The epitome of art as resulting from automated process can be find in fractals. We will discuss two examples to underline two major components of the automated process; the structure, or skeleton, implied by the automata, and the theoretically possible infiniteness of its application in time and space. Fractals are geometrical figures defined by Mandelbrot in Les objets fractals (1975) in an attempt to describe the geometry of nature. These objects are often use by iterated processes and are self-similar for certain scale factors. Similar to the comments by Mandelbrot in his article Fractals and an Art for the Sake of Science, we can distinguished between two types of fractals, the organic and inorganic ones. The organic fractals identifies by and obvious similarity with nature whereas inorganic share the structural quality of independence of scale but keep evident traces of man’s hands.
Inorganic fractals usually results from a defined automated iterated process. For instance, the Koch curve is obtained by infinitely adding triangles on the middle thirds of each segments of its constitution. Surprises arise when one realises the same figure can be obtained from different automated processes. The Thue-Morse sequence is a sequence of zeros and ones built iteratively in such a way to avoid any triplet repetitions. It is constructed by the infinite concatenation of the complement of a binery sequence. The sequence is constructed as follow: 01, 0110, 01101001, 0110100110010110 etc.
It has been shown it was intrinsically related to the Koch curve: by assigning directions values to the digits of the sequence, it is possible to obtain the iterations of the Koch curve (Ma and Holdener, 2005). In a similar fashion, by assigning another set of instruction to the digits it has been demonstrated that the same Thue-Morse sequence can serve to obtain a tamil kolam (types of ritual figure drawn with sand or rice powder) (Allouche, Allouche and Shallit, 2006). The three entities, the Thue-Morse sequence, the Koch curve and this particular kolam are simply different interpretation of a common genetic code hidden in their automated iterative processes. (Figure 8) An automated process could, therefore, generate three or more different objects that we, from a visual point of view, consider distinct.
Figure 8: A Kolam and its equivalence as Koch curve and Thue-Morse sequence
Finally, we present a set of fractals named Julia sets and the Mandelbrot set. Again, it all roots back to the idea of representing complex numbers in a plane. Complex numbers are have two components, and real part on which we add an imaginary part, or equivalently a multiple of the square root of -1, denoted i. We usually write them a+bi. To reprensent them on the plane, we give the real component value to the x-axis and the imaginary part to the y-axis. Therefore the point (3,4) represent the complex number 3+4i. This representation helped understanding the way complex numbers multiply themselves and led to studies of conformal mappings as previously seen.
With this coordinate equivalence for a complex number, we can represent complex numbers on the plane as vectors with a length and a direction. The new vectors obtained from the multiplication has an angle equals to the sum of the previous vectors and a length equals to the product of their length. As a result, a number bigger than one will spiral out to infinity if multiplied by itself an infinite number of times. At the end of the First World War, the Academy of Science of Paris promised a prize for the better paper on complex numbers’ dynamic. From his hospital room where he cured his injuries, Gaston Julia wrote many important papers on the topic. He defined his set by the set of complex numbers not diverging to infinity when iterated in rational functions. For C a nonzero complex constant, the Julia set of quadratic forms f(z) = z² + C forms a fractal. Indeed, at the time, Julia did not have the tools to visualize the complexity of these sets. When the computer entered universities in the 60’s and 70’s, researchers started to code programs that would automatically generates Julia sets. The results started to evoke, even if only slightly, how rich wew the images Julia was trying to draw decades ago.
Figure 8: Julia Set
Nowadays, colors are added to these figures to produce marvelous pictures. On top of having to compute a great number of points in the complex plane in order to obtain a single picture, by a step by step focusing figures on the border of these sets we can obtain fractal zooms. In theory, these zooms could produce infinitely many different forms and could last forever, such is the complexity of Julia sets. (Figure 9) A French mathematician, decided two classify the Julia sets into connected and disconnected ones. His classification led him to another infinitely complex set now dubbed the Mandelbrot set on which infinite zooms are also possible. The exploration of fractals led mathematicians into trying to define three dimensional versions of Julia sets and the Mandelbrot set. Difficulty arises and multiplication for complex numbers are to be represented. Since the complex numbers are defined on two dimensions, the real and imaginary one, the complex multiplication can be visualised in the plane, the representation of two complex numbers would need four dimensions. Fortunately, Paul Nylander have found a way to represent such mapping in three dimensions and three dimensional fractals based on this operation have arised, as for example the Mandelbox and the Mandelbulb. (Figure 10) As for the planar versions of these fractals, three dimensional fractals are never fully seen since they are infinitely intricate. Although, with the arrival of 3D printers, there’s been many attempt the represent some fractals such as the Sierpinski triangle. As well, it worths mentioning Tom Beddard’s work with fractal sculptures with lasers.
Figure 9: Inside of the Mandelbox, image by Krzysztof Marczak
Fractals apply naturally to arts, but they can also find specific technical application. For instance, there are many attempt to construct virtual landscapes based on fractal oriented programs. This tradition find its roots in the work of Voss who was developing programs to generate infinite maps, which has been done as well by Mandelbrot.
Again, the question of author seems problematic. The multi-level architecture behind these zooms starts with the definition of complex numbers, the idea of the complex plane, the studies of Julia, the long story behind computers and their programs, and then gigantic calculations made by the computers to generate a fractal zoom. The choice of the zoom’s point and the applications of specific colors is what is left to the last person involved in line, the artist. In that case, what is the fractal, where does it stands between discovery, invention and piece of art. Mandelbrot put it in these words: ‘’ Thus fractal art seems to fall outside the usual categories of ‘invention’, ‘discovery’ and ‘creativity’.’’ (Mandelbrot 1993, 14)
If again, no simple solution can be drawn from these various examples, they can still be linked to the idea of author. The idea of authorship is not only to refer to the existence of a creative process, but as well to contain a certain mark, a certain signature proper to the author. Of course, in the previous examples, elements of signature could be grasp at different level, in the choice of topic for a conformal mapping photograph, in the program’s style of coding lines, in the idea behind the proofs a the different theorems leading to these constructions. Since traces of authorship could be found at all these level, it shows that the notion transcend the simple binary separation between what is art and what is patentable. I urges as well that research centers such like universities to allow more permeability between areas of sciences and arts and include more classes on cross-disciplinary classes where students and researchers from both groups can meet and work together not only to solve problems, but to propose new ones as well.
Félix Lambert (First ideas presented at Harvard in 2013, first draft for this paper finished may 2015)
 For the detailed history, the reader is referred to Michèle Audin’s work: Fatou, Julia, Montel: The Great Prize of Mathematical Science of 1918.Springer, 2011.
Allouche, Gabrielle, Jean-Paul Allouche et Jeffrey Shallit. 2006. « Kolam indiens, dessins sur le sable aux îles Vanatu, courbe de Sierpinski et morphismes de monoïde ». En Ligne : Annales de L’Institut Fourier, Tome 56, n°7, p. 2115-2130. Consulté le 07/02/12. http://aif.cedram.org/item?id=AIF_2006_56_7_2115_0
Arnold, Douglas N. and Johnathan Rogness. 2008. « Möbius Transforms Revealed ». Notices of the American Mathematical Society, Vol. 55, Nu. 10, p. 1226-1231.
Audin, Michèle. 2011. Fatou, Julia, Montel : The Great Prize of Mathematical Sciences of 1918, and Beyond. New York: Springer, Lecture Notes in Mathematics 2014, History of Mathematics Subseries.
Bouton, Charles. 1902. « Nim, a Game with a Complete Mathematical Theory ». Annals of Mathematics, Second Series, Vol. 3, no. 1. P. 35-39.
Calaprice, Alice, Ed. 2000. The Expandable Quotable Einstein. Princeton: Princeton University Press.
Frampton, Hollis. 1970. Zorns Lemma. In A Hollis Frampton Odyssey, DVD 1, 59 min. Criterion Collection 2012.
Ma, Jun and Judy Holdener. 2005. « When Thue-Morse Meets Koch ». Fractals: Complex Geometry, Patterns, and Scaling in Nature and Society, vol. 13. n°3, p. 191-206.
Mandelbrot, Benoît. Les Objets Fractals : Formes Hasard et Dimension, 4th Ed. Paris: Flammarion, 1995.
Mandelbrot, Benoït. 1993. ‘’Fractals and an Art for the Sake of Sciences’’. In Michel Emmer Ed. The Visual Mind: Art and Mathematics. Cambridge: MIT Press, p.11-14.
Munkres, James R. Topology. 2nd Ed. New Jersey: Prentice Hall, Inc., 2000.
Schattschneider, Doris. 1992 (1990). Visions de la Symétrie: Les Cahiers, les Dessins Périodiques et les Oeuvres Corrélatives de M.C. Escher. Traduit de l’américain par Marie Bouazzi. Paris : Éditions du Seuil.
Smit, B. de and H.W. Lenstra Jr. 2003. « Artful Mathematics: The Heritage of M.C. Escher». Notices of the American Mathematical Society, Volume 50, nu 4, p. 446-451.
Stillwell, John. Geometry of Surfaces. New York: Springer-Verlag, 1992.