Topology and interiority: Folding space inside

Author: Johan Voordouw Year: 2018 Source type: chapter (Ch. 37 of The Interior Architecture Reader, ed. Gregory Marinic, Routledge) Raw file: raw/chapters/voordouw-2018-topology-and-interiority.md

Summary

Voordouw's chapter names interiority, in relation to topology, as "an internalized centralization of space, a condition with an inherent spatial intimacy" — and diagnoses its methodical erosion in contemporary practice, where "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." Focusing on topological surfaces, the chapter reconstructs topology's architectural career — Banham's 1955 usage subservient to form, Deleuze's The Fold via Lynn, Cache, and Eisenman, Spuybroek's deep surface, Carpo's "digital revolution" — and identifies the central paradox: "Modernism was transparent but enclosed. Contemporary topological surfaces are opaque but their thresholds are porous." The chapter's programme is to re-establish interiority through topological geometries themselves: "Topology must simply fold in such a manner that deforms surface to form space" — retaining porosity while rescinding ambiguity, demarcating the interior, and defining "the exterior [as] something 'out there.'"

Key Claims

Interiority defined as centralization: "Interiority, in relation to topology, is understood as an internalized centralization of space, a condition with an inherent spatial intimacy" — and this centralization "facilitates and encloses social activities" (opening). The definition is spatial-centralizing, not psychological: the chapter's interiority is a built condition before it is a state of mind.
The threshold-to-gradient evolution: the formal boundary of the interior has, "in the past century theoretically and formally evolved from a defined threshold to a porous, ambiguous gradient" — with the gradient itself later identified as the "zone that prevents knowing" ("Knowing comes from definition; the gradient that is continually blurred is also the zone that prevents knowing").
Erosion by connectivity: in current discourse of "seamless connectivity," boundaries are "not calcified or contained but pushed, transgressed, penetrated, and erased"; "The interior is erased by its continual link to the outside. This architectural erasure has taken root since modernism."
Transparency as ethic, not just aesthetic: the erosion 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" — the modernist erasure was doctrinal, not merely technological.
Lefebvre's ambiguous continuity (The Production of Space, p. 87): "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" — and Voordouw's update: "As we become more connected, and as practice becomes more digital, this continuity will be more ambiguous."
Teyssot's porous thresholds (A Topology of Everyday Constellations, p. 266): thresholds "rarely constitute a barrier" and are "increasingly porous" — posited for public space but, Voordouw argues, "it equally applies to the interior of contemporary spaces."
The erasure's strongest formulation: "The outside is not only something to let in, there is no inside to be outside of." Via Grosz (Architecture from the Outside, p. 66): "it is not that the inside is the opposite of the outside as much as the outside represents the real." Architectural space "remains in a virtual state of 'becoming,' while the city and environment already pulsate with life."
Topology's mathematical contradictions with architecture: mathematical 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's origin dating: the term was first published in Johann Benedict Listing's "Vorstudien zur Topologie" (1847); the Möbius strip was defined eighteen years later (1865). Euler's "Seven Bridges of Königsberg" — the earliest topological deduction — was solved twenty years after Leibniz's death: "while topology has roots in the Baroque, its clarification was articulated later."
Banham's 1955 priority ("The New Brutalism"): Banham used topology for the Smithsons' Sheffield project — the "connectivity" (topology) of circulation, inside/outside, and penetration, historically "subservient to platonic geometry (building form)." His brick/billiard-ball deformation anecdote shows "the banality of topology." In Banham, "topology is separate from form, subservient in the architectural hierarchy of circulation… In the coming decades, these two considerations would merge."
The Deleuzian entry and its mediators: topology entered the contemporary lexicon through Deleuze's The Fold: Leibniz and the Baroque, brought to a broader architectural audience by Bernard Cache, Peter Eisenman, and Greg Lynn in the mid-1990s — most notably Lynn's Architectural Design March/April 1993 issue "Folding in Architecture" and Cache's "Earth Moves."
The historiographic correction: the fold's Baroque association may stem from Deleuze's use of Wölfflin's Renaissance und Barock, but "architectural topology could exhibit the somber massiveness, movement, painterly effects, and totalizing unity of Wölfflin's Baroque… [and yet] is not inherently Baroque in form." Conclusion: "Topology as an architectural form and practice is a continuation of postmodernism, not a 'Baroque' development of modernism."
Eisenman's process-folding: Eisenman's original position "defined 'folding' as a process – not a product – that induced a notion of movement and continual variation." Within a decade the definition expanded to "continual smoothness as a reaction to changing external events… a flexible form, malleable to specific context and environments, not as representations but as undulating (folding) landscapes" — gaining "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"; Spuybroek later expressed topology as "the undulating deep surface… that could blister, filo, or sponge," a continuation of Frei Otto's minimal surfaces and tensile structures.
Carpo's digital revolution (The Alphabet and the Algorithm): "It [is] a revolution with no identifiable history and no predetermined destination" — which "might explain its broad use and ambiguity." Topology is "the embodiment of the new computer age," but the connection of computers to topological geometry "was built on a truism but generalized into a fallacy": the computer facilitates complex form but "does not presuppose curvilinear forms."
The form/space gap: "Through computation, there is an increased gap between form (which is logically, parametrically, defined) and space, which is now a secondary result of form." On the Möbius strip: "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."
The chapter's pivotal chiasmus: "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."
The Lefebvrian revolution test (The Production of Space, p. 54): "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" — the topological imperative measured against space-production and found wanting.
Landscape as metaphor: topological surfaces "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"; "as a field it astutely expresses environmental condition without resorting to environmentalism. However, this inhabitation must be internalized… For topology to express space, its surface must enclose."
Benjamin at the threshold: Benjamin's nineteenth-century Paris was "both the end of the Beaux Arts facade and the rise of the steel frame and its growing transparency"; Giedion's "mask" — the classical facade as visible expression while architectural "truth" lay hidden in the steel behind the stone.
The dollhouse image: "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 continuous surface's untenability: "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." Inverting Benjamin's arcade "from interiority to exteriority… more aptly forms the lineage towards transparency and nomadism" (Deleuze & Guattari); "The arcade as wandering path expands from line to network to field, from infrastructure to surface."
The two-front affront: "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."
Structured space: "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."
Ingold's quantitative line (Lines, p. 155): "the hegemony of the straight line is a phenomenon of modernity, not of culture in general"; the straight line is "an index of quantitative rather than qualitative knowledge" — "To quantify mathematically is to know materially but not to understand culturally or spatially." Corollary: "The interior was so easily erased as its discourse was qualitative and its boundaries increasingly vague."
Lefebvre's enjoy-and-modify test (The Production of Space, p. 35): "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."
The porosity/ambiguity distinction: "Interiority as a defined inside to a surrounding outside can retain the porosity but rescind the ambiguity that has previously undermined its articulation." The programme: "Topology must simply fold in such a manner that deforms surface to form space."
The Dom-ino critique: "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."
Giedion vs. Deleuze on interior: Giedion's first space conception is volumes in space, "outside looking at the object" (ancient Egypt, Sumeria, Greece lacked interior because the space "contained no windows to connect to the outside"); Deleuze's monad is "completely enclosing inside the object, with no connection to the outside" — "The monad is the autonomy of the inside, an inside without an outside." The monad's 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 topological hole: "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."
Ingold's meshwork (Lines, p. 103): "zones of entanglement" — "a meshwork of interwoven lines – there are not insides or outsides, only openings and ways through" — describing as much Greg Lynn's "felt" (the seminal early-computation metaphor) as Spuybroek's "Gothic logic" of lines "weaving and bifurcating to form surfaces."
Gregotti's impossibility of closure (via Cache's Earth Moves): "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" — "the decentralized vantage point from which architecture is viewed."
Cache's smooth ground: "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."
The two historical 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… nor… 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."
Latent hierarchy: "In mathematical terms, points on a topological surface are not equal. Therefore, in architectural terms, a hierarchy lies latently embedded in the folds."
The two failing 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."
The enclosed-topology exemplars: Spuybroek's H2Oexpo (1993–97) "offers an enclosed topology… It combines ambiguity to facilitate new disparate modes of experience with defining a tectonic and space on the inside"; the Teshima Art Center (Nishizawa with engineer Mutsuro Sasaki): "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"; SANAA's Rolex Learning Center explores "the undulating plane as habitable surface"; 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."
Counterpoint, not defense: "Interiority is not so much a defense against the outside as a counterpoint to recognize its boundaries and clarify its variability."
The threshold's knowability (Leenhardt, p. 169, via Voordouw): "Threshold is a zone formed by precise tectonics, an area of knowledge and even of knowability." "The threshold is an 'interval between things,' not a continuity seemingly without end."
The new étui: "This new étui is not so much the hermetic shell of modernity as the materialized network of the digital age" — a topological interiority that 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."

Connections

Concepts: topology, topological-interiority, threshold-to-gradient, interiority, boundaries, enclosure, inside-outside-dialectic, objectile, performative-envelope, hermetic-seal, digital-architecture, computational-design, parametricism, postmodernism, liminality, smooth-and-striated-space, surface-mathematics, arcades, primitive-hut, space-shape, dematerialization

Authors: voordouw-johan, marinic-gregory (editor), deleuze-gilles, lefebvre-henri, benjamin-walter, giedion-sigfried, banham-reyner, eisenman-peter, spuybroek-lars, le-corbusier

Debates: static-vs-dynamic-architecture (the "variable but immobile" critique of the dynamic pole), postmodernism-vs-modernism (topology as postmodernism's continuation)

Related source notes: Vanucci 2018 (the envelope as interiority's agent — Voordouw's continuous surface is the envelope's degenerate case), Hannah 2018 (the objectile and the fold's genealogy), Anderson & Lovell-Anderson 2018 (surface mathematics — the interior-scale counterpart whose volume-forming surfaces Voordouw's topological surfaces lack), Bachelard 1958/2006 (the inside/outside dialectic Voordouw re-poses as gradient), Macken 2018 (the Reader's preceding interiority register), Goulthorpe 2002/2006 (the Aegis Hyposurface — the actuated surface ancestor), Dubbeldam 2002/2006 (the fold as boundary-crease)

Direct Quotes

"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."

"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."

"Boundaries in recent discourse are not calcified or contained but pushed, transgressed, penetrated, and erased."

"The interior is erased by its continual link to the outside. This architectural erasure has taken root since modernism."

"The outside is not only something to let in, there is no inside to be outside of."

"Mathematically, topology deals with scaleless deformation undermining a critical dimension of architecture; topologies are geometrical relationships devoid of scale and measure."

"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 as an architectural form and practice is a continuation of postmodernism, not a 'Baroque' development of modernism."

"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."

"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."

"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."

"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."

"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."

"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.'"

"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."

"In mathematical terms, points on a topological surface are not equal. Therefore, in architectural terms, a hierarchy lies latently embedded in the 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."

"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."

"Interiority is not so much a defense against the outside as a counterpoint to recognize its boundaries and clarify its variability."

"Knowing comes from definition; the gradient that is continually blurred is also the zone that prevents knowing."

"This new étui is not so much the hermetic shell of modernity as the materialized network of the digital age."

"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." (Lefebvre, The Production of Space, p. 54, quoted)

"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" (Lefebvre, The Production of Space, p. 87, quoted)

"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" (Lefebvre, The Production of Space, p. 35, quoted)

"The monad is the autonomy of the inside, an inside without an outside." (Deleuze, The Fold, pp. 31–32, quoted)

"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." (Deleuze, The Fold, pp. 31–32, quoted)

"it is not that the inside is the opposite of the outside as much as the outside represents the real" (Grosz, Architecture from the Outside, p. 66, quoted)

"It [is] a revolution with no identifiable history and no predetermined destination." (Carpo, The Alphabet and the Algorithm, p. 110, quoted)

"the hegemony of the straight line is a phenomenon of modernity, not of culture in general… Since the straight line can be specified by numerical values, it becomes an index of quantitative rather than qualitative knowledge." (Ingold, Lines, p. 155, quoted)

"a meshwork of interwoven lines – there are not insides or outsides, only openings and ways through" (Ingold, Lines, p. 103, quoted)

"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." (Gregotti, via Cache, Earth Moves, pp. 68–70, quoted)

"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." (Cache, Earth Moves, pp. 25–26, quoted)

"Threshold is a zone formed by precise tectonics, an area of knowledge and even of knowability." (Leenhardt, p. 169, quoted)

Open Questions

The chapter's programme — "Topology must simply fold in such a manner that deforms surface to form space" — is stated as a criterion, not a method. What folding operation spatializes a surface without reverting to "the static compartmentalization of rooms"? Is there a computable definition of a fold that encloses (as H2Oexpo and Teshima do) versus one that merely undulates?
If "the surface must spatialize, the interior must be demarcated, and the exterior must be defined as something 'out there,'" is the resulting architecture still topological — or does demarcation reintroduce exactly the scale, measure, and discreteness that mathematical topology excludes? Can an interior be topological and demarcated at once?
"It remains an open question how one might both enjoy and modify" (Lefebvre's test): the chapter shows topological surfaces stifling modification ("While it is variable, it remains immobile"). Would a spatializing fold answer Lefebvre's demand — or is modification-in-dwelling the one demand the continuous surface structurally cannot meet?
The two failing edge conditions (landscape-blend and violent cut) "deal poorly with the boundary of the outside" — is there a third edge condition, and would it be a threshold (Leenhardt's "interval between things") in topological form? The Rolex Learning Center's walkway that "periodically clarif[ies]" is offered as the beginnings of one — is a line of clarification sufficient demarcation?
The chapter circumvents "the media/transarchitecture lens of early computation to focus predominantly on geometry" (note 1). Does the geometric account transfer to the media-driven erosion of interiority (the smartphone interior, the data-bound envelope), or does the gradient's cause lie elsewhere than geometry?
If "hierarchy lies latently embedded in the folds" (points on a topological surface are not equal), what social or programmatic order does the fold secretly encode — and who reads it? (Cf. Hillier & Hanson's boundary-as-category-and-control.)
The chapter dates topology's architectural relevance from Banham (1955) and its computational formulation from the 1990s, arguing it is "a continuation of postmodernism." Does this periodization survive Carpo's counter-claim that the digital revolution has "no identifiable history" — and does the fold's postmodernism re-open the question of what postmodernism's continuation now is?