(from The Interior Architecture Reader, edited by Gregory Marinic, 2018 Routledge)

Chapter 37

Topology and interiority.

Folding space inside

Johan Voordouw

Interiority, in relation to topology, is understood as an internalized centralization of space, a condition with an inherent spatial intimacy. Interiority is increasingly eroded in contemporary architectural practice; this enclosure has been methodically worn away.$^1$ The formal boundary of the interior, the interface of the inside to the outside, has in the past century theoretically and formally evolved from a defined threshold to a porous, ambiguous gradient. This chapter will focus on a specific aspect of this erosion, namely topological surfaces, and explore the characteristics of this gradient and speculate on strategies to re-establish interiority through topological geometries.

In a larger, meta context, current discourse of contemporary milieu is described in terms of seamless connectivity. This connectivity blurs the boundaries that have defined the social and cultural environment. Boundaries in recent discourse are not calcified or contained but pushed, transgressed, penetrated, and erased. Spatially, this continual assault on the

boundary is having a subversive effect on the interiority of architecture. The interior is erased by its continual link to the outside. This architectural erasure has taken root since modernism. It is not only the pervasive transparency of the curtain wall that “lets the outside in.” Transparency was an aesthetic and an ethic, opening up to a new world.$^2$

Before the digital turn, the “visible boundaries, such as walls or enclosures in general, give rise for their part to an appearance of separation between spaces where in fact what exists is an ambiguous continuity,” as described in Henri Lefebvre’s seminal book The Production of Space.$^3$ As we become more connected, and as practice becomes more digital, this continuity will be more ambiguous. Teyssot states that thresholds rarely constitute a barrier and that these thresholds are increasingly porous.$^4$ While Teyssot posits this condition in relation to public space, it equally applies to the interior of contemporary spaces. The increased openness of the interior (open space) runs concurrent to the increased porosity of the interior to the outside. Furthermore, the hybridity and ambiguity of programmatic thresholds have undermined the ability to define and enclose space; yet we reside inside.

Today’s inside/outside relationship is a stark opposition to modernity, when the facade was a wall of defense to the menacing city. Daily activities hidden behind the facade now play out on an open surface. The face of the building melting to become the field, the classic facade once hiding, and the topological field offers no such coverage. The outside is not only something to let in, there is no inside to be outside of. In her observation about Deleuze, Elizabeth Groz states that it is not that the inside is the opposite of the outside as much as the outside represents the real.$^5$ Architectural space remains in a virtual state of “becoming,” while the city and environment already pulsate with life. What are the implications of interiority if we “live” in a continual exterior? How do we define interiority as a primacy of place when space is continually linked physically, virtually, and digitally to the complex network beyond? If interiority is a centralization of space, and if this centralization facilitates and encloses social activities, how does architecture deal with topological structures that

are both internally and externally porous and social structures that are in a continual state of virtual connection?

Topology

The term “topology” was first published in a paper entitled “Vorstudien zur Topologie” by Johann Benedict Listing in 1847. The Mobius strip was defined eighteen years later, in 1865. Topology and architecture have a number of inherent contradictions. Mathematically, topology deals with scaleless deformation undermining a critical dimension of architecture; topologies are geometrical relationships devoid of scale and measure. Through deformation, different shapes are geometrically the same, which undermines the critical importance of form; topology studies the surface of geometry – it has no space. Topology remains critical to current computational architecture. To understand how topology has retained relevancy, it is critical to understand how the definition has morphed through its short but intensive use in architecture. By no means the first mention of topology in relation to architecture, Reyner Banham used the terms in his essay “The New Brutalism” to describe the Smithson’s Sheffield project, stating that the “connectivity” (topology) of the circulation, inside/outside, and penetration have always been important to architecture but had historically been subservient to platonic geometry (building form). Banham described the banality of topology when he stated that both a brick and a billiard ball could become the same topological shape through simple deformation. In Banham’s article, topology is separate from form, subservient in the architectural hierarchy of circulation, playing an important but secondary role to architectural form. In the coming decades, these two considerations would merge. Topology entered contemporary lexicon through Deleuze’s book Le pli: Leibniz et le Baroque (The Fold: Leibniz and the Baroque). Bernard Cache, Peter Eisenman, and Greg Lynn brought Deleuze’s book to a broader architectural audience in the mid-1990s in subsequent articles for Architectural Design magazine, most notably in Lynn’s March/April 1993 issue entitled “Folding in Architecture” and Cache’s “Earth Moves.”

Deleuze’s The Fold focused on Leibniz, his philosophical work, and his contributions to differential calculus. However it was Euler’s mathematical dilemma, “Seven Bridges of Konigsberg,” that was considered one of the earliest examples of topological deduction. Twenty years separate Leibniz’s death and the Konigsberg solution. Therefore, while topology has roots in the Baroque, its clarification was articulated later.

The fold as a formal practice in architecture later became synonymous with the Baroque. (This might stem from Leibniz’s life’s work and from Deleuze’s use of Wolfflin’s seminal book Renaissance und Barock.) While Deleuze cites the Baroque’s separation of facade and the autonomy of the interior, this was only a facet of the larger characteristics that define Wolfflin’s (ecclesiastical) Baroque architecture. While architectural topology could exhibit the somber massiveness, movement, painterly effects, and totalizing unity of Wolfflin’s Baroque topology, as Banham’s anecdote illustrates, is not inherently Baroque in form.$^9$ In relation to architecture, this shifts the development of topological surfaces to a different historical lineage. Topology as an architectural form and practice is a continuation of postmodernism, not a “Baroque” development of modernism.

Contemporary interest in topology stems from either the precision of its mathematical background, the ambiguity of its philosophical appropriation, its fluid formal characteristics, or a hybrid combination of all three. Eisenman’s original position defined “folding” as a process – not a product – that induced a notion of movement and continual variation.$^10$ Within a decade of Deleuze’s translated text, the definition of topology had expanded to appropriate its continual smoothness as a reaction to changing external events. It was understood as a flexible form, malleable to specific context and environments, not as representations but as undulating (folding) landscapes.$^11$ Here, topology gained the programmatic ambiguity and hybrid activity that would define its present articulation in practice. Lynn’s blobs were an early expression of this programmatic growth and fusion, undermining the predominance of form, a last reaction against modernism.$^12$ More recently, topology was expressed as the undulating deep surface, as Lars Spuybroek defined, that could blister, filo, or sponge, its architectural

articulation a continuation of Frei Otto’s critical work on minimal surfaces and tensile structures.$^{13}$

The architectural practicalities of topology are connected to the development of suitable modeling software and computation. Topology in architecture is an inherent result of the “digital revolution” that has taken shape since the mid-1980s. Mario Carpo notes the unique peculiarity of the digital revolution and its postmodern staging point, “It [is] a revolution with no identifiable history and no predetermined destination.”$^{14}$ This might explain its broad use and ambiguity. While Carpo states that topology is the “embodiment of the new computer age,” the connection of computers to topological geometry was built “on a truism but generalized into a fallacy”; while the computer facilitates complex form its does not presuppose curvilinear forms.$^{15}$ When Greg Lynn introduced “folding in architecture,” the reference point was a broad index of interests that swept topology, morphology, and the like to herald a new “digital turn”; beyond the formal qualities of curvilinear form topology is the study of surface.$^{16}$ As surface, which might become continuous to define form, its spatial study is elusive. At its architectural core is how we define the surface that encloses space to become the inside. Whether this interiority is the corporeal surface of the body (skin) or the material surface of an architectural skin, the boundary needs both depth and enclosure.$^{17}$

Through computation, there is an increased gap between form (which is logically, parametrically, defined) and space, which is now a secondary result of form. Modernism was transparent but enclosed. Contemporary topological surfaces are opaque but their thresholds are porous. The opaqueness defines the surface but does little to define the interior; the continuity of the surface prevents the (programmatic) definition of the interior and the porosity prevents its (spatial) enclosure.

With specific topological surfaces, such as the Mobius strip, the interior and exterior oscillate as the surface folds over and loops around. This is compelling programmatically but has inherent spatial issues. If the surface is given thickness, it gains a form but not a “space.” Space is now a result of form, not a means to form. This is consequential because to

understand contemporary interiority we must understand digital practice on its own terms and in its own time. The topological imperative seems to foster the requirement that “A revolution that has not produced a new space has not realized its full potential; indeed it has failed in that it has not changed life itself but has merely changed ideological superstructures, institutions, or political apparatuses.” As a “revolution” with no identifiable beginning and no inherent direction, it affords us the opportunity to consider and focus on the present. Current topological considerations, even when vertically stacked, inhabit the surface as ground. It invokes the rolling hills of the landscape; it is a metaphorical expression of the environmental concerns of the outside world as continuity of inner life. In this respect, interiority of topological structures is critical to the current architectural milieu because it exemplifies the twenty-first century’s most pressing concerns. While it is poor in illustrating the interconnected network of (social) communication, as a field it astutely expresses environmental condition without resorting to environmentalism. However, this inhabitation must be internalized. For architecture to better express the issues of the outside, it must find a counterbalance within itself. For topology to express space, its surface must enclose.

Erasure of interiority

Walter Benjamin sat at a threshold. The interiority of nineteenth-century Paris was very specific to its time and place. His time was both the end of the Beaux Arts facade and the rise of the steel frame and its growing transparency. It might be purposeful to distinguish between these two spaces, the heavy wall and the light frame, and how in contemporary practice these two elements of architectural enclosure have evolved. The “mask,” as Giedion described the classical facade, was the visible expression of classical composition while the architectural “truth” lay hidden in the steel structure behind the stone.

Topology undermines both of these defined conditions. The porosity of the topological boundary erodes the solidity of the facade. With

the facade removed, the interiority that once contained the modern apartment is opened up like the back of a dollhouse, exposed to the outside. The separation of the facade from the interior is untenable in topological surfaces where the walls, floor, and ceiling are varying deformations of the same continuous surface. Furthermore, if we invert Benjamin’s glass-covered, steel-framed arcade from interiority to exteriority, it more aptly forms the lineage towards transparency and nomadism that, as stated by Deleuze and Guattari, define the spaces of the later twentieth century.$^{20}$ The “truth” of structure is once more hidden, embedded in the continuous surface that facilitates inhabitation. With topology, the arcade as wandering path expands from line to network to field, from infrastructure to surface. The interior is therefore affronted from two directions: one is the erosion of the facade from the outside, the second is the increased transparency and openness within space itself. “Open space” is increasingly synonymous with the outside.

Space is not found, it is structured, and from this structure interiority is constructed. To consider the ground as the plane of inhabitation ignores the volume needed to physicalize human presence. Structured space argues for a tectonic response that undermines the hegemony of surface. Tim Ingold writes, “the hegemony of the straight line is a phenomenon of modernity, not of culture in general.” He continues, “Since the straight line can be specified by numerical values, it becomes an index of quantitative rather than qualitative knowledge.$^{21}$ This might be true of computational surfaces as well. To quantify mathematically is to know materially but not to understand culturally or spatially. The interior was so easily erased as its discourse was qualitative and its boundaries increasingly vague. As Lefebvre writes, “all ‘subjects’ are situated in a space in which they must either recognize themselves or lose themselves, a space which they may both enjoy and modify.”$^{22}$ The definition of interiority need not revert to the static compartmentalization of rooms for space to be constructed, nor does the infinitely folded surface that ideally defines topology need to fragment to staidly define the floor, walls, and ceiling. Topology must simply fold in such a manner that deforms surface to form

space. Interiority as a defined inside to a surrounding outside can retain the porosity but rescind the ambiguity that has previously undermined its articulation. In its present construction, for all the dynamism of topological surfaces, it remains as staid and static as the Dom-ino system of Le Corbusier. While it facilitates passive meandering, it stifles modification. While it is variable, it remains immobile. It remains an open question how one might both enjoy and modify, but for that condition to exist the surface must spatialize, the interior must be demarcated, and the exterior must be defined as something “out there.”

Centralizing space It might be a contradiction that architecture is concerned with space when for such long periods of its practice architecture was unconcerned with the interior. Laugier’s primitive hut was open to the elements, its roof and columns a vague definition of an interior. There is a difference between the architectural history of Giedion and the philosophy of Deleuze when they speak of space and interior, respectively. Giedion speaks of the first space conception as volumes in space, outside looking at the object. He cites the lack of interior in ancient structures (Egypt, Sumeria, and Greece) because the space contained no windows to connect to the outside.$^{23}$ To Deleuze, the interior (monad) is completely enclosing inside the object, with no connection to the outside. Deleuze notes, “The monad is the autonomy of the inside, an inside without an outside.”$^{24}$ Deleuze further separates the enclosing walls from the interior when he continues:

It [monad] has as its correlative the independence of the facade, an outside without an inside. Now the facade can have doors and windows – it is riddled with holes – although there may be no voids, a hole being only the site of a more rarefied matter. The doors and windows of that matter open or even close only from the outside and onto the outside.$^{25}$

This hole is topological. There is no void because the topology is a surface. The hole in the surface has no void because there is no volume. Giedion’s first conception has no space, Deleuze’s monad is space but no architecture, his facade has surface but no space. In both conditions, architecture struggles to materialize.

There are many other lateral anecdotes that describe the erosion of interiority. Tim Ingold’s “zones of entanglement” are particularly interesting because they describe a “meshwork of interwoven lines – there are not insides or outsides, only openings and ways through.” This tangentially describes Greg Lynn’s “felt,” which was a seminal metaphor in early computation. This meshwork describes as much Spuybroek’s “Gothic logic” of lines weaving and bifurcating to form surfaces as it does the lines of an interconnected system that define a facet of today’s contemporary practice. Continually, there is negation of the center (Euclidean) to the expansiveness of the landscape (quasi-Cartesian). “The emphasis is on continuity of the landscape, a smoothness, with a notable reciprocal influence of the environment on building.” Tangential continuation through this expanding field might arrive at Gregotti’s advice to start an architectural work at the geographical scale. Relayed in Bernard Cache’s Earth Moves, Gregotti states, “beyond the built frame there is the site, but beyond that still, there is the outside. Geography is not the surroundings of the building, but rather the impossibility of its closure.” This linkage might be incidental, but it describes the decentralized vantage point from which architecture is viewed. As Cache intimates,

The ground plane rarefies the surface of the earth in order to allow human activities to take shape. As these surfaces become increasingly smooth and continuous, their grip is reduced to a minimum. The stairway becomes a ramp and the ground falls away.

Topological interiors

There are a number of examples that describe topological interiors. Kiesler’s Endless House (1924–50), lifted off the ground, was a separation of space from ground. Conversely, Jacques Gillet’s Sculpture House (1962) was cave-like and contained. Interiority is not an argument for pure separation, recreating Giedion’s first space conception to replace Deleuze’s neo-Baroque. Furthermore, it is not an argument for a calcified interior, the staid rooms of classicism hidden behind the massive facade. It merely argues for a defined interior to partner and counter the pervasive exteriority that defines much of the current discourse.

In mathematical terms, points on a topological surface are not equal. Therefore, in architectural terms, a hierarchy lies latently embedded in the folds. The question remains whether topology is used to form a porous surface, folding out to merge with the exterior, or conversely begins to wrap and form a new interior. The outside will always be present, but interiority in architecture must be nurtured and preferred if space is to be constructed and experienced. If the center of topologies is critical, the definition of the edge is equally important. Topological surfaces currently have two edge conditions: one is blended into the landscape; the other is a violent cut of the surface, extruded like the side of a cliff revealing its folds. Both conditions deal poorly with the boundary of the outside. In the former, the outside merges in; in the latter the transparent edifice puts all interiority on show. Here Lars Spuybroek (NOX) offers a direction forward. The 1993–97 H2Oexpo offers an enclosed topology. The pavilion constructs a surface for active engagement. It combines ambiguity to facilitate new disparate modes of experience with defining a tectonic and space on the inside. The Teshima Art Center, by Ryue Nishizawa and structural engineer Mutsuro Sasaki, offers a more recent example (Figure 37.1). The pure “nothingness” of space that encapsulates the public inside a covered space beautifully frames the outside. This interiority is not for protection. It is simply a definition and difference from the outside in all its poetic subtlety. Additional works by SANAA at the Rolex Learning Center at the Ecole Polytechnique Fédérale de Lausanne explore the undulating plane as habitable surface, and Toyo Ito’s

Taichung Metropolitan Opera House is a topological surface that defines both vertical as well as horizontal space over a series of connected floors.

Interiority is not so much a defense against the outside as a counterpoint to recognize its boundaries and clarify its variability. “Threshold is a zone formed by precise tectonics, an area of knowledge and even of knowability.” Knowing comes from definition; the gradient that is continually blurred is also the zone that prevents knowing. Tectonics speak to the material condition and the finite limitations that distinguish between constructed and unconstructed. The threshold is an “interval between things,” not a continuity seemingly without end. This articulation could suggest the wandering path that traverses the floor surface of SANAA’s Rolex Learning Center, the ambiguity of the undulating surface periodically

clarifying in the line of a walkway, momentarily materialized to denote scale, direction, and purpose (Figure 37.2).

If the interior is the defining characteristic of architecture, then it might be this clarity upon which to shape and wrap practice. This new étui is not so much the hermetic shell of modernity as the materialized network of the digital age. This new topological interiority would not only ask new questions of embodiment, but also would use the digital systems that are pervasively augmenting and fabricating architectural practice to privilege the “real” not only outside, but in any place that the body chooses to wander and reside.

Notes

1. Interface is used in conventional terms “where two things meet,” although the term was selected to play with the idea of the computer interface. An early interest of digital architecture was the use of media, internet, and projection in eroding the boundary of interior and exterior space. This chapter circumvents the media/transarchitecture lens of early computation to focus predominantly on geometry.

2. Maria Rita Perbellini and Christian Pongratz, Natural Born CAADesigners: Young American Architects (Basel: Birkhauser, 2000), p. 9.

3. Henri Lefebvre, The Production of Space, trans. Donald Nicholson-Smith (Malden, MA: Blackwell Publishing, 1991), p. 87.

4. Georges Teyssot, A Topology of Everyday Constellations (Cambridge, MA: MIT Press, 2013), p. 266.

5. Elizabeth Groz, Architecture From the Outside: Essays on Virtual and Real Space (Cambridge, MA: MIT Press, 2001), p. 66.

6. Basic topology information from conversations and e-mail correspondence with Kyle Harvey of Carleton University, School of Mathematics and Statistics (February 2015) and the history of topology from e-mail correspondence with Tom Archibald of Simon Fraser University (May 2015).

7. Reyner Banham, “The New Brutalism”, The Architectural Review 118, 708 (December 1955): 355–361.

8. The Alphabet and the Algorithm (Cambridge, MA: MIT Press, 2011), pp. 85–86.

9. Heinrich Wolfflin, Renaissance and Baroque, trans. Kathrin Simon (London: Fontana Collins, 1964), as represented by the four chapters and content in part one.

10. Mario Carpo, The Alphabet and the Algorithm (Cambridge, MA: MIT Press, 2011), pp. 87–89.

11. Maria Rita and Pongratz, Natural Born CAADesigners, p. 91.

12. Ibid., p. 24, 25.

13. Lars Spuybroek, NOX: Machining Architecture (London: Thames & Hudson, 2004), p. 12.

14. Carpo, The Alphabet and the Algorithm, p. 110.

15. Ibid., p. 85.

16. Ibid., p. 89.

17. Groz, Architecture From the Outside, p. 32.

18. Lefebvre, The Production of Space, p. 54.

19 Teyssot, A Topology of Everyday Constellations, p. 266. 20 Ibid., pp. 99–101. 21 Tim Ingold, Lines: A Brief History (London: Routledge, 2007), p. 155. 22 Lefebvre, The Production of Space, p. 35. 23 Sigfried Giedion, The Eternal Present: The Beginnings of Architecture (London: Oxford University Press, 1964), p. 521. 24 Gilles Deleuze, The Fold: Leibniz and the Baroque, trans. Tom Conley (London: Continuum, 2006), pp. 31–32. 25 Ibid. 26 Ingold, Lines, p. 103. 27 Greg Lynn, “Architectural Curvilinearity: The Folded, the Pliant and the Supple”, Architectural Design (Folding in Architecture) 63, 3/4 (March/April 1993): 15–24. 28 Spuybroek, NOX: Machining Architecture, p. 163. 29 Alicia Imperiale, New Flatness: Surface Tension in Architecture (Basel: Birkhauser, 2000), p. 31. 30 Bernard Cache, Earth Moves (Cambridge, MA: MIT Press, 1995), pp. 68–70. 31 Ibid., pp. 25–26. 32 Jacques Leenhardt, “Le passage comme forme d’expérience: Benjamin face à Aragon”, in Walter Benjamin et Paris, ed. Heinz Wismann (Paris: Cerf, 1986), p. 169. 33 Ibid., p. 169.