"If financial markets aren't efficient, then what are they? According to the 'fractal market hypothesis', they are highly unstable dynamic systems that generate stock prices which appear random, but behind which lie deterministic patterns." (Steve Keen, "Debunking Economics: The Naked Emperor Of The Social Sciences", 2001)
"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."
"Do I claim that everything that is not smooth is fractal? That fractals suffice to solve every problem of science? Not in the least. What I'm asserting very strongly is that, when some real thing is found to be un-smooth, the next mathematical model to try is fractal or multi-fractal. A complicated phenomenon need not be fractal, but finding that a phenomenon is 'not even fractal' is bad news, because so far nobody has invested anywhere near my effort in identifying and creating new techniques valid beyond fractals. Since roughness is everywhere, fractals - although they do not apply to everything - are present everywhere. And very often the same techniques apply in areas that, by every other account except geometric structure, are separate." (Benoît Mandelbrot, "A Theory of Roughness", 2004)
"In plain English, fractal geometry is the geometry of the irregular, the geometry of nature, and, in general, fractals are characterized by infinite detail, infinite length, and the absence of smoothness or derivative." (Philip Tetlow, "The Web’s Awake: An Introduction to the Field of Web Science and the Concept of Web Life", 2007)
"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
"The economy is a nonlinear fractal system, where the smallest scales are linked to the largest, and the decisions of the central bank are affected by the gut instincts of the people on the street." (David Orrell, "The Other Side Of The Coin", 2008)
"A mathematical fractal is generated by an infinitely recursive process, in which the final level of detail is never reached, and never can be reached by increasing the scale at which observations are made. In reality, fractals are generated by finite processes, and exhibit no visible change in detail after a certain resolution limit. This behavior of natural fractal objects is similar to the exponential cutoff, which can be observed in many degree distributions of real networks." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)
"Fractals are self-similar objects. However, not every self-similar object is a fractal, with a scale-free form distribution. If we put identical cubes on top of each other, we get a self-similar object. However, this object will not have scale-free statistics: since it has only one measure of rectangular forms, it is single-scaled. We need a growing number of smaller and smaller self-similar objects to satisfy the scale-free distribution." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)
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