Engineering is too important to wait for science. - Benoit Mandelbrot

"No one is alone in this world. No act is without consequences for others. It is a tenet of chaos theory that, in dynamical systems, the outcome of any process is sensitive to its starting point-or, in the famous cliche, the flap of a butterfly's wings in the Amazon can cause a tornado in Texas. I do not assert markets are chaotic, though my fractal geometry is one of the primary mathematical tools of "chaology." But clearly, the global economy is an unfathomably complicated machine. To all the complexity of the physical world of weather, crops, ores, and factories, you add the psychological complexity of men acting on their fleeting expectations of what may or may not happen-sheer phantasms. Companies and stock prices, trade flows and currency rates, crop yields and commodity futures-all are inter-related to one degree or another, in ways we have barely begun to understand. In such a world, it is common sense that events in the distant past continue to echo in the present."

The next thing which surprised us very much, is that both for Julia sets and even more so for the Mandelbrot set, the complication was not, how to say, arbitrary, and almost everybody found the impression that these shapes were hauntingly beautiful. These shapes resulted from the most ridiculous transformation, z²+c, taken seriously, respectfully and visually. And people thought at first that they were totally wild, totally extraterrestrial, but then after a very short time, they came back and said, "You know, I think they remind me of something. I think they're natural. I think they are like perhaps nightmares or dreams, but they're natural." And this combination of being so new, because literally when we saw them nobody had seen them before, and being the next day so familiar, is still to me extraordinarily baffling.

"My current best model of how a market works is fractional Brownian motion of multifractal time. It has been called the Multifractal Model of Asset Returns. The basic ideas are similar to the cartoon versions above-though far more intricate, mathematically. The cartoon of Brownian motion gets replaced by an equation that a computer can calculate. The trading-time process is expressed by another mathematical function, called f(\propto), that can be tuned to fit a wide range of market behavior. My model redistributes time. It compresses it in some places, stretches it out in others. The result appears very wild, very random. The two functions, of time and Brownian motion, work together in what mathematicians call a compound manner: Price is a function of trading time, which in turn is a function of clock time. Again, the two steps in the model combine to produce a "baby" far different from either parent."