Series Lecture notes in mathematics Springer-Verlag Online Available online. Science Library Li and Ma. More options. Find it at other libraries via WorldCat Limited preview. Contents I The Great Prize, the framework. The book is based partly on new, unpublished sources. New biographical information is given on the little known mathematician that was Pierre Fatou.
How did the injury of Julia during WW1 influence mathematical life in France? To u de sta d the a gu e ts de o st ati g the thi k ess of Ma del ot 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.
Fatou, Julia, Montel
It was already the case with the Mandelbrot set which was compared to the fingerprint of god Stewart and Clarke, , but names of these new fractals underline more clearly the link they share with our conception of God and the space embedding it1. 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 by Handelman Mandelbrot , 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.
The first example comes from Scotland based artist Tom Beddard. 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 2. 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. 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 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.
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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.
On his deviant art page, we can find a small story using the Mandelbulb as a frightening asteroid where a lost souls is landed4. Figure Mandelbulb detail by Krzysztof Marczak. The journey into the quest of sublime goes further with the exploration of fractals in n-dimensional spaces. 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.
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.
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 oke aspe t does t appea e e he e, ut et so e fa tasti i ages a d zoo s a e to e fou d o the web. May Barrallo, Javier. Bois-Reymond, Paul du. Peter Matelski. Burns, Aidan. Histoire de la ligne. Paris : Flammarion.
Cristea, Ligia L.
And Bertan Steinsky. En Lig e. Delahaye, Jean-Paul. Edgard, Gerald. Measure,Topology and Fractal Geometry. Undergraduate texts in Mathematics. New York: Springer-Verlag. Edgard, Gerald A. Classics on Fractals. Eglash, Ron. African Fractals:Modern computing and Indigenous Design. New Jersey: Rutgers University Press, Fuchs, Dmitry and Serge Tabachnikov.
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Gamelin, Theodore W.. Complex Analysis. Guillen, Michael. Traduit de l a glais pa Gilles Minot. DOI Kraft, Roger L. Lei, Tan.