INTRODUCTION Architectural Curvilinearity: The Folded, the Pliant
and the Supple / Greg Lynn
In 1993, Greg Lynn guest-edited an issue of Architectural Design
dedicated to an emerging movement in architecture: folding.
Lynn, a Los Angeles-based architect/educator with a background
in philosophy and an attraction to computer-aided design, was
the ideal person to organize this publication and, in effect, define
the fold in architecture, a concept that generated intense interest
during the remainder of the decade.
In his contributory essay, ''l\rchitectural Curvilinearity: The
Folded, the Pliant and the Supple," Lynn ties together a variety
of sources-including the work of Gilles Deleuze, Rene Thom,
cooking theory, and geology-to present an alternative to
existing architectural theory and practice. He states that since
the mid-1960s architecture has been guided by the notion of
contradiction, whether through attempts to formally embody
heterogeneity or its opposite; in short, postmodernism and decon
structivism can be understood as two sides of the same coin. Yet,
for Lynn, "neither the reactionary call for unity nor the avant-garde
dismantling of it through the identification of internal contradic
tions seems adequate as a model for contemporary architecture
and urbanism." Rather, he offers a smooth architecture (in both a
visual and a mathematic sense) composed of combined yet dis
crete elements that are shaped by forces outside the architectural
discipline, much as diverse ingredients are folded into a smooth
mixture by a discerning chef. This new architecture, what Lynn
calls a pliant, flexible orchitecture, exploits connections between
elements within a design instead of emphasizing contradictions
or attempting to erase them all together. Of equal importance
is that this architecture is inextricably entwined with external
forces, both cultural and contextual. Architects deploy various
.. ,JJiJIIIiIIIIliii.
,j'"'ngies-including a reliance on topological geometry and
"'u"al software and technologies-in the creation of their designs,
II,,' Ihe resulting works tend to be curvilinear in form and inflected ....Ih the particulars of the project and its environment.
In addition to Lynn's essay, Folding in Architecture, as the
A/. hitectural Design issue was titled, included other texts by fig
,,·os such as Deleuze, Jeffrey Kipnis, and John Rajchman, and
'''presentative projects by architects like Peter Eisenman, Frank
( inhry, and Philip Johnson. This list of distinguished collaborators
("fl' weight to the publication, intimating that the phenomenon
"I the fold was already entrenched within architectural design.
If Indeed it was, Folding in Architecture cemented the shift in
(lfchitectural thought by identifying and highlighting this new
mchitecture of smoothness. The importance of Lynn's special
.,sue of Architectural Design was underscored by its reprinting in
2004 as "a historical document,"1 complete with new introductory
nssays analyzing and situating the original publication as a guid
Ing force within twenty-first-century architectural discourse.2
Notes
Helen Castle, "Preface," in Folding in Architedure, ed. Greg Lynn (London: Wiley-Academy. 2004), 7.
2 See Greg Lynn, "Introdudion," in Folding in Architecture, 8-13; and Mario Carpo, "Ten Years of Folding," in Folding in Architecture. See also Branko Koleravic, ed., Architecture in the Digital Age: Design and Manufacturing (New York: Spoon Press, 2003), 3-10.
30 31
GREG LYNN
ARCHITECTURAL
CURVILINEARITY:
THE FOLDED, THE PLIANT
AND THE SUPPLE
First appeared in Architectural Design 63, no. 3/4 (1993): 8-15·
Courtesy ofGreg
For the last two decades, beginning with Robert Venturi's
Complexity and Contradiction in Architecture,' and Colin Rowe and Fred Koetter's Collage City,2 and continuing through Mark Wigley and Philip Johnson's Deconstructivist Architecture, archi tects have been primarily concerned with the production of
heterogeneous, fragmented and conflicting formal systems. These practices have attempted to embody the differences within and between diverse physical, cultural, and social con
texts in formal conflicts. When comparingVenturi's Complexity
and Contradiction or Learning from Las vegas with Wigley and Johnson's DeconstructionArchitecture it is necessary to overlook many significant and distinguishing differences in order to
identify at least one common theme. Both Venturi and Wigley argue for the deployment of dis
continuous, fragmented, heterogeneous, and diagonal formal strategies based on the incongruities,juxtapositions and opposi tions within specific sites and programmes. These disjunctions
,nult from a logic which tends to identify the potential con 1I.IIIit'lions between dissimilar elements. A diagonal dialogue Iwlween a building and its context has become an emblem
I", Ihe contradictions within contemporary culture. From the ', •.1It' of an urban plan to a building detail, contexts have been IIl1l1ed for conflicting geometries, materials, styles, histories, ,.lId programmes which are then represented in architecture as 11111' mal contradictions. The most paradigmatic architecture of t t1l' last ten years, including Robert Venturi's Sainsbury Wing of
'hI' National Gallery, Peter Eisenman's Wexner Center, Bernard I \chumi's La Villette Park or the Gehry House, invests in the ,II ('hitectural representation of contradictions. Through con
II ad iction, architecture represents difference in violent formal • 1111 t1icts.
Contradiction has also provoked a reactionary response ,,, formal conflict. Such resistances attempt to recover unified
.1Il'hitecturallanguages that can stand against heterogeneity. I !Ility is constructed through one of two strategies: either by '\'('onstructing a continuous architectural language through historical analyses (Neo-Classicism or Neo-Modernism) or by ldt'ntifying local consistencies resulting from indigenous cli
mates, materials, traditions or technologies (Regionalism). rhe internal orders of Neo-Classicism, Neo-Modernism and
Ilt'gionalism conventionally repress the cultural and contextual discontinuities that are necessary for a logic of contradiction. III architecture, both the reaction to and the representation of heterogeneity have shared an origin in contextual analysis. Both Iheoretical models begin with a close analysis ofcontextual con
ditions from which they proceed to evolve either a homogeneous or heterogeneous urban fabric. Neither the reactionary call for IInity nor the avant-garde dismantling of it through the identifi t':ltion of internal contradictions seems adequate as a model for l'Ontemporary architecture and urbanism.
GREG LYNN 33 32
In response to architecture's discovery of complex, dis parate, differentiated and heterogeneous cultural and formal contexts, two options have been dominant; either conflict and contradiction or unity and reconstruction. Presently, an alter native smoothness is being formulated that may escape these dialectically opposed strategies. Common to the diverse sources of this post-contradictory work-topological geometry, mor phology, morphogenesis, Catastrophe Theory or the computer technology of both the defense and Hollywood film industry are characteristics of smooth transformation involving the intensive integration of differences within a continuous yet het erogeneous system. Smooth mixtures are made up of disparate elements which maintain their integrity while being blended within a continuous field ofother free elements.
Smoothing does not eradicate differences but incorporates3
free intensities through fluid tactics of mixing and blending. Smooth mixtures are not homogeneous and therefore cannot be reduced. Deleuze describes smoothness as "the continuous variation" and the "continuous development ofform."4 Wigley's critique of pure form and static geometry is inscribed within geometric conflicts and discontinuities. For Wigley, smoothness is equated with hierarchical organisation: "the volumes have been purified-they have become smooth, classical-and the wires all converge in a single, hierarchical, vertical movement. "5 Rather than investing in arrested conflicts, Wigley's slipperi ness might be better exploited by the alternative smoothness of heterogeneous mixture. For the first time perhaps, complexity might be aligned with neither unity nor contradiction but with smooth, pliant mixture.
Both pliancy and smoothness provide an escape from the two camps which would either have architecture break under the stress ofdifference or stand firm. Pliancy allows architecture to become involved in complexity through flexibility. It may be
34 ' ARCHITECTURAL CURVILINEARITY
pn""ihlc to neither repress the complex relations of differences "Ih fixed points ofresolution nor arrest them in contradictions, btu "ustain them through flexible, unpredicted, local connec Ikllls. To arrest differences in conflicting forms often precludes ....ny of the more complex possible connections of the forms of .rc:hilecture to larger cultural fields. A more pliant architectural k'""ibility values alliances, rather than conflicts, between ele "'t"IlIS. Pliancy implies first an internal flexibility and second a dc-,ll'ndence on external forces for self-definition.
If there is a single effect produced in architecture by folding, .. will be the ability to integrate unrelated elements within a new tunlinuous mixture. Culinary theory has developed both a prac tklll and precise definition for at least three types of mixtures. l'lll' first involves the manipulation of homogeneous elements; bt"lIling, whisking and whipping change the volume but not .ht" nature of a liquid through agitation. The second method ur incorporation mixes two or more disparate elements; chop Pill){, dicing, grinding, grating, slicing, shredding and mincing rvisl'erate elements into fragments. The first method agitates • "ingle uniform ingredient, the second eviscerates disparate Ingredients. Folding, creaming and blending mix smoothly multiple ingredients "through repeated gentle overturnings wilhout stirring or beating" in such a way that their individual l'hnracteristics are maintained.6 For instance, an egg and choco tilt' are folded together so that each is a distinct layer within a c:nntinuous mixture.
Folding employs neither agitation nor evisceration but a .upple layering. Likewise, folding in geology involves the sedi IUl'ntation of mineral elements or deposits which become _lowly bent and compacted into plateaus of strata. These strata Afl' compressed, by external forces, into more or less continu IIUS layers within which heterogeneous deposits are still intact in varying degrees of intensity.
GREG LYNN 35
A folded mixture is neither homogenous, like whipped cream, nor fragmented, like chopped nuts, but smooth and
heterogeneous. In both cooking and geology, there is no pre liminary organisation which becomes folded but rather there are unrelated elements or pure intensities that are intricated
through ajoint manipulation. Disparate elements can be incor porated into smooth mixtures through various manipulations
including fulling: "Felt is a supple solid product that proceeds altogether dif
ferently, as an anti-fabric. It implies no separation of threads, no intertwining, only an entanglement of fibres obtained by full ing (for example, by rolling the block of fibres back and forth).
What becomes entangled are the microscales of the fibres. An aggregate of intrication of this kind is in no way homogeneous; nevertheless, it is smooth and contrasts point by point with the
space of fabric (it is in principle infinite, open and uninhibited in every direction; it has neither top, nor bottom, nor centre;
it does not assign fixed or mobile elements but distributes a continuous variation}."?
The two characteristics of smooth mixtures are that they
are composed of disparate unrelated elements and that these free intensities become intricated by an external force exerted upon them jointly. Intrications are intricate connections. They
are intricate, they affiliate local surfaces of elements with one another by negotiating interstitial rather than internal connec
tions. The heterogeneous elements within a mixture have no
proper relation with one another. Likewise, the external force that intricates these elements with one another is outside of the
individual elements control or prediction.
Viscous Mixtures
Unlike an architecture of contradictions, superpositions and accidental collisions, pliant systems are capable of engendering
36 ARCHITECTURAL CURVILINEARITY
unpH'dicted connections with contextual, cultural, program
nutl ito, structural and economic contingencies by vicissitude. Vldssitude is often equated with vacillation, weakness8 and Intll'l'isiveness but more importantly these characteristics are
trt'(IUcntly in the service of a tactical cunning.9 Vicissitude is • (IUality of being mutable or changeable in response to both '.vourable and unfavourable situations that occur by chance. Vldssitudinous events result from events that are neither arbi
trAry nor predictable but seem to be accidental. These events .rt' made possible by a collision of internal motivations with
""Il'rnal forces. For instance, when an accident occurs the vl"1 i m s immediately identify the forces contributing to the acci dt'1l1 and begin to assign blame. It is inevitable however, that nu single element can be made responsible for any accident
." I hese events occur by vicissitude; a confluence of particular Inlluences at a particular time makes the outcome of an acci d('111 possible. If any element participating in such a confluence
(If local forces is altered the nature of the event will change. In A Thousand Plateaus, Spinoza's concept of "a thousand vicis
alludes" is linked with Gregory Bateson's "continuing plateau uf intensity" to describe events which incorporate unpredict
"hie events through intensity. These occurrences are difficult to ICIl'alise, difficult to identify. 10 Any logic of vicissitude is depen tll'nt on both an intrication of local intensities and the exegetic
prcssure exerted on those elements by external contingencies,
Nl'ither the intrications nor the forces which put them into rela lion are predictable from within any single system. Connections
hy vicissitude develop identity through the exploitation of local iltljacencies and their affiliation with external forces. In this fil'nse, vicissitudinous mixtures become cohesive through a log-ic ofviscosity.
Viscous fluids develop internal stability in direct propor lion to the external pressures exerted upon them. These fluids
GREG LYNN 37
behave with two types of viscidity. They exhibit both internal cohesion and adhesion to external elements as their viscosity increases. Viscous fluids begin to behave less like liquids and more like sticky solids as the pressures upon them intensify. Similarly, viscous solids are capable of yielding continually under stress so as not to shear.
Viscous space would exhibit a related cohesive stabil ity in response to adjacent pressures and a stickiness or adhesion to adjacent elements. Viscous relations such as these are not reducible to any single or holistic organisation. Forms of viscosity and pliability cannot be examined outside of the vicissitudinous connections and forces with which their defor mation is intensively involved. The nature of pliant forms is that they are sticky and flexible. Things tend to adhere to them. As pliant forms are manipulated and deformed the things that stick to their surfaces become incorporated within their interiors.
Curving Away from Deconstructivism
Along with a group of younger architects, the projects that best represent pliancy, not coincidentally, are being produced by many of the same architects previously involved in the valorisa tion of contradictions. Deconstructivism theorised the world as a site of differences in order that architecture could represent these contradictions in form. This contradictory logic is begin ning to soften in order to exploit more fully the particularities of urban and cultural contexts. This is a reasonable transition, as the Deconstructivists originated their projects with the inter nal discontinuities they uncovered within buildings and sites. These same architects are beginning to employ urban strategies which exploit discontinuities, not by representing them in for mal collisions, but by affiliating them with one another though continuous flexible systems.
38 ARCHITECTURAL CURVILINEARITY
Just as many of these architects have already been inscribed ...hl" a Deconstructivist style of diagonal forms, there will .",1), hc those who would enclose their present work within • Nto-Baroque or even Expressionist style of curved forms. "JW("vcr, many of the formal similitudes suggest a far richer *kJJCil' of curvilinearity"11 that can be characterised by the Involvement of outside forces in the development of form. If Inltrnally motivated and homogeneous systems were to extend In .tmight lines, curvilinear developments would result from the IIM'urporation of external influences. Curvilinearity can put into ..hllion the collected projects in this publication [Architectural ,"',ign 63], Deleuze's The Fold: Leibniz and the Baroque and Rene Tholn's catastrophe diagrams. The smooth spaces described by these continuous yet differentiated systems result from cur ..linear sensibilities that are capable of complex deformations In rl'sponse to programmatic, structural, economic, aesthetic, political and contextual influences. This is not to imply that Intl'nsive curvature is more politically correct than an unin volved formal logic, but rather, that a cunning pliability is often Inure effective through smooth incorporation than contradic lion and conflict. Many cunning tactics are aggressive in nature. Whether insidious or ameliorative these kinds of cunning con IIt'ctions discover new possibilities for organisation. A logic of rurvilinearity argues for an active involvement with external ('wnts in the folding, bending and curving of form.
Already in several Deconstructivist projects are latent sug J(estions of smooth mixture and curvature. For instance, the (iehry House is typically portrayed as representing materials Hnd forms already present within, yet repressed by, the subur hlln neighbourhood: sheds, chain-link fences, exposed plywood, trailers, boats and recreational vehicles. The house is described liS an "essay on the convoluted relationship between the conflict within and between forms ... which were not imported to but
GREG LYNN 39
emerged from within the house."" The house is seen to provoke conflict within the neighbourhood due to its public representa tion of hidden aspects of its context. The Gehry House violates the neighbourhood from within. Despite the dominant appeal of the house to contradictions, a less contradictory and more pliant reading of the house is possible as a new organisation emerges between the existing house and Gehry's addition. A dynamic stability develops with the mixing of the original and the addition. Despite the contradictions between elements pos sible points of connection are exploited. Rather than valorise the conflicts the house engenders, as has been done in both academic and popular publications, a more pliant logic would identify, not the degree ofviolation, but the degree to which new connections were exploited. A new intermediate organisation occurs in the Gehry House by vicissitude from the affiliation of the existing house and its addition. Within the discontinuities of Deconstructivism there are inevitable unforeseen moments
of cohesion. Similarly, Peter Eisenman's Wexner Center is convention
ally portrayed as a collision of the conflicting geometries of the campus, city and armoury which once stood adjacent to the site. These contradictions are represented by the diagonal collisions between the two grids and the masonry towers. Despite the dis junctions and discontinuities between these three disparate systems, Eisenman's project has suggested recessive readings ofcontinuous non-linear systems ofconnection. Robert Somop3 identifies such a system of Deleuzian rhizomatous connections between armoury and grid. The armoury and diagonal grids are shown by Somol to participate in a hybrid L-movement that organises the main gallery space. Somol's schizophrenic analy sis is made possible by, yet does not emanate from within, a Deconstructivist logic of contradiction and conflict. The force of this Deleuzian schizo-analytic model is its ability to maintain
40 . ARCHITECTURAL CURVILINEARITY
multiple organisations simultaneously. In Eisenman's project Iht' tower and grid need not be seen as mutually exclusive or in (,()lItradiction. Rather, these disparate elements may be seen I" distinct elements co-present within a composite mixture. I%mcy does not result from and is not in line with the previous luc:hitecturallogic of contradiction, yet it is capable of exploit Ing many conflicting combinations for the possible connections Ihilt are overlooked. Where DeconstructivistArchitecture was seen II) t'xploit external forces in the familiar name of contradiction And conflict, recent pliant projects by many of these architects t'xhibit a more fluid logic of connectivity.
Immersed in Context
The contradictory architecture of the last two decades has t'volved primarily from highly differentiated, heterogeneous ('ClI1texts within which conflicting, contradictory and discon Iinuous buildings were sited. An alternative involvement with Iwterogeneous contexts could be affiliated, compliant and con Iinuous. Where complexity and contradiction arose previously (rom inherent contextual conflicts, present attempts are being made to fold smoothly specific locations, materials and pro Jtmmmes into architecture while maintaining their individual Idt'ntity.
This recent work may be described as being compliant; in II state of being plied by forces beyond control. The projects are
folded, pliant and supple in order to incorporate their nmtexts with minimal resistance. Again, this characterisation lihould not imply flaccidity but a cunning submissiveness that Is l'apable of bending rather than breaking. Compliant tactics, Mll'h as these, assume neither an absolute coherence nor cohe ,.ion between discrete elements but a system of provisional, Intl'nsive, local connections between free elements. Intensity d{'scribes the dynamic internalisation and incorporation of
GREG lYNN 41
external influences into a pliant system. Distinct from a whole organism-to which nothing can be added or subtracted intensive organisations continually invite external influence within their internal limits so that they might extend their influence through the affiliations they make. A two-fold deter ritorialisation, such as this, expands by internalising external forces. This expansion through incorporation is an urban alternative to either the infinite extension of International Modernism, the uniform fabric of Contextualism or the con flicts of Post-Modernism and Deconstructivism. Folded, pliant and supple architectural forms invite exigencies and contingen cies in both their deformation and their reception.
In both Learning from Las Vegas and Deconstructivist Architecture, urban contexts provided rich sites of difference. These differences are presently being exploited for their abil ity to engender multiple lines of local connections rather than lines of conflict. These affiliations are not predictable by any contextual orders but occur by vicissitude. Here, urban fabric has no value or meaning beyond the connections that are made within it. Distinct from earlier urban sensibilities that general ised broad formal codes, the collected projects develop local, fine grain, complex systems of intrication. There is no general urban strategy common to these projects, only a kind of tactical mutability. These folded, pliant and supple forms of urbanism are neither in deference to nor in defiance of their contexts but exploit them by turning them within their own twisted and
curvilinear logics.
The Supple and Curvilinear
1 supple\adj [ME souple, fr OF, fr L supplic-, supplex
submissive, suppliant, lit, bending under, fr sub +plic (akin to plicare to fold)-more at PLY] u: compliant often
42 . ARCHITECTURAL CURVILINEARITY
to the point of obsequiousness b: readily adaptable or
responsive to new situations 2a: capable of being bent or
folded without creases, cracks or breaks: PLIANT b: able to
perform bending or twisting movements with ease and
grace: LIMBER c: easy and fluent without stiffness or
awkwardness. 14
At an urban scale, many of these projects seem to be some where between contextualism and expressionism. Their supple rorms are neither geometrically exact nor arbitrarily figural. I:or example, the curvilinear figures of Shoei Yoh's roof struc tures are anything but decorative but also resist being reduced to a pure geometric figure. Yoh's supple roof structures exhibit /I logic of curvilinearity as they are continuously differentiated Ill'cording to contingencies. The exigencies of structural span Il'ngths, beam depths, lighting, lateral loading, ceiling height lind view angles influence the form of the roof structure. Rather than averaging these requirements within a mean or mini mum dimension they are precisely maintained by an anexact ),l·t rigorous geometry. Exact geometries are eidetic; they can bl' reproduced identically at any time by anyone. In this regard, they must be capable of being reduced to fixed mathematical (Iuantities. Inexact geometries lack the precision and rigor nec t'ssary for measurement.
Anexact geometries, as described by Edmund Husserl,15
nrc those geometries which are irreducible yet rigorous. These geometries can be determined with precision yet cannot be reduced to average points or dimensions. Anexact geometries often appear to be merely figural in this regard. Unlike exact geometries, it is meaningless to repeat identically an anexact geometric figure outside of the specific context within which It is situated. In this regard, anexact figures cannot be easily translated.
GREG LYNN 43
http:awkwardness.14
Jeffrey Kipnis has argued convincingly that Peter Eisenman's Columbus Convention Center has become a canonical model for the negotiation of differentiated urban fringe sites through the use of near figures. '6 Kipnis identifies the disparate sys tems informing the Columbus Convention Center including: a single volume of inviolate programme of a uniform shape and height larger than two city blocks, an existing fine grain fabric of commercial buildings and a network of freeway inter changes that plug into the gridded streets of the central business district. Eisenman's project drapes the large rectilinear vol ume of the convention hall with a series of supple vermiforms. These elements become involved with the train tracks to the north-east, the highway to the south-east and the pedes trian scale of High Street to the west. The project incorporates the multiple scales, programmes, and pedestrian and auto motive circulation of a highly differentiated urban context. Kipnis' canonisation of a form which is involved with such spe cific contextual and programmatic contingencies seems to be frustrated from the beginning. The effects of a pliant urban mix ture such as this can only be evaluated by the connections that it makes. Outside of specific contexts, curvature ceases to be intensive. Where the Wexner Center, on the same street in the same city, represents a monumental collision, the Convention Center attempts to disappear by connection between intervals within its context; where the Wexner Center destabilises through contradictions the Convention Center does so by subterfuge.
In a similar fashion Frank Gehry's Guggenheim Museum in Bilbao, Spain covers a series of orthogonal gallery spaces with flexible tubes which respond to the scales of the adjacent roadways, bridges, the Bilbao River and the existing medieval city. Akin to the Vitra Museum, the curvilinear roof forms of the Bilbao Guggenheim integrate the large rectilinear masses of
44 . ARCHITECTURAL CURVILINEARITY
!Cullery and support space with the scale of the pedestrian and IlUtomotive contexts.
The unforeseen connections possible between differenti illl'" sites and alien programmes require conciliatory, complicit, pliant, flexible and often cunning tactics. Presently, numerous IIrchitects are involving the heterogeneities, discontinuities and tli fferences inherent within any cultural and physical context by IIligning formal flexibility with economic, programmatic and Ii' ructural compliancy. A multitude ofpli based words-folded, pliant, supple, flexible, plaited, pleated, plicating, complicitous, t'Ompliant, complaisant, complicated, complex and multiplici tous to name a few-can be invoked to describe this emerging urban sensibility of intensive connections.
The Pliant and Bent
pJiable\adj [Me fr plieirto bend, fold-more at PLY] 1a: supple
enough to bend freely or repeatedly without breaking b: yield
ing readily to others: COMPLAISANT 2: adjustable to varying
conditions: ADAPTABLE, syn see PLASTIC, ant obstinate.17
John Rajchman, in reference to Gilles Deleuze's book Le pli has already articulated an affinity between complexity, or plex words, and folding, or plic-words, in the Deleuzian paradigm of "perplexing plications" or "perplication.'"8 The plexed and the plied can be seen in a tight knot of complexity and pliancy. Plication involves the folding in ofexternal forces. Complication involves an intricate assembly of these extrinsic particularities into a complex network. In biology, complication is the act ofan embryo folding in upon itself as it becomes more complex. To become complicated is to be involved in mUltiple complex, intri (.'ate connections. Where Post-Modernism and Deconstructivism resolve external influences of programme, use, economy and
GREG LYNN 45
http:obstinate.17
advertising through contradiction, compliancy involves these external forces by knotting, twisting, bending, and folding them within form.
Pliant systems are easily bent, inclined or influenced. An anatomical "plica" is a single strand within multiple "plicae." It is a multiplicity in that it is both one and many simultane ously. These elements are bent along with other elements into a composite, as in matted hair(s). Such a bending together of elements is an act ofmultiple plication or multiplication rather than mere addition. Plicature involves disparate elements with one another through various manipulations of bending, twist ing, pleating, braiding, and weaving through external force. In RAA Um's Croton Aqueduct project a single line following the subterranean water supply for New York City is pulled through multiple disparate programmes which are adjacent to it and which cross it. These programmatic elements are braided and bent within the continuous line of recovered public space which stretches nearly twenty miles into Manhattan. In order to incor porate these elements the line itself is deflected and reoriented, continually changing its character along its length. The seem ingly singular line becomes populated by finer programmatic elements. The implications ofLe pli for architecture involve the proliferation of possible connections betweenfree entities such as these.
A plexus is a multi-linear network of interweavings, inter twinings and intrications; for instance, of nerves or blood vessels. The complications of a plexus-what could best be called complexity-arise from its irreducibility to any single organisation. A plexus describes a multiplicity of local connec tions within a single continuous system that remains open to new motions and fluctuations. Thus, a plexial event cannot occur at any discrete point. A multiply plexed system-a com plex-cannot be reduced to mathematical exactitude, it must
46 ARCHITECTURAL CURVILINEARITY
bet described with rigorous probability. Geometric systems have I distinct character once they have been plied; they exchange Axed co-ordinates for dynamic relations across surfaces.
Alternative types of transformation
I ,Iscounting the potential ofearlier geometric diagrams ofprob Ibility, such as Buffon's Needle Problem,'9 D'Arcy Thompson provides perhaps the first geometric description of variable ddormation as an instance of discontinuous morphological development. His cartesian deformations, and their use of flex Ible topological rubber sheet geometry, suggest an alternative to the static morphological transformations of autonomous architectural types. A comparison ofthe typological and trans rormational systems of Thompson and Rowe illustrates two rlldically different conceptions of continuity. Rowe's is fixed, ('xact, striated, identical and static, where Thompson's is dynamic, anexact, smooth, differentiated and stable.
Both Rudolf Wittkower-in his analysis of the Palladian villas of :194920-and Rowe-in his comparative analysis of Iltllladio and Le Corbusier of:194721-uncovera consistent organ isational type: the nine-square grid. In Wittkower's analysis of twelve Palladian villas the particularities of each villa accumu late (through what Edmund Husserl has termed variations) to generate a fixed, identical spatial type (through what could best be described as phenomenological reduction). The typology of this "Ideal Villa" is used to invent a consistent deep structure underlying Le Corbusier's Villa Stein at Garche and Palladio's Villa Malcontenta. Wittkower and Rowe discover the exact geo metric structure of this type in all villas in particular. This fixed type become a constant point of reference within a series of variations.
Like Rowe, Thompson is interested in developing a math ematics of species categories, yet his system depends on a
GREG LYNN 47
dynamic and fluid set of geometric relations. The deformations of a provisional type define a supple constellation of geomet ric correspondences. Thompson uses the initial type as a mere provision for a dynamic system of transformations that occur in connection with larger environmental forces. Thompson's method of discontinuous development intensively involves external forces in the deformation of morphological types. The flexible type is able to both indicate the general morphological structure of a species while indicating its discontinuous devel opment through the internalisation of heretofore external forces within the system. 22 For instance, the enlargement of a fish's eye is represented by the flexing of a grid. This fluctuation, when compared to a previous position of the transformational type, establishes a relation between water depth and light inten sity as those conditions are involved in the formal differences between fish. The flexing grid of relations cannot be arrested at any moment and therefore has the capacity to describe both a general type and the particular events which influence its devel opment. Again, these events are not predictable or reducible to any fixed point but rather begin to describe a probable zone of co-present forces; both internal and external. Thompson presents an alternative type of inclusive stability, distinct from the exclusive stasis of Rowe's nine-square grid. The sup ple geometry of Thompson is capable of both bending under external forces and folding those forces internally. These trans formations develop through discontinuous involution rather
than continuous evolution. The morphing effects used in the contemporary advertis
ing and film industry may already have something in common with recent developments in architecture. These mere images have concrete influences on space, form, politics, and cul ture; for example, the physical morphing of Michael Jackson's body, including the transformation of his form through various
48 . ARCHITECTURAL CURVILINEARITY
surgeries and his surface through skin bleachingand lightening. These physical effects and their implications for the definition of gender and race were only later represented in his recent video Black & White. In this video multiple genders, ethnicities lind races are mixed into a continuous sequence through the digital morphing of video images. It is significant that Jackson is not black or white but black and white, not male or female hut male and female. His simultaneous differences are charac leristic of a desire for smoothness; to become heterogeneous yet continuous. Physical morphing, such as this, is monstrous hecause smoothness eradicates the interval between what Thompson refers to as discriminant characteristics without homogenizing the mixture. Such a continuous system is neither an assembly of discrete fragments nor a whole. 23 With Michael Jackson, the flexible geometric mechanism with which his video representation is constructed comes from the same desire which aggressively reconstructs his own physical form. Neither Ihe theory, the geometry or the body proceed from one another; rather, they participate in a desire for smooth transformation. Form, politics, and self-identity are intricately connected in this process of deformation.
A similar comparison might be made between the liq uid mercury man in the film Terminator 2 and the Peter Lewis House by Frank Gehry and Philip Johnson. The Hollywood special effects sequences allow the actor to both become and disappear into virtually any form. The horror of the film results not from ultra-violence, but from the ability of the antagonist 10 pass through and occupy the grids of floors, prison bars, and other actors. Computer technology is capable of constructing intermediate images between any two fixed points resulting in a smooth transformation. These smooth effects calculate with probability the interstitial figures between fixed figures. Furthermore, the morphing process is flexible enough that
GREG LYNN . 49
http:whole.23
http:system.22
multiple between states are possible. Gehry's and Johnson's Peter Lewis House is formulated from multiple flexible forms. The geometry of these forms is supple and can accommodate smooth curvilinear deformation along their length. Not only are these forms capable of bending to programmatic, structural and environmental concerns, as is the roof ofShoei Yoh's roof struc tures, but they can deflect to the contours and context of the site, similar to Peter Eisenman's Columbus Convention Center and RAA Urn's Croton Aqueduct project. Furthermore, the Lewis House maintains a series of discrete figural fragments such as boats and familiar fish-within the diagrams of D'Arcy Thompson, which are important to both the morphing effects of Industrial Light and Magic and the morphogenetic diagrams of Rene Thorn, Gehry's supple geometry is capable of smooth, heterogeneous continuous deformation. Deformation is made possible by the flexibility of topological geometry in response to external events, as smooth space is intensive and continuous. Thompson's curvilinear logic suggests deforma tion in response to unpredictable events outside of the object. Forms of bending, twisting or folding are not superfluous but result from an intensive curvilinear logic which seeks to internalise cultural and contextual forces within form. In this manner events become intimately involved with particular rather than ideal forms. These flexible forms are not mere rep resentations of differential forces but are deformed by their environment.
Folding and Other Catastrophes for Architecture
3 fold vb [ME folden, fro OEfoaldanj akin to OHGfaldan to fold, Gk di plasios twofold] vt 1: to lay one part over another
part, 2: to reduce the length or bulk of by doubling over,
3: to clasp together: EN1WINE, 4: to clasp or embrace
50 ' ARCHITECTURAL CURVILINEARITY
closely: EMBRACE, 5: to bend (as a rock) into folds, 6: to
incorporate (a food ingredient) into a mixture by repeated
gentle overturnings without stirring or beating, 7: to bring
to an end.24
Philosophy has already identified the displacement presently occurring to the Post-Modem paradigm of complexity and con tradiction in architecture, evidenced by John Rajchman's Out oj the Fold and Perplications. Rajchman's text is not a mani fl'sto for the development of new architectural organisations, but responds to the emergence of differing kinds of complex ity being developed by a specific architect. His essays inscribe spatial innovations developed in architecture within larger intellectual and cultural fields. Rajchman both illuminates I'l'ter Eisenman's architectural practice through an explication of Le P!i and is forced to reconsider Deleuze's original argu ment concerning Baroque space by the alternative spatialities of Eisenman's Rebstock Park project. The dominant aspect of the project which invited Rajchman's attention to folding was the employment of one of Rene Thorn's catastrophe diagrams in the design process.
Despite potential protestations to the contrary, it is more than likely that Thorn's catastrophe nets entered into the archi tl'cture of Carsten Juel-Christiansen's Die Anhalter Faltung, Peter Eisenman's Rebstock Park, Jeffrey Kipnis' Unite de Habitation at Briey installation and Bahram Shirdel's Nara Convention Hall as a mere formal technique. Inevitably, architects and philosophers alike would find this in itself a catastrophe for all concerned. Yet, their use illustrates that at Icast four architects simultaneously found in Thorn's diagrams a formal device for an alternative description of spatial com plexity. The kind of complexity engendered by this alliance with Thorn is substantially different than the complexity provided by
GREG LYNN 51
either Venturi's decorated shed or the more recent conflicting forms of Deconstructivism. Topological geometry in general, and the catastrophe diagrams in particular, deploy disparate forces on a continuous surface within which more or less open systems of connection are possible.
"Topology considers superficial structures susceptible to continuous transformations which easily change their form, the most interesting geometric properties common to all modifi cation being studied. Assumed is an abstract material of ideal deformability which can be deformed, with the exception of disruption. "
These geometries bend and stabilise with viscosity under pressure. Where one would expect that an architect looking at catastrophes would be interested in conflicts, ironically, architects are finding new forms of dynamic stability in these diagrams. The mutual interest in Thorn's diagrams points to a desire to be involved with events which they cannot predict. The primary innovation made by those diagrams is the geometric modelling of a multiplicity of possible co-present events at any moment. Thorn's morphogenesis engages seemingly random events with mathematical probability.
Thorn's nets were developed to describe catastrophic events. What is common to these events is an inability to define exactly the moment at which a catastrophe occurs. This loss of exacti
tude is replaced by a geometry of multiple probable relations. With relative precision, the diagrams define potential catas trophes through cusps rather than fixed co-ordinates. Like any simple graph, Thorn's diagrams deploy X and Y forces across two axes of a gridded plane. A uniform plane would provide the potential for only a single point of intersection between any two X and Y co-ordinates. The supple topological surface of Thorn's diagrams is capable of enfolding in multiple dimensions. Within these folds, or cusps, zones of proximity are contained.
52 . ARCHITECTURAL CURVILINEARITY
As t he topological surface folds over and into itself multiple pos "ible points of intersection are possible at any moment in the /. dimension. These co-present Z-dimensional zones are pos liible because the topological geometry captures space within lis surface. Through proximity and adjacency various vectors of force begin to imply these intensive event zones. In catastrophic ('vents there is not a single fixed point at which a catastrophe occurs but rather a zone of potential events that are described hy these cusps. The cusps are defined by multiple possible inter IIl·tions implying, with more or less probability, multiple fluid Ihresholds. Thorn's geometric plexus organises disparate forces ill order to describe possible types of connections.
If there is a single dominant effect of the French word pli, II is its resistance to being translated into any single term. It is precisely the formal manipulations of folding that are capable of incorporating manifold external forces and elements within form, yet Le pli undoubtedly risks being translated into archi Il'cture as mere folded figures. In architecture, folded forms risk quickly becoming a sign for catastrophe. The success of the IIrchitects who are folding should not be based on their ability 10 represent catastrophe theory in architectural form. Rather, Iht, topological geometries, in connection with the probable ('vents they model, present a flexible system for the organisa lion of disparate elements within continuous spaces. Yet, these
"mooth systems are highly differentiated by cusps or zones of (·o-presence. The catastrophe diagram used by Eisenman in the I{l'bstock Park project destabilises the way that the buildings meet the ground. It smoothes the landscape and the building hy turning both into one another along cusps. The diagrams used by Kipnis in the Briey project, and Shirdel in the Nara (:onvention Hall, develop an interstitial space contained simul laneously within two folded cusps. This geometrically blushed surface exists within two systems at the same moment and in
GREG LYNN 53
this manner presents a space of co-presence with multiple adja cent zones of proximity.
Before the introduction of either Deleuze or Thom to archi tecture, folding was developed as a formal tactic in response to problems presented by the exigencies of commercial develop ment. Henry Cobb has argued in both the Charlottesville Tapes and his Note on Folding for a necessity to both dematerialise and differentiate the massive homogeneous volumes dictated by commercial development in order to bring them into rela tion with finer grain heterogeneous urban conditions. His first principle for folding is a smoothing of elements across a shared surface. The facade of theJohn Hancock Tower is smoothed into a continuous surface so that the building might disappear into its context through reflection rather than mimicry. Any poten tial for replicating the existing context was precluded by both the size of the contiguous floor plates required by the developer and the economic necessity to construct the building's skin from glass panels. Folding became the method by which the surface of a large homogeneous volume could be differentiated while remaining continuous. This tactic acknowledges that the existing fabric and the developer tower are essentially of differ ent species by placing their differences in mixture, rather than contradiction, through the manipulation ofa pliant skin.
Like the John Hancock Building, the Allied Bank Tower begins with the incorporation of glass panels and metal frame into a continuous folded surface. The differentiation of the folded surface, through the simultaneous bending of the glass and metal, brings those elements together in a continuous plane. The manipulations of the material surface proliferate folding and bending effects in the massing of the building. The alien building becomes a continuous surface of disappearance that both diffracts and reflects the context through complex manipulations of folding. In the recent films Predator and
54 ' ARCHITECTURAL CURVILINEARITY
I'r('dator II, a similar alien is capable of disappearing into both urban and jungle environments, not through cubist camou Unge lS but by reflecting and diffracting its environment like an octopus or chameleon. The contours between an object and lIS context are obfuscated by forms which become translucent, rellective and diffracted. The alien gains mobility by cloaking its volume in a folded surface of disappearance. Unlike the "deco rnted shed" or "building board" which mimics its context with a singular sign, folding diffuses an entire surface through a Ilhimmering reflection of local adjacent and contiguous particu Inrities. For instance, there is a significant difference between a limall fish which represents itself as a fragment of a larger fish Ihrough the figure of a large eye on its tail, and a barracuda whieh becomes like the liquid in which it swims through a dif fused reflection of its context. The first strategy invites deceitful detection where the second uses stealth to avoid detection. Similarly, the massive volume of the Allied Bank Tower situates hsclfwithin a particular discontinuous locale by cloaking itself In a folded reflected surface. Here, cunning stealth is used as a way of involving contextual forces through the manipulation of II surface. The resemblance offolded architecture to the stealth bomber results not from a similarity between military and archi I('ctural technologies or intentions but rather from a tactical disappearance'6 of a volume through the manipulation of a sur fnce. This disappearance into the fold is neither insidious nor innocent but merely a very effective tactic.
Like Henry Cobb, Peter Eisenman introduces a fold as a method of disappearing into a specific context. Unlike Cobb, who began with a logic of construction, Eisenman aligns the fold with the urban contours of the Rebstock Park. The repetitive Iypologies ofhousing and office buildings are initially deployed un the site in a more or less functionalist fashion; then a topo logical net derived from Thom's Butterfly net is aligned to the
GREG LYNN 55
perimeter of the site and pushed through the typological bars. This procedure differentiates the uniform bars in response to the global morphology of the site. In this manner the manifestation of the fold is in the incorporation of differences-derived from the morphology ofthe site-into the homogeneous typologies of the housing and office blocks. Both Eisenman's local differen tiation of the building types by global folding, and Cobb's local lUlU11l~ across constructional elements which globally differen tiates each floor plate and the entire massing of the building are effective. Cobb and Eisenman "animate" homogenous organi sations that were seemingly given to the architect-office tower and siedlung-with the figure of a fold. The shared of folding identified by both Eisenman and Cobb, evident in their respective texts, is the ability to differentiate the inherited homogeneous organisations of both Modernism (Eisenman's siedlung) and commercial development (Cobb's tower). This differentiation of known types of space and organisation has something in common with Deleuze's delimitation of folding in architecture within the Baroque. Folding heterogeneity into known typologies renders those organisations more smooth and more intensive so that they are better able to incorporate disparate elements within a continuous system. Shirdel's use ofThom's diagrams is quite interesting as the catastrophe sec tions do not animate an existing organisation. Rather, they begin as merely one system among three others. The convention halls float within the envelope of the building as they are sup ported by a series of transverse structural walls whose figure is derived from Thom's nets. This mixture of systems, supported by the catastrophe sections, generates a massive residual pub lic space at the ground floor of the building. In Shirdel's project the manipulations of folding, in both the catastrophe sections and the building envelope, incorporate previously unrelated ele ments into a mixture. The space between the theatres, the skin
56 . ARCHITECTURAL CURVILINEARITY
und the lateral structural walls is such a space of mixture and intrication.
With structure itself, Chuck Hoberman is capable of transforming the size of domes and roofs through a folding structural mechanism. Hoberman develops adjustable struc tll res whose differential movements occur through the dynamic transformation of flexible continuous systems. The movements of these mechanisms are determined both by use and struc ture. Hoberman's structural mechanisms develop a system of smooth transformation in two ways. The Iris dome and sphere projects transform their size while maintaining their shape. This flexibility of size within the static shape of the stadium is (:apable of supporting new kinds of events. The patented tiling patterns transform both the size and shape of surfaces, developing local secondary pockets of space and enveloping larger primary volumes.
So far in architecture, Deleuze's, Cobb's, Eisenman's and Hoberman's discourse inherits dominant typologies of organ isation into which new elements are folded. Within these activities of folding it is perhaps more important to identifY those new forms of local organisation and occupation which inhabit the familiar types of the Latin cross church, the siedlung, the office tower and the stadium, rather than the disturbances visited on those old forms of organisation. Folding can occur in both the organisations of old forms and the free intensi ties of unrelated elements as is the case with Shirdel's project. Likewise, other than folding, there are several manipulations of elements engendering smooth, heterogeneous and intensive organisation.
Despite the differences between these practices, they share a sensibility that resists cracking or breaking in response to external pressures. These tactics and strategies are all compli ant to, complicated by, and complicit with external forces in
GREG LYNN 57
manners which are: submissive, suppliant, adaptable, con tingent, responsive, fluent, and yielding through involvement
and incorporation. The attitude which runs throughout this collection of projects and essays is the shared attempt to place seemingly disparate forces into relation through strategies
which are externally plied. Perhaps, in this regard only, there are many opportunities for architecture to be effected by Gilles Deleuze's book Le plio The formal character tics of pliancy
anexact forms and topological geometries primarily-can be more viscous and fluid in response to exigencies. They maintain
formal integrity through deformations which do not internally cleave or shear but through which they connect, incorporate and affiliate productively. Cunning and viscous systems such
as these gain strength through flexible connections that occur by vicissitude. If the collected projects within this publication do have certain formal affinities, it is as a result of a folding out of formalism into a world of external influences. Rather than
speak of the forms of folding autonomously, it is important to maintain a logic rather than a style of curvilinearity. The formal
affinities of these projects result from their pliancy and ability to deform in response to particular contingencies. What is being
asked in different ways by the group of architects and theorists in this publication is: How can architecture be configured as a complex system into which external particularities are already
found to be plied?
58 ARCHITECTURAL CURVILINEARITY
Notes
Robert Venturi, Complexity and Contradiction in Architecture (New
York: Museum of Modern Art Papers on Architecture, 1966).
Two ideas were introduced in this text that seem extremely relevant
to contemporary architecture: typological deformation and the
continuity between objects and contexts. Both of these concepts
receded when compared with the dominant ideas of collision cities
and the dialectic of urban figure/ground relationships. Curiously, they
illustrate typological deformations in both Baroque and early modern
architecture: "However, Asplund's play with assumed contingencies
and assumed absolutes, brilliant though it may be, does seem to
involve mostly strategies of response; and, in considering problems
of the object, it may be useful to consider the admittedly ancient
technique of deliberately distorting what is also presented as the
ideal type. So the reading of Saint Agnese continuouslyfluctuates
between an interpretation of the building as object and the building as
texture . .. Note this type of strategy combines local concessions with a
declaration of independence from anything local and specific." 77.
See Sanford Kwinter and Jonathan Crary, "Foreword," Zone 6:
Incorporations (New York: Urzone Books, 1992), 12-15.
.1 (rilles Deleuze, A Thousand Plateaus: Capitalism and Schizophrenia
(Minneapolis, Minn.: University of Minnesota Press, 1987),478.
Mark Wigley and Philip Johnson, DeconstructivistArchitecture: The
Museum ojModernArt, New York (Boston: Little, Brown, 1988), 15.
" Marion Cunningham, The Fannie Farmer Cookbook, 13th edition (New
York: AlfredA. Knopf, 1990),41-47.
Deleuze, Plateaus, 475-6.
H An application ofvicissitude to Kipnis' logic ofundecidability and
weak form might engender a cunning logic of non-linear affiliations.
This seems apt given the reference to both undecidability and
weakness in the definition of vicissitudes.
II Ann Bergren's discussions of the metis in architecture is an example
of cunning manipulations of form. For an alternative reading of these
tactics in Greek art also see Jean-Pierre Vernant.
I II Deleuze, Plateaus, 256.
I 1 This concept has been developed by Leibniz and has many resonances
with Sanford Kwinter's discussions of biological space and epigenesis
GREG LYNN 59
as they relate to architecture and Catherine Ingraham's logic of the
swerve and the animal lines of beasts of burden.
12 Wigley, DeconstructivistArchitecture, 22.
13 See "0-0" by Robert Somol in the WexnerCenterfor the VisualArts,
special issue ofArchitectural Design (London: Academy Editions,
1990). 14 Webster's New Collegiate Dictionary (Springfield, Mass.: G&C Merriam
Company, 1977), 1170 15 Edmund Husserl, "The Origin ofGeometry" in Edmund Husserl's Origin
ofGeometry:An Introduction, trans. Jacques Derrida (Lincoln, Neb.:
University of Nebraska Press, 1989).
16 See Fetish, ed. Sarah Whiting, Edward Mitchell, and Greg Lynn (New
York: Princeton Architectural Press, 1992), 158-173.
17 Webster's, 883. 18 Rajchman identifies an inability in contexualism to "Index the
complexifications ofurban space." John Rajchman, "Perplications:
On the Space and Time of Rebstock Park," in Unfolding Frankfurt
(Berlin: Ernst &Sohn Verlag, 1991), 21.
19 Asimilar exchange, across disciplines through geometry, occurred
in France in the mid-18th century with the development ofprobable
geometries. Initially there was a desire to describe chance events
with mathematical precision. This led to the development of a
geometric model that subsequently opened new fields of study in other disciplines. The mathematical interests in probability of the
professional gambler Marquis de Chevalier influenced Comte de
Buffon to develop the geometric description of the Needle Problem.
This geometric model of probability was later elaborated in three
dimensions by the geologist Dellese and became the foundation for
nearly all of the present day anatomical descriptions that utilise serial
transactions: including CAT scan, X-Ray, and PET technologies. For
a more elaborate discussion of these exchanges and the impact of
related probable and anexact geometries on architectural space refer
to my [A. Krista Sykes] forthcoming article inNYMagazine no. 1 (New
York: Rizzoli International, 1993). 20 RudolfWittkower, Architectural Principles in the Age ofHumanism
(New York: WWNorton, 1971). 21 Colin Rowe, Mathematics ofthe Ideal Villa and Other Essays
(Cambridge, Mass.: MIT Press, 1976).
60 . ARCHITECTURAL CURVILINEARITY
l1. For an earlier instance of discontinuous development based on
environmental forces and co-evolution, in reference to dynamic
variation, see William Bateson, Materialsfor the Study ofVariation:
Treated with Especial Regard to Discontinuity in the Origin ofSpecies
(Baltimore: Johns Hopkins University Press, 1894).
l.l Erwin Panofsky has provided perhaps the finest example of this kind
of heterogeneous smoothness in his analyses of Egyptian statuary and
the Sphinx in particular: "three different systems ofproportion were
employed-an anomaly easily explained by the fact that the organism
in question is not a homogeneous but a heterogeneous one." l4 Webster's, 445.
l;, In Stan Allen's introduction to the work of Douglas Garofalo forthcoming in Assemblage 19 (Cambridge, Mass.: MIT Press,
1992) a strategy of camouflage is articulated which invests surfaces
with alternatives to the forms and volumes they delimit. The
representation of other known figures is referred to as a logic of
plumage. For instance, a butterfly wing representing the head of a bird invites a deceitful detection. This differs from the disappearance of a surface by stealth which resists any recognition.
l(, This suggests a reading of Michael Hays' text on the early Mies van der
Rohe Friedrichstrasse Tower [unbuilt] as a tactic of disappearance by
proliferating cacophonous images of the city. Hays' work on Hannes
Meyer's United Nations Competition Entry is perhaps the most critical
in the reinterpretation of functional contingencies in the intensely
involved production of differentiated, heterogeneous yet continuous space through manipulations ofa surface.