Vertical Divider

The classics of postmodernism postulated the relationship between shapes and thought a long time ago. In Margins of Philosophy, Jacques Derrida established a famous precedent of such thinking, as did as well the tandem composed by Gilles Deleuze and Félix Guattari. The line extends in time; the potential of thinking through shapes was elaborated, closer to my time, by Nikolaus Gasterer in the project "Drawing a Hypothesis" that associated artists and thinkers issued from the postmodern tradition of cultural theory. My own interest in this type of reflection has been motivated by my search for glimpses of extra-cultural thought. Certainly, the examples of paradigmatic shapes that Derrida brought forth, such as his Greek tympan, are eminently cultural; what is more, he explored his connection between the philosophical, the organic and the technical in a way that rather reinforced than tried to abolish the solidity and unity of cultural transmission. In Tympan, various ways of learning and various types of skills form a synergistic whole. Taking such experiments for the starting point, I wished to reach quite different conclusions. For me, shapes and morphogenesis (the process of creating forms and shapes) are tantalisingly situated on the frontier of cultural and non-cultural, technical production, ritualised play and organic growth.
My first Pinwheel, a short experimental text concerning litholatry, the human instinct of ritual circumambulation of stones, appeared on my blog in 2012. Since then, a slow maturation of this idea and the search for its utmost form of visual-discursive expression has been in course. In this essay, I merge three presentations concerning thinking on and by shapes that I delivered in 2016, contrasting the expanding spiral with quite dissimilar pattern of crisscrossing lines and developing the quite well-known and perhaps not so original idea of diagram issued from the postmodern tradition of Derrida, Deleuze and Guattari. The diagrams of weaving or fish network production, based on lines and knots, seem to be not less cultural, and even more primordial, than Derrida's ancient drum; at the same time, they are so widespread among the cultures of man that one can claim their transcultural validity. On the other hand, if referred to patterns of natural morphogenesis, it seems to be man's own signature. Meanwhile, the spiral growth around a centre appears to be as plant-like as weaving is man-like. This is perhaps the reason why this formal pattern seems to acquire ritualistic validity across a variety of human cultures. In my personal cultural memory, it is associated with spring and Easter, that was still the proper time of pin-wheels, reduced to the role of mere children's toy, in Poland where I was born. At the same moment of the year's cycle, eggs used to be decorated with swastika-like motives, well known across what a century ago the scholars identified as the Arian, or Indo-European, cultural area. Yet the circumabulation of the Black Stone, the whole "divine geometry" of Mecca form a Semitic counterpart. Pin-wheels, if one is to search for them, are to be found worldwide; but I doubt if they are universal in human cultures at the same title as weaving is universal. They appear in any culture, because man is exposed to vegetable morphogenesis anywhere in the world. It is the pattern of opening flowers and growing leaves, associated with spring, regeneration, and perhaps also the general notions of becoming, development, growth, light, good, powerful and divine. As a child, I direly desired pinwheels; as an adult, I was surprised to see similar toys, made of stick and a single leaf, in the hands of children in West Africa. What is more, the particular geometry of all these bidimensional and tridimensional shapes resuming the idea of a whirling movement, celebrating circulation around a centre, is also reflected in movements, both assuming and denying their ritualistic value. 
The transcultural experience is rendered by the simile of pinwheels that are re-centred in such a way that they form a common axis. Those superposed pinwheels establish a celebrating unity in which an infinity of cultural codes turn around a common axis, multiplying a common pattern issued from very deep zones of symbolic existence of man, beyond any particular cultural formulation.
The emergent topology of the symbolic space results from the interference of various cultural codes in the peculiar conditions brought about by the globalization. It is possible to translate this abstract concept of symbolic space into actual, i.e. urbanistic and architectonic spaces. What is more, such a translation is a crucial element in city planning dedicated to new global centres; it is also required while creating spaces dedicated to institutions inscribed in their tissue. I think in particular about the transcultural museum as an institution hosting and mediating the interference of various interpreting codes and systems of cultural memory.
The first part of this essay presents the basic problem, i.e. the emergence of a new level of symbolic complexity, beyond “cultures” as we understood them traditionally. This new complexity is due to the interference of various cultural codes in the individual consciousness of the inhabitant, in the interaction between the inhabitants, and finally, in the macro-space of the city, as well as its specific micro-spaces (such as museums). The concept of emergent symbolic complexity is referred, in the first place, to the "knots" in the globalized networks, emergent metropolises – I will use such examples as Dubai, Kuala Lumpur, the futuristic campus of Masdar, etc. I use idiomatically the concepts of knots and central, defining voids. 
The topological inspiration may be used for creation of actual spaces (architecture for institutions such as transcultural museum, the space of transcultural city in which the museum is inscribed as a particular knot of the symbolic/real space). By topology I mean actual mathematical concepts, such as Möbius stripe and Klein bottle. Bringing those mathematical inspirations closer to humanities, I come close to other experiments (Deleuze & Guattari, Slavoj Zizek) dealing with the proprieties of Place (empty space, exposition room, the desert).
The conclusive step is to stress the importance of the void in constructing non-knotted spaces. I sketch the idea of the empty centre and the pin-wheel as a model of whirling dynamics associated to the central emptiness. It is an example of a possible solution to the problem of translating the transcultural symbolic space.

condensed weaving

The main topic of the exposition organised in the Irish Architectural Archive in June 2016 was formed by two acts of a common nature: the signing of the Covenant and Declaration documents at Belfast City Hall and the reading of the Proclamation document at the General Post Office in Dublin. The common nature of this acts, both of them dealing with writing, puts them in a binary relationship that might eventually remain static. But it doesn't. In fact, a cascading proliferation of relationships takes place.
The Covenant, signed by half a million of men and women, in protest against the Third Home Rule Bill, became the topic of Kipling's poem “Ulster 1912”, that opens at yet another text, a biblical allusion taken from Isaiah:


"Their webs shall not become garments,
neither shall they cover themselves with their works"


Their webs shall not become garments... This allusion brought about by the imperial poet in the heat of the historical moment – the poem was first published in “The Morning Post” on April 9th, 1912 – explicitly negates the proliferating potential of the event. Yet the acts of writing provoke one another, call one another into being. In 1914, a British Covenant, similar to the Ulster Covenant in opposition to the Home Rule Bill, received two million signatures.
In 1916, Easter Proclamation is equally a result of repeated inscription. It was modelled on a similar independence proclamation issued during the 1803 rebellion by Robert Emmet.
Repeated acts of writing call into existence new forms of sense-making: the signature calls into existence the print, the print calls into existence the archive, the archive calls into existence the diagram. Metaphorically speaking, webs do become garments, because the condensing connectivities progressively build up thicker, more solid meanings.
This process of calling into existence has a scientific name: it is the emergence of complexity. Just to quote a standard definition from Wikipedia: “emergence is a process whereby larger entities, patterns, and regularities arise through interactions among smaller or simpler entities that themselves do not exhibit such properties”. In other words, this is a process by which multiplication is beyond a simple quantitative increase, it leads to a new quality. We don't have the repetition of the same, but we have a constitutive difference. Based on repetition of the simpler elements, emergence brings about a qualitative change. New kind of units are no longer to be reduced to the simple elements that existed at the beginning. One might say that emergence is the passage from quantity to new quality. In natural sciences, we can see it when the proliferation of simple cells calls into existence a multicellular organism – something that differs qualitatively from the same number of separate cells.
Even for a scientist, there is probably something miraculous about any case of emergence, because of the impredictable nature of both the moment and the outcome of any such process. The same could be said about the emergence of a historical event out of a multiplicity of simple acts that have by themselves no power of changing the course of the history, such as signatures of individual subjects. How many of them are needed to make an effect? In the Ulster Covenant they were half a million. Yet it is not a stable, fixed value. Many times people tried to make such events happen again, to bring the emergence about again and again. Usually such intents fail. Another five hundred thousand signatures do not provoke exactly the same effect any more. The emergence remains out of control, it cannot be repeated with a predictable outcome. Yet the mechanism is still in place, the former petition fosters new ones, as it is shown by Robin Price. The result of the repeated act remains in the domain of essential uncertainty, yet it creates a “conflation of past and present”, as it is said in the commentary to the exposition, a conflation that so to say keeps our present permanently in an open-ended state of instability that means a possibility of change.
The emergence forms chains – that are not the chains of controllable repetition with predictable outcomes, but chains of qualitative difference. As I said, the signature calls into existence the print , the print – as a form of proliferation – calls into existence the archive, the archive calls into existence the diagram. The Covenant brings about the Proclamation. Or rather, the repetition of the act of writing / signing causes a text-event, the text-event causes its own multiplication. The Proclamation gives rise to the archive filled with other texts, images, connectivities brought about by the emergent text-event. In this exposition, the Covenant brings into existence a drawing machine by Edmund Eva and George Baldwin that transforms digitized signatures by following different algorithmic instructions and re-describes the generated data. And finally, the historical outcome is enacted as “Diagramming the Assemblage of the Proclamation Text”.
The important question in this moment is to evaluate the status of such a procedure as a “next step” in relation to the archive as a static deposit, a mere accumulation – something rather geological than biological.
Thus, this next step puts in evidence the necessity of the diagram making sense of the archive, as a form of the exploration of the deposited strata. The archive without a diagram forms a bulk, a mass in which the connections need to be traced: both inside, in the very bulk of the accumulated material, and outside, creating channels of communication with the external world. The archive must be mapped in its inner complexity and in its external inscription. I believe the diagram forms an emergent level of complexity that adds, or builds, or calls into existence new meanings that were absent in the original archive.
Those channels leading outside the deposit (the inert bulk of the archive) are the connections to external forms, such as the architecture of the City Hall where the proliferating signing acts took place (this is what we can observe in the works of Maxim Surin and Robert Anderson, in the attempt at “Reconstructing Scenographies”). The ornamental forms of architecture are thus connected to the scriptural forms of signature – an arbitrary link is established, based neither on causal relation nor on logical relation, but on a coincidental connectivity of place and event that opens space for a new layer of sense, added to the existing meaningfulness of the world in which we move. This is a meaning that appears as superposed in relation to the previously existing layer of meaning, in which the classical architectonic form is a metaphor of a state order (and, as we see on the anti-Home rule satirical postcard, the ruin of architecture signifies the disruption of the civic order). When this former layer of significance is thematized in the art work by Niamh McDonnell. The art work creates a kind of extended reality, in which the tissue of sense is thicker, more substantial, more garment-like for our improved existence.
The outcome of this exposition is thus something more than just a way of recounting the Irish history. The aim is neither to commemorate nor to revive the events, but to establish a new connection that will transform the web into a garment, a mere network into a tissue that might cover our symbolic bodies, create a way of actual living in the aspect that has been proposed as the starting point of this experiment: to reconsider the “acts that inscribe the subject’s relation to the state”.
The emergence rescales the individual existence through writing. A single man or woman, just as Kipling stated it, stand “before an Empire's eyes”. It doesn't actually matter how such an act is evaluated – the perception differs according to the ideological standpoint. Nonetheless, what Kipling observes with remarkable lucidity is the moment of a peculiar confrontation that is no longer that of infinitely small against infinite oppressive power. Both elements appear as strangely commensurate in the condensing web of singular occurrences.
Even for a historian, there must be something miraculous in any such emergent event. Let's come back to the Deleuzian concept of clinamen as a way of breaking through the linearity of predictable sequence of causes and effects:


The clinamen, as the minimum angle, has meaning only between a straight line and a curve, the curve and its tangent, and constitutes the original curvature of the movement of the atom. […].


3. One no longer goes from the straight line to its parallels, in a lamellar or laminal flow, but from a curvilinear declination to the formation of spirals or vortices on an inclined plane: the greatest slope for the smallest angle. From turba to turbo; in other words, from bands or packs of atoms to the great vortical organizations. [...]1


Each individual signature for the Covenant is an example of such a modification of the course of events at “the minimal angle”, creating the vortice of change out of infinitely small and helpless human atoms that should apparently remain subdued to the prevailing mass of the Empire.
Apparently, the history should have remained in its own wheel-rut. Yet a modified trajectory appeared, some unpredicted factor interfered with the inertial course; the history jumped out of its groove. Still speaking the Deleuzian language, such a change of trajectory, such a nomadic interference of human “atoms” implies a passage from a “striated (metric) space” to “smooth (vectorial, projective or topological) space”, from a space measured and controlled by the Empire to one that may be occupied freely, “without being counted”, by proliferating individual wills leaving their (in)significant inscription.

1Gilles Deleuze, Felix Guattari, A Thousand Plateaus, trans. Brian Massumi, Bloomsbury, 2004, p. 421.

emergent topologies

The way how the advent of the digital changed our perception of the world deserves our attention, even if seemingly we know the main lines of the story. Yet in fact few things are obvious in it. The invention of the computer and the increased possibilities of computation led not only to the creation of the world wide web and all the changes it brought to our everyday life, but also to new mathematics and new possibilities of geometrization of the real. The discovery of the fractals, with the ground-breaking research of Benoit Mandelbrot, was a crucial step towards closing the gap between the abstract and essentially static world of geometric figures and the dynamic world of such phenomena as cloud formation, crystallization and even – although no modern living organism is based exactly on the fractal principle – biological forms such as the familiar green cauliflower, or broccoflower, often cited as a nearly-perfect, three-dimensional natural fractal. They are not only infinitely more varied in shape than the squares and triangles with which classical mathematics dealt, but also permit to grasp the crucial dimension of becoming.
I would say that the advent of this new geometry banished one of the main symbolic forms of the so-called Western civilization: the linear perspective as a strict, binding system subordinated to unmoving, individual, locatable eye. The perspective with a single vanishing point as a way of representing the real became the dominating bias of thinking about architectonic and urbanistic spaces till the dawn of the modernity and the invention of alternative ways of painting (such as cubism or, in fact, the innovation brought about by Cézanne) and, as a consequence, of thinking about the space.
It is by no way an accident that I speak about the ways of representing space as crucial and defining for the ways of conceptualizing it and utterly of creating real spaces. I actually want to insist on this point. The importance of “going digital” is to be seen not only as the appearance of new tools for architectonic design. It is, in the first place, a new way of representing space, powerful enough to help us grasp the dynamic properties of systems, in much the same way as the fractal helps us grasp mathematically such properties as growth, change and complexity arising from extremely simple source formulae.
As an example on which I would like to focus in this presentation, I chose a basic architectural or urbanistic problem such as that of organizing a central space: yet another obsession of the Renaissance and, for me, one of the source problems of modernity. I speak about architecture and urbanism at the same time, because the gist of the question is to be found in both: how to provide a movability in a centrally organized space with multiple axes of symmetry. What makes this problem unsolvable is the interference of the two desired symbolic forms: the linear perspective orientating the subject towards a vanishing point and the proliferation of axes of symmetry. The solution of course had been given, for example in such buildings as the Pazzi Chapel by Brunelleschi, yet only in approximative terms: the building is after all orientated by its portico and, in the interior, by the privileged position of an altar, while the ideal realization of this interference of perspective and centrality would induce either a spinning movement of the subject or its immobilization in the central point of a perfectly circular space. This is why central architectonic spaces, attempted once again in the Palladian model of villa, are in fact quite rare in the history of architecture. They seem somehow confusing and uninhabitable in their mirror-image self-sufficiency.
Yet a liveable solution is quite close at hand if we admit missing the target, if we abandon the frontality of the linear perspective as a way of conceptualizing space. Deleuze and Guattari use the concept of clinamen as a way of breaking through the linearity:


The clinamen, as the minimum angle, has meaning only between a straight line and a curve, the curve and its tangent, and constitutes the original curvature of the movement of the atom. […].


3. One no longer goes from the straight line to its parallels, in a lamellar or laminal flow, but from a curvilinear declination to the formation of spirals or vortices on an inclined plane: the greatest slope for the smallest angle. From turba to turbo; in other words, from bands or packs of atoms to the great vortical organizations. [...]1


The transformation of the square as a static principle of spatial organization and the emergence of the vortex is also beautifully exemplified on stage, namely in Quad, a television play by Samuel Beckett, also called a “ballet for four people”. The actors walk silently around and diagonally across the square space, illustrated on the scheme included in the first edition of the play in 19842.

This particular organization of the stage space may be treated as a perfect illustration of the clinamen principle of vortical organization, although the overall interpretation of Beckett play remains open. Through cyclical dis-encounters, the utmost isolation of the figures seems to be broken. The personages never confront each other face to face. Clinamen, deflection, missing the point becomes the way of living with their overwhelming otherness and strangeness in relation to each other, accentuated by different colours of their robes and the percussion instruments used to give each of them a distinct auditive leitmotiv along the play. Against their essential strangeness, the central void deflecting their steps creates a movable share of space and a basic modality of cohabitation.
This is where I meet the main scope of this presentation. The problem I would actually like to address is one of the consequences of the globalization. What I would actually like to understand and even influence is the topology of the symbolic space that results from the interference of various cultural codes. What I am actually concerned with is the emergence of the vortical organization of the contemporary metropolis that is no longer multicultural; it becomes transcultural, i.e. it marks the transition into a new mode of human existence, in which the human individual is no longer immersed in a single, coherent network of meanings defined by what we call a culture. In other words, in this radically new, transcultural state of the symbolic space, symbols are no longer stabilized inside cultures, that might be heretofore defined as relatively stable constellations of meaningful elements. The cultural orders form multiple, changing and unpredictable patterns of interference, just like independent, yet interconnected sets of circular waves on the surface of the water. As a consequence, symbolic units are no longer contextualized by their relatively stable, crystallized cultural contexts, but appear as indefinite part of those ever-changing patterns of interference. This proliferation of symbolic configurations leads to the emergence of a new level of complexity and creates a new dimension of the symbolic space, situated beyond any locatable, intersubjective cultural formation.
The transcultural condition of man is, just as we saw in Beckett's play, marked by radical alienation, since the culture left behind used to serve as a grid to construct inter-subjective meanings. Losing cultural inscription, humanity becomes so to speak unmitigated, naked, devoid of cultural armour that used to attenuate the shock of the encounter. The transcultural man could be thus compared to a mollusc taken out of his shell. The transcultural city is a space in which divergent rules of politeness superpose and interfere to generate ever-changing patterns of behaviour devoid of stabilized meaning. The clinamen deflects the lethal frontality of unmediated clashes.
Returning once more to the classical work by Deleuze and Guattari, one might say that in the transcultural metropolis we encounter the ideal of “smooth (vectorial, projective or topological) space” as opposed to the “striated (metric) space” predetermined by culture. As the authors of A Thousand Plateaus state, in the first case “space is occupied without being counted” and in the second case “space is counted in order to be occupied”.3 The exploited metaphor of nomadic and sedentary occupation can be translated into the coexisting moods of cultural (residual) and transcultural (emergent) ways of inhabiting the symbolic space which is beyond any possibility of crystallization in cultural norm.
The crucial aspect is to fathom the possibilities of translating this individually conceived, culturally inexpressible symbolic space into actual, i.e. urbanistic and architectonic spaces. Such a translation should be a crucial element of city planning dedicated to new global metropolises; it is also required while creating spaces dedicated to institutions inscribed in their tissue. In particular, I think about the transcultural museum as an institution hosting and mediating the interference of various interpreting codes, rather than just one, defined system of cultural memory.
The topological properties of the museum space are pertinently defined in the essay on Coca Cola as the objet petit a by Slavoj Žižek4. Similarly to a diet, caffeine-free Coke, the object exposed in the museum is always beyond our grasp. The essay refers only at a surface level to the capitalist economy, conditioning the permanent dissatisfaction and surplus-greed of the producer/consumer. At a deeper level, the Lacanian economy of desire that Žižek calls to his cause establishes a particular topology of space surrounding the object, making it constantly slip away from our grip, as if it fell into a black hole of infinite gravity. What is more, this dynamics of desire and permanent slipping away creates a movement in a paradoxical continuum analogous to the Möbius strip and its development, the so-called Klein bottle, mathematically defined as a closed surface with only one side. The dis-encounter with the object, slipping away from the “striated” space of cultural meanings, creates, once again, a vortical organization, a deflection of the subject avoiding the frontal encounter with significance that might otherwise be stabilized in the material object understood as a “semiophore” (term created by Krzysztof Pomian) – a stable vehiculum of meanings recognized by a given community. The spacial curve of the transcultural museum reflecting the clinamen principle induces a circling move, a quad ballet of the transcultural individual and the object that is no longer contemplated frontally, but passed by tangentially, missed, declined.
The appearance of the transcultural condition can be understood as an emergence in the strict, scientific meaning of the word: a process whereby larger regularities, patterns or forms of organization spontaneously arise from a multiplicity of interactions among less complex elements. Significantly, new complexity brings about new qualities. Those simpler entities from which the complexity arises do not exhibit the properties that are patent on the higher level of organization. In other words, emergence might be understood as a mysterious passage from quantity to quality. Instead of the repetition of the same, it brings about a constitutive difference.
If we imagine for a moment that the multiplicity of human cultures constitutes the zero level of complexity, the transcultural condition marks an emergence, opening a new quality, irreducible to “cultures” as we understood them traditionally. In the metropolitan space, this new quality is due to the interference of various cultural codes in the individual consciousness of the inhabitant, in the interaction between the inhabitants, and finally, in the macro-space of the city, as well as its specific micro-spaces such as “quad” museums I've described below.
As a final step, the clinamen dynamics lead to the disappearance of the object and the emergence of the void as a paradoxical form of complexity – a void that is infinitely intricate pattern of tensors – geometric objects describing forces and relations. Already in the Žižekian approach quoted above, dealing with the proprieties of “Place” (empty space, exposition room), the object was evacuated of all its material qualities – as it was shown by the example of sugar-free, caffeine-free, taste-free Coke: a drink without a content.
The gist of architectonic and urbanistic art in the transcultural metropolis consists in producing the tensorised or intricate void through the creation of vortices or what I call “knots” in the urban space. The “quad” type of museum, offering an opportunity of dis-encounter with objects immersed in their particular topologies (or with topologies depleted of objects) is an example of a “knot” in the urban tissue, organized around a specific void. The metropolis itself is a “knot” in the global space, organized around a specific void.
The most important observation to make here is the value of significant absence in the formation of this new type of the city, offering the lure of an absolute vacancy, contrary to the such cities as Rome or Paris, overloaded with history – striated spaces, occupied by layers of former presence. Characteristically, the cities of the new type, like Dubai or Kuala Lumpur, offer no pre-existant inscription of man in space. These are cities without historical centres, with a touch of emptiness in the middle.
It should be stressed once again that the void is of crucial importance in constructing “knotted” spaces. This idea of the empty vortical centre and the whirling dynamics associated to the central emptiness is present in a pin-wheel – significantly ancient and universal toy built up through plying a simple square to form curvilinear, declined arms engaging in a whirling movement at the slightest gust of wind. It is in fact a model of transformation leading from static central forms (a square, a cross) to the dynamic ones (the whirling pin-wheel). What this pin-wheel utterly represents is the interference of a square and a wheel, stability and dynamism.
A plural “pin-wheel” city, growing from an empty centre in a proliferation of declined, tangential lines, might exploit the potentiality of a whirling movement as its creative dynamo. Such a new pattern of organization might be a visual and a conceptual experiment at the same time, structuring the experience of the traumatic expulsion from the world of stable cultures that is shared by ever-increasing number of individuals. Complex topologies of urban and architectonic space – exemplified here in the model of the pin-wheel – may bring about new configurations of thought, processed simultaneously in parallel lines of reflection, corresponding roughly to the plurality of displaced cultural standpoints. Engaged into the transcultural whirlpool, those parallel lines of peripatetic movement/reflection are supposed to form a “smooth space” of Deleuzean nomadic experience. The unstriated space of the city would serve as an active, empowering background for the kaleidoscopic, transcultural phenomena, generating a world for transcultural forms of life that are yet-to-come.
As I said, the pin-wheel as a toy and the decorative motives derived from it are extremely widespread across human cultures, adopting everywhere a similar symbolism of birth, growth or creation. At the same time, I would like to call your attention to the way how a new, productive model of thinking in A Thousand Plateaus is built up upon a reference to the origins, to very remote stages of culture. It is also a way of building up a transcultural dimension through systematic descent to the beginnings of cultural becoming of man. It is not by chance that one of chapters starts with a return to Georges Dumézil and his binary system of Indo-European mythology. Similarly, the pin-wheel is a primordial topological model that is still present in our contemporary experience as a toy. As far as I remember from my childhood in Poland, this is an element associated with Easter celebration. It is in fact a primordial symbol of creation and rebirth, graphically represented by the swastika and its derivatives. As it is well known, swastika had not been created by the Nazis; it is a much older and much more universal symbol that we find in archaeological findings and ancient art. It is also abundantly attested in ethnography, including the eggs richly decorated for Easter as a symbol of rebirth. At the same time, the wheel turning in the opposite direction is a symbol of destruction.
Nonetheless, as I believe, its universalism doesn't derive from any hidden structure of the human visual thought. I think it may rather reflect the convergent understanding of the basic fractal forms in nature, such as the spiral pattern of growth in plants. Similarly widespread is the vortical pattern of self-organization. It corresponds to a higher mathematics of life that emerged to our full appreciation with the advent of the digital.

1Gilles Deleuze, Felix Guattari, A Thousand Plateaus, trans. Brian Massumi, Bloomsbury, 2004, p. 421.
2Samuel Beckett, Collected Shorter Plays of Samuel Beckett, Faber and Faber, 1984, p. 293.
3Gilles Deleuze, Felix Guattari, A Thousand Plateaus, trans. Brian Massumi, Bloomsbury, 2004, p. 421.
4Slavoj Žižek, “Coke as objet petit a”, in: The Fragile Absolute – or Why the Christian Legacy Is Worth Fighting For, Verso, 2009, p. 21-39.

geometries of textual space

The aim of this essay is to reflect upon something we usually take for granted: the book as a topological unit. The form of what is technically called a codex (a book constructed of a number of sheets of paper, vellum, papyrus, etc.), as opposed to other forms of textual topologies such as a scroll, is nonetheless one of the most important premises of what we may call the civilization. As one can hardly indicate any alternative to the textual topology of the codex, we hardly notice how far we are influenced by the consequences of such a textual organization, that presupposes a unique, unidirectional flux of narration. There are some cases as well as some experiments of dividing and multiplying this flux, f.ex. when the book contains a text and its translation put side to side, or when you use footnotes, or again when the parallelism of two independent texts is used as a way of conveying intersectional meaning, as you can see in the text under the title “Tympan” included in the Margins of Philosophy by Jacques Derrida1. We might add to this the typographic experiments of modernist poetry, the caligrammes by Guillaume Apollinaire, etc. Nonetheless, none of these experiments obliterated the “default” order of linear textuality.
In Deleuzian terms included in A Thousand Plateaus, the textual space, organized in subsequent verses could be considered as an example of the striated or metric space, a space that “is counted in order to be occupied”. Even the empty space prepared for writing is covered with visible or invisible lines pre-existing its occupation by the text. This notebook prepared for writing is already the contrary of what Deleuze calls “a smooth (vectorial, projective or topological) space”, a space that might be “occupied without being counted”.

It is the difference between a smooth (vectorial, projective or topological) space and a striated (metric) space: in the first case “space is occupied without being counted” and in the second case “space is counted in order to be occupied”.2

A Thousand Plateaus is also essentially a traditional book, a text occupying the metric space of numbered pages. Yet implicitly, the concept of multiplicity is referred, among other aspects, to the textual space, as the book itself is a tentative of constructing a “smooth space” enabling nomadic exploration. In my opinion, the invitation to read this book in any order, included in the initial Author's note, breaks not only with the habits of linear reading, proper to the western tradition, but also with a determined topology of the textual space based upon such presuppositions as the unidirectional flux of the text, divided into rectangular, movable pages as basic visual and topological units, and the subsequent segmentation of this flux into chapters, that tend to coincide with those basic topological units – new chapter opens up a new page, etc. The introduction of sketches is a significant step towards the un-striating of the textual space. Drawing as a way of opening the metric space of writing, returning to the primary topology of a blank page before it is numbered and included in a book, with a presupposition of sequential reading, has been explored by several artists, such as Marc Ngui who created a series of illustrations for A Thousand Plateaus or Nikolaus Gansterer who organized the collective project “Drawing a Hypothesis”, which is an advanced exploration of the possibilities of thinking unrestrained by the striation of the textual space3. Also the aim of my reflection is similar. I want to go on with the Deleuzean reflection concerning the nomadization of the textual space, proposed in A Thousand Plateaus.

But why do I need the textual vortices rather than the striated space of a traditional page? Why striated space is not enough?

The starting point of this reflection is given by the emergence of a new level of complexity. Emergence and complexity are key-words here. I mean by this a radical change, such as, in biology, the radical change when we pass from the unicellular to multicellular organism. Just to quote a standard definition from Wikipedia: “emergence is a process whereby larger entities, patterns, and regularities arise through interactions among smaller or simpler entities that themselves do not exhibit such properties”. In other words, this is a process by which multiplication is beyond a simple quantitative increase, it leads to a new quality. We don't have the repetition of the same, but we have a constitutive difference.
What I'm concerned with is the emergence of a radically new, transcultural state of the symbolic space. In the contemporary, globalized conditions, symbols are no longer stabilized inside cultures, that might be heretofore defined as relatively stable constellations of meaningful elements. Now the cultural orders form multiple, changing and unpredictable patterns of interference; symbols are no longer read inside their relatively stable, crystallized cultural contexts, but inside the ever-changing patterns of interference. This proliferation of symbolic configurations leads to the emergence of a new level of complexity and creates a new dimension of the symbolic space, situated beyond any locatable culture. This is what Michail Epstein called “transculture”. This new state of affairs makes me think about the absolute necessity of finding a multiplied textual space. My reflection on the topology of the textual space should lead to the proposal of a new modality of theory-building, enabling me to grasp this emergent complexity, to cope with it.
Let's come back to the concept of clinamen as a way of breaking through the textual linearity:

The clinamen, as the minimum angle, has meaning only between a straight line and a curve, the curve and its tangent, and constitutes the original curvature of the movement of the atom. […].
3. One no longer goes from the straight line to its parallels, in a lamellar or laminal flow, but from a curvilinear declination to the formation of spirals or vortices on an inclined plane: the greatest slope for the smallest angle. From turba to turbo; in other words, from bands or packs of atoms to the great vortical organizations. [...]4

I would like to call your attention to the way how A Thousand Plateaus is built up upon thinking about the origins, about very remote stages of culture. It is not by chance that this chapter starts with a return to Georges Dumézil and his binary system of Indo-European mythology. There is no time here to explain why in fact this return to the primordial classifications is so important. Once again, lets bring back a kind of primordial topological model that is still present in our contemporary experience as a pin-wheel. Probably most of you are not conscious of the ancient symbolism of this toy. As far as I remember from my childhood in Poland, this is an element associated with Easter celebration. It is in fact a primordial symbol of creation and rebirth, graphically represented by the swastika and its derivatives. As you know, swastika hadn't been created by the Nazis; it is a much older and much more universal symbol that we find in archaeological findings and ancient art. It is also abundantly attested in ethnography, including the eggs richly decorated for Easter as a symbol of rebirth. At the same time, the wheel turning in the opposite direction is a symbol of destruction. While the basic pattern is derived from square, the pin-wheel may be multiplied and become such form as this Armenian symbol of eternity and eternal rebirth.
What this pin-wheel utterly represents is the interference of a square and a wheel, stability and dynamism, sedentarism and nomadism. It illustrates the clinamen in action, the emergence of the vortex.
What is curious and what is the clue of my presentation today is an example of non-linear text-making that I would like to bring about. As I said, the tradition of text-making in the striated space is dominant and nearly universal. In order to find a significant alternative to it we need to search the peripheries. This is why I would like to bring about a very minor tradition of Malay manuscripts. This type of writing is minor even in Malaysia, where the predominant way of producing books doesn't differ from the organization of the textual space that we know. Yet what calls my attention is the presence of the pin-wheel pattern and the emergence of the textual vortex. The writer breaks through the linearity of the text turning the book around, creating a clinamen, a slope, a minimal angle that divides up and multiplies the textual space. At the same time, the space becomes vectorial and topological, non-striated. I don't pretend to delve into the content of those manuscripts; they might also be quite uninteresting for our conference. Those books have mostly basic religious contents, such as muqaddam, a kind of Islamic catechism, or an Arabic grammar useful for a non-Arabic speaking Malay people. I'm interested in those artefacts merely for the sake of their peculiar textual organization. A single key-concept, written in the middle of the page, is surrounded by multiplied zones of writing, glossing simultaneously on various aspects of this central concept. Impossibility of a linear reading is implied in the very distribution of this text in the visual space of the page. At the same time, the book, in order to be read, has to be constantly moved around. Such a simple gesture, as I believe, may symbolize a constant changing of perspective on the key-concept.
Confronting the Deleuzoguattarian notions of multiplicity and “smooth”, open-ended nomad space with the inspiration given by this Malay manuscript, I would like to present the idea of a plural “pin-wheel” text, growing from a given centre as a proliferation of argumentative lines and implying the potentiality of a whirling movement as its creative opening. This new pattern of textual organization is a visual and a conceptual experiment at the same time. New, complex topologies of the textual space – exemplified here in the model of the pin-wheel – may bring about new configurations of thought, processed simultaneously in parallel lines of reflection, corresponding roughly to the plurality of displaced cultural standpoints. Engaged into the transcultural “whirlpool”, those parallel lines of reflection are supposed to form a “smooth space” of Deleuzean nomadic reading. The utmost purpose of all this is to create a new type of artefact, a kind of “post-book” in which the topology of the blank space prepared for writing (actual pin-wheel shape of the sheets of paper) would be adequate to the kaleidoscopic character of the transcultural phenomena. The challenge presented by the emergent patterns of cultural interference requires the final deconstruction of the linearity of the text as a primary medium of the theoretical discourse in the humanities. The final result, as I believe, would be situated somewhere between cultural criticism and an art-work, “generating”, according to the statement included in the call for paper, “a new world”, i.e. preparing a new configuration of space for transcultural texts that are yet-to-come.


1Jacques Derrida, Margins of Philosophy, trans. Alan Bass, Brighton, The Harvester Press, 1982, p. ix – xxix.
2Gilles Deleuze, Felix Guattari, A Thousand Plateaus, trans. Brian Massumi, Bloomsbury, 2004, p. 421.
3Nikolaus Gansterer et al., Drawing a Hypothesis. Figures of the Thought, Springer, 2011.
4Deleuze, Guattari, op. cit., p. 421.

Malay manuscript, 19th century AD. IAMM, Kuala Lumpur.

---

This essay develops the ideas exposed in the following presentations: “Condensed weaving: diagram as an emergent stage of sense-making”, Diagramming the Archive Workshop, The Irish Architectural Archive, Dublin, 17.06.2016; “Emergent topologies: transcultural symbolic space and its architectural translations”. Second International Conference on the Digital Age “Going Digital: Innovation in Art, Architecture, Science and Technology”, Museum of Applied Arts, Belgrade, 2-3.06.2016; ​“Pin-wheels. Proliferating geometries of the textual space”, Deleuze + art. Multiplicities | Thresholds | Potentialities, Trinity College, Dublin, 8-10.04.2016.
​