"One wonders why models selected on their virtues of simplicity prove so attractively applicable." (Benoît B Mandelbrot, "The Fractal Geometry of Nature", 1977)
"The nature of fractals is meant to be gradually discovered by the reader, not revealed in a flash by the author. And the art can be enjoyed for itself." (Benoît B Mandelbrot, "The Fractal Geometry of Nature", 1977)
"The theory of probability is the only mathematical tool available to help map the unknown and the uncontrollable. It is fortunate that this tool, while tricky, is extraordinarily powerful and convenient." (Benoît B Mandelbrot, "The Fractal Geometry of Nature", 1977)
"Topology, which used to be called geometry of situation or analysis situs ('topos' means position, situation in Greek), considers that all pots with two handles are of the same form because, if both are infinitely flexible and compressible, they can be molded into any other continuously, without tearing any new opening or closing up any old one. It also teaches that all single island coastlines are of the same form, because they are topologically identical to a circle." (Benoît B Mandelbrot, "The Fractal Geometry of Nature" 3rd Ed., 1983)
"Science and Art: Two complementary ways of experiencing the natural world – the one analytic, the other intuitive. We have become accustomed to seeing them as opposite poles, yet don’t they depend on one another? The thinker, trying to penetrate natural phenomena with his understanding, seeking to reduce all complexity to a few fundamental laws - isn’t he also the dreamer plunging himself into the richness of forms and seeing himself as part of the eternal play of natural events?" (Benoît Mandelbrot, 1984)
"Why is geometry often described as cold and dry? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in straight line. [...] Nature exhibits not simply a higher degree but an altogether different level of complexity." (Benoît B Mandelbrot, 1984)
"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", Leonardo [Supplemental Issue], 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", Leonardo [Supplemental Issue], 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)
"In everyday language, a fair coin is called random, but not a coin that shows head more often than tail. A coin that keeps a memory of its own record of heads and tails is viewed as even less random. This mental picture is present in the term random walk, especially as used in finance." (Benoit B Mandelbrot, "Fractals and Scaling in Finance: Discontinuity, concentration, risk", 1997)
"Randomness is an intrinsically difficult idea that seems to clash with powerful facts or intuitions. In physics, it clashes with determinism, and in finance it clashes with instances of clear causality, economic rationality and perhaps even free-will. It is easy to acknowledge that randomness can create its peculiar regularities. But it is difficult to acknowledge that such regularities either could be interesting or could arise in physics or finance. As a result, the fact that any statistical model could be effective seems a priori inconceivable and is difficult to acknowledge." (Benoit B Mandelbrot, "Fractals and Scaling in Finance: Discontinuity, concentration, risk", 1997)
"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)
"For a complex natural shape, dimension is relative. It varies with the observer. The same object can have more than one dimension, depending on how you measure it and what you want to do with it. And dimension need not be a whole number; it can be fractional. Now an ancient concept, dimension, becomes thoroughly modern." (Benoît B Mandelbrot, "The (Mis)Behavior of Markets", 2004)
"In science, all important ideas need names and stories to fix them in the memory." (Benoît B Mandelbrot, "The (Mis)Behavior of Markets", 2004)
"The brain highlights what it imagines as patterns; it disregards contradictory information. Human nature yearns to see order and hierarchy in the world. It will invent it where it cannot find it." (Benoît B Mandelbrot, "The (Mis)Behavior of Markets", 2004)
"Unfortunately, the world has not been designed for the convenience of mathematicians." (Benoît B Mandelbrot, "The (Mis)Behavior of Markets", 2004)
"There is no single rule that governs the use of geometry. I don't think that one exists." (Benoît B Mandelbrot, [interview] 2004)
"Georg Cantor claimed the essence of mathematics lies in its freedom. But mathematicians do not pick problems from thin air for the pleasure of solving them. To the contrary, a mark of greatness resides in the ability to identify the most interesting problems in the framework of what is already known." (Benoît B Mandelbrot, "The Fractalist", 2012)
"Most of the world is of great roughness and infinite complexity. However, the infinite sea of complexity includes two islands of simplicity: one of Euclidean simplicity and a second of relative simplicity in which roughness is present but is the same at all scales." (Benoît B Mandelbrot, "The Fractalist", 2012)
"A fractal is a mathematical set or concrete object that is irregular or fragmented at all scales […]" (Benoît B Mandelbrot)
"It is an extraordinary feature of science that the most diverse, seemingly unrelated, phenomena can be described with the same mathematical tools. [...] But the variety of natural phenomena is boundless while, despite all appearances to the contrary, the number of really distinct mathematical concepts and tools at our disposal is surprisingly small. [...] When we explore the vast realm of natural and human behavior, we find the most useful tools of measurement and calculation are based on surprisingly few basic ideas." (Benoît B Mandelbrot)
"Today we can say that the abstract beauty of the theory is flanked by the plastic beauty of the curve, a beauty that is astounding. Thus, within this mathematics that is a hundred years old, very elegant from a formal point of view, very beautiful for specialists, there is also a physical beauty that is accessible to everyone. [...] By letting the eye and the hand intervene in the mathematics, not only have we found again the ancient beauty, which remains intact, but we have also discovered a new beauty, hidden and extraordinary. [...] Those who are only concerned with practical applications may perhaps tend not to insist too much on the artistic aspect, because they prefer to entrench themselves in the technicalities that appertain to practical applications. But why should the rigorous mathematician be afraid of beauty?" (Benoît B Mandelbrot)
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