Fuzzy Metrics

Dana Čupková and Kevin Pratt are the core of EPIPHYTE Lab, an architectural design and research collaborative based in Ithaca, NY. Their work engages the built environment at the intersection of ecology, computationally driven processes, and systems analysis. Dana Čupková is a visiting assistant professor and Kevin Pratt is an assistant professor at the Cornell University Department of Architecture.
Fuzzy: hairy, furry, fluffy, downy, wooly, blurry, unclear, nebulous, hazy, vague, misty, bleary, indistinct, uncertain, ambiguous, unclear, confused.[1]


At first glance it seems that mathematics is all about clarity, while fuzziness implies the opposite: a softness of touch, the sensuality of something enjoyably fluffy, or even cute, covered with furry haze.[2] It seems improbable that fuzziness could be a concept used in applied mathematics as a method of specific problem solving, given that mathematics (especially in the design disciplines) is usually associated with precision, numerical determinism, and the possibility for a singular truth. “Design truth”—a scary thought, one might think, as the glow of numerical matrices encases the design world in an impenetrable, and yet fuzzy, veil of data. But while optimization routines and partial feedback loops wrestle with the beauty of objectified mathematical topologies, the ambiguities encompassed by a contemporary architectural practice embedded in a planetary biosphere are growing exponentially.

There are other ways to interpret fuzzy: blurry, uncertain, confusing.[1] Here we are on firmer (or at least more familiar) semantic ground. So, as a kind of disciplinary reflex to the memetic culture of the 21st century,[3] we have already constructed, analogically, a literalization of blurriness. Hairiness or fuzziness as a formal, experiential, or atmospheric aspect of design has emerged as yet another breed of contemporary production, strongly associated with the notion of style, exemplified by such buildings as British Pavilion at the 2010 Shanghai Expo by Heatherwick Studio. Yet our interest in thinking about fuzziness lies elsewhere. It lies in using computational protocols to investigate a multilayered stochastic network of intuitive and observed patterns. We understand such a matrix as a territory of probability, a layering of intervals, and not as a singular underpinning for globally computable solutions. This is a merged territory between the mathematical and the empirical; a patch system of partially computable protocols and observed concurrences between the built and natural world.

“The map is not the territory”[4]

However, taking such a stochastic approach requires coming to terms with a contemporary shift in the disciplinary territory of architecture. The discipline of architecture constructs itself between the twinned poles of art and science, producing a schizophrenic condition characterized by an attraction to, and repulsion from, both the determinacy of the quanta and the ineffability of the sublime. Of late, the availability and ubiquity of digital methods has driven the profession to adopt tools incubated in departments of engineering and computer science, while at the same time creating a peculiar resistance to explicit modes of analysis. Thus, as the outside pushes in, some notion of disciplinary singularity attempts to reassert itself, and pushes out. One casualty of such resistance has been the systemic assimilation, within the discipline, of ecological thinking in the service of global sustainability, since sustainability necessitates some acknowledgment of the concept of the computable metric as a means to measure the potential impacts of design on the biosphere.

The reluctance to embrace such pressures and accept certain quantitative methods has its roots in a fear of numerical determinism, is husbanded by a habitual obsession with convention, and buttressed by the fear of loss of identity. Yet our fussy disciplinary boundaries need to be more fuzzy and inclusive of new knowledge. In the end, this will depend on understanding that just because phenomena can be quantified (or even qualified) does not mean that their interactions and ultimate effects can be determined with any certainty over time, and further that such uncertainty does not imply illegibility. Ironically, the existence of such a middle ground, between determinacy and disorder, has been accepted and systematically studied in other disciplines—especially in applied mathematics— for decades.

In advanced mathematics the terms “fuzzy system” and “fuzzy set” refer to an approximate mode of reasoning; to a non-binary logic based on the notion of dynamic range, of an interval with elastically modifiable boundaries. Imprecision and gradient are fuzzy. A partial truth has greater value, because it admits to indeterminacy and can thus be used in a probabilistic analysis. The potential for degrees of a multiplicity of simultaneous memberships in disparate groups (intervals) is the basis for this particular expansion on classical set theory. It is, at root, a stochastic vision of reality, in which logical chains branch and intertwine, forming thickets of probable outcomes. Fuzzy logic theory[5] has enabled advancement in the fields of probability theory, data and knowledge mining, pattern recognition, robotics, artificial intelligence, genetic algorithms, non-linear dynamic analysis, forecasting, simulation, and cognitive psychology. Fuzzy sets are a cornerstone of the mathematics of uncertainty. And the world is uncertain. Despite this fact, contemporary building practice tends to describe architecture with certainty as an objects situated in a spatial and cultural field, objects contained within closed systems having definable boundaries. Ecology, on the other hand, posits that all entities within the system have thermodynamic relationships to each other and are bound together in complex systems of energy and information exchange: an ecosystem. And as the boundaries of ecosystems are uncertain, they are open and transient.[6] Without such dynamism there could be no change, no evolution; our planet would be static and cold. The systemic boundaries in ecology are nonexclusive and temporal. Just like in fuzzy sets. But in contrast the production of a particular design necessarily and inescapably dictates distinct boundaries to discreet objects. The process of design, with its dependence on an a priori vision of the future, stands in opposition the mindlessly effective processes of evolution. Thus the question becomes: how can architectural objects be “fuzzed” into the site? How can they nest in hierarchical sets of niche based relationships while maintaining the potential for a plurality of memberships in the future? In other words, how can they be designed to retain the ability to adapt to an uncertain future? One of the issues is that the disciplinary notion of “architectural site” systemically resists this temporality, the fuzziness of an evolving niche. Context is generally reduced to an abstraction of the site conditions, rendered as a culturally constructed understanding of the ground, as either palimpsest, analogic geometry, or infinitely occupiable surface. The necessity to respond and adapt to specific dynamic environments (from the bio-synergetic point of view) requires us to expand the definition of a context, to treat it as a series of nested complex systems embedded into larger patterns of dynamic exchange. To, in effect, construct the cultural through an understanding of ecology, and abandon the fixed idea of place in favor a fuzzy notion of indeterminate and spatially concurrent places.

The Computability of Patched Objecthood

Architecture is by default a component based practice, its efficacy bound by the economics of repetitive construction. Increase in component variation tends to be driven by a desire for distinctiveness and ornament, whereas a decrease in variation (often termed “value engineering”) is all too often driven by the need for economic viability. Due to the evolution of parametric digital tools and techniques one can (easily) produce an unlimited amount of variation, resulting in an expressive diversity of parts. Given the preponderance of formal concerns that seem to be a somatic imperative of contemporary practice, this reliance on digital geometric mutability tends to produce internally self-referential, non-scalar systems with low information content[7] (in that the essential point of differences between parts remains mysterious) and a need for a high degree of construction customization. Ironically, the fundamental disconnect between the abstraction of computational topology and the concreteness of the built environment, when coupled with a capacity for generative multiplicity, all too often leads to projects whose inanity is matched only by a commitment to a heroic, baroque un-constructability.

Yet specific information (such as climate, structural, or material resistance) does suggest a way to produce concrete information and effects through the use of analytic data as input for inquiry. Unfortunately, a current tendency to embed multiple performative and programmatic requirements into singular mutable elements, which require a high degree of specificity, stands in contradiction to the need for adaptation over time. Put more simply, we cannot overcome the “inanity problem” merely by introducing “performance” as a variable that controls one aspect of the multitude. Such an approach leads to a hyper-specificity that has only a low degree of adaptability and limited time-based effects.

Therefore the methods by which we define adaptive strategies on multiple nested and interlinked scales are critical. And how these strategies respond to the notion of energy capital and power usage—as well as a desire for material effects in support of social systems—could demonstrate a different approach in using ecology to anticipate an architecture that is, within a certain lifespan, tightly bound to the specificity of place; yet not hyper-specific in and of itself (in other words, systemically, if not at the component level: fuzzy). This logic anticipates the need for a system of parts having the ability to dynamically transform both during the process of design and construction and throughout its functional lifespan. It inspires us to rethink architectural material and its componentization in more complex way. It suggests a need for biomimetic strategies, using biomimetic process not for the creation of a formal analogy but to develop an understanding of performance, systemic logic, and materiality, where component evolution produces variation through degrees of adaptive mutation in multiple, loosely bound layers, and not within a single, continuous surface. At the most basic level it requires us to recognize that hierarchy, which is a distinguishing feature of all ecosystems[8] will be a necessary organizational principle of architectural eco-logic.

Green Negligee

Some suggestion of how such processes might play themselves out can be gleaned by looking at a project we have named[9] Green Negligee. The Negligee is a modular retrofit system selectively attached to prefabricated concrete panel housing blocs in Petrzalka, Slovakia. The Green Negligee attempts to address social, urbanistic, and ecological problems created by the area’s top-down ciam-inspired compositional and organizational schema, while at the same time dealing with energy consumption issues stemming from the fact that the un-insulated concrete building blocks perform poorly in Slovakia’s temperate climate.

Aerial view of the Green Negligee, EPIPHYTE Lab: Tensile network, component systems, energy harvesting devices, plantings, and communal apartment blocks.
Aerial view of the Green Negligee, EPIPHYTE Lab: Tensile network, component systems, energy harvesting devices, plantings, and communal apartment blocks.

Constructed across the Danube from Bratislava (the de facto capitol of the Slovak Republic) in the sixties and seventies, as a communitarian parallel to the modernist, social democratic model of a “functional” city, Petrzalka is an example of a state-controlled planning process that resulted in rigid programmatic segregation and a compositional urban scheme dominated by the application of a panelized prefabricated concrete system to the task of social engineering. Despite the stigma usually attached to this type of social housing, the real estate itself has become quite valuable due its proximity to the city center and shifts in post-Soviet demographics. The privatization of state owned apartment blocks with centralized utility systems has created ambiguities about the extent of ownership beyond the line of the single apartment. Privatization has also led to questionable ownership of auxiliary spaces and energy systems, neglect, and maintenance problems, as well as imposing on residents the human costs of living in fundamentally unsustainable buildings. The issue of landscape ownership is particularly problematic. Historically the district was used for viticulture; during the era of collective ownership, the strong cultural need to transform the ground within the stark figure of the rectilinear plan was channeled into cultivation, planting, and gardening. In the post-Soviet era, this desire has been suppressed by the loss of coherent ownership of public spaces. Thus, because of the general deterioration of the collective, nature is reasserting itself. Every crack is overgrown and filled with weeds. As a result, the figure-ground is dissolving, adding textural and spatial complexities.

The effect of the weeds is fuzzy. The Green Negligee project takes its inspiration from these weeds. The purpose of the Green Negligee is to leverage the embedded (yet currently suppressed) social desire for collective cultivation and leisure, and the political and economic ad-hoc social conditions that have developed since the fall of communism, to create a system that can be used to engender a sustainable future for the Petrzalka district. It attempts to accomplish this by blurring the hard boundary between the buildings and the landscape with lightweight, secondary facade elements that host a variety of low-maintenance quasi-naturalistic systems, energy harvesting devices and native biotic species that, in their aggregation, create alternatives to installed building service systems, new social spaces, and opportunities for emergent biodiversity.

These social spaces bind the disjointed and atomized physical spaces of the existing blocs into semi-autonomous zones of habitation that nest into larger, pre-existing ecological systems that define the district as a whole.

Given this desire to integrate the designed and ecologically and socially pre-existent, the first step in the design process was to understand the dynamic environment that exists, currently, on the site. To create, in effect, a layered dynamic map of the site that could function, virtually, as a kind of test bed to evaluate potential methods of intervention. This test bed was constructed by aggregating existing site information in a four-dimensional digital model, while running simulations on the model to deduce the existence of conditions that would eventually influence the nature of intervention. In parallel, we gathered information about existing economic and social systems, paying particular attention to the way that modes of ownership in the post-Soviet era have created rather novel structures for collective decision making and programmatic re-appropriation of existing functional spaces. We also surveyed existing building service and structural systems, assembling a picture of the way that the existing engineered systems both meet (or fail to meet) the basic needs of the inhabitants, and how these systems interact with both larger engineered and natural systems that serve and define the entire region.

A few key insights came out of this process. One: that the collapse of a top-down authority has led to atomistic patterns of ownership, and that pressures from the transition to a market-based economy has led to the formation of ad-hoc decision making structures at the scale of the building, as well as localized uncoordinated reprogramming, meaning that efforts that require much coordination at the macro- or site scale are impractical. Two: that the population is beginning to reorganize at the local scale to oppose the co-option of public space by external market forces, but it they lacks any particular physical mechanism to effect its plans. Three: that existing service systems are both inadequate and almost completely dependent on off-site sources and sinks to generate energy and dispose of wastes, meaning that local ecosystem services remain largely untapped. Four: that the physical layout of the buildings and the landscape, in interaction with climatic and hydrological forces, create multiple overlapping zones for opportunistic intervention, which reveal themselves through dynamic mapping processes. In other words, there exists, between figure and ground, a third zone, defined by the ephemeral forces of sun, wind, and water, and uncaptured by existing modes of bureaucratic control: an unoccupied niche.

Paramatric Datascape, EPIPHYTE Lab: Example of dynamic environmental mapping technique used to discover the ephemeral hidden landscapes formed by the interaction of climate and site. These invisible boundary conditions are determined based on radiation mapping and computational  fluid dynamics analysis. They define niches where componentized energy and waste processing systems would be more or less functional and could be deployed. The goal is to define zones of overlap, or gradients, where viticulture, wind harvesting, and gray-water filteration are effective.
Paramatric Datascape, EPIPHYTE Lab: Example of dynamic environmental mapping technique used to discover the ephemeral hidden landscapes formed by the interaction of climate and site. These invisible boundary conditions are determined based on radiation mapping and computational fluid dynamics analysis. They define niches where componentized energy and waste processing systems would be more or less functional and could be deployed. The goal is to define zones of overlap, or gradients, where viticulture, wind harvesting, and gray-water filteration are effective.

Given that, axiomatically, nature abhors a vacuum, we designed a system to fill that niche. Using the dynamic digital map as a base model, we have proposed a hierarchical system of parts that can be implemented at the sub-building scale, and then be aggregated to interlock both with the larger built environment and the existing landscape. The primary component system forms tensile cable networks that create a secondary surface between ground and facade—a host for the functional systems that produce both spatial effect and systematic integration. These networks enable a new form of landscape distribution, subdivision, and ownership as the vertical tensile system transforms into an obliquely malleable catenary landform. This modular net is itself anchored to the existing buildings which, due to their method of construction, are massively over-structured and are thus able to bear significant additional loads and the existing landscape, which is, for the most part, intermittently occupied and given over to automobile parking.

The form of the net is developed using relaxation algorithms applied to boundary conditions between areas of relative environmental and programmatic differentiation revealed by the digital map of the site. Given that in any defined zone a multiplicity of these boundary conditions exist (between areas of lesser and greater turbulence, available radiation, existing view corridors, etc.), particular surfaces and different subdivision patterns can be selected to create terrains that are more or less suited to different purposes—to form enclosures, private winter-gardens, and semi-enclosed public spaces, to be hosts for localized viticulture and other water-purifying horticulture, and to support localized energy-generating devices.

Systems of secondary components populate the nets. They operate according to a patch logic, defined by intervals associated with particular ranges of performance. This allows aggregated systems to intertwine, producing second-order architectural and spatial effects. Potential component distributions are determined through interaction with the digital map, which delineates fuzzy zones of relative effectiveness, and their specific layout derived from matching these potentialities with programmatic and formal intent.

The components are not integrated into the cable network: rather they overlay the network in a loose pattern, meaning that they do not require a geometric specificity that is dependent on the underlying geometry of the net itself. The multiple component types have a myriad of functional overlaps, including the collection of water, the transformation of wind and solar energy into electricity or mechanical energy, and the provision of armatures to support both biotic communities and the growth media they require. Each of the secondary component types has a definite interval of variation, which is again governed by material properties, methods of assembly, and functional constraints. Because the components are relatively small and loosely attached to the net, it is imagined that they could be reconfigured without necessarily disrupting the system as a whole during a functional lifespan.

The components themselves aggregate into larger overlapping systems. They tend to organize themselves sectionally, and spread horizontally to define space. An example of such an aggregation might be: wind-powered devices at higher elevations to take advantage of prevailing winds (which come from the northeast) and power existing building systems and devices like lights and pumps that are part of the network. Shingles, of different types, that congregate on the near vertical surface of the network near the facade to form a secondary system of enclosure to both shade and trap solar radiation. A mixture of trellis and planter components, occurring where the network turns horizontal, which capture and filter both runoff and graywater, host grape vines that allow for localized viticulture, and create a semi-enclosed public space. The component network is married to a landscape that has been reconfigured using berms, swales, and gabions to both reinforce the attributes of the public spaces created and further store and process runoff water descending from the component patchwork. The landscape manipulation also serves to contain new spaces, which are necessary to service the new systems. Such a system could begin life occurring across a single structural bay of a building, and expand over time as both financing and political will become available.

Loose Fit Modularity, EPIPHYTE Lab: The adaptive logic of the overall system negotiates between environmental variables and the collective desires of the building co-op.
Loose Fit Modularity, EPIPHYTE Lab: The adaptive logic of the overall system negotiates between environmental variables and the collective desires of the building co-op.

Formal Study, EPIPHYTE Lab: Relaxation of lightweight tensile system network and pattern distribution logic for vibro-wind components according to wind speed data.
Formal Study, EPIPHYTE Lab: Relaxation of lightweight tensile system network and pattern distribution logic for vibro-wind components according to wind speed data.

Fuzzy Conclusions

It is important to understand that the configuration described above does not so much represent a singular design, which would be repeated ad infinitum across the larger site, but rather a particular scenario that might fit a localized set of conditions and intents. The process of making a “best guess” at a configuration suitable to a particular locale might be best described as making a series of choices at each level of the system hierarchy from a set of fuzzy potentialities—delineated by overlapping sets of analytical and observational data—that emerge from the indeterminate analysis. Because the digital representation of any particular scenario is parametric, the systems can, during the design phase, be “tuned down” and tested, that is, a series of scenarios can subjected to the same analytical procedures that were applied to the site before potential the intervention has been inserted. The hierarchical organization and loose fit between systemic levels also allows for a localized specificity that enables the system to scale all the way down to the human dimension, and be reconfigured over time. For example, imagine if a fairly large horizontal area of the cable net has been populated with trellis and planter components creating an overhead garden. If, in the future, a section of the public space under this system is converted to a playground, the patch of network above it can be covered with shingle, either in place of or above the plants, to form a pavilion of sorts that would enable children to play in the rain. Although this example seems rather prosaic, it is worth remembering that, ultimately, the point of the system is not to create the appearance of adaptability, or an image of green architecture to be sold to prospective investors, but rather to create some kind of, however imperfect, armature that supports potential methods of sustainable living that are developed out of the quotidian process of everyday life itself.

Rather than imagining the intricacies of form, the intent is to construct a system logic imbued with both formal and performative tendencies, while retaining enough malleability to escape the particuarity of a single instantiation. It is, perhaps, this tension between intent (system) and available modes of representation (image) that explains why the discipline of architecture has struggled to assimilate the idea that a project can be carefully designed, and yet resist a definitive tectonic formulation. It is, after all, difficult to apply traditional ways of making architectural judgments (relative proportionality, compositional effect, etc.) about the relative merits of a particular design to a proposal that remains, even during the course of its occupied life, fuzzy. It is here that we come to the most difficult piece of interdisciplinary thinking that, as architects, we have to accept, if we are to seize the territory made available made available to us by the intersection of computational potential and systemic, ecological modes of thought. It is simply this: we have to expand our definition of beauty and to acknowledge potentiality, to see value in systems that can become, outside of our direct control, something remarkable. If you ask a mathematician whether or not this:

(the definitive form of the integral) is beautiful, she will tell you yes. Not because of the way the ink has arranged itself on the page, or even because, as a method, the integral represents a particularly elegant way of solving a particular problem, but because the definitive integral is a fundamental piece of a system (the calculus) that enables anyone to act, freely, as a lever to move the systems that create the world.


The Green Negligee project was in part supported by the Arnold W. Brunner Grant AIA NY Chapter.

Epiphyte Lab’s project participants: Principal Investigators Dana Čupková & Kevin Pratt (Design and Production Team) / Michael Esposito, Travis Fitch, Shujian Jian, Siyuan Ye, Daniel Marino, Karbi Chan Yuet, Jeremy Burk (Analysis Team) / Andrew Heumann, Sebastian Hernandez. Special thanks to Francis Moon, Ephrahim Garcia. and William Jewell for their feedback.


1. English Thesaurus, Microsoft Word (Microsoft 2010).


3. See: Cultural Software: A Theory of Ideology by J.M. Balkin (New Haven: Yale University Press, 1998) and Viruses of the Mind by Richard Dawkins (Seattle: Integral press, 1991).

4. Alfred Korzybski’s original quote (1931) is cited by Gregory Bateson, in Steps to an Ecology of Mind (Chicago, IL: University of Chicago Press, 1972) where he explores notions of subjective representations of objective realities:

5. Fuzzy mathematics were originally discussed by Zadeh Lotfi, in “Fuzzy Sets,” in Information and Control, Vol. 8 (New York: Academic Press, 1965), and “Fuzzy Algorithms,” in Information and Control, no.12, 1968, pp. 94–102.

6. The example of a cattail comes to mind. Is cattail part of a land system or a water system? The answer is either—depending. Depending on water level and degree of rainfall the cattail adapts.

7. “The notion of the amount of information attaches itself very naturally to a classical notion in statistical mechanics: that of entropy. Just as the amount of information in a system is a measure of its degree of organization, so the entropy of a system is a measure of its degree of disorganization.” Norbert Weiner, Cybernetics or Control and Communication in the Animal and the Machine (Boston, MA: MIT Press, 1948).

8. Explained perhaps most eloquently by H.T. Odum, in the fourth chapter of Environment Power and Society (NY: Wiley & Sons, 1971)

9. With posthumous apologies to both F.L. Olmstead and the city of Boston.

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