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. 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 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 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: 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: 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 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: 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 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 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 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 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.
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: 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: 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. 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: 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: Mandelbulb detail by Krzysztof Marczak. ©2010-2014 Krzysztof Marczak
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).
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/
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