Apprising everything around us, from our daily life, architecture, products and urban spaces, we can see topological architecture as a new paradigm. This new mode of thinking eco-logical suggests that humans, like any other constituent component of the earth’s ecology, are continuously in exchange with one another. Any other parameters constituting the system (Rocker, 2011).

The potential in parametric design hence the mathematical aspect of topological architecture, lays in its ability to produce a meaningful and hyper inclusive network consisting of parameters and relationships and at the same time managing variability (Sakamoto & Ferré, 2008 )(Carpo, 2013).

Nonetheless, also criticism is brought upon Topological architecture today. Although, parametric architecture aims for full control of design at all scales (Carpo, 2013), architecture is more complex than a person can singularly ‘map out’ with a computer. Not all relationships can be quantified or geometrical translated. Instead, architectural design requires cultural, social engagement and relevance, and is considered, primarily, a cultural socio-political form, not technological determinism. “It is not the parametric the relentlessness malleability of form, nor is it complexity for its own sake, but rather a complex of complex relationships that produce architecture” (Sakamoto & Ferré, 2008).

THEORETICAL FRAMEWORK

Parametric design can be seen as a discipline that is simultaneously searching for a unified organizational clarity (through diagram, parti, etc.) and visual complexity (Venturi).

With new methodologies, supported by computed (cinematic) techniques, the exploration of architectural form and experience is revised. By using relationships, interaction, diagram, program and movement, unpredictable forms and spaces are unfolding (Savaskan, 2012). “With this methodology, parametrising architecture, creates a new era in the understanding and creating the phenomena of spatiality” (Schumacher, 2009).

“Space can be seen as the purest, irreducible substance of architecture – the property unique to it, that sets architecture apart from all other artistic practices.“ (Forty, 2004). When we look closer at the term ‘spatiality or space’ there was until modernism, little agreement on what ‘space’ means.

Since the eighteenth century, architects used words as ‘volumes, voids, void spaces, loss of space’. Modernist architects, tried to give notion to the word ‘space’ and by 1920 the word was well established as a category in the architectural vocabulary (Carpo, 2013; Forty, 2004).

However, we can still find a large ambiguity within the descriptions of the term ‘space’. The ambiguity of the word starts within the field in which we discuss the term ‘space’, for example philosophy or architecture. While architecture sees ‘space’ as a physical property of dimensions or extent, ‘space’ within philosophy can also be a property of the mind. Thus, on one hand we talk about a ‘physical space’ – produced by architecture, while on the other hand we talk about the ‘lived space’ – a mental construct through which the mind consumes a space (Lefebvre and Denari and Lasdun in Forty, 2004).

The second cause of ambiguity lies within languages, in which the term space has different translations (Forty, 2004). The awareness of the effects of translation upon the meaning of the term ‘space’ should however, also be placed within time. Therefore, we cannot assume the meaning of space is fixed.

Modernisms’ notion of space was based on the concept; ‘universal space’. Parametric architecture however, does not differentiate spaces, but fields. Space is, within parametrical architecture, considered as empty, while fields are considered full or filled with something fluid. “We might think of liquids in motion, structured by radiating waves, laminal flows and spiralling eddies” (Carpo, 2013). Within Topological architecture, topology is a hidden spatial relationship. Topological architecture tries to capture flows, complex conceptions and interactions. This results in fluid non-Euclidian shaped architecture, that creates the impression of ‘seamless fluidity’ related to the natural system (Carpo, 2013; Savaskan, 2012).

Nonetheless, critics come forwards, supposing that the abstractness of the digitally created spaces, contradicts with the spatial understanding and reality of the human body and of buildings. The objection goes, that we do not find ourselves in a non-Euclidean space, and thus should not add topological geometries that are considered as too abstract (in Massumi, n.d.).

Massumi urges, that the objection that topological architecture is too abstract would however dissipate, if we would rethink the connection between the body and lived abstractness. “The life of the body, its lived experience, cannot be understood without reference to abstract-real processual dimensions, which cannot be conceptualized in other than topological terms” (Massumi, 2002).

Describing contemporary theory of architecture has seen the philosophy of space emerge within the context of architectural discourse. Worlds like ‘smooth, ‘space’, ‘abstract’, the ‘Fold’ and ‘fields’ are passed around, but the question remains, what space in Topological architecture actually entails?

RESEARCH OBJECTIVE

‘Space’ has had different meanings through time. It was only until the time of Modernism, that a clear notion of ‘space’ was defined into the vocabulary of architecture.

With the rise of parametric architectural design, a new understanding and creating spatiality evolved. This switch raised the questions; how does our notion of ‘space’ change? Is ‘space’ as described in Modernism still recognisable in Topological architecture of fluid non-Euclidian forms/shapes generated by parameters? and, does space even exist in Topological architecture?

These questions lead to the main objective of this research; changing notions of ‘space’ and space perception of topological architecture (fluid non-Euclidian forms/shapes) created by parametric architecture.

PROBLEM STATEMENT

To what extent does topological architecture – which generates fluid non-Euclidian forms/shapes, created by parametrising architectural design, change our notion of ‘space’?

Within the objective of this research we can distinguish 3 parameters, namely: topology architecture (recognisable on its fluid non-Euclidian forms and shapes), parametricism and space.

When we talk about space we can as we saw within the theoretical framework on one hand talk about production of space and on the other hand the consumption of space. Thus, to understand space within topological architecture generated by parametrising architecture, several steps must be taken, which are here listed.

Firstly, we will create a theoretical framework that will investigate the production of space in ´topological architecture’ which generated the fluid dynamic and non-Euclidian forms and shapes. The objective is twofold; (1) introduce and understand the process of parametrising architecture and (2) understand what space is within topological architecture.

Q1: How can we define Topological architecture?

Q1.A: Definitions of Topological architecture?

Q1.B: How is Topological architecture created?

Q2: How can ‘space’ be defined and understood within Topological architecture?

Secondly, we will investigate the consumption of space in ‘topological architecture’. The objective is how space is consumed/perceived within the physical reality and understand the topological aspects of space Perception.

Q3: How do we perceive space, within the physical reality?

Q3.A: How does a human being perceive a space?

Q3.B: With what geometry can we describe our perceptual experience?

Q3.C: To what extent can we find a topological aspect in space perception?

PART 1: PRODUCTION

Q1: HOW CAN WE DEFINE TOPOLOGICAL ARCHITECTURE?

According to Spuybroek (2008), there is a huge misunderstanding nowadays about the word topology in architecture……

A new generation of form in architecture – morphogenesis (Deleuze). To understand how parametrising architectural design creates Topological architecture, it is important to first create an understanding what Topological architecture is, by looking at different definitions and theories. From these definitions, we will look into the influence of parametrising architectural design on the development/creation of Topological architecture.

Q1.A: Definitions of Topological architecture?

Let us start with a basic definition of the word Topology. Topology, bears comparison with calculus, probability theory and number theory. A handful of mathematically-minded intellectuals in the late seventeenth and early eighteenth century discussed about the first ideas of the new field of Topology, as an Analysis Situs or Geometria Situs (Pointcaré, 1895). The mathematicians Gauss and Riemann, from Germany, conceived Topology (analysis situs) as a science dealing with the logic of quantity, shape and arrangement. Topology looks at systemacy instead of the physical features of an object. In other words; topology looks at relationships (Spuybroek, 2008).

Deleuze and Guattari (1987) refer to these relationship with haecceities – a set of relationships which results in a tactile space which is localized and not delimited, the nomad space (Teyssot, n.d.).

“The model is a vertical one; it operates in an open space throughout which things-flows are distributed, rather than plotting out closed space for linear and solid things. It is the difference between a smooth (vectorial, projective or topological) space and a striated (metric) space” (Deleuze and Guattari, 1986, p.18 and p.125)

…

TOPOLOGY, TIME, MOVEMENT AND PARAMETERS…. Topology raise for the relationship between time and shape. Topological entities are defined with calculus and therefor are composed out of a continuous stream of relative values. Topological surfaces are defined as a flow.

…

Topological geometry is generated by transformations, which can be seen, as continuous alterations that preserve geometric properties, as if it is a flexible and dynamic system able to curve, fold and twist. In other words; topology refers to spatial properties that are unaffected by changes of shape and size. Euclidean geometry considers objects as rigid bodies, whose motions are those of transfer, rotation, and reflection. Topology on the other hand also permits elastic motion of figures, which change shapes by virtue of continuous nonlinear transformation or deformation (Aguiar, n.d.; Emmer, 2005).

….

Topology becomes, when applied to logic, continuity or as Charles S. Pierce calls it; Thirdness. Continuity is a purely relational field without any parts/elements yet. In other words, we can say that the parts/elements are a result, a product of the relationships and not a priori given (Spuybroek, 2008; Pierce …. ).

….

It could be argued that any architecture (however Euclidian it may actually be) could be analysed topologically once the emphasis shifts from what it looks like to how it works. See my essay (on topology) written for Speculative Art Histories in the attachment. (Andrej)

How it looks versus the way it works…

Q1.B: What is the influence of digital technology on the creation of Topological architecture?

Within topology the behaviour of a structure of surfaces is studied. The structure undergoes as explained in Q1.A deformations, which the surface registers in differential space-time leaps of continuous deformations. By inserting differential fields of space and time into an otherwise static structure, topological architecture designs can be made.

Topological architecture, as the means of mapping architectural intention provoked the discovery of mapping parameters, with the help of computers, onto the frozen Euclidean moment in the physical world. As a result, developments of a new architecture, an architecture influenced and modulated by the infinite and provocative possibilities offered by these technological tools, beyond the simple promise of greater efficiency and production capacity, are emerging (Hana Rashid, 2014).

Eisenman utilises the computer to conduct what he calls ‘synthetic movements’. By computing variables and relationships (the known), new possibilities in architectural space and form are opened up as a result (the unknown) (Wong, 2010). For example; The mathematical description of form and space that architects have historically understood, involve mathematical descriptions from which time was eliminated……… However, with the rise of computer CAD software, along with the ability to use time-and force or motion-based modelling techniques – surfaces can be made, which are defined by U and V vector coordinates. By using time as a measure of changes in the form of an object; bending, stretching, twisting and folding of architectural forms with motion based modelling techniques, are now possible (Lynn, 1999; Mark Burry, year).

According to DeLanda in ‘Immanence and Transcendence in the Genesis of Form’, morphogenetic process of these physical assemblages occurs as “abstract machines”. Abstract machines can be defined as systems that control certain parameters, which create dynamic structure generating processes (Teyssot, n.d.).

When we take the diagram as an example of these abstract machines, we can compare the diagram to an; abstract map, a scripted procedure referring to the processes of morphogenesis. The aim is a modulation between natural components, physical elements and architectural design.

Deleuze’s notion on the diagram is based on the multiple forces that work upon the form, as two vectors of differences, referring to the entropic arrow between tension and matter (Teyssot, n.d.). He uses the metaphor of ‘the body without organs’ to refer to the notion of machine and of diagram. The body with no organs, is the notion of matter, which is not yet formed or represented (Deleuze, n.d.).

“Overcoming organized form, one is introduced to matter as a receptacle of forces. Beyond the matter-form opposition, beyond organized form, there is matter as a non-formal mix of forces and materials” (Teyssot, n.d.).

The diagram, deals with fluxes, fluids and functions and concerns the representation of forces that belongs to a stratified formation. The diagram should be seen, as a virtual problem – something that is real but not actual, and not as a permanent structure or form (Sauvagnargues in Teyssot, n.d.).

The concept of the diagram as an abstract machine, offers the tools to map and understand different types of strata in for example institutions, technologies and apparatuses.

These new processes and methodologies associated with history, theory, experimentation and production are radically changing the way we see and think about space. As Hani Rashid in the catalog of the Biennale states: “In one form or another, it is now within the reach of artists and architects to discover and evoke digitally induced spatial deliria in which the merging simulation and effect with physical reality creates the possibility of a sublime digital metamorphosis from thought to its realization” (Rashid, 2009).

Q2: HOW CAN ‘SPACE’ BE DEFINED WITHIN TOPOLOGICAL ARCHITECTURE?

“Topology is a mathematical way of conceiving of TOPOS; the place, a space, all space and everything included in it” (Kantor, 2005).

To understand how space can be defined within Topological architecture a more detailed description of morphogenetic processes is needed. When we are interested in morphogenetic processes, in other words; processes that generate form, we should start with the question, with what fuel these processes are generated.

This starts at the difference between the qualities extensive properties versus intensive properties (Deleuze’s ‘intensities’). While extensive properties can be defined as properties you can divide, such as length, areas, volumes – intensive properties such as speed, temperature, pressure, density and concentration cannot be divided (Perren & Mlecek, 2015).

By putting two intensive qualities that are opposed to each other and then put into contact, you create something that can drive a process. Thus, we can state that intensive differences drive flows and is the most basic form of fuel of a morphogenetic process (DeLanda, 2011).

Intensive properties exhibit critical thresholds – singularities, at which something morphogenic happens. Intensities come into relation with each other through repetition (Deleuze, …. By james Williams) This brings us to the second step in defining space within Topological architecture, which is to understand how Topological thinking works.

It is important to first mention the Virtual. Deleuze explains the interaction of creative experimentation and of changing intensity with the virtual and the actual. Virtual means real but not necessarily actual. In other words, the Virtual is the structure of the space of possibilities, which gives something that’s not actual, reality. To be able to think of these virtual spaces, we need mathematics, or geometry – topology.

Traditionally, in architecture, the abstract space of design is conceived as an ideal neutral space (vacuum) of Cartesian coordinates.

The abstract space in which we design, is traditionally understood as a neutral space (a vacuum), described with Cartesian coordinates. A two-dimensional (curved) space, would be inscribe, with the help of cartesian coordinates, in a space with one more dimensions then the object obtains. Every point of the object contains a set of points (neighbourhoods of points) plus a definition of one point nearby (DeLanda, 2011).

Deleuze rejects this method. According to Deleuze a space should be studies only with local information that’s on the space itself, not global or relative.

Gauss, invented in 1930 this new approach to inscribe an object in space, other than with Cartesian coordinates. By measuring the instantaneous value of curvature at each point, every point becomes a speed (speeds of becoming), that in its turn becomes a field of rapidities or slownesses at which curvature is changing at each point. With this a revolution geometry that was before extensive now became intensive.

This theory, further developed by Riemann, could now describe the N dimensional space as intensive field of rapidity and slowness’s, called a multiplicity of manifold.

It is important to mention that the discovery of this non-Euclidean geometry, the higher dimensions (from the fourth on) and topology, the new idea of space to summarize, is one of the most interesting examples of the profound repercussions that mathematical ideas will have on humanistic culture, art and architecture. This theory had direct consequences on science.

The original Newtonian absolute space, with cartesian coordinates, changed with the developed theory of Gauss and Riemann, into space-time theory, in which space is composed with local information.

Einstein saw real space in these terms and what can be curved is the amount of gravity of objects around it. Space in this sense is not a container, but is shaped by its contents that bend space, which is folded dimensional entity.

Deleuze adopts the earlier term multiplicity, and plays with his notion of the ‘fold’ a major role for setting up the groundwork for spatial understanding in topological architecture. In the notion of the Fold, Deleuze considers every feature a kind of folding, a smoothing, with a single expressive continuum. Folding can be seen, as a means of introducing another concept of space and time within conventionally conceived ‘spatial boundaries’. Spatiality as a ‘becoming’ with no external measures or ends within a complex repetition, no longer restricted to imitation.

Several folds creating a blurring of inside/outside, solid/void and space to space thresholds; reconceptualising traditional architectural notions of spatial connections and separations. Within the fold, the conventional architectural conception of spatial is itself problematized, no longer rendering the repetition of the same but a repetition of differences. An architectural process of spatial conception, where new and unanticipated possibilities occur without predetermined outcomes. A space which is no longer detached from program and event but where the folds become the events themselves as a product of possibilities (Deleuze, 2006).

Spaces in this sense, can be seen, as spaces of possibilities. If we need to think about the virtual (something that is real but not actual) then we need to think how the possibilities (singularities) are structures, in other words how to study the possibilities open to a dynamic system (Delanda, 2010).

At the same time the notion of the spacephase was developed by Pointcare.

Dynamical things can be represented.

Every Attractor had a singularity. This is the tendency of a system to always end up in the same place. F.e. soapbubble

Graphically tendency virtual

Attractor = topological points that become actual but the diagram is the structure of a space of possible states!

Face of possible states given the relative factors!

Every single point is a possibility. But only the attractor is interesting – tendenties are the points with the highest probabilities

Poincare continued his research with a new singularity (periodic singularity) – long term tendency of a system.

Poincare invented topological thinking

Thinking visually about the virtual – thinking about something that is not actual

Adding the geometry of complex systems, fractal geometry, chaos theory and all of the mathematical” images discovered (or invented) by mathematicians in the last thirty years using computer graphics, it is easy to see how mathematics has contributed to changing our concept of space – the space in which we live and the idea of space itself. Because mathematics is not merely a means of measurement in recipes but has contributed, if not determined, the way in which we understand space on earth and in the universe, specifically in regard to topology, the science of transformations, and the science of invariants (Poincaré, 1930).

Possibilities have a structure in the shape of becoming which are structured for a system. Topological thinking allows us to do that.

“The scientific discoveries have radically changed the definition of the word Space, attributing a topological shape to it. Rather than a static model of constitutive elements, space is perceived as something malleable, mutating, and its organization, its division, its appropriation become elastic.” (Imperiale, 2009).

In this way, topology as described, allows for not just the incorporation of a single moment but rather a multiplicity of vectors, and therefore, a multiplicity of times, in a single continuous surface.

Topological surfaces have been recognized as a way, to implement new ideas about space. In 1858 the German mathematician and astronomer August Ferdinand Moebius described for the first time in a work presented to the Academy of Sciences in Paris a new surface of three-dimensional space, a surface that is known today as the Moebius Strip. The Moebius Strip is the first example of a non-orientable surface – that changes the traditional approach to the structure, organization and perception of space. In this sense, the Mobius strip, offer important topological properties of non-orientable surfaces. The architectural interpretation of a non-orientable surface symbolizes spatial relativity between the exterior and interior (Tepavčević & Stojaković, 2014).

Non-orientable surfaces such as the Mobius strip and Klein bottle, were soon accepted as conceptual models with other architects of our time such as Ben van Berkel, Steven Perella, Zaha Hadid, McBride Charles Ryan, Aquilialberg studio and BIG Architects studio.

PART 2: CONSUMPTION

Q3: HOW DO WE CONSUME/PERCEIVE SPACE, WITHIN THE PHYSICAL REALITY?

Q3.A: With what mechanism does a human being perceive a space?

Maurice Merleu Ponty, one of the father of phenomenology, wrote this in his book, “The Primacy of Perception”: “if we are seeking to form an idea of, or to understand the essence of, a spatial figure…we must first perceive it. Then we will imagine all the aspects contained in the figure as changed. That which cannot be varied without the object itself disappearing is the essence.” (Magyar, 2015)

PERCEPTION VERSUS SENSATION

In order, to understand how space perception works, it is important to understand the difference between space in architecture or science and space for the human perception. While in science, space has had different notions (see Q2) for the human experience, one has always ‘known’ space, for example its determination by dimensions. It is important to start by understanding a few differences in how we become aware of the space around us. Let us start with the difference between perception and sensation.

In traditional theories perceiving things, depend on sensations that occur first. A sensation is based on; colours, sounds, touches, odours and tastes, while objects and space depend on perception. For this reason, vision plays a dominant role in the human perception of its surroundings. The visual perception starts by registering information from the surrounding objects and secondly, after initial accumulation of this visual data, process this information and construct them into meaningful information (Gibson, 1950).

Perception as an active process is an important feature of the way we perceive space around us. It is the direct connection of the eye and brain that enables perception. It is the very analysis, decomposition and structuring of the scene that is at the heart of perception (Lawson).

Despite the dispute among philosophers; Nativism assumed that the synthesis was intuitive or innate, while empiricism, explained the synthesis as learned or inferred from past experience, and last Gestalt theory suggested that it is produced by a characteristic achievement of the central nervous system which may be termed sensory organization, – the most accepted view of perception is still that “the percept is never completely determined by the physical stimulus” (Gibson, 1950). The perception goes beyond the stimuli and is superposed on sensations and is in this way essentially subjective and depending on the contribution of the observer. Even though perceptions rely on past experiences of the individual, the sensations tend to be the same for all.

We have seen that sensation and perception are by no means the same and we can state that perception is usually not an analytical process but rather an integrative experience (Gibson, 1950; ).

RATIONAL ANALYSIS VERSUS SENSORY PERCEPTION

The second difference is dividing cognition into rational analysis and sensory perception. Rational analysis (Anderson 1990, 1991a) is an empirical program of attempting to explain why the cognitive system is adaptive, with respect to its goals and the structure of its environment. We argue that rational analysis has two important implications for philosophical debate concerning rationality. First, rational analysis provides a model for the relationship between formal principles of rationality (such as probability or decision theory) and everyday rationality, in the sense of successful thought and action in daily life. Second, applying the program of rational analysis to research on human reasoning leads to a radical reinterpretation of empirical results which are typically viewed as demonstrating human irrationality.

Sensory perception- Descarted considered valueless ….

Whatever meets our senses much exist in space and time. The space and time we presuppose before we sense reality much have innate subjective transcendental ideal. Thus, we can state that space ad time are forms of perception.

We know that nature governed is by principles immutable. But for objects of our senses there are no prior explanations. This rises a controversy between philosophers, regarding space perception. Philosophers believed that visual space perception must be learned, or believed that visual space perception was either intuitions fundamental to the mind itself or an innate feature of the sensations themselves (Gibson, 1950).

The argument made by Kant, which had a different view, was arises from geometry, regarding the nature of space. ‘Space’ as either a real thing that exist independently of our minds or not a real thing that exist independently of our minds. With real we mean space that exist in a mind independent world.

Kant states that space is something our own minds impose onto our representation of our world. Our capacity to sense things, which is the capacity to imposes special structures onto our representations of the world as a transcendental ideal of space. Space is in this way merely a form of our intuition, that is just a structure our own minds impose onto our representations, and space is not a property of things in themselves, as they exist independently of our minds and of our knowledge of them. In other words; space is merely a form of intuition and not a property of things in themselves. This strategy Kant proposes is based on facts of geometry and consists of two premises: 1) Geometrical knowledge is synthetic a priori. Necessary and universally. Its not contingently true and no exceptions. 2) Synthetic a priori knowledge is possible only if space i¬s merely a form of our intuition and not a property of things in themselves (Edgar, 2014).

Thus, our knowledge on geometry is based on nothing but our own mind and space is merely a form of intuition and not a property of things in themselves.

Note: The objection goes that Kant neglects the fact that this Premise does not exclude that space is also an object in the independent world, thus; both intuition and a property of things in themselves.

Q3.B: With what geometry can we describe our perceptual experience?

It is often assumed that the space we perceive is Euclidian. However, this concept has been challenged by many authors. For this reason, we are interested whether our perceptual spatial structure would be better described by Euclidean geometry or with arbitrarily curved Riemannian space, as described in q2.

Let us start with the premises of Kant, earlier mentioned. The premises of Kant seem to be referring to what we call ‘intrinsic geometry of perceptual space rather than ‘extrinsic’ geometry of perceptual space. On one hand, extrinsic geometry refers to the relationship between the structure of the observers’ perception and the actual structure of the physical space, while on the other hand intrinsic geometry, a global set of constraints provides, by which the judgements of a given observer are formally related to one another, irrespective of their relation to the external environment (Buchanan & Lambert, 2005; Fernandez & Farell, 2009).

Because we are dealing with internal geometry of a subject’s perceptual space and not with the relationship between perceived and actual shape, this paper is concerned with the intrinsic geometry of perception. In this, we assume there is a geometry of internal perceptual space in which perception is stable and consistent enough to support such a geometry.

Many approaches can be found regarding the perception of space in architecture. Space sensory perception are in these theories in most cases been linked to the concept of Three-dimensional geometry of the Euclidian spaces (Gibson, 1950). However, we can distinguish two ways in how intrinsic geometry could depart from Euclidean. First, it could stay within the realm of more primitive geometries, which do not have an internal metric structure (Fernandez & Farell, 2009).

On one hand is suggested that Euclidean metric distances in three-dimensial space are not a primary component of an observer’s perceptual experience (Gibson, 1950), which then was further tested and developed by others like Normal and Todd and Klein. Suggested was, that perceptual space can be best described as a more abstract space, in which the concept of distance, as found in Euclidean space, is not defined but a hierarchy of spatial structures that might be available to our perceptual system, from the most concrete to the most abstract, as Euclidean, affine, ordinal, topological and nominal (Merleau-Ponty, 2013).

Thus, although observers can exhibit a conceptual understanding of Euclidean metric structure, the basis of this knowledge might be more cognitive than perceptual.

By a second alternative, the departure from Euclidean geometry is through a change in the metric of the space, so that the concept of distance is still defined but differently from the way it is defined in Euclidean geometry – namely, the Riemannian geometry (mentioned before in Q2).

Luneburg (1947) introduced a Riemannian space of constant curvature as a description for visual space, a model that was further developed by Blank (1958, 1978).

Next to what geometry best describes our perceptual experience, perception is obtained by the action of combining information from the multiple cues available in the sensory input as we have stated in Q3A.B. …

Summing up, there is ample evidence that perceptual space is not Euclidean, though there is still no consensus in the scientific community about this.

Q3.C: The topological aspect of space perception?

…..

Jean Piaget, the late Swiss philosopher and clinical psychiatrist, after hundreds of actual tests with children of very young age, in his book: The Child’s Perception of Space, made the statement, that our psyche, our consciousness, is organized by topological principles (Magyar, 2015).

Experience is no longer in us. We emerge from experience.

the dimension of ‘lived abstractness’ (in Massumian terms’, i.e. architecture in the expanded field. Architecture as produced and architecture (built environment) as consumed, experienced. (Massumi, 2002)

….

CONCLUSION

Q1: How can we define Topological architecture?

Q1.A: Definitions of Topological architecture?

…

Q1.B: How is Topological architecture created?

By computing variables and relationships (the known), new possibilities in architectural space and form are opened. Abstract machines can be defined as systems that control certain parameters, which create dynamic structure generating processes, where the concept of the diagram as an abstract machine, offers the tools to map and understand different types of strata in for example institutions, technologies and apparatuses. These new processes and methodologies associated with history, theory, experimentation and production are radically changing the way we see and think about space.

Q2: How can ‘space’ be defined and understood within Topological architecture?

Intensive differences drive flows and is the most basic form of fuel of a morphogenetic process. Traditionally, in architecture, the abstract space of design is conceived as an ideal neutral space (vacuum) of Cartesian coordinates. However, with the new approach to inscribe an object in space, other than with Cartesian coordinates by Riemann, a revolution geometry that was before extensive now became intensive.

Spatiality as a ‘becoming’ with no external measures or ends within a complex repetition, no longer restricted to imitation. Spaces in this sense, can be seen, as spaces of possibilities.

Adding the geometry of complex systems, fractal geometry, chaos theory and all of the mathematical” images discovered (or invented) by mathematicians in the last thirty years using computer graphics, it is easy to see how mathematics has contributed to changing our concept of space – the space in which we live and the idea of space itself.

Possibilities have a structure in the shape of becoming which are structured for a system. Topological thinking allows us to do that.

Q3: How do we perceive space, within the physical reality?

Q3.A: How does a human being perceive a space?

Whatever meets our senses much exist in space and time. The space and time we presuppose before we sense reality much have innate subjective transcendental ideal. Thus, we can state that space ad time are forms of perception.

We know that nature governed is by principles immutable. But for objects of our senses there are no prior explanations. Although this rises a controversy between philosophers, we can say, our knowledge on geometry is based on nothing but our own mind and space is merely a form of intuition and not a property of things in themselves.

Q3.B: With what geometry can we describe our perceptual experience?

Thus, although observers can exhibit a conceptual understanding of Euclidean metric structure, the basis of this knowledge might be more cognitive than perceptual.

Next to what geometry best describes our perceptual experience, perception is obtained by the action of combining information from the multiple cues available in the sensory input as we have stated in Q3A.B. …

Summing up, there is ample evidence that perceptual space is not Euclidean, though there is still no consensus in the scientific community about this.

Q3.C: To what extent can we find a topological aspect in space perception?

…

DISCUSSION

…