Sharing Paradoxes: Impossible Spaces, Impossible Times and Impossible Facts. The Function of Self-Contradictory Structures in Arts, Sciences and Philosophy. (Part 4)

A first step in dealing with paradoxes is then to accept their double existence as being true and false. Outside academic studies, it is a natural process implied in the appreciation of any narrative art. As it has been described about cinema, there is a point where we accept the false to be true, as if trying to find a proof reducto ad absurdum. This process is called the suspension of disbelief (Walton, 7). Proposed first by the poet Samuel Taylor Coleridge, it has extended in the study of literature, cinema and videogames to name a few. This refers to any action where the false is taken as possibly true in order to appreciate a narration and facilitate the immersion process. Youssef Ishaghpour describes the duality implied as the reality of the image and the image of reality (Ishaghpour, 8-11). The suspension of disbelief is therefore a way to conciliate this duality in order to appreciate the fiction.

The difference when working in a fictional environment rather than in a scientific one is that every time a contradiction or unearthly events appears, it is usually taken as an extension of the perceived diegetic world: when the staircase optical illusion appears in Inception, it is accepted as part of the fictional world. Instead of defying physical rules, it is simply accepted as a new information contained within the fiction. Again, as in Kierkegaard or Bohr’s vision, it is an extension of the paradigm. The same happens for multiple worlds’ diegetic construction such as previously mentioned in movies like ExistenZ or Avalon. The multiple ontological world, very similar in their nature to the Russell’s paradox are accepted as such. The suspension of disbelief catalyses the conceptual acceptation of such constructions and even changes them into interesting and pleasing artistic visions.

Cat'.s Paradox

Figure 7: The cat’s Paradox. Source: Wikipedia

The use of paradoxical constructions taken from science does not end here. Another case comes from quantic physic. Erwin Schrödinger described the nature of some quantic events by the metaphor of a cat in a box. Let say there is a cat and some poison inside a box. There is as well a 50% chances for the poison to be relieved and therefore for the cat to die. The way quantic physics works is that as long as the information about the cat has not been extract from the situation, the cat is in both states: alive and dead. Both states excludes each other and therefore it leads to a paradox that of ‘’ the living and the dead cat mixed or smeared out in equal parts.’’ (Schrödinger) What modern physics proposes as a solution the acceptation of both state for a certain period of time. This process is known as quantic bifurcation. Even if this is very difficult to accept as being true for neophytes, when transferred to fiction it leads to acceptable and interesting narrative constructions. An example of a movie using this type of multi-linear time frame is Source Code by Duncan Jones. In this movie, a soldier is sent multiple times in the past to prevent a terrorist attack. After failing multiple times, he achieves his goal and life continue normally in this new independent timeline. The use of quantic bifurcation appears in multiple science-fiction movies and communities of fans are sketching schemas to understand the structure behind the film. Movies like Primer by Shane Carruth and Looper by Rian Johnson have generate numbers of complicated charts using quantic bifurcation in order to explain these narrations. (Figure 8)

Print

Figure 8: Looper movie chart by Rick Slusher. Source: Film.com

Another paradoxical time construction that has caused many problems is the time loop. Circular construction of time was accepted by many cultures around the world: Egyptians had circular time named Neheh (Assman, 137), the tzolkin, the Mayan calendar based on cycles of 260 days (Falcón, 19-21) and Hinduism has constructions of multiple intricate circles (Eliade, 134-136). This vision does not conciliate with the European linear construction of time, but it still easily apply to fiction.

Time being both in the future and the past appears in various fictional cases. First of all, whenever there is a time loop a cyclic time has to be accepted. Movies like Terminator (Cameron, 1984), Before the Rain (Manchevsky, 1994) or Chin Chin el Teporocho (Retes, 1976) all present this cyclic time construction. Indeed, time loops can be multiple and quantic bifurcation might again apply.

In the quantic bifurcation case as in the circular time construction, the paradoxical construction induces multivalued time states, discrete moment can be different but at a same time distance from a specific moment, the bifurcation point or those previous, or they can stand both in the future and in the past of a referential moment. It can also be seen as a specific case of multiple ontological states, as previously described, but with the specificity that the ontological state is defined by a time value.

It can be presented in a more mystic way as in the movie Voyage in Time by Tarkovsky and Tonino Guerra. In this case, the movie shows the directors talking about the film they will make about a trip they once had. The anecdotes supposedly in the past appears as well in front of the camera and therefore the time of the movie is triple, it stands for the past, the present and the future as in Three Sundays in a week, but without the logical explanation.

Perhaps the most well-known results about paradoxes is Gödel’s incompleteness theorem. After Cantor and Russell discoveries, logicians have tried to build a perfect and complete system for logic. The project happened to be more problematic than expected and new set of axioms surfaced. The outstanding result obtained by the mathematician Kurt Gödel changed radically the conception of logic and left the community in crisis. The incompleteness theorem states that no matter how many axioms we add to a logic system, there will always appear some statements that will be undecidable, meaning it will be impossible to prove them right or wrong (Nagel, 19-20). This is a perfect example of Paz’s perspective of grasping dual objects as such instead of considering them as problematic undefined concepts to reach a better understanding of it. In this case, the conclusion obtained by paradox is that paradoxes are inherent part of complex logical systems.

Paz’s consideration encompasses a big range of logical instance and, as seen previously, they apply to a wide variety of paradoxical objects: from optical illusions to narrative charts passing through quantum physics. It still does not hold for a type of undecidable statements. Some facts are not necessarily true or false; they stand somewhere in between as a result of incomplete definitions. They work as ambigrams but instead of offering mainly a finite amount of elements, they offer a continuous range of possible information. Such problems are common in everyday life since more situations are not clearly defined. For instance, we can pretend the sky is blue but it can’t be proven without adding precisions to the statement offered; at night the statement does not hold for instance. A relatively new branch of mathematics dedicates itself to such logical system. The idea behind this fuzzy logic, as it is coined, is to attribute truth values that varies continuously between the usual zero and one (Kandell). Therefore allowing any probability of truth ranging from zero to 100%. Such logical system coincides with perspective of quantum physics allowing diverse states with various probabilities. It is the case for instance for electrons in the atomic model were they navigate through a probabilistic area instead of following a precise trajectory.

Finally, paradoxes can appear within humoristic or philosophical functions. The twist are often used in usually called intellectual humour such as Woody Allen’s work. In Allen’s quote from Annie Hall ‘’ The food here is terrible and the portions are too small’’, the double statement stands in the contradiction that, in fact, if the food is terrible there is no reason to ask for more, but complaining about small portions implies asking for more food. This kind of construction can be found as well in Annie Hall: ‘’ Life is full of mystery, loneliness, and suffering –and it’s all over much too soon’’. The role of the paradox is then, in this case, to release a tension constructed around the paradoxical statement. In this situation, the contradiction, or double truth value, stands as a sign that the joke has reached its climax. The contradictory aspect of the logic involves is to be read as a sign to character does not make sense anymore, therefore the humoristic relief. The humour can follow as a comment on a paradox: way before Russell, Lewis Carroll underlines that no one can contain himself because of excitement because nothing can contain itself (Benayoun, 84). These considerations follow the seriousness of Ambrose Bierce’s definition of logic as the art of thinking within human capacities (Benayoun, 113), the presence of this limit is in itself both humoristic and a relief.

Paradox Humour

Figure 9: Paradox humour

In a broader perspective, the same applies to koans, small stories or statement present in the zen tradition. The sentences serve to increase doubt and questioning. The simple logic behind the koan ‘’What is the sound of one hand clapping’’ is similar; it states the possibility and impossibility of the referred sound. Possible since it is stated there is a clap sound and the impossibility by the uniqueness of the hand producing it. In this case, the paradoxical information serves again to release tension. The same holds for the koan: If you have a stick, I’ll give one to you, if you don’t I’ll steal it from you. The tension is released with acceptation to work outside a strict logical frame, to accept our humanity as proposed by Bierce.

This work outside logic may serve as well, paradoxically, for theological arguments. To understand we have to go back to the unliftable rock paradox. God, being almighty, should be able to create an unliftable rock, but then if he can lift the rock he is not almighty. An easy solution to this problem is to state that God’s work beyond human based logic.

As we have shown, the contradictory dialectic raising from paradoxes only cause problem within its own paradigm of binary logical values as being true or false. When grasped as specific concepts gathering both values, or, even infinitely many values ranging from absolutely true to absolutely false, many applications can be found. Accepting such condition standing in between these poles is what allows us to enjoy narratives in different ways; first to enhance the emotive effect of a diegetic world by accepting the ontological quality of fiction as being an image of reality that is itself included in and presented as a simulacra of reality, secondly as to define science fiction or fantastic narratives as legitimate by extending the accepted diegetic world. Logical statements sharing both truth and false value are integral parts of human scientific and cultural knowledge[1]. It is hoped that more research to consider paradoxes in their social appearances will be provided as to understand better their functions as a fundamental part of human thinking rather than solely as odd mythological thoughts gravitating in the abstract spheres of philosophy and logic.

Felix Lambert

First version September 2015

[1] As it is the case in Dialetheism. For a good review of this as a philosophe, the reader is invited to read the entry in the Stanford Encyclopedia of Philosophy by Francesco Berto.

 

Mediagraphy

Alexander, Amir. 2014. «Guldin et les indivisibles de Cavalieri». Pour La Science, n˚ 440, p.70-73.

Allen, Woody (real.) 1977. Annie Hall. USA: Rollins-Joffe Productions. DVD

Assman, Jan. ‘’Le Temps Double de l’Égypte Ancienne’’. Pour La Science, n˚ 397,2010.136- 141. Print

Baltrušaitis,Jurgis. Aberrations : Les Perspectives Dépravées-1. Paris : Flammarion, 1995. Print

Baltrušaitis,Jurgis. Anamorphoses : Les Perspectives Dépravées-II. Paris : Flammarion, 1996. Print

Benayoun, Robert. Les dingues du nonsense.Paris : Balland,1984. Print

Berto, Francesco. Dialetheism. Entry in the Stanford Encyclopedia of Philosophy. Online: http://plato.stanford.edu/entries/dialetheism/.

Biederman, Irving. ‘’Recognition-by-Components : A Theory of Human Image Understanding.’’ Psychological Review, Vol. 94, No. 2, 1987. 115-147. Print

Boll, Marcel. Histoire des mathématiques.Paris: Presse Universitaire de France, 1968. Print

Browne, Cameron. ‘’Impossible Fractals’’. Computer & Graphics, 31(4), 2007. 659-667. Online: http://eprints.qut.edu.au/15013/1/15013.pdf

Cantor, Georg. Contributions to the Founding of Theory of Transfinite Numbers. New York: Dover Publications, 1952. Print

Carruth, Shane (real.) Primer. 2005. États-Unis: ERBP. DVD

Cronenberg, David (real.) EXistenZ. 1999. Canada et Grande Bretagne : Alliance Atlantic Communications, Canadian Telivision Fund, The Harold Greenberg Fund, The Movie Network, Natural Nylon Entertainment, Serendipity Point Films, Téléfilm Canada et L’Union Générale Cinématographique. DVD

Delahaye, Jean-Paul. ‘’Labyrinthes de longueur infinie’’. Pour la Science, n˚318 (Avril), 2004. 90-95. Print

Delahaye, Jean-Paul. ‘’Ambigrammes’’. Pour la Science, n˚32 (Septembre ), 2004.90-95.Print

Falcón, Maricela Ayala. ‘’Tiempos mesoamericanos, calendarios mayas’’. Artes de México, vol 107, El Arte del Tiempo Maya, 2012. 18-25. Print

Garibay, Tito. El albur: Método introductorio. México, D.F.: Grupo Editorial Tomo, S.A. de C.V., 2008.

Grimaldi, Ralph P.. Discrete and combinatorial mathematics; an applied introduction, 5th Ed.    Boston: Pearson Addison Wesley, 2004. Print

Hofstadter, Douglas. Gödel Escher Bach: Les Brins d’une Guirlande Éternelle. Paris : Interéditions,         1985. Print

Hofstadter, Douglas.’’On Self-Referential Sentences‘‘. Metamagical Themas: Questing fort the Essence of Mind and Pattern. New York: Basic Books, 1985, p. 5-24. Print.

Jones, Duncan ( real.) Source Code. 2011. États-Unis: Vendome Pictures et The Mark Gordon Company. DVD

Kandel, Abarham. Fuzzy Mathematical Techniques with Applictions. Addison-Wesley Publishing company.

Koch, Hel von. “ On a Continuous Curve without Tangent Constructible from Elementary Geometry .“ Traduit du français par Ilan Vardi. Dans: Classics on Fractals, édité par Gerard Edgar. New York : Addison-Wesley Publishing Company, 1993: p. 25-45.

Kunzmann, Franz, Franz-Peter Burkard et Franz Wiedmann. Atlas de la philosophie.Paris : La Pochothèque, 1999. Print.

Labelle, Jacques et Armel Mercier. Introduction à l’Analyse Réelle. Montréal: Modulo, 1993.. Print

Mandelbrot, Benoît. The Fractal Geometry of Nature.New York: W.H. Freeman and Company,1977.

McHale, Brian. Postmodernist Fiction. New York: Methuen,1987. Print

Moore, Ruth. Niels Bohr: The Man, His Science, & the World They Changed.New York: Knopf, 1966. Print

Nagel, Ernest, James R. Newman, Kurt Gödel, Jean-Yves Girard. Le théorème de Gödel. Traduis de l’anglais et de l’allemand par Jean-Baptiste Scherrer. Paris :Éditions du Seuil, 1989. Print

Nolan, Christopher (réal.). Inception. 2012. États-Unis et Grande-Bretagne : Syncopy. DVD

Paz, Octavio. Conjunciones y disyunciones. Distrito Federal : Editorial Joaquín Mortiz, 1991. Print

Peano, Giuseppe. ‘’Sur une courbe, qui remplit toute une aire plane’’. En ligne : Mathematiche Annalen, Vol. 36, n° 1, p. 157-160, 1890. Consulté le 10-04-2012. DOI : 10.1007/BF01199.438.

Pelletier, Francis Jeffry. 2000. «Review of mathemtics of fuzzy logic». The Bulletin of Symbolic Logic 6 (3), p. 342-346.

Poe, Edgar Allan. Ne pariez jamais votre tête au diable : et autres contes non traduits par Beaudelaire. Traduction, préface et notes par Alain Jaubert. Paris : Gallimard, 1989. Print

Reinhardt, Fritz et Heinrich Soeder. Atlas des mathématiques. Paris: La Pochothèque, 1997. Print

Sagan, Hans. Space-Filling Curves. New-York : Springer, 1994. Print

Schrödinger, Erwin. 1935. «The Present Situation in Quantum Physics». Trans. John D. Trimmer. The original article in Naturwiessenschaften, 23 (49), p. 807-812. Traduction in Proceedings of the American Philosophical Society, 123, p. 323 -338

Slaatté, Howard A. A Re-Appraisal of Kierkegaard. Boston: University Press of America, 1995. Print

Stewart, James. Analyse : concepts et contextes Volume 1. Fonctions d’une variable. Traduction de la première édition par Micheline Citta-Vanthemsche. Paris : De Boeck Université. Print

Tarkovski, Andrei, Tonnino Guerra (real.) 1983. Tempo di viaggio. Italy: Genius s.r.I. and RAI Radiotelevisione Italiana. DVD

Vidal-Rosset, Joseph. Qu’est-ce qu’un paradoxe?. Paris: Librairie Philosophique J.Vrin, 2004. Print

Walton, Kendall, L. «Fearing Fictions». The Journal of Philosophy, Vol.75, No.1 (Jan., 1978), p. 5-27. Print

Advertisements

Sharing Paradoxes: Impossible Spaces, Impossible Times and Impossible Facts. The Function of Self-Contradictory Structures in Arts, Sciences and Philosophy. (Part 3)

We can classify visual paradoxes into two categories: the simply self-contradictory ones and the ambigrams. In looking at most of Maurelius C. Escher’s optical illusions, the spatial design seems to be both true and false. As pointed out by Biederman, since the shapes preserve adequate angle constructions, the objects seems to be credible. That is a direct result of the importance of line intersection in object recognition. Nevertheless, the overall conceptualisation of the object seems not acceptable as being true (Biedermann, 135-140). In this case, these visual paradoxes are simply self-contradictory because the macroscopic veracity statement contradicts the microscopic ones (as in the two sentence: The next sentence is true. The last sentence is false (Hofstadter, 19)) Many artists have found multiple interesting ways to produce these types of paradoxes. Escher indeed is well known for such constructions, but we can name as well graffiti artist Damien Gilley, Dutch artist Ramon Bruin and Istvan Orosz (Figure 3) Having met quite a wide popularity, it is normal that these simply self-contradictory visual objects have appeared in various situations. For instance, Penrose’s triangles has appear on post stamps, tattoos and many everyday objects.

Inception stair scene

Figure 4: Inception.s stairs scene by Christopher Nolan

These self-contradictory visual objects can contribute solving other artistic problems. In Christopher Nolan’s movie Inception the infinite stairs illusion is used and as real instance within the diegetic world to trap the enemy. In this case, it is a use of a visual paradox to solve of narrative problem. (Figure 4) On his hand, Cameron Browne have found interesting ways to merge the optical illusion construction with another old problem; the paving of the plane (Browne, 2007). Browne has constructed infinite patterns of self-contradictory visual objects that can be used to fulfill the entire plane. He worked as well with contradictory fractal structures (Figure 5)

Camron Browne.jpg

Figure 5: Impossible Fractal by Cameron Browne. Source: http://www.cameronius.com/graphics/impossible-fractals-figures/

Ambigrams are figures that show two incompatible information at the same time, inasmuch a paradox, they work on a scale more nuanced than the dichotomic paradoxes[1]. The figure of a young-old lady is a popular example of ambigram due to H.H. Hill (Delahaye, 91). In this image we can actually perceive two different portraits. One pictures a young lady and one offers the profile view of an old lady. It is a paradox since it contradicts itself, not in the previous case straightforward manner, but by ricochet. If it is a young lady, then it is impossible to be the old lady at the same time and vice versa. In the other hand, we can interpret the image as containing two informations, two different images. Ambigrams, working on a larger scale; they can contain more information. For instance, in figure 5 one can find six apparitions of the word palindrome, all put upside-down and to be read in both directions.

The advantage of using a word like ambigram is that it underlines an interesting property of paradoxes, that fact of being containing multiple statement that would usually not appear in general in a coherent manner. To go further in this sense, we have to go back to an analysis made by Mexican sociologist and writer Octavio Paz. In an analysis of complementary and dual concepts existing in various societies, Paz stresses the importance of considering such pairs as a whole by focusing on the relation between them. For instance, body and no-body are not to be considered as specific meanings except to express contraries (Paz, 55). This perception of duals as a whole can lead to interesting results when applied to paradoxes.

Palindrome

Figure 6: Ambigram

[1] We mean by this that the information is not straight opposite, like true and false, up and down, etc.

Narrative sculptures: graph theory, topology and new perspectives in narratology

“If there is one thing in mathematics that fascinates me more than anything else (and doubtless always has), it is neither “number” nor “size”, but always form. And among the thousand-and-one faces whereby form chooses to reveal itself to us, the one that fascinates me more than any other and continues to fascinate me, is the structure hidden in mathematical things.”

A. Grothendieck. Récoltes et Semailles

There have been many attempts to model narratives from a structural point of view. From these numerous models we want to preserve a macroscopic vision that allows a quick and simultaneous understanding of various important elements of the story, which we call, following Labov’s and Wilensky’s definitions, narrative points (1). Models mapping the general structure of the story can be found, for instance, in the work of Marie Laure Ryan where both diegetic and possible events are represented and where narrative points are related by vectors. In order to preserve this telescopic view and superpose its logic with McCloud’s notion of infinite canvas (2), which will be defined in the body of this text, an option is to start with the notion of a parametric curve. Before doing so, an overview of the pragmatic motivation that led to this research is needed.

The motivation behind this exploration is taken from an interest in mathematics and an increasing amount of narratives using complex time structures and story representations. Movies like Primer (2005) by Shane Carruth lead to the construction of various charts in attempts to understand the hidden time structure (3). Source Code (Jones, 2011) and Looper (Johnson, 2012) are other examples that created the need for such macroscopic representation and many other films, like Cronocrimenes, Triangle (Smith, 2009) and the Terminator suite are cases that could have led to similar practices. In the case of the movie Looper, a three dimensional version of the chart has been produced, bringing to light wider possibilities (Figure 1).

NS Figure 1

Figure 1: Movie chart for the movie Looper by Rich Slusher.

Comic artists have explored this path in some isolated cases, either in the use of bigger expositional space (4), or as the juxtaposition of various three dimensional objects, like the booklets in Chris Ware’s Building Stories (2012). This work constitutes a box containing many booklets that can be read in different order. This creates multiple combinations for the reader exploring the diegetic world. Another interesting example, comparable to a mutoscope or other early cinematic devices, is the three dimensional cyclical structure of Julius Coretin Acquefaques, prisonnier des rêves: Le décalage by Marc-Antoine Mathieu. In this case, when leaving the story at the end of the comic, they actually enter the story again to loop the cycle.

The model presented in this paper is a first exploration in the variety of different surfaces that might be used in further narrative experimentations as well an attempt to establish the basis of a formal narrative tool for academics and artists. Therefore, the author wishes to open discussions in defining narratives and hopes to inspire artists in exploring the challenges offer by this model.

One of the key elements of our model is the use of curves with the continuous stretch of time maintained across them. Even if it seems natural nowadays to represent time with a line, its extensive use in various models results from many different traditions. In our model, these influences are mainly the following: history charts, the construction of the real number line based on Dedekind and Cantor’s work, and the use of parametric curves with time as the general parameter. We will discuss these three influences briefly.

For most of the Middle Ages, time was mainly represented on timetables. Around the beginning of the 19th century, time flux started to be embedded within natural metaphors like lightning and rivers (5). These two examples are important since they allow the time frame to branch out simultaneously. Various time lines could be traced out of single elements. In mathematical terms, these structures are equivalent to oriented graphs, and more precisely to oriented trees, since cycles do not exist in these structures.

For its part, the concept of continuity led to multiple complications and was not well defined until the topology of the real line was properly described. We owe much to the work of Weirstrass, Dedekind and Cantor for this definition and understanding. This dense continuous line of values serves as well in defining parametric curves, curves based on a continuous parameter, usually the time. These curves can be used to represent various types of motion, for instance, the movement of particles in space.

The first trick to make use of mathematical models to represent time frames is to base diegetic time on parametric curves. As a building strategy, this enables various constructions of diegetic time structures. First of all, it allows the concatenation of many line segments as it happens in the time charts discussed above, therefore constructing structures like tree graphs. A simple example of a narrative based on that idea is Griffith’s movie Intolerance, in which different independent stories flow separately (6). Examples can also be found in the work of artists like Chris Ware or Jason Shiga, or in the hypercomics based on McCloud’s infinite canvas such as Daniel Merlin Goodrey’s work (7).

The concatenation of various time segments allows the construction of multi-cyclic time structures as well. This kind of structure is not in itself a novelty; in some mythologies, cyclic time is accepted as the general topology of time frames, and some even make use of many intricate cyclic times as in the Tzolkin and Vedic time constructions. In extending parametric curves into graph theoretical frameworks, we can obtain infinity of cyclic graphs where cycles may be intersecting or independent. This application naturally allows a wide variety of already proven theorems to apply to narratology. For instance, observing the underlying structure of a graph might allow us to determine the number of possible cycles, each of them being a possible reading path.

Because cycles are naturally embedded on a flat surface, some considerations about the implied spaces become important. The Jordan curve theorem states that any simple closed curve separates space in exactly two sections, the interior and exterior of the closed curve, or equivalently, of a cycle. As a result, constructing a cyclical story leads to the creation of these inside and outside spaces that might be used later for a semantic purpose.

In Reinventing Comics, Scott McCloud coined the term infinite canvas to represent the possibility of extending comics infinitely in all directions of a plane. His website specifies that it provides the perfect conditions for a type of comic he names hypercomics. Looking back at mathematical definitions of planes and surfaces, it seems clear that various types of infinities are involved in the notion of an extended version of the infinite canvas.

First, in terms of the continuum defined by Cantor, a plane is dense since it follows from the product of two continuous axes. This implies that infinite zooms are possible at any point on a plane, and as such, on any compact surface (8). To understand this implication, we have to look at a category of curves called space-filling curves, or Peano curves after Giuseppe Peano who first proposed such an example. Space-filling curves are iterated curves that, at their limits, fill a whole part of the plane. (Figure 2) Indeed, many other examples have been provided by other mathematicians in order to provide extra characteristics, as for instance Moore’s curve that is a closed space-filling curve. The density of the plane implies that the breakdown of iterated narrative into infinitely smaller scales is possible. This density leads to possible infinite zoom, fractal-like, construction as found in Marc-Antoine Mathieu’s first and third tomes of his Julius Corentin Acquefaques serie.

Figure 2: A Space-filling curve

The second way in which the canvas is infinite arises first when we allow the plane to be infinite in all directions. In mathematical term, it means the surface is not compact because it would be impossible to cover a plane with a finite amount of bounded sets. From a representative point of view it means it could never be entirely seen, in particular, not in a finite amount of images. In this case, this is why McCloud claims that the infinite canvas naturally supports digital comics. Although true, we suggest the infinite canvas presents even more value with the infinite amount of shapes we can allow the canvas to have.

Also, the canvas does not have to be contained simply in the plane. For instance, as suggested visually in McCloud (9) and in the diegetic world of French author Marc-Antoine Mathieu (10), comics could be presented on spheres (11). The use of different properties of the sphere can lead to a variety of narratological compositions in link with the intrinsic properties of the sphere: the presence of loxodromes, the covering groups different from the wallpaper groups and so on.

In addition, as proposed by many artists, from Alan Moore in Promethea to Jim Woodring in a side project (12), passing by members of the OuBaPo collective, the use of a Möbius strip as the canvas leads to interesting constructions. These can be used as objects existing within the diegetic world as in Moore, or directly as a support inducing a specific topology within the diegesis as in Woodring’s case.

Indeed, any sculptural surface may offer interesting options for narrations and a complete survey of such an approach should be done. In our case, we would like to focus on surfaces that have been studied from a mathematical point of view. The reason is that many theorems shed light on hidden properties that enable us to imagine interesting narratives and limiting ourselves to a sculptural point of view would have prevent us from finding and using these properties. The variety of surfaces is infinite and a list of inspiring surfaces can be found in the fields of differential geometry, differential topology, and knot theory. For instance, as a result of their definition, minimal surfaces seem pleasing to embed stories. It involves the possibility of working on some surfaces of infinite area spreading in different axes, as with Sherk’s surface and Costa’s surface (Figure 3), or even with self-intersecting sections, as in the case of Henneberg’s surface.

NS Figure 3

Figure 3: Costa’s surface. Source: Mathworld.com

Compact surfaces also lead to interesting possibilities. In topology, the study of surfaces is bound to the analysis of characteristics which are preserved when surfaces are torn and stretched. Such invariants are coined topological invariants. An example is the number of holes present in the surface. For instance, the sphere contains no holes, but the torus has one; therefore the two surfaces are fundamentally different. On the other hand, the sphere and the cube are classified as the same surface since they both have no holes. This argument leads to a classification for compact surfaces depending on the number of holes involved. As it turns out, all compact orientable surfaces are torus of genus n, meaning a torus with n holes, for n a positive integer These will become useful in the next section.

Orientability is another characteristic that helps refining surface classification. Orientable roughly means they possess an inside and an outside and it is impossible to move smoothly from the inside to the outside. For instance, it is impossible to move on the sphere and end up being inside the sphere without piercing a hole. The Möbius trip is a simple example of non-orientable surface since by smoothly moving along the surface it is possible to end up on the other side of the departure point. In constructing sculptures, non-orientable surfaces lead to some difficulties. For instance, the Klein bottle invented by German mathematician Felix Klein in 1882 cannot be embedded in our three-dimensional world without self-intersecting (Video); it is only possible in four or more dimensions. This makes the visualisation of these surfaces more difficult, but a general classification is still possible.

The class of infinite compact non-orientable surfaces are all equivalent to spheres with a certain number of Möbius strips glued to holes in them (the edge of the Möbius strip is equivalent to a circle, therefore when cutting a circular hole on the sphere it becomes possible to glue the strip’s edge along the edge of the hole). The more complex the non-orientable surface, the more dimensions one needs to avoid self-intersections. Even if it seems very hard to work on these surfaces as a possible infinite canvas, shortcuts exist. There is a way to represent any compact surface, orientable or not, with their fundamental polygons which can easily be represented on the plane. These polygons are simplified maps for these surfaces; to obtain a surface, it suffices to fold its edges by respecting so pair connections or edge directions. Indeed, the writing on non-orientable compact surfaces that aren’t embeddable in three dimensions might be done in a virtual environment, or directly on the equivalent fundamental polygon. The figure below shows the construction of the Klein bottle from its fundamental polygon. (Figure 5)

NS Figure 5

Figure 5: Klein botte’sfundamental polygon.

As a result, the infinite canvas is infinite as well in the number of dimensions a non-orientable surface holding a story could ‘’naturally’’ exist without self-intersecting. Indeed, the use of computers can be a handy tool in constructing such narratives.

The next question we need to address is the following: why would we want to work with parametric curves on this collection of surfaces? The answer comes from the field of topological graph theory. The Polish mathematician Kasimierz Kuratowski and the Russian mathematician Lev Pontryagin proved independently the necessary and sufficient conditions to be able to embed a graph on the plane without crossing edges. It states a graph is planar if and only if it does not contain the subgraphs K₃,₃ or K₅. (Figure 6)

NS Figure 6

Figure 6: The obstruction set for the plane

In constructing comics on parametric curves based on graphs containing one of these would inevitably leads to edges crossovers. Indeed, such overlapping can always be dealt with, as in the case of Chris Ware diagram comics, but the point here is to explore the possibilities provided by restricting ourselves to planar embeddings. To give a pragmatic application, we know the two aforementioned graphs can be drawn on the torus or the Möbius strip without having edge overlapping, it means they have planar embedding for the torus. It follows that it is possible to draw planar stories on such graphs if we use the torus as the canvas. (Figure 7)

NS Figure 7

Figure 7:Toroidal embedding of K₅

The study of topological graph theory led to the discovery that different surfaces don’t share the same obstruction groups, i.e. the set of graph making the planar embedding impossible, such as K₃,₃, and the K₅, in the case of the plane. We know for instance that the Möbius plane has 35 such graphs (Archdeacon, 1980), and the Torus has more than 16 000! On the other hand every finite graph can find a planar embedding in some compact orientable surfaces with at least n holes for a certain n values, and same holds for non-orientable surfaces and a certain number of Möbius strip glued to the sphere.

Another result is that the presence of cycles leads to different amount of bounded spaces. In other words, if the Jordan curve theorem holds for the sphere, it is not true in general. Already in the case of the torus, construction of longitudinal and transversal cycles leads to a single bounded space; it does not hold for torus with n holes neither.

The construction of narrative on these extended infinite canvases, such as non-orientable surfaces, minimal surfaces and so on, is what we call narrative sculptures because their structures are deeply linked to the surfaceskno hosting them. The main goal in constructing narrative sculptures is the research for new narratological challenges. An optimised use of this involves considerations of the following distinctive properties of narrative sculptures: the possible use of complex multi-cyclic time curve constructions, the use of different spaces the cycles are bounding and the possible semantical implications in our world, or in a digital equivalent to it.

We present two examples, expressing challenges brought by simple constructions. The K₅ graph has a planar embedding on the torus. . It can as well be constructed by the union of two cycles by taking a cycle being the outside pentagon and the second one being the star shape in the middle. We could construct a highly ‘’twisted’’ story as following. Through the double cycles, we could describe the interactions of two individuals at desynchronised moments of their life cycles. The complications and self-containing elements of the story could then be reinforced by presenting it on a trefoil knot, which is simply a torus but embedded differently in three dimensions. (Figure 8) Of course, many other options since the torus can find multiple embedding in four dimensions that could lead to interesting narrative sculptures (13).

NS Figure 8

Figure 8: Trefoil knot by Jos Leys. Source: josleys.com

The graph K₅ also possesses a planar embedding on the Klein bottle. It would then possible to construct a complex science-fiction comic. First the multiple desynchronised elements present on the two cycles would bring an intricate time structure. Then, different bounded area could hold their proper images and symbolism related to the story. Finally, the Klein bottle canvas leads to a hyper-fictional statement since the canvas itself could not be properly constructed in our world. The same holds for the infinite collection of surfaces that aren’t embeddable in three dimensions without self-intersecting. (14)

In conclusion, we have seen that by merging various paradigms and concepts from narrative theory, the infinite canvas and mathematical knowledge about surfaces and graphs, we can define highly complex narrative structures that we coined narrative sculptures. Such constructions not only leads to new narratological and artistic challenges, but it can bring new questioning about the way we first, understand stories, and secondly how we teach narratology. In the first case, experiments in cognition could help understanding the effect of dealing with highly complex but still visually clear narratives in our learning process. In the latter case, it evokes the possibility of including some mathematical notions in teaching narratology or even information design.

Félix Lambert

Notes

1- Ryan, p. 150-151

2- McCloud, 2000.

3-An example can be found at http://movies.yahoo.com/blogs/the-projector/incredibly-detailed-primer-timeline-210027548.html

4-Gravett, p. 136-137

5-Rosenberg and Grafton, p.143-149.

6-Eisenstein, p. 397

7- http://e-merl.com/

8-It should simply be understood in this case of surfaces of finite area.

9-McCloud, 1993

10-Mathieu, 2004

11-McCloud also suggest writing on the cube in Reinventing Comics.

12-https://www.fantagraphics.com/rarities-and-miscellany-by-various-artists/moebius-strip-comic-by-jim-woodring-video-photo-animation.html

13-Séquin, 2012

Mediagraphy

Bibliography:

Ball, David M. et Martha B. Kuhlman Éditeurs. 2010. The Comics of Chris Ware: Drawing Is a Way of         Thinking. Mississippi: University Press of Mississippi.

Barr, Stephen. 1989. Experiments in Topology. New York: Dover Publications.

Bondy, J.A. et U.S.R. Murty. 2008. Graph theory. New York: Springer.

Cates, Isaac. 2010. ‘’Comics and the Grammar of Diagrams’’. In The Comics of Chris Ware: Drawing Is      a Way of Thinking. Edited by David M. Ball and Martha B. Kuhlman. Mississippi:    University Press               of Mississippi.

Delahaye, Jean-Paul. 2008. ‘’Une propriété cachée des graphes’’. Pour la Science, n˚366 (Avril), p. 92-97.

Di Liddo, Annalisa. 2009. Alan Moore: Comics as Performance, Fiction as Scalpel. Mississippi:      University Press of Mississippi/ Jackson.

Falcón, Maricela Ayala. 2012. ‘’Tiempos mesoamericanos, calendarios mayas’’. Artes de México, vol.       107, El Arte del Tiempo Maya, p. 18-25.

Gagarin, A., W. Myrvold et J. Chambers. 2005. ‘’Forbidden minors and subdivisions for toroidal graphs   witn no K₃,₃’s’’. Dans Electric Notes in Discrete Mathematics Vol. 22, p. 151-156.

Genette, Gérard. 1972. Figures III. Coll. Poétiques. Paris: Éditions du Seuil, 1972.

Glover, Henry H., John P. Huneke and Chin San Wang. 1979. ‘’103 Graphs That Are Irreducible for the     Projective Plane’’. Journal of Combinatorial Theory, Series B 27, p.332-370.

Gravett, Paul. 2013. Comics Art. New Heaven: York University Press.

Gross, J.L. and T.W. Tucker. [1987] 2012. Topological graph theory. New York: Dover Publications.

Groupe Acme. 2011. L’Association: Une utopie éditoriale et esthétique. Paris : Les Édtions Nouvelles.

Kuratowski, Casimir. 1930. ‘’Sur le problème des courbes gauches en topologie’’. Fundamenta Matematicae, Vol. 15, p. 272-283.

Lickorish, W.B. Raymond. 1997. An Introduction to Knot Theory. New-York: Springer.

Mathieu, Marc-Antoine. 1991a. Julius Coretin Acquefaques, prisonnier des rêves: L’origine. Paris: Éditions                 Delcourt.

———— 1991b. Julius Coretin Acquefaques, prisonnier des rêves: La qu…. Paris: Éditions             Delcourt.

———— 1993. Julius Coretin Acquefaques, prisonnier des rêves: Le processus. Paris: Éditions     Delcourt.

———— 1995. Julius Coretin Acquefaques, prisonnier des rêves: Le début de la fin. Paris: Éditions            Delcourt.

———— 2004. Julius Coretin Acquefaques, prisonnier des rêves: La 2,333e dimension. Paris: Éditions      Delcourt.

———— 2013. Julius Coretin Acquefaques, prisonnier des rêves: Le décalage. Paris: Éditions Delcourt.

McCloud, Scott. 1993. Understanding Comics. New York: HarperPerrenial.

———— 2000. Reinventing Comics. New York: Paradox Press.

McHale, Brian. 1987. Postmodernist Fiction. New York: Methuen.

Munkres, James R. 2000. Topology. 2nd Ed. New Jersey: Prentice Hall.

Pickover, Clifford. 2006. The Möbius Strip : Dr. August Möbius’s Marvelous Band in Mathematics, Games,                 Litterature, Art, Technology, and Cosmology. New York: Thunder’s Mouth.

Rosenberg, David and Anthony Grafton. 2010. Cartographies of Time: A History of the Timeline. New      York: Princeton Architectural Press.

Ryan, Marie-Laure. 1991. Possible Worlds, Artificial Intelligence, and Narrative Theory. Indianapolis:       University Bloomington & Indianapolis Press.

Séquin, Carlo. 2012. ‘’Topological Tori as Abstract Art’’. Journal of Mathematics and the Arts, Vol. 6, Nu. 4, December, p. 191-209.

Stillwell, John. 1992. Geometry of Surfaces. New York: Springer-Verlag.

Termes, Dick. 1994. ‘’The Geometries behind my Spherical Paintings’’. In The Visual Mind: Arts and                 Mathematics. Edited by Michele Emmer. Cambridge: MIT Press, p. 243-248.

Ware, Chris. 2000. Jimmy Corrigan: The Smartest Kid on Earth. New York: Pantheon Books.

——— 2003. Quimby the Mouse. Seattle: Fantagraphics Books.

——— 2012. Building Stories. New York: Pantheon Books.

Websites:

https://www.fantagraphics.com/rarities-and-miscellany-by-various-artists/moebius-strip-comic-by-jim-woodring-video-photo-animation.html

http://e-merl.com/hypercomics

http://scottmccloud.com/4-inventions/canvas/index.html

Filmography:

Carruth, Shane (réal.). Primer. 2005. États-Unis: ERBP. DVD. 77 min.

Johnson, Ryan (réal.). Looper. 2012. États-Unis et Chine: Endgame Entertainment. DVD. 119 min.

Jones, Duncan (real.). Source Code. 2011. États-Unis: Vendome Pictures et The Mark Gordon Company.               DVD. 93 min.