Keywords

1 Topology

The analysis of topology as a contemporary mathematical discipline requires a transition from the term place to the term space because mathematics does not recognize places with their contextual particularities but examines and describes abstract mathematical spaces and everything they comprise. The relevant literature in the field of mathematical topology explains that, in general, topology studies the properties of geometric shapes that are preserved under continuous deformation, such as connectedness or compactness, i.e., mathematical topology makes no distinction between two shapes or two spaces if it is possible to shift from one to other under continuous deformation. When it comes to these spaces, it is irrelevant whether something is large or small, round or square, if it can be changed by stretching or bending, for example. The difference between the two spaces is primarily related to the components that remain unchanged under the deformation [1].

Sergei Petrovich Novikov emphasizes that it is even intuitively clear that knowledge about the geometric properties of shapes does not end with data about their metric characteristics, such as length, height, angles, etc., i.e., that “there remains something beyond the limits of the old geometry” [2]. Regardless of the length, the line can be open, closed, tied in a knot, several lines can be chained in different ways, volumes can have holes, etc. These and similar properties of geometric shapes, but also of various mathematical objects that do not have geometric realizations, are characterized by the fact that they do not change during deformations without interruption. Some typical examples of topological spaces are the Moebius strip, Klein’s bottle, tori, various knots, and similar objects.

During the 19th century topology was developed by several mathematicians, among others, Karl Friedrich Gauss, Bernhard Riemann, etc. but it is considered that topology as an autonomous branch of mathematics was founded by Henri Poincaré at the end of the 19th century. In the following decades, its internal problems were solved, so that in the second half of the 20th century there would be a more serious breakthrough of topological methods into modern physics and chemistry, but also a more general interpretation of topology through the discourses of the social and humanistic sciences.

It is certain that the mathematical definition of topology, when separated from the main field of research, is difficult to understand and cannot represent a basis for further analysis of the appearance of topology in the architectural discourse. Partly it can be explained by the fact that it is a scientific field that requires more complete and greater mathematical knowledge, the subject of research is far from the perceptible world, and therefore it is difficult to explore its visuality. In odred to comprehend the evolution of topology in architectural discourse, one must look deeper into the history of science, especially mathematics.

2 Toward Architectural Discourse

Morris Kline indicates that the first thoughts on topology can be found in the works of Gottfried Wilhelm Leibniz, in the book “Characteristica geometrica” from 1679, in which Leibniz introduces the term Analysis situs /position/, in order to opposite size and form, emphasizing the lack of an adequate term when talking about form [3]. Also, in the letter to Christiaan Huygens, Leibniz points out that: “We need another strictly geometrical analysis which can directly express situm in the way algebra expresses the Latin magnitudem” [4]. In order to understand his idea to differentiate the properties of geometric shapes by position and by measurement, it is important to take into consideration the fact that at the same time, Leibniz worked on the invention of calculus. It is also known that, as a branch of modern mathematics, topology initially arose from the study of geometric problems, but its methods are based on Georg Cantor’s theory of sets as well as on modern algebra. The roots of topological phenomena can be found in Euler’s work on seven Königsberg bridges from 1736, but the first truly modern fundamental concepts of topology were given by Henri Poincaré in 1895 in one of the most significant classical works of mathematics - “Analysis situs” [5].

As more important for this research, it is the draft overview of relevant literature of the history of mathematics that shows that the development of topology runs parallel to the achievements that will distance mathematics from the sensory world. Between 1830 and 1850, Nikolai Ivanovich Lobachevsky and János Bolyai published the first model of non-Euclidean geometry based on the understanding that Euclid’s fifth postulate is independent and that there can be logically non contradictory geometries containing an opposite one [6]. In the 19th century, Bernhard Riemann developed another kind of geometry based on the generalization of Gauss’s concept of “curvature”. He also stated that information about points in space need not necessarily be obtained using the coordinate system, the ultimate transcendental space of the Cartesian system, but that it is possible to determine for each point its local properties contained in the space itself. Riemann thus clarified that mathematical objects can be released of the external reference system, i.e., they can be defined as fields of local information. For the broader interpretation of observed reality, a proof such as Beltrami-Klein’s from 1868, which equalised two geometries, one that belong to the real world of human perception and one that does not, meant the absolute relativization of reality as people understood it so far.

Mathematics philosopher Stephen Francis Barker points out that when we talk about the curvature of space, we must not assume or imagine a visual representation of curved space. Although separated from observable reality, the consequences of discovering these geometries were fundamental to the epistemological status of mathematics and for its wider intellectual influence [7]. In general, the development of topology, along with other mathematical achievements from the same period, indicated that the prevailing philosophical platform of Immanuel Kant was being undermined, in which mathematics had a special status as the essence of all natural sciences but had to be applied. Zvonimir Šikić, in his book on the new philosophy of mathematics, discusses the problem of the relationship between the abstractness of mathematics and the reality of nature, i.e., the applicability of contemporary mathematics in reality, and emphasizes that the culmination of this concept can be found in the philosophical platform of Immanuel Kant at the end of the 18th century. It is based on the idea that abstract mathematics is always directed to the description of nature because mathematical knowledge is specific as knowledge of the a priori forms of space and time, which are also components of reality [8]. Despite being intuitive, mathematics was still necessary for Kant to refer to the sensible world. In this context, the new way of thinking that accompanied the discoveries did not rule out the applicability of mathematics. However, the discipline was no longer prioritizing it. As a result, over time it stopped being a priority for all sciences that rely on mathematics, and ultimately for the overall understanding and perception of the world that surrounds us.

The methodology of applying mathematical concepts to a broader range of knowledge often draws on specific knowledge in various fields, and Arkady Plotnitsky defines it with the term “quasi-mathematics”[9]. Although he does not question the philosophical influence of mathematics on the development of civilization, he states that quasi-mathematics enables the dissemination of certain mathematical concepts and principles which, although originating from it, are not exclusively defined by its tools and, as such, become possible and applicable outside its disciplinary framework. With the term quasi-mathematics, Plotnicky explains the difference in the interpretation of algebra, geometry, and topology in a general sense. He interprets algebra as the ultimate concept of formalization, be it the formalization of systems in the natural sciences, conceptual systems as in logic or philosophy, or the language system that exists in linguistics. In this sense, “algebra” means a set of specific formal elements and their relations. On the other hand, “geometry” and “topology,” although both deal with questions of space, are distinguished by their mathematical origins, “geometry” arises from the measurement of space as geo-metry. In contrast, “topology” ignores quantities and deals exclusively with the structure of space (topos) and the essence of the form of a shape.

Such reflections have shown that different transitions of concepts from mathematics to other discourses, and therefore to architectural, where possible, whether it is about exact application or flexible appropriation of notions. With the previously presented broader image of mathematics in the field of science, it becomes clear that the path from topology to architecture has become open. During the nineties of the twentieth century, this will become particularly significant in architectural theory.

3 Topological Deformability in Architectural Theory

Even though the dominant architectural style in the most of the twentieth century – Modernism was based on the standard elements of Euclidian geometry, there were examples that architects were familiar with the more organic, freely deformed architectural form, but that was never referenced in the topological terms. However, the small number of buildings and significant research work during this period indicate that architects did not have a aspiration to include topology in the dominant movements of architecture.

At the beginning of the nineties of the twentieth century, with the appearance of adequate digital tools in the architectural design process, the conditions for more extensive research of modern mathematical theories of space arrested. Thus topology has started to become an integral part of the architectural design methodology, and therefore the architectural theory. The first attempts to record and analyze the term topological deformation in the architectural theory appear in the historical and theoretical overviews of contemporary architecture using more general term, topological architecture.

Mario Carpo explains the new architectural avant-garde at the beginning of the new millennium, known as topological, as an architectural response to the new digital technologies that were flourishing at the time. “Topological” architecture, as it was called then, was seen for a while “as the quintessential embodiment of the new computer age - and we all remember the excitement and exuberance that surrounded all that was digital between 1996 and 2001” [10]. Branko Kolarević uses the same term while classifying digital architecture: “This new fluidity of connectivity is manifested through folding, a design strategy that departs from Euclidean geometry of discrete volumes represented in Cartesian space, and employs topological, “rubber-sheet” geometry of continuous curves and surfaces” [11].

The similar term topological tendencies in architecture were introduced by Guiseppa Di Christina in her doctoral dissertation “Architecture and topology: for a theory of space in Architecture” in 1999 at the Faculty of Architecture in Rome [12], where topological tendencies were explained as “the topologizing of architectural form according to dynamic and complex configurations that lead architectural design to a renewed and often spectacular plasticity, in the wake of the baroque and of organic expressionism.” Furthermore, she started defining the appearance of topology in architectural design in the domain of creating dynamic variations of form. The focus of her research is directed towards the formal vocabulary of buildings, where topological methods are primarily used to achieve the desired dynamics of the architectural form. Di Cristina also indicates a theoretical problem related to the question of to what extent the forms obtained by the dynamic process of topologizing are dynamic in the domain of architectural work. As the main protagonists of this, for her progressive tendency, she cites Peter Eisenman, Greg Lynn, Daniel Libeskind, and Bahram Shirdel, as well as the influence of the theoretical works of Bernard Cache, Jeffrey Kipnis, Brian Massumi, and other authors, crucial for the development of topological architectural forms.

As seen from the beginning, the use of the term topological deformability in architectural discourse pointed to the problem of formulating a comprehensive definition, because the interpretations were constantly shifted between the field of architectural theory of form and the field of architectural design theories. The first half of the 1990s was evidently dedicated to the “fascination with topological objects”, where for example, the project for the Guggenheim Museum in Bilbao from 1997, by architect Frank Gehry, was cited as a typical example of using the “deformation made possible by flexibility of topological geometry” with “forms that bending, twisting and folding” [13]. Moreover, the term topological architecture [14] is mentioned in some historical reviews, even as a strategy to create the new contemporary architectural paradigm, or a new architectural style.

As architectural criticism advanced with these tendencies, concerns about the idealization of form were raised. The majority of theoreticians and authors who influenced the development of the term topological deformability in architectural theory at the end of the nineties were directly confronted with the criticism of the idealization of geometry, that is, that placed deformability as a representative of the idea of diversity is placed exclusively under the framework of phenomenology. Referring to Di Cristina’s research, Michael Speaks underlines that the topological form technique, which is based on continuity and movement, is entirely negated by the finitude of the end product [15], additionally moving the problem into the domain of the experience of the architectural space. Mario Carpo sees it as a cause-and-effect relationship between digital technologies and complex geometry. He emphasizes that generalization has led to delusion and that many projects with computer-generated formal characteristics have become inconspicuous, almost banal architectural objects, and the use of digital tools, as well as the reference to topology, did not give objects validity. Antoine Picon emphasizes an additional problem arising from the topological treatment of form, which refers to the aesthetic valorization of deformed amorphous architectural forms. He sees part of the problem in the lack of an established aesthetic evaluation system for evaluating the aesthetic characteristics of new forms and another part in the process of their creation, which he underlines with the question of what in the process of form transformation determines when it will end [16]. Similar observations are made by Michael Meredith when he says that the result of using the topological method during the nineties is reflected in isolated physical and aesthetic models, which do not have broader significance but remain within their framework [17].

However, in the end of the ninties with moving away from the theory of form more towards the theory of design, some other interpretations of the term topological deformability have been developed, which will influence the architectural projects on a much deeper and more significant level. One of the basic definitions was given by the architectural theorist Kostas Terzidis in the book Expressive Form, a conceptual approach to computational design, introducing the term “topological operations” which includes twisting, stretching and compression of the architectural form, excluding cutting and tearing. Any type of operation that deforms the form by hollowing out, creates two topologically different entities, which leads him to the conclusion that “topology should be used in order to achieve the unity of the form, because it preserves the integrity of the endlessly transformed geometry” [18]. He implies that certain formal properties remain unchanged, even when the geometric shape undergoes intense distortions, resulting in the loss of its metrical and projective properties. Apart from mathematical precision, the great importance of Terzidis’ definition lies in the clear distancing from traditional architectural methodologies that were based on addition and substitution of forms. Similar explanations of the topological method speak of a departure from the Cartesian geometric model in architecture towards a more complex, non-linear logic of space, with which it is possible to express the flexibility and continuity of an endless number of variations.

The transition from “making form” to “finding form” occures at the moment when the question of curvature is left aside, and along with the complex network of influences mentioned above, the topological deformability has become an integral part of architectural design methodology. This phase of development implied that the topological deformability should be considered as a comprehensive spatial system, where topology is understood as a flexible structure formed by specific and clear relations, which remains unchanged as a result of transformation and deformation. To design topologically, it was implied to emphasize specific relations or certain “conditions” which are key to the logic of organization, whereby geometry is flexible when it comes to dimensions, distances, or form. By the end of the nineties of the twentieth century, the topological deformability was no longer interpreted as the geometry of architectural objects, nor its prototype, but as a demonstration of certain geometrical principles. Topological thinking implies that spaces are not about a specific form but rather about relations. The authors explain this by stating that topological principles can be manifested through various forms where “the concept of continuity is obtained only by applying algorithmic logic” [19].

Over time, these types of definitions resulted in a more diverse understanding of topological deformability, leading to a more liberated and broader interpretation of the term in various contexts. It will turn out that evident heterogeneity in use, without a clear system or unique definition, has spread the term far beyond the limits of architectural discipline to many contiguous scientific fields, showing clearly the inherent architectural interdisciplinary dimension. The more the term was used in the domain of design methodology and less in the domain of form design, the easier and faster it started to increase its social impact.

4 Transformation of the Thinking Modality

As demonstrated, the outcome of the shift in the scientific paradigm from a determined and stable to temporal and complex, which resulted in a distinct comprehension of the contemporary cultural context, brought the increasing complexity in diverse domains of architectural theory and practice. Although criticized from many aspects, its formalism, lack of relevant space logic, a fixation on digitalization etc. the topological deformability as methodology appear in the architectural discourse as a response to the more comprehensive scientific and cultural context.

On the one hand, the complexity of the spatial structures encountered by the users of the built objects undoubtedly influenced their relationship to the space, as architectural form remains inseparable from the way we experience the world, which involves our senses and perceptions. The complex relationship of people to architecture is an intimate and longstanding one, and it is strongly linked to the relationship of the human body to the wider cultural context. This relationship has evolved over time to reflect the philosophical and architectural discourses that shape both.

On the other hand, with the development of computer technologies, there has been a change in our notion of materiality. Our age can be defined as a flow of information, and architecture captures this flow, creating more complex conceptions and interactions through the space. In this regard, it cannot be ignored that aspects that influenced the occurrence of topological deformability in architectural theory were strongly supported by the development of digital technologies at the end of the twentieth century. It is clear that only with the development of technological tools has architecture become interested in these types of complex spatial relations, primarily through the research of the medium itself - software. Upon examining the chronological progression from craft to engineering and to the digital design of virtual or natural spaces, it is evident that the study of medium has consistently dominanted architectural practice. However, when discussing topological deformability in architecture, is it simply dealing with the medium, or is it something else?

The overall picture of the emergence and evolution of the concept of topological deformation from mathematical abstraction to the creation of architectural space suggests that the essence of the influence originated from a change in the thinking modality. The phenomenon of topological spatial structures where the sole relevant factor is their deformability refers to the idea that it is possible to tolerate, but initially endure, the most diverse types of deformation. This research argues that topological deformability through architecture opened the way to the essential acceptance of the different, to the changing the idea of otherness. If architecture, together with natural sciences, contributes to the creation of a specific system of world perception, then the predominant role of topological deformability in architecture is to serve as an instrument of expressing plural social realities.

5 Idea of Otherness

Otherness can be viewed as an articulation of diversity as well as a definition of differences. According to Jean-Francois Staszik, difference belongs to the realm of fact, and otherness to the realm of discourse. The notion of otherness mainly examines the idea of a criterion that allows humanity to be divided into two groups: one that embodies the norm and whose identity is valued, and another that is defined by its faults and devalued [20]. Hence, the concept of otherness is attributed less to the distinction between the other and the other person than to the perspective and discourse of the individual who perceives the other as such. Since topological spaces deal with relations and connections with a given spatial context and not a specific form, it is clear that a particular topological construction can manifest itself through numerous forms. It is more spatial relation than a spatial determinism. Furthermore, through the relation with topological spaces, one can become aware of the variability of form and, therefore, the possibility of the existence of other forms. It is possible to create an idea of a space that is subject to change, which can lead to a different enviroment for users. If such spatial structures also belong to the multifunctionality, where numerous activities are interwoven or possible, then the idea of finality and certainty of the space is changed.

Moreover, it is possible to assume that there is not only duality, the opposition between self and other, but that many spaces in many forms with the most diverse activities are possible. To put it differently, despite the inherent tendency of humans to make categorical distinctions, the categories themselves and meanings associated with them are social construction rather than natural processes. Therefore, it is possible that topologically designed spaces may open the way to the diversity of multifunctional deformable spaces rather than to construct a new architectural typology.

According to the theory, the notion of otherness originated in a spatial form, arosing from the idea of difference that is associated with the geographical nature of segregation. This approach implies that groups are divided into territories or spatial units with clearly defined boundaries that are difficult or impossible to exceed. However, topological constructions fundamentally change the relationship between the outside and the inside because, for those types of spaces, it is impossible to determine their boundaries. The only relevant characteristic of the structure is its ability to deform. In general, the notion of the boundary of space in mathematics, even when viewed chronologically, is closely associated with the notion of distance between two elements. The idea of metrical space corresponds more closely to the idea of Euclidean space, as it relies on understanding of spatial relations, such as the notion that the distance between two points is always positive. At a higher level of abstraction, the distance between two elements must be understood as a transition from one element to another, which, in the context of topology, is continuous.

As previously demonstrated, in architecture, the interpretation of purely mathematical definitions moves away from the original model. Hence, the treatment of the relationship between exterior and interior, wherein one can simultaneously be both the exterior and interior of the architectural space, requires that the users permanently change their relationship to the space. In the architectural discourse, the hierarchical treatment of the structure can be precisely discerned through the outside-inside relationship. With a model like the Möbius strip, the boundaries and images of a hierarchical structure are weakened. At the end, it becomes evident that one of the primary characteristics of topological spaces is the ability to blur the distinction between territorial boundaries, as well as to examine the traditional spatial duality between interior and exterior, employing methodologies that involve integreating the structure with the immediate environment.

6 Conclusions

This paper examined a widespread problem in architectural discourse, the denotation of concepts from other scientific fields, which, due to the interdisciplinarity of architecture, have become an integral part of the design process.

The beginning of the paper provided a brief overview of the term topological deformability within its native field - mathematics. Since topology is difficult to understand and appropriate, it has remained highly abstract to the architectural discourse. Nonetheless, taking into account its potential for expansion into other scientific fields, a multitude of interpretations appeared, along with a multitude of topological propositions that expanded the boundaries of topology beyond its native field, thereby bringing it closer to architectural theory and practice.

With an undeniable social impact, architecture traces the way of topology to a broader audience, with a specific impact on understanding the potential of the term deformation. Endless changes of form were understood as a potential to perceive transformability and to accept the differences in architecture. The significance of deformability was heightened, and it was imperative to clarify the limitations imposed on society as those that society ought to be able to tolerate. The paper examined how topological deformability in architecture, in a very subtle way, teaches us about and how to accept the differences.