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When I publish in academic journals, as a scientist must, I often stir intense controversy. Each time, I have listened to the critics, rephrased my claims, gone back to my study to think and to my computers to analyze, and devised better, more-accurate models. Result: progress. Unavoidable side-effect: an element of complication. Indeed, I did not conceive of just one model of price variation, but several. Starting in 1963 and 1965 I devised two separate but incompatible models of behavior, succeeding at last in reconciling them in 1972.

As he relates it: It happened so fast I was not conscious of it. I would say to myself: This construction is ugly, let’s make it nicer. Let’s make it symmetric. Let’s project it. Let’s embed it. And all that, I could see in perfect 3-D vision. Lines, planes, complicated shapes. Ever since, pictures have been his special aids to inspiration and communication. Some of his most important insights came, not from elaborate mathematical reasoning, but from a flash recognition of kinship between disparate images—the strange resemblance between diagrams concerning income distribution and cotton prices, between a graph of wind energy and of a financial chart.

The creative essence of fractal geometry is to combine the formal and the visual. The ready intuition of fractal pictures has, today, made the subject a college course at Yale and other universities, and a popular addition to many high school math courses. But among “pure” mathematicians, Mandelbrot’s approach was initially criticized. Not rigorous, they chided; the eye can mislead. But, Mandelbrot rejoins, observation often led him to conjectures that have stimulated and challenged the most skilled mathematicians; many of these problems remain unsolved.

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