Sharing Paradoxes: Impossible Spaces, Impossible Times and Impossible Facts. The Function of Self-Contradictory Structures in Arts, Sciences and Philosophy. (Part 2)

Russell’s paradox is an example that shows how thinking about paradoxes can be fruitful. Bertrand Russell studied set theory at a time when it was still in a naïve simple form at a time the vary basics of logic were studied in philosophy and mathematics. He realised the idea of set was really powerful but was still problematic since it involved the following paradox equivalent to an extension of Grelling’s paradox: separate all sets into two categories, the sets including themselves and the sets not including themselves (Reinhart, 23). As in in Grelling’s case, the paradox appears when we try to classify the set of all sets not including themselves. The paradox works in the exact same fashion as Grelling’s one. Finally, Russell added some axioms to the naïve set theory and included levels of inclusions for a theory called type theory (Vidal-Rosset, 17). A set defined at a certain level of inclusion cannot be taken as a set of a lower level. Therefore, the question of including the set of all sets not including themselves within itself or its dual is not a legitimate question since it becomes a prohibited operation.

Although this paradox has been solved in a logical point of view, it still holds in many fictional construction[1]. In constructions that Brian McHale, based on Gerard Genette’s theorical work on narratology, defines as being on different ontological levels. McHale provides with many examples of novels and short stories based on this idea. Many science-fiction movies are also based on this principle of multi-stage inclusions: it is the case for EXistenZ by David Cronenberg or Avalon by Mamoru Oshii. In both cases, as underlined by McHale for one of Robbe-Grillet’s short stories Project for a Revolution in New-York, the reader cannot identify the ontological level on which the action is happening (MacHale, 117). This narratological strategy helps staging a Daedalus that extends to many ontological levels and reinforces the impression of being lost. This state of delocalisation of the protagonist has a double impact: the loss of referent implies by the fact that they can’t identify on which ontological level they are, and by secondly by erasing the value of the initial referent as the real initial and authentic state. This construction of infinite ladder of diegesis is constructed from confounding a set with a higher (or lower) level set.

The section of mathematics dealing with set, their properties and their axiomatic use as a starting point for mathematics is called set theory. Set theory also had to deal with other conceptual problems at the time and this had to do with bijective functions: correspondence between the elements of two sets where an object is related to one and only one object on the second set (Grimaldi, 279). As an example, we can set such a function between the set of positive integers and the set of negative integers by linking every number to its negative equivalent. No matter how many numbers there are, an infinity as a matter of fact, we will always be able to construct the function. The problem arises when we put in relation sets that seem to clearly have a different number of elements. This time, we can create a bijective function between every positive integers and every positive even integers. In this way, one is linked to two, two is linked to four, and three is linked to six and so forth. Since we have an infinite number of integers we will always have enough to construct the relation. This is counterintuitive since all the elements of the set of even numbers are present in the set of integers but the converse is false. We can set other kinds of strange relations between finite length segments and infinite length segments. The two-dimensional stereographic projection provides such an example. We proceed as follow: we set a circle on an infinite line and from the top of the circle we trace rays that cross the circle at a point and then continues until it hits the line. By proceeding as such and scanning the rays on 180 degrees, every point on the circle will be linked to a point on the line and this time the reverse holds. Even more surprising, it has been shown that the line segment can be put in relation with the square which at the time seemed very curious since both geometrical objects did not even share the same number of dimensions: the line is one dimensional whereas the square is two dimensional (Sagan, 115).

These paradoxical constructions led Georg Cantor to the creation of the transfinite numbers theory. Cantor defined different types of infinity: the countable and uncountable (Cantor, 1976) Countable simply means they can be put in a bijective relation with the natural numbers. As mentioned previously, the set of even numbers is countable for that very reason. The two sets, natural numbers and even numbers are then of the same cardinality because they are both infinite countable. The same logic holds for the circle and the infinite line: both of them are infinite uncountable and therefore are of the same size. The expansion of the theoretical frame for infinite numbers explained as well the natural relation between the line and the square. In 1635, Bonaventura Cavalieri already proposed the idea that plane figure were made of infinitely many line segments (Alexander, 70). With the work of Peano (1890), the idea of filling the square with a single curve spread widely and many mathematicians proposed such curves. These curves where not bijections, they were in fact surjections; points of the square were actually covered many times by the same curve.

These curious objects also brought light on other concepts that were taken for granted like the idea of dimension. As a result, many definitions for dimensions have been proposed and objects called fractals have been found having non integer dimensions. For instance, the Koch curve, a well-known fractal, has dimension 1.2619 (Mandelbrot, 36). The Koch curve is famous as well for being paradoxical to the notion of continuity. When Cauchy developed the concept, he believed that continuity implied derivability, i.e. the existence of a tangent line (Wallace, p. 187). Bolzano and Weirstrass constructed such curves, but von Koch, unsatisfied by the too analytical model of these curves decided to construct geometrically his now famous curves with the property of being continuous but nowhere differentiable. (von Koch 1904-1905) Various other examples from that time shared similar paradoxical value over the canonical comprehension of continuity and dimensions. They participated in the birth of the notion of fractal geometry by French mathematician Benoît Mandelbrot in 1975. Again, redefining paradigm from arising paradoxes led to improvement of various theory in mathematics.

Another common way to use paradoxes appears in mathematical proofs. In a proof technique called reducto ad absurdum a statement that seems to be false is taken to be truth. The proof holds if a contradiction with the hypothesis is to be found. A common example of such proofs is Aristotle proof that √2 is not a rational numbers. It concedes the number a rational form and by dividing by all possible cases of appearance of even or odd occurrences for the numerator and denominator, contradictions appears in all cases. The hypothesis is therefore impossible (Boll, 31-32). Euclid showed in a similar way that there is an infinite amount of prime numbers. He started by setting the highest prime number on the theoretically finite list and then show he could construct in bigger number not divisible by any of the finite prime number list (Grimaldi, 222). The list here could be very long but the result would be the same: paradoxes can be use actively in search of knowledge. It maps this knowledge by defining areas of impossibility, therefore implying areas of certainty.

This strategy holds for other scientific area. In neuroscience, paradoxes often helps understanding the way the brain gather and compute information. A typical case is the Ames room, named after the American ophthalmologist Adelbert Ames Jr. In this particular construction, a room is deformed in such a way that, viewed from a specific perspective, it seems to be a normal room. In that sense, it is a three-dimensional anamorphosis (Baltrušaitis, 7). When looking at people in the room from that specific point of view, it seems that they are changing sizes like balloons. The Ames rooms creates a falsifical paradox: we clearly know that these people are not changing size, but our brain is unable to actively interpret the visual information as such. This experiment shows how our brain treats information and take some shortcuts for granted instead of computing them all the time. In this case, the granted fact is that rooms are rectangular prisms regulated by right angles. (Figure 2)

Elm's room 1

Ames Room

The importance of the intersection of lines in our perception has been proved again to be of prime importance by Biederman in 1987. In his study, he shows how the brain is faster in recognizing drawings from which no intersection segments have been erased compared to drawings where totally random segments have been withdrawn. By this mean, Biederman also explains how work optical illusions, objects that we can define as visual paradoxes.

Istvan Orosz

Figure 2: Istvan Orosz Source: pour la science 2005, nu 330.

[1] We do not imply here that these narratives are a direct result of Russell’s work, we simply state that overall shape of its structure is similar to these narratives. More detailed historical researches could lighten this question of direct implication.

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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).

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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.

Fractal zooms and infinite spaces: the Unbearable Quest for the Sublime

Even if fractals are omnipresent in nature, we have had to wait until the last century to call attention to their existence in the mathematical literature and acknowledge their importance. Fractals have received international attention and have motivated myriad in-depth studies. They have reached wide recognition in popular culture and are now considered some of the most beautiful mathematical, and in a larger sense, visual wonders. In recent decades, a multitude of videos have appeared on the internet, categorized as Fractal Zooms. In this article, we are interested in understanding what relates these videos to the Sublime. An historical review of the subject provides the basics for comprehending their definitions and characteristics. Some very important basic fractals are presented first, such as the Von Koch curve, and their definitions allow us to apprehend a better view on more modern and complex curiosities such as the Julia sets and the Mandelbrot set. We then turn our attention towards Kandinsky’s theory of art and finally to neuropsychology in order to reach a better understanding of the multiple processes involved in looking at fractal zooms, and therefore, better capturing the cathartic experience of fractal zooms.

Historic; from a blurred definition to tangible examples

An absolute and precise definition of fractals has still yet to be found. The term “fractal” was first adopted by the French mathematician Benoît Mandelbrot in his book Les Objets Fractals: Formes, Hasard et Dimensions first published in 1975. Mandelbrot argued for the necessity of keeping the word partly undefined since any precise definition would inevitably exclude the several examples and the counter-examples that would fall outside of such a narrow definition; the definition would have to be constantly reviewed (159-160). Nevertheless, many characteristics seem to have found consensus as being part of what would be a potential definition. Self-similarity, invariance of scale and fractional dimension are such properties which definitions will be elaborated on in the body of this work. Examples will speak for themselves.

                The first fractal to be conscientiously defined was the Apollonius gasket around 200 BC. Apollonius of Perga, well known for his work on conics, in his study of tangent circles proposed a system of infinitely decreasing tangent circles (tangencies). The infinite number of circles of decreasing sizes permits this geometrical construction to be considered as a fractal. It was not until 1525 that Albrecht Dürer developed a similar construction in his Four Books on Measurement. His idea was to fill a pentagon with other smaller pentagons and so on infinitely many times. These examples laid in obscurity for a while and only in the 19th century, fractals started flourishing in the mathematical field, doomed to simply exist as counter examples for major theorems and conjectures. That will be the case for the functions constructed by Bernhard Riemann in 1961 and Karl Wierstrass in 1872 (Lemoir-Gordon, 14).

Derivabilityis defined as the existence of a tangent line at a point, characteristic believed to be implied by the continuity of a function. The Riemann and Weirstrass functions have been of great importance in showing that continuity does not imply derivability even if the converse holds. A function carrying this property would have to abruptly change direction at every point. To define his function, Riemann used an infinite sum of sinusoidal functions, but due to the high complexity of the constructed function, the first proof of absence of derivative was only given in 1916 by G.H. Hardy (Weirstrass, 3-9). For his part, Weirstrass built his infinitely broken function over an infinite sum of cosine functions and showed it was nowhere derivable (Weirstrass, 5-7). As a matter of fact, a similar function seems to have existed already in 1831, offered by Bolzano. The authors Matin Jašek (1922), Voytěch Jarník (1922) and Karel Rychlík refer to it in their articles. Unfortunately, this function fell into oblivion and only Weirstrass’ function made history.

Figure 1: Weirstrass function

Figure 1: Weirstrass function. Source: Wikipedia

Weirstrass argument for his function was mainly analytical, and unsatisfied by this non intuitive method, Helge von Koch took the challenge of constructing a continuous non derivable function geometrically. As seen previously, this statement is equivalent to finding a curve that allows no tangent; that is only made of peaks. In his 1904 and 1906 articles, the von Koch curve is explicitly defined as an iteration of triangle inclusion over a line segment. Recently, a beautiful proof of the absence of derivative has been given by Šime Ungar using convergence of suites (2007, 61-66). This fractal is now a well-known object and many experiments have been proposed to give similar curves: gluing von Koch curves together we can construct the von Koch snowflakes, and changing the triangle iteration for some other regular polygon iteration leads to some complications like self-intersection (Keleti and Paquette, 2010).

These broken curves forced the mathematicians to redefine the concept of continuity and infinity. Both curves, the Weirstrass function and the von Koch curve, have an infinite length for every segment. This polemic fact and few other counter intuitive ones were shocking the mathematical community at the time of their discovery, and some mathematicians simply refused to work on these curves. The famous French mathematician Henri Poincare went as far as referring to them as the Galerie des monstres. The concept of infinitely broken was still evolving and ready to bring new challenges.

Another fractal that made history is the Cantor set. This set had been defined as early as 1875 by Henry Smith from Oxford University, by Paul du Bois Reymond in 1880 and Vito Voltera in 1881. Georg Cantor defined the Cantor set as part of his great work on different types on infinities. He used it to reach a better understanding of continuity and density. The set is constructed as the recursive deletion of the middle third of a segment. The cantor dust, as named by Mandelbrot, is the limit of that set when iteration is repeated at infinity. This set has a topological dimension higher than 0 but less than 1 and possesses no length (Edgard, 2). As for the von Koch curve, is it easy to give an explicit formula for the iterative function defining it. Some generalisations of the set have been proposed as well by removing other fractions than the ones in the third or the middle segment. These experiments even led Roger Kraft to develop a measure to be able to compare different Cantor sets in size (Kraft, 1994).

Figure 2: Asymetric Cantor set

Figure 2: asymetric cantor set by Tsang. Source: Wikipedia

A fundamental characteristic of certain fractals is to possess a fractional dimension oppositely to the dot, the line and the plane that each possesses respectively 0, 1 and 2 dimensions. Starting from Caratheodory’s work, Felix Hausdorff elaborated a definition apt to describe the size of these figures freshly arrived at the mathematical pantheon. Though this notion is slightly too complex to dwell on in the context of this article, it suffices to understand that non integers are allowed as a dimension of geometrical objects. So a fractal can have a dimension of 0,6309 as the symmetric Cantor Set, or 1,2619 as the Koch Curve.

The examples we’ve seen so far represent this principle well. Starting with a line of dimension 1 for the Koch curve, we add an infinite amount of segments. The result is a geometrical object with a dimension between 1 and 2. Starting from the same line, we withdraw an infinite amount of segments and get the Cantor set of a dimension less than 1. We could do the same for a square and withdraw some smaller squares or from a cube and withdraw smaller cubes to get, respectively, the Sierpiński carpet (1915) and the Menger sponge. Although most fractals have non integer dimension, some have the strange property of ending up exactly at 1 or 2. For instance, starting with a line, we can break the line and intricate it in such a way that it fills, completely, a part of the plane. The first ones to offer such a curve were Giuseppe Peano (1890) and David Hilbert. Each of them wanting to bring little variations Moore (1900), Lebesgue and Osgood (1903) did the same. (Delahaye, 2004, p.90-95)

Figure 3: Peano Curve

Figure 3: Peano curve by Antonio Miguel de Campos. Source: Wikipedia

It is worth mentioning the existence of three dimensional fractals. In general, they share similar properties with some lower dimensional fractals. The Menger sponge, based on the Sierpiński carpet provides a good example. Again, fractals can have fractional dimension or fulfill an entire volume. Yet again, in the tradition of challenging conventions some have been found as counter-examples and followed by some new discoveries such as the Alexander horned sphere. Mathematicians’ passion in abstraction brings objects and theorems to n dimensional fractals, for n any real integer.

Figure 4: Sierpinsky gasket and Menger sponge

Figure 4: Sierpinski gasket by Paul Bourke Source: http://paulbourke.net/fractals/carpet/ and Menger sponge by Niabot Source: wikipedia

Fractal and the hauling semantic

We are now interested in the perceptive experiences related to fractals. The first one is indeed the understanding of its infinitely broken nature. This is in general a result of the iterated process hidden behind these objects. We saw the Koch curve was constructed as an addition on smaller and smaller triangles to obtain a completely broken line, but only within a certain range. The Von Koch curve possesses an infinite length but still lies inside a bounded area due to the process involved in its construction. Some other fractals can be obtained as an infinite Brownian broken trajectory, which is a randomly infinitely broken line. The old conception of line, or curve as previously conceived and understood since antiquity is therefore to be revised. Not only is a line not necessarily what it was intuitively before, but even, as we’ve seen, the complex and so well thought conception of continuity as patiently built by mathematicians over many centuries was to be shaken, even if, luckily enough, not to be redefined.

Indeed, fractals enable non mathematician as well to face peculiar experiences. As mentioned by Manlio Brusatin , the sublime is a broken and zigzagging line, a trouble of sensitive soul (Brusatin, 132) which evokes the first semantic impact of fractals. Fractals as lines are terribly broken geometric objects, therefore amplifying Brusatin’s conception to greater extends. This characteristic challenges Brusatin’s notion of sublime not merely as a result of an infinite higher complexity, but as well as its possible extension to an infinite number of dimensions. Nevertheless, the case in three dimensions is of note. From these fractal objects, there can still rise an overall simple structure, and bringing them into three dimensions, we realise our world might not be as simple as we commonly perceive it. This is where fractals find another root in their mysterious behaviour: namely, their relation with the real world.

In an article written by Richardson (Mandelbrot, 1995), we find the spark of such a discovery. In his article, the author intends to approximate the real, exact and absolutely precise length of Britain’s coast. Surprisingly enough, the answer given was simple. It is infinite, and so is any coast or coastal segment in the world. The logic is based on scaling, just as for the Koch curve. Let’s say we approximate the coast by a regular polygon of side n, we get a value L. Changing each segment for two smaller ones of length m < n, the segments mould themselves more accurately on the coast. In consideration of triangle inequality, we see that by additionally adding each new segment, we obtain the total length approximation for the new polygon, which is bigger than L, the first approximation. Iterating the process we get an infinite coast, a line of infinite length just as the Weirstrass function or the Koch curve.

Figure 5

Figure 5: Polygonal approximation by Alexander Polesnikov Source: http://cs.joensuu.fi/~koles/approximation/Ch3_1.html

In his 1975 book, the French mathematician Benoît Mandelbrot explains that this argument can be applied to any object in nature. The world is then everywhere infinitely broken. Constructing fractals is thus reconstructing nature, and so is their deconstruction. From this point of view, the Earth is in fact of infinite area, and fractals can therefore be used to model the land.

The work of Voss has been discussed in the same book by Mandelbrot. Voss generated pictures of artificial pieces of land from fractal based algorithms. The results were stunning. Resemblance with maps was not as shocking as the fact that any zoom on a coastal area would bring infinitely as remarkably convincing landscapes. The similarity between fractal produced images and some parts of our world as we see it are amazing, and a great deal still waits to be discovered.

Figure 6: Voss infinite map

Figure 6: Voss infinite landscape Source: Mandelbrot 1975

Fractals may find attributes and exemplifications in nature, but some even deeper connexions can be retraced. Just as π seems to have found its way in all parts of science and arts, fractals blooms naturally from different human made constructions.

In southern India, the Tamil people keep geometrical drawings made from rice powder. These drawings, or kolams, are bound to spiritual beliefs. Kolams vary in shape, size and pattenrs. One kolam brought forth a peculiar interest for Gabrielle Allouche, Jean-Paul Allouche and Jeffrey Shallit in 2006 (2115-2130). Studying the Kolam of figure 7, by a assignment of values 0 and 1 to the direction taken by the curve when one follows the curve, the researchers found a very well-known suit known under the name of Thue-Morse suit. The kolam can be constructed by taking the Thue-Morse digits as instructions. Strange fact, considering the suite was first invented for a completely unrelated purpose.

The Thue-Morse suit is remarkable for avoiding any repetition of any triplets using only the alphabet 0,1. The construction goes as follow: from any previous `word` (string of digits), we start by taking the whole word, and then concatenating its opposite on the next iteration. For example, starting with the word 0 the Thue-Morse suit gives the following construction: 0, 01, 0110, 01101001, 0110100110010110, 01101001100101101001011001101001 etc. Having no repetition means that taking any section of any size in this infinite word, the following part of same size will differ by at least one digit (Delahaye, 90-95). The fact that such a suit appears in a kolam is surprising, but even more surprising is the fact, discovered by Jun Ma and Judy Holdenerin 2005 that the same suits can be used to construct the Koch curve. Starting with a line segment and putting the next one in an angle determined by the digit 0 or 1 of the Thue-Morse suit, they obtained the very same curve as Helge von Koch exactly a century before (Jun and Holdener, 191-206). The Thue-Morse suit is then a genetic code for both a kolam and the Koch curve: only the interpretation differs.

Indeed, this example is not the sole appearance of fractals in other geometric form: for instance the Sierpiński triangle has been found in the Pascal triangle (Fuchs and Tabachnikov) as well as a sea shell. For its part, the Cantor set has been found in Julia sets (Audin), which we are to study next.

Figure 7: Kolam, Koch curve and the Thue-Morse Sequence

Figure 7: Tamil Kolam from Allouche’s pape and Koch curve from Jun and Holdenberg’s paper and the Thue-Morse sequence

Towards the sublime

We have now understood, partially, the complex uprising of fractals in the mathematical literature. Our goal now shifts to understanding another object that exists as part of more popular culture; fractal zooms. These videos are available online and can be enjoyed by many. To understand the great complexity behind these videos we have to go back again, far in the past.

Complex numbers stood aside from mathematical studies for a long time. For instance, these numbers appear if we try to find the solution for the equation x²+1 = 0, this leads us to obtain √(-1). For a long time, this seemed impossible for most mathematicians. The number √(-1) has been called i for imaginary. Imaginary numbers are built with two components. The real component, being a real number, and the imaginary part which is a multiple of i. Complex numbers can therefore be written as a + bi where a and b are real numbers and i = √(-1).

In 1797, the idea of representing these numbers in a plane came to Norwegian cartographer Casper Wessel. The x axis and y axis would respectively be the real and imaginary components of the complex number (Lemoir-Gordon). This enable us to represent the complex numbers as couplets (a,b). This plane, the complex plane, was hiding many surprises.

At the end of the 19th century, several studies bloomed concerning the transformations of the complex plane. To understand these, let us see few basic examples. To add 1 to every number of the complex plane is the same as translating the whole plane one unit to the right because the x axis is the real value component of complex numbers. Adding i everywhere would be translating the plane one unit up.

Several other transformations can be constructed with multiplication. Multiplying complex numbers, as it have been defined by mathematicians, amount to the same as considering only the length and angle of a vector. Complex numbers being coordinates (a,b), we can picture them as vectors from the origin to the point (a,b) of the complex plane. This line, called a vector, possesses a length and an angle with respect to the x axis. Multiplying two complex numbers proved to be the same as multiplying their lengths and adding their angles in order to get the resulting complex number.

Figure 8: complex number multiplication

Figure 8: Complex numbers multiplication

We are now able to understand the following function: f(z)= z². With z as a complex number, we are simply taking every complex number of the plane, and multiplying it by itself. From the previous definition of multiplication, every number is mapped to a number that possesses the square of its length and the double of its original angle. We can now iterate the function. This means we take the whole new plane obtained and put the values again in our function, therefore taking every new complex number and multiplying it by itself. With enough time and motivation we can iterate that function infinitely many times.

At the beginning of the 20th century, some mathematicians such as Henri Poincarré started to analyze the behavior of iterated fractional functions, of which f(z)= z² represents a basic example, on a certain area of the plane (Audin, 2011). In 1915, during the Great War, the French Academy of Science launched a contest which offered a 3,000-franc award to the best paper on the behavior of iteration of rational functions over the whole complex plane in an attempt to follow Poincaré’s work. In 1918, a disfigured soldier named Gaston Julia won the concourse with a paper he partially wrote in his hospital bed in Paris (Audin, 2011).

The main protagonists for the competition were Gaston Julia and Pierre Fatou, names that are now honored for the sets they worked on, the Julia sets and the Fatou sets. We’ll focus on the Julia sets since Fatou sets have a similar complementary definition. To understand them, let’s use our example f(z)= z². If we take a dot inside the unit disc in the complex plane, that is, the disc of radius 1 centered at the origin, by iterating the multiplication of its length, the value becomes closer and closer to 0, spiraling around it since the original length was smaller than 1. We say that its length converges to 0. If we take a value on the boundary, the length always keeps the value 1 and the dot spins around the unit circle. If we take a dot outside the unit disc, its length grow bigger and bigger to move towards infinity as we keep on iterating. This point is said to diverge. Given a rational complex function, the Julia set is the set of points not diverging after iterating infinitely many times. For our example, the Julia set is simply the unit disc.

This very simple case is the iteration of the function f(z) = z²+c where c = 0. More difficult situations arise when the c value is taken to be other complex values. Works published by Fatou and Julia concerned general facts about the sets and boundary of points diverging after the iterations. The shapes of these sets varied greatly depending on the c values. Some sets seemed connected, some other ones constituted of many islands, and yet other ones seemed to be a fine powder on the plane. It was difficult to analyze their overall behavior, but some classifications, such as the one made at the time by Salvatore Pincherle, could still be made around the idea of connectivity: if the set would be formed of only one piece, it would be connected, otherwise it would be disconnected. The most disconnected ones, made as a fine powder, were revealed to be topologically equivalent to the Cantor set. The other sets were extremely difficult to display and publications of the time included almost no pictures (Audin, 2011). Pen and paper were not sufficient to represent the deepness and rich complexity of the Julia and Fatou sets.

Figure 9: Julia Set by Gaston Julia

Figure 9: Julia set by Julia Source: Audin (2011) ©Archives de l’Academie des sciences

In the 1970s, the arrival of computers would drastically change the way mathematics would be seen and applied. The pre computer era would define mathematics as a science of absolute precision but suddenly this new tool and the astronomical calculations it enabled would make this science take an experimental path as well. Teaching in Paris at the time, John Hubbard and Adrien Douady undertook, in concert with Sullivan from the Institu des Hautes Études Scientifiques to produce pictures of non-diverging sets of points for iterated functions of degree 2. These were Julia sets for functions such as f(z) = z²+c . (Lei)These experiments in turn led to images they could barely have foreseen, images of which only a minuscule glance could have been reached by Julia and Fatou. Another mathematician, Benoît Mandelbrot, would take over this work and popularise what has been defined by Arthur C. Clark as the most complex shape ever created by men (Stewart and Clarke, 2004).

Figure 10: Mandelbrot Set

Figure 10: Mandelbrot set as a map Source: Audin (2011)

During his visit to France, Hubbard and Sullivan showed Mandelbrot the images obtained from their computer. At the time, Mandelbrot had already studied the sets: his grandfather had incited him to seek them out in Fatou’s and Julia”s papers years ago when Mandelbrot was searching for a Ph.D. topic. Previously, however, he had never tried to picture these sets. The next year, his entrance to the IBM laboratory allowed him access to powerful computers from which he extracted, for the first time, a wide range of printed Julia sets. Trying to organize these, he created a set constructed around similar characteristics as Pincherle concerning the connectivity of the Julia sets. Even if the Mendelbrot set found its first historical appearance in paper related to special projective linear groups signed by Robert Brooks and J. Peter Matelski in 1980, the merit of an independent discovery, and subsequent wide popularisation of it, belongs to Mandelbrot.

The Mendelbrot set is defined as the set of values for which the associated Julia set is connected. That is, if we fix a complex value c, and the related iteration of f(z) = z²+c gives a set of non-diverging points that happens to be connected, then the point c of the complex plane belongs to the Mandelbrot set. The Mandelbrot set is then a map of the connected Julia sets. An important theorem states that only the behavior of the origin, the point (0,0), is important to know if the Julia set is connected. The Julia set is disconnected if and only if the origin diverges in the iteration process. This result would faster the production of pictures of the Mandelbrot set. The colors found on the pictures of the Mandelbrot Set one would find on the internet indicates the speed at which the origin diverges, where each of the black dots stands for the connected Julia sets.

Figure 11: Mandelbrot set by Matelsky and Brook

Figure 11: Mandelbrot set by Matelski and Brook (1980)

Since its discovery, many studies have helped to understand the Mandelbrot set, and many important facts were revealed. We now know it is connected and quasi similar, which means it contains almost identical but increasingly smaller copies of it spread densely on its boundary. Furthermore, Shishikura demonstrated that the boundary is so twirled in on itself that it is of Hausdorff dimension 2, just like the Peano curve. That explains why we can zoom everlastingly on its boundary and still get beautiful complex shapes. These zooms that we can now easily find on the internet are what interest us as we try to understand why these are so shocking, almost cathartic. As already underlined by Rothstein, the Sublime can already be found in Cantor’s definition of various size of infinity (Rothstein, p.188), the Mandelbrot set and the collection of all fractals provide a visual equivalent to this.

The incommensurable 

Figure 12: Mandelbrot set (details)

Figure 12: The LotusFort of Seahorses by Ingvar Kullberg   © Ingvar Kulberg. Source: http://klippan.seths.se/ik/frholmes/english.html

The concept of infinity is difficult to handle and it indubitably leads to controversy. Therefore, even if fractal zooms are in theory infinite, we’ll focus for this section on the finite aspect of them. It is also practical since only finite zooms can be found on the internet. Part of the traumatic experience of a fractal zoom comes from its unbearable sense of immensity. The real size of the represented picture even after a finite time zoom is simply incomprehensible to us. To help us we will turn to National Aeronautics and Space Administration. To compare, the farthest object in the universe perceived by man is seated at 13,2 billion light years ,that is 1,2488256 x 10²⁴ meters from the earth. Starting from a 10 centimeters long Mandelbrot set image on a computer screen, the size of the final set at the end of the greatest zoom we could find on Youtube, which is of 2 exponent 3039 times its size, surpasses by far the distance mentioned above (calculated by NASA in 2010). To give an approximate impression, let’s remember the impact of squaring a number. Squaring 10 gives 100, squaring 100 gives 10 000, and squaring the last one gives 100 billion. Thus, the greater the number is, the more important is the impact of squaring the number. The movie Powers of Ten (1977) helps visualizing these numbers.

To get the size of the line crossing the final Mandelbrot set for side to side we started with at 10 centimeters, we have to get the value obtained by NASA in 2010 and square it between 4 and 5 times. The size of the final object is simply unbearable and this is why the zoom provides such an intense vertigo in which we are totally lost. And yet the size is only one aspect of the traumatic experience: the shapes of it, its colors, are what complete the intolerability of this entrancing experience.

Kandinsky and the hypercoloumns

In 1926, Wassily Kandinsky published of treaty on lines and dots in the plane. He offered certain definitions and classifications of these objects as well as the emotional impact of the different types, sizes, and dispositions of lines and dots mixed with colors. Describing this work, Brusatin said it is though using the rhythm of expressive geometry that these objects provoke perceived sonorities and synesthesia (156). For instance, Kandinsy explains that red is associated to diagonal lines and yellow is for free straight lines (1970, 77). These observations are clearly made from a synesthetic viewer.

Many have suggested that synesthesia is at least partly influenced by social schemes. That is, the connections made by the brain’s synapses are either reinforced or curbed by interactions with other individuals. As a result, most people lose these synesthetic synaptic connections, but vestiges can still be found in most people’s perception of the world. As an easy and evocative example let’s take the two pictures here.

Figure 13

Figure 13: Tic-tac and Bubbla Source: Wikipedia, synesthesia article

One of these figures is called Bubbla and the other one Tic-Tac. Our propensity to relate acute angles with sounds like t or hard c influence us to name the one on the left Tic-Tac, while the round shaped one seems more eligible for Bubbla. The connection between a visual object and sounds is not a logical or natural one but a social construction. The same applies to colors with the appellation of warm and cold. A single picture, or painting as those by Kandinsky, can therefore evoke a wide variety of emotions since they are constructed with many colors and lines. Being abstract art, and thus non-figurative, it avoids a clear semantic result. To understand the impact of a fractal zoom we now need to look closely at the visual system.

In 1982, after many experiments on the cells of the visual cortex, Hubel and Wiesel proposed a model for the primary visual cortex constitution. In their model, big structures called hyper-columns are associated as the receptacle for all the incoming information of a very precise visual field’s area. Inside these columns, the cells are tuned for color and orientation, meaning they’ll only react to specific colors and line orientation within a small angle range. A simple picture on the retina will only stimulate certain cells in these hyper-columns and the information will be gathered in the higher visual cortex until the semantic process of the information, or the result, is object recognition and its corresponding and at times subtle emotion, such as in the case of a Kandinsky painting.

Figure 14

Figure 14: Hypercolumns Source: Wolfe, 2009.

In the case of zooms made on the boundary of the Mandelbrot set, the zoom being applied quickly in almost all cases, the amount of different images presented to the retina is highly rich in mutating shapes and colors. Therefore, every section of the retina is constantly being bombarded with new shapes and colors; it continuously stimulates many different neurons in the hyper columns. The visual cortex thus fires very much information towards the semantic processors of the brain, such as the fusiform cortex, that is then challenged, in vain, with making sense of this saturated information. The information also goes to the limbic system though the ventral stream, which is involved in the treatment of emotion. Considering synesthesia, all lines and colors tend to create their own specific emotions. The gathering of all such, at times contradictory information is probably the reason why focusing at a fractal zoom is a charged experience.

Semantic and knowledge

The lack of an easy emotional reading for the fractal zoom could incline one to search for a deep semantic sense of the picture or the object. As pointed out by Mandelbrot, it seems delicate just enough that the fractals can be interpreted as a pattern for shapes in nature. Such a point of view easily evokes spiritual motivations. How can nature be self-constructed so well? Otherwise, who created such intriguing geometrical objects? This almost theological and cathartic perspective quickly hit a wall where the meaning, and even more, the understanding of such objects as fractals is relinquished to a higher spiritual world and the viewer stays in the state of the sublime, overwhelmed logically and emotionally. The other venue left, then, is to try to understand the mathematical construction that led to this set and, moreover, the different theorems surrounding the beast.

This is where new problems arise. In the understanding of the definition of sets as Julia sets, Fatou sets and Mandelbrot set happen to be fairly accessible even without a mathematical background. Nevertheless, even really specialize knowledge of complex numbers’ arithmetic, great mathematical abilities and hard work doesn’t provide sufficient tools for one to construct these sets and display them. As we have seen, almost no pictures were found in the papers by Fatou and Julia. The few drawing provided failed to be accurate and detailed and, decades later, computers had to be used for this tedious task. That is only to obtain a picture, understanding the theorems about the Julia sets, the Fatou sets and, indeed, the Mandelbrot set, is quite another challenge.

To go through a whole proof about as simple facts as the connectivity of the Mandelbrot set, the reader has to master complex numbers analysis for the use of derivative and integrals over complex valued functions and a great deal of results in the aforementioned field, such as Schwarz’s lemma, Poisson’s Integral formula and results on harmonic functions. The use of meromorphic functions may involve familiarity with non-Euclidian geometry and different metrics, like the chordal distance, to reach theorems by Marty and Picard. To understand the arguments demonstrating the thickness of Mandelbrot set’s boundary, one has to accustomed himself with non-integer dimensions and a great deal of topological results. A quick review of a book like The Mandelbrot Set: Theme and Variations reveals that the same goes for most results in the field. This immense and tedious mathematical background certainly creates a strong deterrent to most people to deeply understand facts that permit the fractal zoom to exist, even though these zooms are easily accessible via internet. Yet, this gigantic gap between the viewer and the understanding of the Mandelbrot set enshroud the set with a mystical aura leading the spectator to a cathartic sensation. This forced distance to the object tops the mixed feelings and vertigo already underlined, leaving them with a blurred idea about the greatness of the object presented but surely with overwhelming emotions.

In this case of extreme complexity, there is no surprise in finding fractals related to some deities. Such is the case for the Buddhabrot and the Brahmabrot. These fractals result from various ways to represent the Mandelbrot set in the complex plane. They appear as a type of new gods in a pantheon of a science driven era. It was already the case with the Mandelbrot set which was compared to the fingerprint of god (Stewart and Clarke, 2004), but names of these new fractals underline more clearly the link they share with our conception of God and the space embedding it (1).

Figure 15: Buddhabrot

Figure 15: Buddhabrot. Source: Wikipedia

Extension in 3D

Naturally, mathematicians wanted to expend the fractals to the third dimension. As previously seen, some simple fractals like the Cantor dust or the Sierpinski carpet found logical three dimensional equivalent. It is indeed the fact as well with the the idea of mapping landscapes. Many such constructions provided realistic landscapes as early as 1974 by Handelman (Mandelbrot 1993, 13). Creating realistic landscape representing the great power of nature and its complexity is already a first step in trying to grasp the Sublime with the third dimension. Yet again, it seems that objects that are closer to be discovered, such as the Mandelbrot set, than to be used to copy naturalistic landscapes lead to more sophisticated surprises.

The possibilities offered by more and more powerful computers has reached a point where they enable, as with the two-dimensional equivalent, to present and materialise the sublime by using the same concepts and possibilities and in the previous examples analysed. We now present two such cases where such tendencies collides.

Figure 16: Fabergé fractal by Tom Beddard

Figure 16: Tom Beddard’s Fabergé fractals. Source: http://thecreatorsproject.vice.com/blog/faberg-fractals-are-intangible-geometric-wonders

The first example comes from Scotland based artist Tom Beddard. Beddard, already familiar with fractal generating programs and three dimensional modelisation from his background in physic from university of St-Andrews, created the Fabergé fractals, in tribute to the famous Russian jeweler. If these fractals are not expending in space, they still offer a peculiar notion of infinitely detailed shapes. Some beautiful videos exposes such shapes in constant transformation.

The second case includes a series of different examples and comes from a generalisation of algebra for complex numbers. Because we use two dimension to represent a complex numbers, the representation of n-dimensional complex numbers would imply 2n dimensions. To represent the equivalent of the Cartesian product of two complex numbers we would then need 4 dimensions. Mathematicians have tried to solve this by developing different definition for the product of complex numbers and represent higher dimension fractals arising from complex numbers. Such examples includes Rochon’s Tetrabrot , Tom Lowe’s Mandelbox and Paul Nylander’s Mandelbulb.

Although it is possible to imagine objects similar to the Mandelbrot set in three dimension, there is a problem with their formal construction. The algebra of complex numbers is well defined in 2 dimensions, but it turn out that an equivalent cannot exist in three dimensions. In order for an element to have an inverse element with respect to the operation of division, the space would need to have a dimension that is a power of 2 (2). It is the case for instance for the quaternions developed by Hamilton in order to find complex and for the octonions that hide some symmetries for 4 dimensional objects. The three dimensional attempts to recreate the Mandelbrot might not lead to any proper construction, nevertheless they still provide an extension of the sublime invoked in the two dimensional version. The various zooms offered by digital arstists such as Krzysztof Marczak, Arthur Stammet and many others proved to include all the elements of the planar fractals that leads to overwhelming feeling provoked by these objects. The specificity of this feeling involved has even been use for narrative purpose by Daniel White, the mathematician that constructed the equations behind the Mandelbulb and used in higher polynomial degree by Nylander. On his deviant art page, we can find a small story using the Mandelbulb as a frightening asteroid where a lost souls is landed (3).

Figure 17: Madelbulb (details)

Figure 17: Mandelbulb detail by Krzysztof Marczak. ©2010-2014 Krzysztof Marczak

Source: http://krzysztofmarczak.deviantart.com/art/Mandelbulb-power-4-163319841

Indeed, these are only the fractals we are able to represent. The journey into the quest of sublime goes further with the exploration of fractals in n-dimensional spaces. Such exploration can be made with books like Kenneth Falconer’s Fractal Geometry: Mathematical Foundations and Applications.

The quest for the Sublime, which started in our case with the simple exploration of simple two dimensional geometric object, leads to an unbounded perception of space, both as infinitely small and broken and as incommensurable and embeddable in any number of dimensions. As well precised by Rothstein again, «it makes the imagination seem inadequate while giving our understanding an almost ecstatic sense of having apprehended what should be beyond its containing powers» (Rothstein, p. 187).

Conclusion

The understanding of the experience related to fractal zoom as we now can easily find on internet, needs to be seen as the result of a long path from which much information and various points of view are gathered. First, through multiple examples such as the Koch curve or the Peano curve, we have seen that the emergence of the concept of fractal in the mathematical literature was by itself shocking for the community. Many concepts like continuity, dimensionality and infinity needed to be revisited, and new definitions had to be proposed. We also have underlined that some fractal images were far too complex to be pictured by humans without computer assistance; which had been indispensable to produce accurate images of the Julia, Fatou and Mandelbrot sets. Aware of the difficult trajectories to reach fractal images, and therefore fractal zooms, we were then ready to focus on the different aspects that make the screening of such zooms a traumatic experience.

We first underlined the mystical aspects of fractals by looking at some very surprising properties that places these fractals between one another and some other human created geometric constructions. We then looked at fractals as preponderantly curious shapes, and more so, as being the canvas for shapes which occur in nature, revealing more of their mystical aspects. The overwhelming size of shapes created by fractal zooms was then used to show why these zooms can be hard to handle since it forces the viewer to situate himself in a space impossible to imagine or seize. After explaining the construction of the Mandelbrot set, we were ready to show via synesthetic and neuropsychological arguments why the reception of the images contained in the fractal zooms are related to the Sublime, creating series of chaotic emotions. Finally, referring back to the mathematical background on which these fractals, especially the Julia sets and the Mandelbrot set, are constructed, we could see how semantics, or a more decent comprehension of fractals and fractal zooms is unreachable for the common spectator, deepening the gigantic gap between the spectator and the geometrical objects.

All of these aspects redefine the fractal zooms as objects of the Sublime: the screening is emotionally twofold, the spatial construction of the object is incomprehensible and the logical aspects are very difficult to reach. Our incomprehension is difficult to handle since it seems to have some implications in the creation of nature itself, and that very incomprehension found certain mind-blowing applications like fractal image compression. Some more developments bloomed in the last few years concerning the construction of three dimensional Mandelbrot set using a new way to compute complex numbers in four dimensions. This shape, the Mandelbulb, is a new creature as fascinating as its two dimensional acolyte and already, 3D fractal zooms on the web are available. These zooms still seem incomplete since the infinitely broken aspect doesn’t appear everywhere, but yet some fantastic images and zooms are to be found on the web.

1- Lori Gardi, who coined the term Buddhabrot, was actually looking for a proof of God in the Mandelbrot set (http://www.butterflyeffect.ca/Close/Pages/Buddhabrot.html)

2-For a more complete description the reader can explore the following site http://blog.hvidtfeldts.net/index.php/2011/06/distance-estimated-3d-fractals-part-i/

3-http://dspwhite.deviantart.com/art/The-Eternal-Dream-149572608

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