I spent half my life, roughly speaking, doing the study of nature in many aspects and half of my life studying completely artificial shapes. And the two are extraordinarily close; in one way both are fractal.
BENOIT MANDELBROTI spent half my life, roughly speaking, doing the study of nature in many aspects and half of my life studying completely artificial shapes. And the two are extraordinarily close; in one way both are fractal.
BENOIT MANDELBROTThe theory of chaos and theory of fractals are separate, but have very strong intersections. That is one part of chaos theory is geometrically expressed by fractal shapes.
BENOIT MANDELBROTA formula can be very simple, and create a universe of bottomless complexity.
BENOIT MANDELBROTUnfortunately, the world has not been designed for the convenience of mathematicians.
BENOIT MANDELBROTOrder doesn’t come by itself.
BENOIT MANDELBROTClouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.
BENOIT MANDELBROTOne couldn’t even measure roughness. So, by luck, and by reward for persistence, I did found the theory of roughness, which certainly I didn’t expect and expecting to found one would have been pure madness.
BENOIT MANDELBROTThe straight line has a property of self-similarity. Each piece of the straight line is the same as the whole line when used to a big or small extent.
BENOIT MANDELBROTA cloud is made of billows upon billows upon billows that look like clouds. As you come closer to a cloud you don’t get something smooth, but irregularities at a smaller scale.
BENOIT MANDELBROTThe most complex object in mathematics, the Mandelbrot Set … is so complex as to be uncontrollable by mankind and describable as ‘chaos’.
BENOIT MANDELBROTIf you assume continuity, you can open the well-stocked mathematical toolkit of continuous functions and differential equations, the saws and hammers of engineering and physics for the past two centuries (and the foreseeable future).
BENOIT MANDELBROTNow that I near 80, I realize with wistful pleasure that on many occasions I was 10, 20, 40, even 50 years ahead of my time.
BENOIT MANDELBROTThere are very complex shapes which would be the same from close by and far away.
BENOIT MANDELBROTBoth chaos theory and fractal have had contacts in the past when they are both impossible to develop and in a certain sense not ready to be developed.
BENOIT MANDELBROTHumanity has known for a long time what fractals are. It is a very strange situation in which an idea which each time I look at all documents have deeper and deeper roots, never (how to say it), jelled.
BENOIT MANDELBROTThink not of what you see, but what it took to produce what you see.
BENOIT MANDELBROT