactive feedback (e.g., via sensors and actuators), thereby problematising the classical notion of cybernetics as closed systems. This also means that it is problematic to conceive of models and their performances in purely empirical terms, for there is now a potentially infinite number of variations of those models. In Parisi's words, “[t]he mode of existence of algorithms no longer merely responds to models that simulate material bodies: instead it constructs a new kind of model, which derives its rules from contingencies and open-ended solutions” (1–2).

3.3. Responsive Techniques and *Soft Architecture Machines* (late 1960s, 1975)

Parisi's observation describes a current tendency in developments that started during the late 1960s, when architectural spatial design problems started to be explored by applying concepts and methods originating in cybernetics to architectural problems. In particular, Nicholas Negroponte^{11} and Myron Krueger among others explored responsive techniques in integrated approaches to human-computer interaction, bringing various media forms (audio, visuals, light) into a single framework. These are computer scientists, who are not architects by training, but rather skilled people interested in solving particular problems of *man–machine interactivity* we may now retrospectively think of as architectural with computers. The idea of empowering users with computers, rather than force upon them the values of a machine expert, is one such idea particularly important to Negroponte, who posited “responsive architecture” as the “natural product of computing power into built spaces and structures” (Sterk 2003, 407). But while the central ideas are still prominant and guiding in many of its current offspring fields, technological advances in the fields of artificial intelligence and robotics have changed its techniques and possibilities of production dramatically.

Responsive architectures are those that use sensors to measure actual environmental conditions, enabling structures to *adapt* their properties (e.g., form, shape, colors) according to those discrete responses via actuators – as in cybernetic systems. Incorporating technologies into *structural systems* has provided architects with the ability to tie the shape of a structure directly to its environment; a type of architecture that has the ability to alter its form in response to changing conditions (Sterk 2003, 407). Moreover, in accordance with the second order concept of cybernetics positing all observers are participant observers, the *concept* of participation has become central to its control model. Negroponte recognised this in his seminal book *Soft Architecture Machines* (1975), which builds on previous insights from *The Architecture Machine* (1970), and provides, among other things, an argument for “intentionalities” to be understood in their contexts (1975, 60–63) as well as a proposal for “computer-aided participatory design”, revitalising an “indigenous architecture” that is responsive to the local needs and desires of its users (1975, 102–123). With regards to the concept of intentionalities, his text offers some critical thoughts on notions of participation and modelling of user intentionalities on multiple levels as illustrated by a phrase like “the computer's model of the user's model of its model of him” (Wardrip-Fruin and Montfort 2003, 353). In particular, it is interesting to recall his notion of participation because it is intended as an approach to remove “the architect and *his* [*sic*] previous experience as intermediaries between my needs (pragmatic, emotional, whimsical, etc.) and my house” (357, emphasis in original). The desires of the architect, in other words, are conceived as conflicting with those of the participant or “user” (i.e., an organism reduced to its discrete computable activity functions). With regards to the concept of participation, Negroponte's text contains many diagrams of this “indigenous architecture” – a negotiation between “local activity” and “global forces”, “which act in a very real sense as elements of town planning and which ensure an overall unity” (Wardrip-Fruin and Montfort 2003, 359) – represented as a model, based on “personal stories” that can be visualised or mapped using graph diagrams, preceding house plans generated from these graphs. The design of buildings thereby turns into the design of *structures for human activity* that form the basis of both architecture and human-computer interaction (353). It is in these graphed structures that separate and reconnect architectural form to foundational activity model that we begin to see a glimpse of conceptual heritage from Euler's solution to the Königsberg bridge problem in 1736.

Being one of the early pioneers pursuing artistic and technological goals, Myron Krueger engaged with man–machine interaction in a similar spirit as Negroponte. As Noah Wardrip-Fruin and Nick Montfort wrote, “Krueger has worked and written in areas called ‘responsive environments’ and ‘artificial reality’, which includes (from Krueger's point of view) the various inventions referred to as *virtual reality*.” (2003, 377, emphasis in original). His concern is not so much with things that happen to be interactive, but rather with exploring what interactivity itself could be like; an “art of interactivity” asserting that “response is the medium” (1977, 430). In particular, the notion of the artist as a “composer” of intelligent, *real-time* computer-mediated spaces, or “responsive environments”^{12} lies at the heart of his contribution. This is illustrated by his maze programme, which he calls a “composed environment”, a “carefully composed sequence of relations designed to constitute a unique and coherent experience” (427). In order for such a composed environment to work correctly, a control system is required that “includes hardware and software control of all inputs and outputs as well as processing for decisions that are programmed by the artist” (430). The concepts and methods encountered with the implementation of these responsive techniques have enabled a reconceptualisation of space as “environments” for a participant's behaviour;^{13} a Euclidean matrix or grid of sensors to register discrete activities and movements in a bounded absolute space, which is the setting or quite literally the stage for personal action sequences – or “patterns of living” as Negroponte would have it – which in turn informs a *response* aimed to help the human architect who must (at least in Negroponte's view) *correct* – in the cybernetic sense of the term – his or her “course” to avoid foisting personal values through expertise upon others in favour of insights derived from a particular concept of participation or interaction. Architecture, in this sense, is a form of social statement (Negroponte), and its design can be approached simultaneously as aesthetics and engineering (Krueger). A notion of control thus emerged that incorporates both an aesthetic and engineering dimension in which *control* and *composition* are deeply entangled, and in which the computer's role is to “aid” or “support” the participant's behaviour. This means that the defining boundaries separating the artist, architect, and engineer are reconfigured, a change that is further catalysed as computational algorithms increasingly enter into these assemblages of techniques, rationales, and practices.

3.4. Cellular Automata and Generative Algorithms (1982, early 2000s)

Jumping again, now roughly 20 years ahead, brings us to a place with more concrete applications of the calculation of topological and morphogenetic concepts as well as with continuations of concepts and methods preceding those. Two new notions of computation stand out in particular as playing a key role in a plethora of applications for generative algorithms in digital design and architecture: first, the computation of *cellular automata* to automatically generate structures or let programmes produce shapes according to evolutionary design modelling (e.g., Fernando 2014), and second, the computation of topological invariants to ultimately pre-empt and reprogramming events in real-time before they can happen. Calculation, then, involves far more than arithmetic and numbers, including techniques that model processes of transformation from some finite input into some other finite output. As such, these techniques of calculation or functions may be understood as mediators (in Latour's sense of the term, actors that make a difference) within a given network, and may also have inscribed into them certain ideas about time, space, expressiveness and control.

Although John von Neumann and Stanisław Ulam originally developed the concept of cellular automata in the 1940s – at a time when the range of possible computations was broadened significantly – as simple (i.e., elementary) models for studying biological processes like self-reproduction (von Neumann 1966; Wolfram 1982), it is Stephen Wolfram's contribution to the development of the concept that is of interest to the story of this paper. In particular, a 1982 article titled “Cellular Automata as Simple Self-Organising Systems”, and the later publication of his magnum opus *A New Kind of Science* in 2002. At the core of both publications is the profound discovery that cellular automata are not just discrete models for studying the *computability* of biological (evolutionary) processes; but instead are capable of complicated behaviours, producing *complex patterns* with “universal features”, which may be derived using statistical mechanics (1982, 1985, 2002). As he explains, “[e]ven though the underlying rules for a system are simple, and even though the system is started from simple initial conditions, the behaviour that the system shows can nevertheless be highly complex” (2002, 28). In other words, a finite set of simple rules may quickly derive complex emergent patterns, the possible applications of which far exceed modelling biological processes of self-reproduction, and now includes many other kinds of (natural) phenomena that appear to us as complex. This development is of interest not just to physicists, mathematicians, or biologists, but now includes those who identify as architects or computer scientists for at least two different reasons: for the possibility of generating complexity in form, with its roots in the ornamental arts appearing for example as repetitive or nested patterns (cf. 43), or for the more profound possibility of modelling behaviours themselves using simple programmes, thereby coupling the dynamic discrete modelling of behaviours to the generative capacity of architectural form composed of elementary “particles” and resulting in the use of structural metaphors like “swarm architectures”. These particles are positioned within a time-structure in such a way that with each “generation” or “step” (i.e., a discrete unit of *constant time*, making the running time of a programme independent of the size of the problem) its position along a spatial axis is proportionally effected. Thus, the algorithmic coupling of these two axes of a discrete model (where the former has primacy over the latter), in combination with the coupling of a model to a physical shape conforming to changes in the modelling space, provide architectural designers and engineers with the ability to easily and remotely control the production of dynamic shapes by manipulating – either manually or automatically – the control points, weights and knots (cf. Kolarevic 2001, 117; Lynn 1993).^{14} Besides generating structures, shapes and forms, ornaments, and so forth, this sensory-motor coupling opens new *conditions of interactivity* between software and behaviour, as initially explored by such people as Negroponte and Krueger. These conditions must be included in the programming of an infinite series of anticipated scenarios; that is of all possible states of a system, or its state space.^{15} In short, cellular automate enabled these conceptual innovations by introducing metaphors and models from evolutionary biology (e.g., adaptivity, emergent behaviours, speciation, reproduction, self-organisation) into a problem space where computers were increasingly involved to extend calculation capacities.

Besides the conception of a *programmable* generative capacity of architectural form (or indeed of any form with topological properties) through the anticipation of possible states of a system, the introduction of metaphors and models from evolutionary biology into computational architectural design also put the concept of *morphology* at the centre. Morphology in architectural shapes has enabled two different innovations. On the one hand it paved the way for a computational concept of *morphogenesis*; literally the beginning of shape. In these cases, cellular automata are able to provide homogeneous structures which may be programmed using generative algorithms for exploring algorithmic solutions through the simulation of experiments rather than pure geometrical or algorithmic subjects (e.g., Khabazi 2011; Terzidis 2003, 2006, 2009). For example, a morphogenesis according to a continuous morphology of space would result in a structure where walls, ceilings, and floors are continuous into each other, undermining the defining feature of these surfaces and their symbolic functions (e.g., distinguishing surfaces and spaces from each other). Common to such approaches for finding a form is the almost exclusive use of *topological geometries*, referring to incidence geometries, the natural geometric operations of which are continuous maps with respect to their topologies (Kolarevic 2001, 119; Kotnik and Weinstock 2012; Kotnik 2013).^{16} It is here that we find the emergent aesthetic of complexity in architectural ornament and form, especially in the design of architectural *skin* (i.e., surface). On the other hand, morphology in parallel with topological concepts also provided a metaphor for thinking time-structure in a way that is continuous with its spatial form, namely *homeomorphism* or *topological isomorphism*. When a shape (*morphe*) is isomorphic, it means that two different shapes may be considered equal as long as they share those properties that define its morphism. These are, in other words, its topologically invariant properties. The computational concept of homogeneity in cellular automata is thereby made compatible with the topological concept of homeomorphism, enabling the generative algorithm and the shape of a responsive environment (i.e., its topological geometry) to be continuous. Within this new kind of mechanistic worldview, algorithms do not just execute some initial set of instructions, but may also derive new instructions from open-ended solutions and contingencies (through the integration of sensory and motor systems). This means that topological computation has provided urban design with the capacity “for longterm planning to become open to revisions, updates, real-time inputs and contingencies” (Parisi 2012, 167). This space of real-time sensory–motor interaction, however, is also far from exhaustive with respect to the set of all possible signals to respond to, which makes the pursuit to observing and registering emergent complex behaviours – not merely of human “participants” with their own “patterns of living”, but behaviours of any kind – into a prime objective.

4. Two Vectors of Variation within Computational Architectural Design and Engineering

Having first gathered different threads or clusters of ideas regarding space, time, and expressiveness and control, the next step is to examine how these particular concepts and methods are mobilised or converge in topological computation as technique or practical rationale, tied in with practices of contemporary computational architectural design and engineering that give it meaning and purpose. Within the methodological framework set out by this paper, this means to explore the “vectors of variation, heterogeneity, and fermentation” (Rieder 2012) of topological calculation and its computation, and to do so while being attentive to the various *degrees of sophistication* they may take when applied. Where and how is the algebraic topology of continuity today? And how can its expressive power be characterised – what can we actually do with it? The technique is first and foremost a *knowledge technique*, for in order to control one first has to capture – or measure – what and when something is in the first place. In fact, topological computation captures a particular kind of spatial knowledge encompassing all the continuity and complexity of the terrain. In this sense, it designates more than a theory of plasticity or calculating invariant properties; it also concerns the deeper philosophical assumption of space as something having empirically discernable properties that may be measured, manipulated, and controlled through mathematical formalism. Although topological computation has a variety of applications both within and outside architectural design and engineering (e.g., the emerging field of computational topology, which is devoted to the study of efficient algorithms for solving topological problems), the following paragraphs describe an approach to localise topological computation within “parametric practice” (Gerber 2007). Parametrics refers to a set of techniques that leverage computational power and (interactive) algorithms to automatically manipulate or generate forms,^{17} thereby offering flexibility throughout the design process, and the possibility for rapid digital prototyping (Leach 2014; Kocher et al. 2014; Davis 2013; Jabi, Woodbury, and Johnson 2013; Dino 2012; Rocker 2011; Woodbury 2010; Schumacher 2011, 2009; Gerber 2007; Woodbury, Williamson and Beesley 2006).

If we conceive of the computer as *algorithmic machine* – a machine, such as a Turing machine, *transforming* input data (*x*) into output data (*f ^{(x)}*) according to some

*computable mathematical function*(

*f*) – then a key question for computing concerns the kind of functions that can be computed, and what a computable function is precisely (Kotnik 2010)? This transformation from input into output data requires a function – such as a continuous function – with a set of valid input, a parameter space, and a set of possible variations in its output. Based on developments in architecture in particular since the 1990s, several “degrees of computational awareness” (Kotnik 2006, 6) may be distinguished in the introduction – or “conscious overcoming of the traditional level of representation in the use of the computer as design tool” – of affordable computational power to architectural design. According to Toni Kotnik, these include an operative level, a parametric level, and an algorithmic level (2006, 6; Di Angelo, Ferschin, and Paskaleva 2012). Firstly, the operative variation utilises the computer “for modelling in a pre-defined geometric way. That is implemented geometric operations of the software in use are explored in an architectural context in order to deform the classical formal language of architecure by means of controlled transformations” (Kotnik 2006, 7). Here, it is the increasing computational power in particular that opens new possibilities for geometric modelling. Secondly, the parametric variation has emerged from the closer examination of NURBS geometry – a Non-Uniform Rational B-Spline model used for generating B-spline and Bézier curves and surfaces – which has shifted interest “away from drafting and modelling towards a more mathematically based view on architectural design”. Since the NURBS object is defined by a local parameter space of control points and weights, this model offers the designer flexibility and a range of new structural possibilities for design, replacing fixity with variability, singularity with multiplicity. As Kotnik observes, the use of parameters in contemporary architecture is particularly popular for programming time as primal parameter of a structure, including time-based techniques like morphing, keyframe animation, kinematics, force fields, or particle systems (10). Thirdly, the algorithmic variation develops algorithmic descriptions of the design using suitable software tools, and includes such things as procedural modelling and implemented shape grammars (Di Angelo, Ferschin, and Paskaleva 2012, 109). This approach deploys techniques involving parameters, shape grammars, scripts and programming to generate building models. However, beyond mere description of empirical evidence, algorithms increasingly operate in a

*prescriptive mode*when they are programmed to generate or change their own instructions in response to registered event, thereby actively intervening or changing the event itself. In this case, the generative capacity of self-evolving structures governed by rules in combination with interactive systems of measurement and physical sensory-motor response amount to a model of governmentality – the conduct of conduct (Foucault 1982) – deeply entangled with performance and behavioural patterns. Thus, topological computation – or the computatable functions of an algebraic topology of continuity – is located at the core of algorithmic machines, governing the specific relationship or rules of transformation from input data into output data.

One approach to localise topological computation in terms of its application is through solution-driven approaches to generative design achieved computationally through parametric and algorithmic processes. Topological calculation is used, for instance, in parametric modelling environments that serve to support spatial feedback in parametric systems coupled with generative spatial design (e.g., Coorey and Jupp 2014). For example, such a design or explorative logic may involve synthesis models for generating designs, analysis models, evaluative models for design filtering, and optimization models (Alfaris and Merello 2008). In this example of Alfaris and Merello's Generative Multi-Performance Design System (GMPDS), the synthesis models capture design intentions, can be described in algorithmic or parametric terms, and “[b]ecause the design is composed as a system of relationships, large populations of design solutions can be generated by varying the parameters” (Coorey and Jupp 2014, 278). This opens the door to a notion of improvement, or indeed of a “best” variation. As a particular iteration of the digital design of algorithmic architecture, “permutation design” (Terzidis 2014), for instance, regards problems of design as problems of permutations, which means the designer is called upon to design components and order elements in a set of solutions. The “best design” is one that remains after all other variations have been eliminated, thereby assuring an *optimum design*. The analysis model calculates the spatial analysis metrics to determine characteristics of a design solution. These include “space syntax gamma measures to drive the spatial analytics model, including integration, control, and topological depth as well as geometric measures including area, length, width, and proportion” (Coorey and Jupp 2014, 279). The evaluative model then assesses the *fitness* and is crucial in guiding the search for the “best” solution. Finally, the optimisation model enables the search for the optimim design, and involves cycling through the permutation design space. Thus, this process continuously reconfigures a *graph network* consisting of nodes (spaces with spatial properties) and relationships to produce alternative and optimum solutions by randomising the locations of spaces, while maintaining their topological integrity. In effect, this conceptual framework allows for a theoretically infinite number of variations for allocating spaces. Although the specific configurations and workflows of these models may vary significantly, this example does help to understand the “technical logic” (Rieder 2012) undergirding these parametric systems or structures.

Another approach to further localise topological computation in terms of its application is by looking at specific methods used for generating curved surface structures in architecture. NURBS models, for instance, are widely used in Computer-Aided Design (CAD). In brief, a NURBS curve or surface is described by “its degree, a set of weighted control points, and its knot vector” (Felkner, Chatzi, and Kotnik 2013, 16; Meng and Yin 2011).^{18} Using a two-dimensional grid of control points, NURBS surfaces can be created, and by using a three-dimensional matrix of control points, complex organic shapes can be created. NURBS surfaces are used, for instance, in connecting separate *patches* that are fitted together during construction (e.g., of the hull of a boat) in such a way that no boundaries or corners occur; the surface is *geometrically continuous*. But since NURBS curves and surfaces are mathematical functions, it is also possible to discuss the derivatives of the geometrical surface with respect to the parameters. These derivatives can then be *parametrically continuous*, referring to the smoothness of the parameter values for a curve (one dimension), surface (two dimensions), or shape (three dimensions). NURBS curves and surfaces are useful because they are invariant under continuous deformation (i.e., they have topological properties), and therefore allow a high level of freedom of design, and a great variety of shapes to be generated from them, while maintaining a constant mathematical form. Typically, each control point of the curve is computed by calculating a weighted sum of a specified set of control points, where the weight of each point varies according to the governing parameter (i.e., they are derivative). The outcome of interpolating the values between each point and its surrounding points results in a continuous shape such as a polygon (two dimensions) or polyhedra (three dimensions). Further, NURBS and NURBS-like models have also been applied to very specific problems such as particle swarm optimisation for the architectural design of truss structures with aesthetic criteria (Felkner, Chatzi, and Kotnik 2013), generating continuous membrane surfaces to control the transfer of loads in any kind of surface structure (Bahr and Kotnik 2010), or for building statically optimised shell structures based on stress field analysis. In each case, shape is a generated after going through an analytical function, an evaluation function, and subsequent optimisation by an objective function such as a search for an equilibrium solution for the internal loads of a structure under a specific load condition (Bahr and Kotnik 2010, 4).

These and other developments have led to an emerging aesthetics of continuity and complexity, characterised by computational geometries (e.g., Voronoi diagrams, A* search algorithms), rule-based systems (e.g., L-Systems, shape grammars), self-organising systems (e.g., cellular automata, swarm systems), optimisation (e.g., genetic algorithms, “springs systems”, fitness functions), and so forth (Kotnik 2006). The algebraic topology of continuity is at the core of these developments in the digital realm of binary information, for in these cases the relation between different forms of data is governed through the computation of topological invariants (e.g., a continuous function; Parisi 2012). Clearly there are many overlaps, repetitions and imitations of present concepts and methods with those retrieved previously. These tend to support one another and thereby gradually produce the outlines for more general methods: from simple procedures of calculation to the lengthy chains of operative procedures supported by interactive environments and infrastructures able to deal with the associated computational demands, required to design and engineer digital and computational architectures. In this context, topology offers nothing less than “a model for mapping the dynamics of time as well as space, allowing the rigorous description of events, situations, changing cultural formations and social spatiaizations” (Shields 2012, 43). Indeed, a new computational relationship to space and time mediated by computable topological functions. Moreover, if, as Kolarevic also argues, deformations in this modelling space need to conform to changes in the geometry of the modelling space (2001, 119), then the algorithms that store and provide instructions to these models are effectively equated with the generative capacities of matter itself to evolve. Parsing emergent complex behaviours characterised by continual variation and uncertainty thus becomes a key challenge in constructing interactive mechanisms of control that operate by pre-empting change and prescribing or reprogramming anticipated events. In this sense, the appeal of a notion of control flattened “onto a topological matrix of continual co-evolution” (Parisi 2012, 171) ultimately relies on the assumption that a finite set of parameters (i.e., control points) may be instructed to enact a state space that is capable of dealing appropriately with the various anticipated scenarios.

5. Conclusions

Taking software as an object in-the-world, the preceding analysis enquired how we may understand the techniques of topological calculation and their computation as particular kinds of mediators with the capacity to introduce new and specific modes of ordering space and time-structures by reconfiguring or modelling their relationships differently. Accordingly, the objective has been to excavate the historical a priori of the object; an approximation to defining its contours of demarcation (i.e., the demarcating criteria) that define its vectors of variation within a particular domain of application where these techniques find a practical aim or purpose. Analysing such vectors of variation will require an attentive attitude to the subtle nuances and differences, for within a domain of application a set of concepts and methods may overlap, repeat and imitate one another to gradually produce a more general method. Within the fields of architectural design and engineering, for instance, parametric practices mobilise concepts and methods differently according to a specific pattern of objectives, which is first and foremost a question of contingent power and knowledge relations, and which enables particular cultures or ideas of order to emanate from them. To make this argument, the concept of cultural techniques was deployed to make a distinction between calculation and computation. This perspective offers an analytical sensitivity that helps to understand techniques according to their degrees of sophistication when applied. That is, they may appear in their simplest elementary forms or in lengthy operative chains implicating a wide range of other actors to support their procedures (e.g., processing units, data storage), and acknowledging these differences is a way to deconstruct the “technical logic” (Rieder 2012) undergirding such operations. In order to investigate these techniques in this way, two axes of analysis were used: a vertical axis (i.e., writing a “history of the present”) and a horizontal axis (i.e., a close reading of the technique as regards its vectors of variation). Taken together, they offer a conceptual matrix to consider changing cultures or ideas of order emanating from this new computational relationship to space and time mediated by computable topological functions. It is a new way of thinking logistics, or organising data and movement of data as much as organising people and the movement of people in urban spaces towards purposes involving security, communication, managing traffic flows, and so forth. Thus, as Parisi also observed, “culture is not simply produced by computation, but emanates from new modes of calculating, ordering, classifying, predicting, profiling, constraining, and rescripting data” (2013, 266).

What takes shape in these reconfigurations is the exploration of computational methods of formal exploration and expression, of formal properties as sources for ordering systems, and a separation between the empirical and the analytical through the interactive or responsive. Mathematical formalisation is burdened with the task to appropriately coordinate these processes and models, objects and shapes, rules and instructions. In a more specific sense, topological computation has enabled and introduced different workflows and activities, requiring different tools and skillsets, a different set of aesthetic principles and imaginaries, new shape grammars and geometries, continuous skins, surfaces and membrane structures, generative design methods, and a particular concept of optimisation based on such things as measurements, analytics, evaluative and optimisation models. Although far from exhaustive, the discussed examples indicate a more general drive towards exploring and engineering *optimised relations* between the human body and its movement in a continuous relation to the built environment, as well as with the natural world. While former can technically be implemented in the form of *interactive algorithms*, the latter relies on the engineering of systems according to biological, evolutionary, and natural dynamic systems. As a consequence of this concept of continuity, the orientation of the body in space and time, and the concept of *distance* are problematised. In contrast to, say, a classical aesthetics according to the golden ratio based on Leonardo da Vinci's Vetruvian Man, or Leon Battista Alberti's invention of perspectival drawing as geometrical instrument of artistic and architectural representation, the contemporary computational aesthetics of continuity and complexity orients itself differently to geometrical concepts and methods (e.g., they are incidental and derive from their topological descriptions). This reorientation is profound, and means that many spatial concepts based on geometrical models will need to be reconsidered. For example, what does it mean to talk about a body or its contours or skin in non-geometrical terms, and what does this mean for existing techniques, rationales and practices that subjectivise and work with bodies or territories (i.e., biopolitical implications; e.g., medical practitioners operating on human bodies, property insurance on a building, city governance)? Moreover, in the same way that distances are reconceptualised as continuous relations, actions and effects are also brought together in the form of interactive algorithms, thereby making every captured action count. How do we know the effects our actions have in these built environments or in the natural world? To some extent, simulation techniques try to work with this problem in putting actions within an integral context so that the effects of these actions may also be measured, analysed, evaluated, and controlled. It is therefore important to also consider the new types of knowledge – such as spatial knowledge encompassing all the continuity and complexity of the terrain, or statistical types of knowledge that derive from topologically-defined surfaces like Henri Poincaré's *hairy ball theorem* (it states that any continuous tangent vector field on the surface of a sphere must have a point where the vector is zero, even if we do not precisely know where this point is) – that are available through topological calculation and its computation and that drive these models in their applications.

6. Endnotes

1. This is to state from the outset that I do not intend to stick to an *a priori* framing of software as object (e.g., as code). Instead, in asking what it means to take software as an object of cultural analysis in a particular setting, I hope to overcome this limitation and then make assessments based on my own investigation – to see how it distinguishes itself according to its domains of application.

2. Henceforth topological computation will refer to a particular kind of operationalisation of mathematical calculation described in procedural terms by algorithms that enable one to manipulate or work with topological concepts such as continuity, connectivity, and boundary, Following Thomas Macho's claim that cultural techniques (*Kulturtechniken*) – like calculation, measuring and counting – are always older than the concepts that are generated from them (2003, 179), the operative chains of calculation precede their computation. Thus, their distinctions are primarily historical.

3. For each variation, a plethora of examples is available online, or reprinted as static images in publications, articles in periodicals, and so forth. In addition, Kostas Terzidis has listed some of the early attempts exploring computational mechanisms for exploring formal systems, resulting in topological geometries, hypersurfaces and hyperbodies, blobs and folding architectures: “beginning with Marcos Novak's “Computational Compositions” (1988), William Mitchell's *Logic of Architecture* (1990), and Peter Eisenman's “Visions Unfolding” (1992), and continuing through John Frazer's An *Evolutionary Architecture* (1995) and Greg Lynn's *Animate Form* (1999) to name a few” (2003, 2). Typically, these are structures, or components, that seem to defy laws of gravity in their weight distributions, or where ceilings, walls, and floors are unified in a continuous morphology of space.

4. Here I am referring to its narrow definition in contrast to planning. As Kostas Terzidis has proposed, “Design is a term that differs from, but often is confused with, planning. While planning is the act of devising a scheme, program, or method worked out beforehand for the accomplishment of an objective, design is a conceptual activity involving formulating an idea intended to be expressed in a visible form or carried into action. Design is about conceptualization, imagination, and interpretation. In contrast, planning is about realization, organization, and execution. Rather than indicating a course of action that is specific for the accomplishment of a task, design is a vague, ambiguous, and indefinite process of genesis, emergence, or formation of something to be executed, but whose starting point, origin, or process often are uncertain. Design provides the spark of an idea and the formation of a mental image. It is about the primordial stage of capturing, conceiving, and outlining the main features of a plan and, as such, it always precedes the planning stage.” (Terzidis 2007).

5. It should be noted that topological concepts and methods have a series of applications outside of architectural design and engineering (e.g., Potier, Spicher, and Michel 2013; Miyake 2010; Sau et al. 2010; Raussendorf, Harrington, and Goyal 2007; Freedman et al. 2002; García López and Sabbah 1998).

6. For an overview of prominent figures and current debates around cultural techniques, see: Geoffrey Winthrop-Young, Ilinca Iurascu, and Jussi Parikka, eds. *Cultural Techniques*. Spec. issue of *Theory, Culture & Society* 30.6 (2013). Print.

7. Parisi refers to these techniques in the context of what she calls “programming culture”, or “[t]he computational mediation of all modes of cultural expressions (from the design of space to sound design, from the construction of socialities to the development of management and communication) that produces a specific aesthetics defined by the changing capacities of software, hardware, and interfaces to order data” (2013, 266).

8. The title of the original publication reads “*Solutio problematis ad geometriam situs pertinentis*”, which translates to “The solution of a problem relating to the geometry of position”. More info on this publication is available online at: http://eulerarchive.maa.org/pages/E053.html.

9. Euler himself discovered the first of these invariant properties, now known as the “Euler characteristic” (*X ^{(S)}*). The formula for calculating this property is as follows:

*V*−

*E*+

*F*≡ 2, where

*V*represents the vertices,

*E*the edges, and

*F*the faces. The formula can be proved by looking for an invariant property, and finding a procedure in which that property is demonstrated to be unchanged. For example, when one continually deforms a shape by repeatedly taking away one edge and one face, the outcome of the formula would be left unchanged (e.g., Flood 2014).

10. As a critical footnote, Wiener noted that “Vitalism has won to the extent that even mechanisms corrrespond to the time-structure of vitalism; but . . . this victory is a complete defeat, for from every point which has the slightest relation to morality or religion, the new mechanics is fully as mechanistic as the old” (1961, 44).

11. Negroponte was also responsible, together with Jerome Wiesner, for initiating the renowned Architecture Machine Group at MIT in 1967, which would later grow into its Media Lab in 1985.

12. One of Krueger's best-known systems, *VIDEOPLACE*, is an “artificial reality” that surrounds the participating users, and responds to their movements and actions within a bounded space. As he describes the project: “VIDEOPLACE is a conceptual environment with no physical existence. It unites people in separate locations in a common visual experience, allowing them to interact in unexpected ways through the video medium. The term VIDEOPLACE is based on the premise that the act of communication creates a place that consists of all the information that the participants share at that moment” (1977, 429). Earlier iterations of his “artificial reality systems” include *GLOWFLOW*, *METAPLAY*, and *PSYCHIC SPACE*.

13. For the interested reader, Krueger provided eight different approaches to thinking the environment–participant pair in responsive terms in his text (1977, 430–431).

14. Branko Kolarevic has provided an overview of some of these new digital approaches to architectural design (digital architectures) based on computational concepts such as “topological space (topological architectures), isomorphic surfaces (isomorphic architectures), motion kinematics and dynamics (animate architectures), keyshape animation (metamorphic architectures), parametric design (parametric architectures), and genetic algorithms (evolutionary architectures). . . . New categories could be added to this taxonomy as new processes become introduced based on emerging computational approaches.” (2001, 117–118; Kolarevic 2000).

15. This is illustrated by the new avant-garde for architecture and urban design denoted by such terms as “parametricism” (Schumacher 2009) or “permutation design” (Terzidis 2014). These architects think not in units of buildings, but in much larger units from urban environments to complete metropolitan regions, of which smaller units like walls, bridges, soundwalls, and so forth then become integral parts (Parisi 2013, 97, footnote 15).

16. As Kolarevic continues. “[t]he capacity of digital architectures to generate new designs is therefore highly dependent on designer's perceptual and cognitive abilities, as continuous, dynamic processes ground the emergent form, i.e., its discovery, in qualitative cognition. Their generative role is accomplished through the designer's simultaneous interpretation and manipulation of a computational construct (topological surface, isomorphic field, kinetic skeleton, field of forces, parametric model, genetic algorithm, etc.) in a complex discourse that is continuously reconstituting itself – a ‘self-reflexive’ discourse in which graphics actively shape the designer's thinking process. It is precisely this ability of ‘finding a form’ through dynamic, highly non-linear, indeterministic processes that gave the digital media a critical, generative capacity in design. Even though the technological context of design became thoroughly externalized, its arresting capacity remains internalized (McCullough 1996)” (2001, 119–120).

17. While parametric shapes rely on parameters that communicate internally to generate forms, interactive algorithms communicate with parameters to respond to users, in which case an element of physical or sensory-motor interaction is usually required.

18. As Juliana Felkner, Eleni Chatzi, and Toni Kotnik note, “[t]he development of NURBS curves and surfaces was pioneered in the 1950s by the French engineers, Bézier and Casteljau, in their research for a sound mathematical description of free-form structures” (2013, 16).

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Subtitle: A Historical and Conceptual Investigation of a Cultural Technique and Its Vectors of Variation

Author.affiliation: Graduate School of Humanities, Faculty of Humanities, University of Amsterdam; Dept. of Media Studies, Faculty of Humanities, University of Amsterdam

Instructor.affiliation: Dept. of Media Studies, Faculty of Humanities, University of Amsterdam

*in-the-world*. By conceptualising topological calculation and its computation as

*cultural techniques*, it is argued that techniques can and should be critically analysed in their own right as

*mediators*with the capacity to introduce new and specific modes of ordering space and time-structures by reconfiguring or modelling their relationships differently. Building on previous insights by Luciana Parisi on algorithms in architectural and interaction design (2012, 2013), two lines of enquiry are developed to demonstrate what such an approach might look like: first, topological computation is situated within a larger conceptual space or genealogy of concepts and methods that developed across a variety of fields and disciplines. Second, I examine how these particular concepts and methods are mobilised or converge in specific instances of topological computation as technique or practical rationale, tied in with practices of contemporary computational architectural design and engineering that give it meaning and purpose. Rather than taking an

*object*as the starting point of enquiry, the approximation to defining its contours of demarcation as an object in-the-world is taken as the main

*objective*. This crosscutting approach to software and culture through techniques invites us to be skeptical of bifurcations – between the arts and sciences, the theoretical and empirical, the abstract or formal and concrete, the digital and physical, the structural and aesthetic, the skeleton and skin – and forces us to rethink concepts and methods based on geometrical assumptions.

Keywords: cultural techniques, topological calculation, invariance, control, digital design, computational architecture

Length.reading: 56 mins

*Analysis Situs*(1736, 1895); 3.2.

*Cybernetics*and Neocybernetics (1948, 1970s); 3.3. Responsive Techniques and

*Soft Architecture Machines*(late 1960s, 1975); 3.4. Cellular Automata and Generative Algorithms (1982, early 2000s); 4. Two Vectors of Variation within Computational Architectural Design and Engineering; 5. Conclusions; 6. Endnotes; 7. References

Date.evaluated: 13 Mar. 2015

Date.publishedonline: 15 Mar. 2015