Topology

Definition

Topology names two things in architecture that Voordouw's chapter holds apart and then joins: the mathematical study of surfaces and their properties under deformation, and the architectural practice of continuous, undulating form that grew from Deleuze's The Fold and computational design. Mathematically — the term was first published in Johann Benedict Listing's "Vorstudien zur Topologie" (1847), the Möbius strip defined eighteen years later (1865), Euler's "Seven Bridges of Königsberg" the earliest topological deduction — 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." These are the inherent contradictions topology carries into architecture: a geometry without scale, a form-theory that undermines form, a surface-study with no space.

Architecturally, Voordouw's historiography runs: Banham (1955) used topology in "The New Brutalism" for the Smithsons' Sheffield project — the "connectivity" of circulation, inside/outside, and penetration, then "subservient to platonic geometry (building form)"; his brick/billiard-ball deformation anecdote displays "the banality of topology," yet "In the coming decades, these two considerations would merge." Deleuze's The Fold: Leibniz and the Baroque entered the lexicon through Bernard Cache, Peter Eisenman, and Greg Lynn in the mid-1990s Architectural Design — most notably Lynn's March/April 1993 "Folding in Architecture" and Cache's "Earth Moves." Eisenman 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," acquiring "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's "undulating deep surface… could blister, filo, or sponge," continuing Frei Otto's minimal surfaces and tensile structures. Voordouw's historiographic correction: "Topology as an architectural form and practice is a continuation of postmodernism, not a 'Baroque' development of modernism" — the fold's Baroque association stemming from Deleuze's use of Wölfflin rather than from any inherent Baroque form.

The concept's architectural crisis is 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, interior and exterior oscillate as the surface folds — "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." Hence the 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 concept's repair is topological-interiority: "Topology must simply fold in such a manner that deforms surface to form space."

Key Thinkers

• Johan Voordouw — the concept's architectural historian and diagnostician: the Banham origin (topology subservient to form, then merged), the Deleuzian entry via Lynn/Cache/Eisenman, the postmodernist periodization, the form/space gap, and the opaque-but-porous paradox (2018).

• Gilles Deleuze — The Fold: Leibniz and the Baroque (1988/1993), the philosophical source: the monad as "the autonomy of the inside, an inside without an outside," with a facade "riddled with holes" where "a hole [is] only the site of a more rarefied matter" — the hole Voordouw reads as topological. See objectile.

• Reyner Banham — the first architectural usage (1955): topology as circulation-connectivity, subservient to form, illustrated by the brick/billiard-ball deformation — "the banality of topology."

• Peter Eisenman — the process-folding position: "folding" as "a process – not a product – that induced a notion of movement and continual variation."

• Greg Lynn — the popularizer and blob-theorist: the 1993 AD "Folding in Architecture" issue; blobs as "programmatic growth and fusion, undermining the predominance of form, a last reaction against modernism"; "felt" as the seminal early-computation metaphor (not yet paged — see log).

• Bernard Cache — Earth Moves (1995): the ground plane that "rarefies the surface of the earth… 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 objectile's source. See objectile (not yet paged — see log).

• Lars Spuybroek — the deep-surface practitioner (NOX): "the undulating deep surface… that could blister, filo, or sponge," continuing Frei Otto; his H2Oexpo (1993–97) as the enclosed-topology exemplar; the "Gothic logic" of lines weaving and bifurcating to form surfaces.

• Mario Carpo — the media historian: topology as "the embodiment of the new computer age," the digital revolution "with no identifiable history and no predetermined destination," and the truism-fallacy correction — the computer facilitates complex form but "does not presuppose curvilinear forms" (not yet paged — see log).

• Frei Otto — the form-finding ancestor: minimal surfaces and tensile structures as the pre-digital ground of the deep surface (see form-finding).

• Heinrich Wölfflin — the misattributed ancestor: Renaissance und Barock as the source (via Deleuze) of the fold's Baroque association, which Voordouw separates from topology's actual formal lineage (not yet paged — see log).

• David Erdman — the urban-scale counter-usage: topology lifted from surface to city — the "topological city" with its "highly continuous, centrifugal qualities (and the economies and politics that drive them) is a rare and dying breed"; the built-out twenty-first-century city will instead "grow inward, … reconfigure [its] interior," with design pulling "centripetal[ly]" (alteration "pulls and tucks architecture into shape") — topology's continuity diagnosed as the condition whose time has passed, not the one to repair (2018). See topological-interiority for the repair-program Voordouw proposes at surface scale.

Related Concepts

• topological-interiority — the concept's interior programme: interiority as centralization, re-established by folding that encloses; the erasure of interiority as the topological surface's unresolved condition.
• threshold-to-gradient — the boundary condition the topological surface instantiates: the porous, ambiguous gradient that defines surface without defining interior.
• objectile — the fold's object-level concept (Deleuze/Cache/Eisenman): "a temporal modulation that implies a continual variation of matter" — the philosophical twin of architectural topology.
• digital-architecture — the enabling medium: "Topology in architecture is an inherent result of the 'digital revolution' that has taken shape since the mid-1980s" (Voordouw); Carpo's "embodiment of the new computer age" with its truism-fallacy correction.
• parametricism — the logical form-side: "form (which is logically, parametrically, defined)" as the term space has become secondary to.
• performative-envelope — the envelope-theory relative: Vanucci's hyperarticulated envelope differentiates the skin to produce interiority; Voordouw's continuous topological surface is the degenerate case where walls, floor, and ceiling are "varying deformations of the same continuous surface" and interiority goes undefined.
• surface-mathematics — the interior-scale counterpart: Anderson & Lovell-Anderson's shape < surface < volume ladder makes volume "the three-dimensional space enclosed by a collection of surfaces" — precisely the volume-forming the topological surface lacks.
• smooth-and-striated-space — the Deleuzian smoothness: the "continual smoothness" of the topological field and Cache's falling-away ground as the smooth space of the landscape-continuum.
• boundaries — the concept's boundary-form: the topological boundary as gradient — "the outside merges in" (landscape-blend) or "the transparent edifice puts all interiority on show" (violent cut).
• hermetic-seal — the contrast case: modernism's "transparent but enclosed" against the topological "opaque but porous" — two failing enclosures in opposite directions.
• inside-outside-dialectic — the division topology dissolves: on the Möbius strip interior and exterior "oscillate as the surface folds over and loops around."
• interiority — the concept's stake: "an internalized centralization of space, a condition with an inherent spatial intimacy," whose erosion topology both exemplifies and could repair.
• static-vs-dynamic-architecture — the dynamism question: "for all the dynamism of topological surfaces… While it is variable, it remains immobile."
• postmodernism — the periodization: topology as postmodernism's continuation rather than modernism's Baroque development.
• form-finding — the Otto lineage: minimal surfaces and tensile structures as the deep surface's structural ancestor.
• arcades — the inverted lineage: Benjamin's arcade read "from interiority to exteriority," the wandering path expanding "from line to network to field, from infrastructure to surface."

Source Support

Sources in the wiki that discuss this concept:

• Topology and interiority: Folding space inside (Voordouw 2018) — the concept's source: the Listing/Möbius/Königsberg dating; the Banham 1955 origin ("connectivity" subservient to platonic geometry; the brick/billiard-ball banality); the Deleuzian entry via Lynn, Cache, Eisenman; Eisenman's process-folding; the blobs and the deep surface; Carpo's digital revolution and truism-fallacy; the form/space gap; the opaque/porous chiasmus; the two failing edge conditions; the enclosed-topology exemplars (H2Oexpo, Teshima Art Center, Rolex Learning Center, Taichung Metropolitan Opera House); "Topology must simply fold in such a manner that deforms surface to form space."

• Event-space (Hannah 2018) — the fold's ontological register: Deleuze's objectile, "a temporal modulation that implies a continual variation of matter"; the avant-garde precedence claim and the non-digital thesis — the philosophical companion to Voordouw's historiography of the same texts. See objectile.

• Thing-Shapes (Dubbeldam 2002/2006) — the responsive-surface register: the crease and the folded wrapper as the boundary-condition cousin of the topological surface (see crease, thing-shapes).

• Transforming Interior Volumes (Anderson & Lovell-Anderson 2018) — the mathematical counterpart: surfaces mathematically defined to produce interior volume (shape < surface < volume) — the volume-forming capacity Voordouw's continuous topological surfaces lack. See surface-mathematics.

• Altered (e)states (Erdman 2018) — the urban-scale counter-register: the "topological city" (centrifugal, continuous, outward-growing) declared "a rare and dying breed" against the twenty-first-century city that is "built out, episodic, layered, and interiorized" — with the centripetal/centrifugal pair transferred from geometry to urbanism ("Alteration could be called a more 'centripetal,' rather than 'centrifugal,' approach to design"). Where Voordouw's chapter repairs the topological surface's missing interior, Erdman's declares the continuous urban topology economically and physically finished — "cities will have to consider how to adapt and alter their existing structures to grow inward." The two chapters frame the wiki's topology cluster from opposite ends: the fold that must form space, and the continuity whose era is over. See alteration, topological-interiority, interior-urbanism.

Open Questions

• "Topology must simply fold in such a manner that deforms surface to form space" — what is the spatializing fold's computable definition, and is it still topology in the mathematical sense (scaleless, measureless) once it demarcates?
• Banham's subservient topology (circulation) and form merged "in the coming decades" — is the merger complete, or has the digital turn re-separated them ("an increased gap between form… and space")? Is the present condition the second banality — continuity without space?
• The chapter's two exemplars of enclosed topology (H2Oexpo, Teshima) are pavilions and art centers. What is the topological interior's domestic or programmatic form — can a dwelling be an enclosed topology?
• If "hierarchy lies latently embedded in the folds" (points on a topological surface are not equal), what order does the fold encode, and can it be designed rather than merely inherited?
• Does Voordouw's postmodernist periodization of topology survive contact with Hannah's avant-garde precedence claim for the evental — or do the two corrections together imply the fold has three genealogies (avant-garde event, Baroque philosophy, postmodern form) that architecture conflated?