On Fractals III

"Fractals are geometric shapes that are equally complex in their details as in their overall form. That is, if a piece of a fractal is suitably magnified to become of the same size as the whole, it should look like the whole, either exactly, or perhaps after a slight limited deformation." (Benoît B Mandelbrot, "Fractals and an Art for the Sake of Science", 1989)

"Fractal geometry appears to have created a new category of art, next to art for art’s sake and art for the sake of commerce: art for the sake of science (and of mathematics). [...] The source of fractal art resides in the recognition that very simple mathematical formulas that seem completely barren may in fact be pregnant, so to speak, with an enormous amount of graphic structure. The artist’s taste can only affect the selection of formulas to be rendered, the cropping and the rendering. Thus, fractal art seems to fall outside the usual categories of ‘invention’, ‘discovery’ and ‘creativity’." (Benoît B Mandelbrot, "Fractals and an Art for the Sake of Science", 1989)

"What were the needs that led me to single out a few of these monsters, calling them fractals, to add some of their close or distant kin, and then to build a geometric language around them? The original need happens to have been purely utilitarian. That links exist between usefulness and beauty is, of course, well known. What we call the beauty of a flower attracts the insects that will gather and spread its pollen. Thus the beauty of a flower is useful - even indispensable - to the survival of its species. Similarly, it was the attractiveness of the fractal images that first brought them to the attention of many colleagues and then of a wide world." (Benoît B Mandelbrot, "Fractals and an Art for the Sake of Science", Leonardo [Supplemental Issue], 1989)

"Some fractals come close to qualifying as chaos by being produced by uncomplicated rules while appearing highly intricate and not just unfamiliar in structure. There is, however, one very close liaison between fractality and chaos; strange attractors are fractals." (Edward N Lorenz, "The Essence of Chaos", 1993)

"Fractals are patterns which occur on many levels. This concept can be applied to any musical parameter. I make melodic fractals, where the pitches of a theme I dream up are used to determine a melodic shape on several levels, in space and time. I make rhythmic fractals, where a set of durations associated with a motive get stretched and compressed and maybe layered on top of each other. I make loudness fractals, where the characteristic loudness of a sound, its envelope shape, is found on several time scales. I even make fractals with the form of a piece, its instrumentation, density, range, and so on. Here I’ve separated the parameters of music, but in a real piece, all of these things are combined, so you might call it a fractal of fractals." (Györgi Ligeti, [interview] 1999)

"Mathematical fractals are generated by repeating the same simple steps at ever decreasing scales. In this way an apparently complex shape, containing endless detail, can be generated by the repeated application of a simple algorithm. In turn these fractals mimic some of the complex forms found in nature. After all, many organisms and colonies also grow though the repetition of elementary processes such as, for example, branching and division." (F David Peat, "From Certainty to Uncertainty", 2002)

"Wherever we look in our world the complex systems of nature and time seem to preserve the look of details at finer and finer scales. Fractals show a holistic hidden order behind things, a harmony in which everything affects everything else, and, above all, an endless variety of interwoven patterns. Fractal geometry allows bounded curves of infinite length, as well as closed surfaces with infinite area. It even allows curves with positive volume and arbitrarily large groups of shapes with exactly the same boundary." (Philip Tetlow, "The Web’s Awake: An Introduction to the Field of Web Science and the Concept of Web Life", 2007)

"Fractals' simultaneous chaos and order, self-similarity, fractal dimension and tendency to scalability distinguish them from any other mathematically drawable figures previously conceived." (Mehrdad Garousi, "The Postmodern Beauty of Fractals", Leonardo Vol. 45 (1), 2012)

"One of the most important artistic properties of fractals is the randomness  governing the process of making them. Each fractal is essentially generated by a basic formula and one or more gradients that identify the colors of the fractal. Sometimes, however, fractals are generated by tens of different formulas and gradients." (Mehrdad Garousi, "The Postmodern Beauty of Fractals", Leonardo Vol. 45 (1), 2012)

"The concept of infinity embedded in fractals' identity provides  an infinity of possibilities to explore in  a single image. The repetition of a formula is the key to becoming more familiar with it. When trying a completely new formula, all fractal artists are engaged in the same activity - a random playing  around." (Mehrdad Garousi, "The Postmodern Beauty of Fractals", Leonardo Vol. 45 (1), 2012)