Gilles Deleuze's ideas on non-Euclidean narrative: a step towards fractal narrative
German A. Duarte
Écrire n'a rien à voir avec signifier,
mais avec arpenter, cartographier,
même des contrées à venir 
 Until the second half of the twentieth century, linguistic and formalist theories highly influenced the understanding of both visual media and the spaces derived from them. In particular, theories of linguistics were applied to the study of filmic narrative structures in formalist critical theories. However, this type of formalistic analysis has been employed less frequently since Deleuze's works on cinema were published in the 1980s . Since Deleuze's intervention, it has become less common to approach cinema purely as a language – in both French meanings of langage and langue. Indeed, the search for a 'cinematographic syntax' – understood during the formalist period as a series of rules or principles governing the structure of a language, rather than its Greek meaning of 'arrangement' (σύνταξις) – was replaced by the search for narrative spaces governed by geometrical rules, including narrative spaces shaped by non-Euclidean geometries.
 This paper will investigate how Deleuze, through his works on cinema, moved cinematographic narrative away from the sphere of linguistics and how in this way he developed a rapprochement of audiovisual narrative to geometrical spaces, mainly to non-Euclidean geometries. In fact, it is the contention of this paper that Deleuze's theory is deeply influenced by the effect video technology in the 1980s had on the spatial organization of audiovisual narratives. In addition, if one takes Deleuze's theories on cinematographic and audiovisual narrative as a starting point of investigation, it is possible to identify the influence of Mandelbrot's fractal geometry on the structure of audiovisual narrative spaces. Significantly, in the last twenty years, digital technology has made the development of fractal geometry possible, thereby directly changing the ways in which audiovisual narrative spaces are organized.
 In this paper, then, I will first briefly analyze the influence of structuralism on the analysis of cinematographic narrative. Subsequently, I will discuss Deleuze's theorizations on the cinematic medium. In the final section, I will explore how film narrative may be seen as an example of spatial organization which is directly influenced by technological improvements, and how digital technology generates a kind of fractal ordino in audiovisual narrative.
 Cinematographic technology has long been understood as the result of a long chain of advances in visual media, from the invention of the camera obscura to the streamlining of the photographic technique. Deleuze departs from this dominant history of the cinematic medium. He suggests that the cinematographic technique finds its prehistory in instantaneous photography, in the equidistance of the snapshots materialized by the film as a surface, and in the mechanism from which the images are mechanically driven (Lumière's invention of claws).  This accurate revision can be read as Deleuze's desire to release the cinematographic image from its Euclidean legacy. Like Bergson, Deleuze contends that cinema operates under a completely new mode of spatial organization. In fact, Deleuze theorizes, in Cinéma 1 and Cinéma 2, that even if this new spatial organization were to derive from a machine that operated according to Euclidean laws, it is able to develop a space in 'becoming' and to exert a non-Euclidean ordino. In fact, he compares the narrative space developed by some directors to non-Euclidean spaces, e.g. Bresson's films with Riemannian spaces, Robbe-Grillet's films with quantic spaces, and Resnais's films with topologic spaces.  Consequently, in Deleuze's analysis, filmic narrative once again acquires the status of geometry, a status that was previously lost due to the influence of linguistics on cinematographic analysis. However, Deleuze did not arrive at this result without a number of precedents; indeed, following in Bergson's footsteps, a few scholars had previously attempted to highlight the geometrical character of cinematographic practices.
 An example is Jean Epstein's L'intelligence d'une machine, published in 1946. Here, the author investigates cinematography as a spatial organization directly derived from the geometrical imposition of the camera. In Epstein's analysis, the camera, provided with its own intelligence, creates a particular space liberated from the 'hierarchy' of natural things. In his words, "Tous les systèmes compartimentés de la nature se trouvent désarticulés."  The capacity of the cinematographic camera to destroy the dogma of the irreversibility of life was for Epstein an important factor that required long study and placed the analysis of film narrative more in spatial terms than in linguistic ones. According to Epstein, the inexistence of entropy in filmic space becomes not only a special narrative instrument but also a different way to analyze life, to bring about some new phenomenological inputs and confront mankind with them.  According to Epstein, the cinematographic camera was erroneously considered a machine "to reform and popularize the theater" instead of a philosophical instrument that could offer a new way to analyze the world, including society. 
 Despite these first attempts, the intensive search for a grammar for filmic narratives eclipsed the study of the cinematic medium as a geometrical form. In fact, linguistic analyses dominated theories of film narrative. It seems that this current of thought was mainly inspired by Algirdas J. Greimas's Sémantique structurale, in which he notes that the process of signification cannot occur without the interaction between two terms subjected to a relation that bonds them into a single unit. In film theory, this hypothesis was supported by the Kuleshov effect.  It is probable that the Kuleshov effect was used to support a link between linguistic and filmic narrative through which Greimas's formalist theories could find a scientific approach to film narrative studies.
 The formalist 'mistake' consisted most probably in adopting Sassure's sémiologie not as a field of study that includes linguistics but as a field of study included in linguistics. Therefore, under the influence of French formalism, film narrative was approached as a language, or as a visual Esperanto, as showcased in the work of Christian Metz. In fact, Metz, during the first phase of his intellectual production, attempted to demonstrate that film narrative responds to linguistic analysis. He also attempted to theorize how a fixed relationship between signifying and signified was constructed by certain visual effects used in film narratives. For example, Metz's formalistic analysis interpreted the frame as a word, the shot as a sentence, a close-up as a synecdoche, and some cinematographic effects as punctuation marks. But his perfect analogy between filmic narrative structure and linguistics could not successfully overcome two challenges: the absence of double articulation in film narrative, and the impression of reality generated by the photographic medium. 
 The opposition between Deleuze's and Metz's theories is clear. In fact, Deleuze theorized that reference to a linguistic model in film narrative is unnecessary.  Hence, he highlighted again the geometrical character of cinematography. In order to do so, Deleuze undertook a thorough analysis of Bergson's works. This analysis brought him to analyze cinema as a geometry, namely as a spatial organization derived from a visual medium able to mediate the relationship Man – Nature, after becoming a narrative instrument.
 Since Cinema 1. L'image-mouvement, Deleuze has argued that both Pier Paolo Pasolini and Roman Jakobson analyzed framing (cadrage) by means of unnecessary linguistic terms. Instead, Deleuze finds an analogy for framing in the concepts of informatics, rather than linguistics.  He theorizes a narrative structure in which framing is a dynamic spatial construction in 'becoming,' even though it is limited in its geometry and its physical character.  Its dynamic is based on the frame's capacity to divide and link different spaces. These spaces can be in the frame due to the frame's layered structure, derived from its inner structure, composed in sets and subsets, but also in the totality of the succession of images. Deleuze composes a space where the enclosed set, represented by the frame, is linked with the Whole (le Tout) by means of the off-screen (hors-cadre).  As one can see, in this structure proposed by Deleuze, we can find Bergson's concept of the inexistence of totally enclosed systems.
 Through Bergson's theories, which identify Time as an exterior, infinite entity in the continuous process of 'becoming' and theorizes the inexistence of completely enclosed systems,  Deleuze theorizes that filmic narratives are never absolutely closed narrative systems. This allows him to formulate the off-screen (hors-cadre) space in a different manner than in structuralist film theories and, consequently, to theorize a new narrative structure in spatial organization in film. It is also possible to suggest that in this structure, the off-screen space represents an entity in absentia, which builds a folded structure, or a space within the space in presentia. According to Deleuze, every single frame in presentia, to use the above definition, is accompanied and linked to an exterior space that I will call space in absentia, which can serve two roles: the off-screen space may set up a relation to the other frames (the latter understood as spatial sets), or place the frame in relation to the Whole. This encased structure, or to follow the Deleuzian concept, folded structure, finds its existence in the hors-cadre. In Deleuze's words: "(...) un ensemble étant cadre, donc vu, il y a toujours un plus grand ensemble, ou un autre avec lequel le premier en forme un plus grand, et qui peut être vu à son tour, à condition de susciter un nouveau hors-champ, etc. L'ensemble de tous ces ensembles forme une continuité homogène, un univers ou un plan de matière proprement illimité."  Thus, a film's spatial organization acquires a complex structure that can be perceived as a web of heterogeneous entities inseparably associated.  Note that even if this structure is subject to the linearity imposed by the material surface of the film, the many links generated by this encased spatial order do not create an arborescent structure. The geometrical character of cinematographic narrative highlighted by Deleuze can be primarily identified in the single frame of a sequence. In fact, framing is understood as the determination of a relatively enclosed system that constitutes a set composed of subsets.
 Motion pictures put in this space can be understood as the interaction between the space in presentia and the space in absentia (in other words, between paradigm and syntagm or, using informatics terms, between Interface and Database).  Thus, editing provides cinema with the ability to articulate the narrative space of moving pictures and to create a new geometry. According to Deleuze, editing allows a single object to be connected to the Whole and to determine the Whole itself. He not only identified editing as the act of putting frames – a space composed of layers – in succession, he also suggested that editing could be present in the inner dynamics of a single frame, in the inner dynamics of this layered object, thus creating a palimpsest.  As a consequence, editing becomes the essence of the Deleuzian spatial composition because it governs the 'layout' (agencement) of images-movement. Further, according to Deleuze the image-movement is placed in relation to the Whole by means of editing.  However, as mentioned above, Deleuze also posits that the logic of narrative spaces can be non-Euclidean; a film's Gestaltung reflects in some way the 'rhizomorphic structure' already theorized by Deleuze and Guattari (Deleuze–Guattari 1980) some years before. In his view, the objects placed in the cinematographic narrative space, which are seen as a set (agencement), follow a principle of connection and heterogeneity similar to that theorized in the introduction of Mille Plateaux (Deleuze–Guattari 1980), where any point of the set can be connected to any other.  The meaning, in this narrative space, derives from 'semantic chains' of any kind – e.g. linguistic, perceptive, gestural, expressive, and cogitative  – which are linked to one another in many different ways.  The layered composition of the smallest part of the set (agencement) personified by the frame creates, by means of its links to the Whole, a growing system of connections, which is clearly limited by the interaction imposed by the linearity of the film as a surface. However, this new way to understand the narrative structure of the cinematic image allows both the comprehension of the narrative as an opened structure and the comparison of the special narrative organization to non-Euclidean spatial compositions, even though it derives from a Euclidean device: the cinematographic camera. In fact, the openness of the narrative theorized by Deleuze derives directly from the Bergsonian concept of the inexistence of totally enclosed systems. Taking it as a postulate ("il y a toujours hors-champ, même dans l'image la plus close"),  Deleuze describes the cinematographic image as a nesting system that shows, via the single frame, an inner structure in which its existence is confirmed by the hors-champ.
 As demonstrated by Deleuze's research on cinema, an accurate study of technological improvements is required in order to attain an in-depth understanding of the different ways of articulating the narrative space from the beginning of the cinematographic art to the development of audiovisual narrative.  Technological improvements directly influenced the narrative space, especially by means of editing, which represents, according to Deleuze, "the pure vision of a human eye," and provided cinematographic technology with its status as instruments of analysis and geometry. Technology, as prostheses of the human body, allows a better understanding of natural objects and phenomena. Visual media, as technological devices, are instruments that mediate the relationship between object and subject. Thus, in its most advanced state, one can suppose that they can develop a kind of consciousness or intelligence.  In filmic narratives, this phenomenon can be seen when the camera acquires freedom of movement. According to Deleuze, with regard to l'image-perception, the camera, when trying to give the point of view of a character in the diegesis, imposes its own point of view, a point of view in which the vision of the character is transformed and then reflected. Thus, according to Deleuze, the viewer cannot be in front of an image that pretends to be objective or subjective; the viewer is placed in a "corrélation entre une image-perception et une conscience-caméra qui la transforme."  Deleuze goes beyond Vertov's theories about the consciousness of the camera demonstrated in his movement Kino-Pravda. In fact, Deleuze, opposing Vertov's ideas, posits that cinematography does not represent an improved human eye but through editing it represents a human point of view.  This affirmation changes radically the understanding of machines as simple prostheses of the human body. If one considers technology as extensions of our body, one must accept the fact that these prostheses are aimed at improving our sight, in the case of visual media, and in consequence they allow an objective representation of objects and phenomena. However, if one accepts the fact that technology deeply modifies the relationship between object and subject, establishing a kind of relationship in which man is alienated, then one is recognizing a kind of independence of the machine, a kind of consciousness. Thus, Deleuze recognizes that technology represents a way to improve human senses. However, cinematographic technology acquires its consciousness through editing. Hence, following this framework, we may suppose that cinematographic cameras express their own 'intelligence' by imposing a particular relationship between narrative spaces in absentia and narrative spaces in presentia.
 However, I would like to draw attention to a progressive phenomenon highlighted by Deleuze in his two volumes, Cinéma 1 and Cinéma 2. The new narrative structure theorized allows him to suggest a kind of fractalization of the narrative space. This phenomenon is highlighted in his work since L'image-mouvement mainly in the way Deleuze understands the close-up (gros plan).  Deleuze's ideas on close-up converge with Bonitzer's; in fact, Bonitzer argues that a close-up erases the depth of field and, by means of this special phenomenon generated by the space of the close-up, it is possible to identify the existence of shots free from any imaginary connection with the space.  This phenomenon of the fractalization of the narrative space can also be founded in Deleuze's espace quelconque (any-space-whatever). This term, coined by Auger, describes a space that does not appear at first as a real field; it expresses by means of its fractal qualities an infinite space. The any-space-whatever embodies the 'space of the possible,' a sort of virtual space. This space loses its homogeneity, the principle of metric ratio, and the 'natural laws' connecting its parts.
 Deleuze explained both the phenomenon of the fragmentation of the narrative space and the elaboration of non-Euclidean spaces in a filmic narrative through the analysis of Bresson's cinematographic production, using the visual dynamics of Pickpocket (1959) as illustrations of his concept. Further, Deleuze affirms that the law of these new spaces, such as the close-up or l'espace quelconque, is the fragmentation theorized by Bresson, in which he posits that:  "Elle [the fragmentation] est indispensable si on ne veut pas tomber dans la représentation. Voir les êtres et les choses dans leurs parties séparables. Isoler ces parties. Les rendre indépendantes afin de leur donner une nouvelle dépendance." 
 The process of the fractalization of the cinematographic narrative theorized by Deleuze can be compared with the process of the development of non-Euclidean geometries. For instance, it is possible to establish an analogy between the space theorized by Deleuze in film narrative and the multidimensional space theorized by Hermann Grassmann (1878). Grassmann's research constituted groundbreaking investigations into multidimensional geometries as well as being one of the most important steps toward the understanding of non-Euclidean geometries. In fact, Grassmann suggested a shift from the conception of geometry as a study of physics or spatial perception, to a conception of geometry as a study of independent structures or complex sets. In particular, he developed the concept of 'continuous form,' which is characterized by three major phenomena: Erzeugen (generation), Setzen (positioning), and Verknüpfen (bonding).  Thanks to the concepts of Setzen and Verknüpfen, the 'continuous form' becomes a truly 'spatial' concept. Within this spatial arrangement (Ordnung), one deals with an infinite space derived from the 'continuous form,' where a single object in the set is recognized not only by means of its position (Setzen) but also by means of its direct relation (Verknüpfen) to the whole set (agencement). In other words, the object becomes a dimension, a direction. The space proposed by Deleuze is also multidimensional. Further, his idea of spatial organization does not accept a structure that over-encodes or creates a hierarchical axis. In other words, Deleuzian narrative spatial organization does not represent an arborescent structure, which usually presents a hierarchical system in which the object only receives information from a superior object in a direct line. Like a rhizome, Deleuze analyses the narrative structure of cinematic medium as a 'center-less' system where the communication is not hierarchical and where many different signs in their heterogeneity are able to communicate. In fact, he theorizes a space in which the single image becomes a dimension, or 'direction,' in the narrative structure, like in Grassmann's theory.  In fact, Deleuze describes the type of image that establishes a fractal and multidimensional space in film narrative as the image-cristal, a term that recalls Grassmann's studies on crystal surfaces.
 The analogies between the transformation of the narrative space – highlighted by Deleuze in the passage from an image-mouvement to an image-temps – and the development of non-Euclidean geometries become more evident with the new narrative space generated by video technology. In fact, in the early 1980s, the influence of technological improvements on spatial composition in film became more direct thanks to the important impact of video technology on the narrative space. According to Deleuze, video technology represents the absence of exteriority, i.e. a lack of hors-champ in the narrative space,  a conclusion clearly influenced by Bonitzer's analysis. According to Bonitzer, video technology leads to a metamorphosis of the nature of the image; he defines the electronic image as a 'pure surface.' Further, he suggests that the mise en scène in the video space could be likened to the mise en page (page layout) due to the 'lack of depth' in the space. He also states that, through video technology, the image is set free from perspective, allowing it to become a surface that one can 'inlay to infinity' (incrustable à l'infinit).  From a technical point of view, the image-video embodies the continuum. Video lines are composed of points invisible to the viewer. These points are spread over the image in temporal succession and are constituted in time and only to a lesser extent in space. Similarly, the variation of points is an interval.  Thus, video technology technically embodies some of the theories proposed by Deleuze regarding the analog medium. On the one hand, video technology generates an image that produces inner changes through a variation of its minimal components (the points), which are only punctual in character and represent a momentary entity in becoming. Therefore, the video-image is not defined by spatial notions but by the temporality of its intervals and by the method of its reproduction or its inner dynamics. On the other hand, the nature of objects placed in this space radically changes. The dimension of the video-image is not able to contain elements such as geometric figures understood as objects. This dimension contains, in Engell's words, Nicht-einfach-vorhanden-bleiben-können elements (elements that cannot just continue to exist). 
 The space discussed above presents some analogies with another non-Euclidean space, the Sierpinski Carpet, a proto-fractal object.  This proto-fractal was conceived as a 'super object' able to contain huge quantities of information in a space without an area, because of its infinite holes. In fact, the Sierpinski Carpet aimed at containing one-dimensional curves on the plane in the topological dimension. The Sierpinski Carpet, as a video-image, represents a pure surface without depth. One can say that this proto-fractal, like the pure surface of the video-image, can be 'inlaid to infinity' (incrusté à l'infinit). Both the video-image and the Sierpinski Carpet have been defined as spaces without space, as symbolic spaces of data without topos.
 The non-Euclidean ordino in the audiovisual narrative theorized by Deleuze establishes another kind of relationship between the space in absentia and the space in presentia. It is possible to posit that this new relationship became clear with the appearance of video technology in audiovisual spatial organization. As mentioned above, Deleuze focused his research on the interaction between the different spaces that composed a multidimensional narrative space. Thus, the technological improvements became an important factor in the analysis of narrative spatial organization. Indeed, technology determines the way to compose the narrative space as well as the interaction between the narrative space and the viewer. The analysis of audiovisual narrative in spatial terms implies in consequence the analysis of the influence of new technologies on the way of organizing the relationship between paradigma and syntagma. The recognition of the role played by technology – in this case of the video-image – on the narrative act was also analyzed in Eco's Opera aperta. This study, despite being conducted when formalist theories were at their peak, allowed the understanding of film narrative space as a spatial organization. Eco interpreted the 'plot dissolution' (dissoluzione dell'intreccio) in contemporary cinematographic narrative as a consequence of the spatial organization imposed by television, especially live television, which naturalizes narrative 'dead times.'  In Eco's view, the phenomenon of 'dead times,' namely, the presence of non-narrative objects in the plot, shows that the notion of plot as derived from Aristotelian poetics was understood in contemporary audiovisual narrative as just an exterior organization of events that guides the pathos. Eco goes further and interprets this phenomenon of 'dead times' as the birth of an open narrative space. The apertura is in this sense made real by means of the negation of the plot, a negation that recognizes the world as a 'knot' of possibilities.  The phenomenon highlighted by Eco could also be read, in linguistic terms, as the transformation of the fixed relationship between Paradigm and Syntagm. Through this transformation, the Paradigm is externalized and some 'non-narrative' elements usually present in Paradigm start to be expressed in Syntagm. In informatics terms, one might say that the Database radically modifies its relationship with the Interface. This phenomenon could be interpreted as a 'fractalization' of the narrative space, which creates a pure narrative surface in which the distance between objects in presentia and objects in absentia simply disappears. Eco theorizes the same 'space without space' of Bonitzer, Deleuze and Engell. Eco's theories enable the understanding of changes to space that has been subjected to the use of digital technology in the organization of the narrative audiovisual space. He theorizes a kind of narrative Super Object, like the Sierpinski Carpet, an object that is able to contain enormous quantities of information, thus also making allusions to a navigable space.
 Thanks to the development of digital technology, Deleuze's and Eco's philosophical concepts of filmic narrative space have finally been realized. In our digital context, i.e. in the period of post-photographic technology, the Deleuzian layered space of the single frame is realized by the range of possibilities given by digital photography. Digital photography is not just a technology to capture images but also to "construct images ('unseen' data from remote sensors and other cameras) and generate images (from raw numbers); it treats them, stores them, associates them, disburses them, and transmits them into a media flow."  As a consequence, the layered space is no longer a mere allusion to a spatial structure linked by means of hors-cadre. Thanks to post-photographic technology, the depth of field of an image or hors-cadre is no longer the sole articulation that enables the development of semantic chains. Post-photographic technology allows the development of an encased structure, a sort of palimpsest, which characterizes the hypertextual phenomenon generated by digital encoding. Further, the digital medium blurs the distinctions between Paradigm and Syntagm. Indeed, the possibility of creating a navigable space in which the surface (Syntagm) contains all the information in absentia (Paradigm), or a space in which the semantic elements in absentia are reachable through hypertextual links, could be perceived as the technical representation of the exteriorization of the Paradigm as theorized by Eco.
 The phenomenon described by Eco became stronger when the logic imposed by the digital medium started to govern the cinematographic narrative space. In the 1990s, some films displayed a different type of interaction between Paradigm and Syntagm (i.e. between Database and Interface). By way of illustration, in Peter Greenaway's The Pillow Book (1996), the narrative act is not the result of a linear selection  but the creation of a potentially navigable, non-linear space. The 'fractal spatial composition' of the film, which results from the first phase of compositing video, creates the same relationship between layers that can also be found in fractal objects. Finally, with the advent of computer technologies, the Deleuzian multidimensional narrative and the creation of non-Euclidean narrative spaces underwent radical changes again. As stated earlier, the digital medium technically made real Deleuze's hypotheses regarding film narrative. The developments do not stop there: narrative space also began to exhibit certain characteristics shared by fractal geometry.  In fact, the instrument that enabled the development of fractal geometry (i.e. the computer) also started to govern the audiovisual spatial organization.  Fractal geometry also directly influenced the aesthetics of film narrative. During the 1980s, with the diffusion of fractal geometry, fractal forms exerted some influence upon architecture  and the dramatic arts.  With respect to audiovisual media, the influence of the aesthetic derived from fractals was particularly strong in the field of music videos, for example in Michel Gondry's video Let Forever Be (The Chemical Brothers, 1999) and Zbigniew Rybczynski's Let's Make Lots of Money (Pet Shop Boys, 1985). This notwithstanding, the most important contribution of fractal geometry is that it provided another way to analyze nature and, consequently, to represent it. As with every visual medium, the digital medium became an instrument for the analysis of natural objects and phenomena and developed a new space for scientific analysis, which became in turn a narrative space. By means of fractal geometry, nature began to be represented according to a spatial organization that accepted its irregularity and created a special relationship between the individual unit and the entire arrangement. In a fractal arrangement, the many links, while maintaining their heterogeneous characteristics, produce a homogenization of the complex. In other words, all the parts fit together and stand in direct relation to the whole without losing their specific properties.
 Fractal geometry was developed thanks to computers and these same devices have started to govern the audiovisual narrative space. The single Deleuzian frame thus became a 'knot' of hypertextual links. It assumes more the shape of a fractal object than that of a layered space as proposed by Deleuze.  At this point, the beholder is in front of a fractal object in which Interface becomes Database in a multidimensional space. Therefore, it is now legitimate to propose not just an aesthetic of Interface but also of Database.  In fact, in a narrative space in which the previously described interaction between Database and Interface is possible, the inner organization of information, as well as the role of Interface programming, plays an important aesthetic role, and allows highly expressive possibilities from both sides. 
 Video technology, as a transitional technology or, in Spielmann's words, a hybrid technology  between analog and digital media, can be understood not only as the instrument that helped to make Deleuze's and Eco's theories regarding narrative space a reality, but also as the technology from which certain proto-fractal objects in the audiovisual space originated. The digital medium thus made it possible to organize nature according to newfound geometrical laws. In a sense, geometries and visual media share similar aims, as Deleuze demonstrated in his studies. Both attempt to create new spaces and dimensions where natural objects can be translated according to specific laws in order to enhance our analysis of both nature and natural phenomena. Presently, the narrative act in the audiovisual field is perceived as a spatial organization that might be strongly influenced by fractal geometry beyond pure aesthetics. One could even consider fractal geometry as an episteme that governs the narrative act. The latter can be interpreted as the 'cartography' of the geometrical narrative space and therefore as the geodesy of an infinite informative object.
 Deleuze's works certainly contributed to the understanding of the contemporary narrative in the audiovisual media as a pure spatial organization. As we can see, the parallel that Deleuze created between non-Euclidean geometries and cinematographic narrative allowed not only the comprehension of the narrative act in the audiovisual media as the organization of a space that can manifest fractal characteristics but also as a space that is directly influenced by the consciousness of the machine. Thus, nowadays, when narrative space is completely governed by digital technology, one can assume that digital technology not only transmit a fractal order in its pure aesthetic meaning but also in the logic of the narrative act, in which the narrative attempts to map an infinite object in dévenir.
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 Deleuze 1985, p. 169.
 Epstein 1946, p. 6.
 Epstein explains this unique capacity of the cinematographic camera as follows: "Une innombrable expérience a préparé le dogme de l'irréversibilité de la vie. Toutes les évolutions, dans l'atome et dans la galaxie, dans l'inorganique, dans l'animale et dans l'humain, reçoivent, de la dégradation de l'énergie, leur sens irrévocablement unique. L'accroissement constant de l'entropie est ce qui empêche les rouages de la machine terrestre et céleste de jamais se mouvoir à rebours. Aucun temps ne peut remonter à sa source; aucun effet ne peut précéder sa cause. Et un monde qui prétendrait s'affranchir de cet ordre vectoriel ou le modifier, paraît physiquement impossible, logiquement inimaginable. Mais, voici que, dans un vieux film d'avant-garde, dans quelque burlesque, on voit une scène qui a été enregistrée à l'envers. Et le cinématographe, tout à coup, décrit avec une claire exactitude un monde qui va de sa fin à son commencement, un antiunivers que, jusqu'alors, l'homme ne parvenait guère à se représenter." Epstein 1946, p. 7.
 Epstein 1946, p. 9.
 See Balázs 2001.
 Pasolini, by rejecting the impression of reality and arguing that cinema is in fact reality, attempted to give to the cinematographic image a double articulation. In fact, it was the aim of his theory which affirms that a shot is a moneme and the objects placed in the frame are cinemes, the linguistic equivalent of which is the phoneme. See Pasolini P.P. 1972.
 "La référence au modèle linguistique finit toujours par montrer que le cinéma est autre chose, et que, si c'est un langage, c'est un langage analogique ou de modulation. On peut dès lors croire que la référence au modèle linguistique est un détour dont il est souhaitable de se passer." Interview realized by Bonitzer and Narboni, published in Cahiers du cinéma, in 1983. (no. 352).
 Deleuze 1983, p. 23.
 See Deleuze 1983, p. 24.
 This Whole corresponds to Bergson's concept of Le Tout, which is defined by means of the relationship between objects. According to Bergson, the relationship is not an intrinsic property of the objects themselves, but an exterior phenomenon. In other words, Le Tout describes how the capacity of an object to be related to another object is not an object's intrinsic property but an attribute pertaining to the Whole, which represents an immutable and universal entity toward which every object is projected. In Bergson's view, the Whole is Time, a universal and indivisible entity to which every object and movement is connected.
 "Nous verrons que la matière a une tendance à constituer des systèmes isolables, qui se puissent traiter géométriquement. C'est même par cette tendance que nous la définirons. Mais ce n'est qu'une tendance. La matière ne va pas jusqu'au bout, et l'isolement n'est jamais complet. Si la science va jusqu'au bout et isole complètement, c'est pour la commodité de l'étude. Elle sous-entend que le système, dit isolé, reste soumis à certains influences extérieurs." Bergson 1907, p. 503. (10).
 Deleuze 1983, p. 29.
 I would like to link this concept to Morin's description of complexity in which he defined complexity as follows, "Au premier abord, la complexité est un tissu (complexus: ce qui est tissé ensemble) de constituants hétérogènes inséparablement associés: ell pose le paradoxe de l'un et du multiple. Au second abord, la complexité est effectivement le tissu d'événements, actions, interactions, rétroactions, déterminations, aléas, qui constituent notre monde phénoménal." Morin 2005, p. 21.
 Arborescent is a term employed by Deleuze-Guattari (Mille Plateaux) to create a conceptual distinction between the rhizomatic structure, which is a non-hierarchical conception where each unity of the complexus is linked with everything, and the arborescent structure, which imposes a hierarchical structure based on binary cuts that only allows unidirectional developments. Both terms come from the field of botany and dendrology. In fact, both make reference to the growth mechanisms developed by nature. The term arborescent makes allusion to the shape or the characteristics of a tree, to its branching structure, and the term rhizomatic makes allusion to the horizontal stem of a plant that grows underground.
 I consider important to quote Deleuze's words on this definition: "On appelle cadrage la détermination d'un système clos, relativement clos, qui comprend tout ce qui est présent dans l'image, décors, personages, accessoires. Le cadre constitue donc un ensemble qui a un grand nombre de parties, c'est-à-dire d'éléments qui entrent eux-mêmes dans des sous-ensembles." Deleuze 1983, p. 23.
 See Manovich 2002.
 Deleuze 1983, p. 46-47. See also Ropars-Wuilleumier 1999.
 Deleuze 1983, p. 46.
 I refer especially to chapter one, Introduction: Rhizome. In particular we can see in the next passage that Deleuze-Guattari mention a kind of structure that recalls the cinematographic narrative structure. "N'importe quel point d'un rhizome peut être connecté avec n'importe quel autre, et doit l'être. C'est très différent de l'arbre ou de la racine qui fixent un point, un ordre. L'arbre linguistique à la manière de Chomsky commence encore à un point S et procède par dichotomie. Dans un rhizome au contraire, chaque trait ne renvoie pas nécessairement à un trait linguistique: des chaînons sémiotiques de toute nature y sont connectés à des modes d'encodage très divers, chaînons biologiques, politiques, économiques, etc., mettant en jeu non seulement des régimes de signes différents, mais aussi des statuts d'états de choses." Deleuze-Guattari 1980, p. 13.
 See Deleuze-Guattari 1980, p. 14.
 It is interesting to quote Deleuze's words with regard to this subject: "Il ne s'agit pas encore de savoir ce que signifie tel signe, mais à quel autre signe il renvoi, quels autre signes s'ajoutent à lui, pour former un réseau sans début ni fin qui projette son ombre sur un continuum amorphe atmosphérique" (Deleuze-Guattari 1980, p. 141). This passage recalls the process of differentiation of the cinematic image theorized by Deleuze in L'image-temps in which he describes the cinematographic image as a "matière signalétique qui comporte de traits de modulation de toute sorte, sensoriels (visuels et sonores), kinésiques intensives, affectives, rythmiques, tonaux, et même verbaux (oraux et écrits)." Deleuze 1985, p. 44. Further he describes the image, making difference between the structure of a language, as a "masse plastique, une matière a-signifiante et a-syntaxique, une matière non linguistiquement formée, bien qu'elle ne soit pas amorphe et soit formée sémiotiquement, esthétiquement, et pragmatiquement." Deleuze 1985, p. 44.
 Deleuze 1983, p. 31.
 This concept is clearly explained by Deleuze in this extract: " C'est en lui-même, ou en tant que tel, que le hors-champ a déjà deux aspects qui diffèrent en nature: un aspect relatif par lequel un système clos renvoie dans l'espace à un ensemble qu'on ne voit pas, et qui peut à son tour être vu, quitte à susciter un nouvel ensemble non-vu, à l'infini; un aspect absolu par lequel le système clos s'ouvre à une durée immanente au tout de l'univers, qui n'est plus un ensemble et n'est pas de l'ordre du visible." Deleuze 1983, p. 30.
 See Fahle 1999.
 Maybe that is the reason why Deleuze did not carry out an in-depth analysis of early cinema, when the camera was fixed and there was no editing, that is to say, when the movement was accomplished just in the inner structure of the frame by means of the movement of actors and objects. See Fahle 1999.
 This phenomenon certainly derives from the dependence that technology generates in its highly-codified nature. Digital technology accentuated this phenomenon. In Flusser's words: "Die technischen Bilder sind Ausdruck des Versuchs, die Punkelemente um uns herum und in unserem Bewusstsein auf Oberflächen zu raffen, um die zwischen ihnen klaffenden Intervalle zu stopfen; des Versuchs, Elemente wie Photonen oder Elektronen einerseits und Informationsbits anderseits in Bilder zu setzen. So etwas können weder die Hände noch die Augen, noch die Finger leisten. Denn die Elemente sind weder faßbar, noch sind die sichtbar oder greifbar. Deshalb müssen Apparate erfunden werden, die für uns das Unfassbare fassen, das Unsichtbar imaginieren, das Unbegreifliche konzipieren können. Und diese Apparate müssen, um von uns kontrolliert werden zu können, mit Tasten versehen sein, Die Apparate sind Voraussetzung für die Erzeugung der technischen Bilder." Flusser 1996, p. 21.
 Deleuze 1983, p. 108.
 On this subject see Vertov 2011.
 Deleuze 1983, p. 117.
 "(...) le gros plan n'arrache nullement son objet à un ensemble dont il ferait partie, dont il serait partie, mais, ce qui est tout à fait différente, il l'abstrait de toutes coordonnées spatio-temporelles, c'est-à-dire il l'élève à l'état d'Entité." Deleuze 1985, p. 136.
 Bonitzer 1982, p. 37. The phenomenon was also analyzed by Balázs in his work Der Geist des Films specially in Die Kontinuität des Raums, p. 60.
 Deleuze 1983, p. 153.
 Bresson 1975, p. 95.
 See Lewis 1975.
 See, in general, Grassmann 1878.
 Deleuze-Guattari 1980, p. 25.
 The transformation of objects represented in the image into vectors establishes an analogy with Grasmann's theories. However this phenomenon had already been studied in Mille Plateux by Deleuze-Guattari. In fact, as we can see in the follow extracts, the rhizomatic structure described in Mille Plateux shares some aspects with the narrative structure of the cinematic medium. In Mille Plateux, they affirm: "Il [the rhizome] n'est pas fait d'unités, mais de dimensions, ou plutôt de directions mouvantes." Deleuze-Guattari 1980, p. 31. The unities in the complexus are described as dimensions, or directions, vectors that embody the nature of dévenir of every system. This phenomenon is presented by Deleuze in his oeuvre on cinema, especially in Cinéma 2. As an example, I can quote Deleuze's analysis of Mizoguchi's long take, which he defines as follows: "Le plan-séquence assure une sorte de parallélisme des vecteurs orientés différemment, et constitue ainsi une connexion des morceaux d'espace hétérogènes, conférant une homogénéité très spéciale à l'espace ainsi constitué." Deleuze 1983, p. 263.
 "Les nouvelles images n'ont plus d'extériorité (hors-champ), pas plus qu'elles ne s'intériorisent dans un tout: elles sont plutôt un endroit et un envers, reversibles et non superposables, comme un pouvoir de se retourner sur elle-mêmes. Elles sont objet d'une réorganisation perpétuelle où une nouvelle image peut naître de n'importe quel point de l'image précédente." Deleuze 1985, p. 347.
 See Bonitzer 1982, p. 41.
 "Tous les trous sont toujours bouchés par ce qui vient affleurer en surface, il n'y a pas de trou puisqu'il n'y a que des incrustations, des fleurs qui viennent éclore à la place des yeux, un nez qui émerge à même la bouche, un lapin dans le pavillon de l'oreille et le tout en musique, muzak." Bonitzer 1982, p. 40.
 See Engell 1999.
 Some analogies arise from this interpretation of the image in becoming. In fact, being the image-video in perpetual 'becoming' even in its inner structure, it is possible to claim that it personifies a continuum where the space does not respond to a spatial mold or to a relationship of form to matter. Thus, it can be possible to analyze the video-image as an objectil, like Deleuze theorized for the anamorphism in his analysis of Leibniz. As described by Deleuze, an objectil cannot be interpreted as a point in space; it is a situs subjected to transformations. Thus, the transformation of the object to an objectil imposes a change of the subject, which Deleuze defined as a superject. Consequently, the relationship between objectil and superject implies a new notion of point of view. In fact, if object becomes objectil, i.e. it follows a group of transformation, the subject becomes superject, i.e. becomes a point of view on a site. See Deleuze 1988.
 The Proto-fractals were developed during the nineteenth century and are mathematical formulae and geometric objects that, at that time, were considered 'monsters of mathematics' because of their infinite structures. They were the result of research focused on both the development of non-Euclidean geometries and the creation of mathematical spaces able to include an infinite number of forms and information. These creations played a fundamental role in Mandelbrot's concept of a new geometry.
 Eco 1962, especially Il caso e l'intreccio. L'esperienza televisiva e l'estetica, pp. 185-198.
 See in more detail Eco 1962, p. 200: "Nel rifiuto dell'intreccio si attua il riconoscimento del fatto che il mondo è un nodo di possibilità e che l'opera d'arte deve riprodurne questa fisionomia".
 Ascott 1992.
 Aristotle, Poetic 1451a.
 Fractal geometry was developed in the 1970s by the French-Polish mathematician Benoît Mandelbrot (1924-2010) and is a mathematical explanation for natural and unnatural phenomena that are considered chaotic. See Mandelbrot 1983 and 1995.
 Initially, fractal geometry was applied to the analysis of movements in the stock market and their curves in order to predict economic phenomena. Mandelbrot, by analyzing huge quantities of information with computers, found that in between the scales of chaotic economic movements there exists a degree of harmony that could represent a possibility to understand and predict those movements. Aiming at understanding those chaotic movements, Mandelbrot used proto-fractal objects like the Koch Snowflake and Cantor Set. (See Peiten-Jürgen-Saupe 1992 and Mandelbrot 1963 and 1967). By means of the first computers, which could analyze huge quantities of information, Mandelbrot could understand, by applying these 'monsters of mathematics,' not only the development of the chaotic movements of the stock market but also a new episteme, which changed humankind's way of organizing and representing space.
 In general, see Bovill 1996.
 In general, see Lavandier 2004.
 It is important to remember that a fractal object is infinite and it is also able to contain infinite quantities of information. This was, in fact, the goal of proto-fractal objects.
 See Caronia 2006.
 See Caronia 2006.
 See Spielmann 2005.