Mandelbrot's early computer images served to confirm his suspicion that Julia sets have fractal properties. He says he began producing recognizable pictures of the Mandelbrot set—which in a sense is a generalized version of all Julia sets—in late 1979. Mandelbrot subsequently displayed images of the set and elaborated on its significance in speeches, papers and books. This discovery and his other work in fractals were also celebrated in the media, in numerous books (notably the best-seller Chaos, by former New York Times reporter James Gleick) and in IBM advertisements.
No one denies that Mandelbrot's pictures and descriptions spurred other mathematicians to study the set. Two prominent examples are John H. Hubbard of Cornell University and Adrien Douady of the University of Paris. In the early 1980's—in the course of proving that tiny "islands" surrounding the main body of the set are linked to it by infinitesimal filaments—they named the set after Mandelbrot. "Mandelbrot was the first one to produce pictures of it, using a computer, and to start giving a description of it," Douady wrote in 1986.
Douady now says, however, that he and other mathematicians began to think that Mandelbrot took too much credit for work done by others on the set and in related areas of chaos. "He loves to quote himself," Douady says, "and he is very reluctant to quote others who aren't dead."
[In the fall of 1989] Steven G. Krantz of Washington University aired some of these grievances in the Mathematical Intelligencer, a quarterly journal. The main point of his article was that fractals, computer-generated graphics and other "popular" mathematical phenomena associated with Mandelbrot have contributed little of substance to mathematics, especially in comparison to the publicity they have garnered.
This view—and its opposite, which holds that Mandelbrot's "popular" work has been a stimulating force in mathematics—had been voiced before. Krantz introduced a new element into the debate, however, by stating that the Mandelbrot set "was not invented by Mandelbrot but occurs explicitly in the literature a couple of years before the term 'Mandelbrot set' was coined." He cited a paper by Robert Brooks and J. Peter Matelski published in the proceedings of a 1978 conference at Stony Brook, N.Y.
Sure enough, the paper contains the famous z2 + c formula and a crude but unmistakable computer printout of the set's basic image. Brooks and Matelski say they did not actually present the paper at the 1978 conference, but they did circulate it as a preprint in early 1979. Brooks, who is now at the University of California at Los Angeles, also presented the paper at Harvard University in the spring of that year. (Mandelbrot, who held an appointment at Harvard then, says he did not hear Brooks speak and only saw the paper years later.) The paper was not published, however, until early 1981.
In a rebuttal to Krantz's article, called "Some 'Facts' that Evaporate upon Examination," Mandelbrot noted that he "fully published" on the Mandelbrot set before Brooks and Matelski did. (Mandelbrot's paper, published in the December 26, 1980, Annals of the New York Academy of Sciences, features a function and image that are variants of those now associated with the Mandelbrot set, which Mandelbrot did not publish until 1982.)
Mandelbrot also suggested that even if Brooks and Matelski's publication had preceded his, they still could not be considered discoverers of the set, because they did not appreciate its significance. "[They were] close to something that was to prove special, but they gave no thought to the picture," he wrote.
Brooks retorted in the following issue of the Intelligencer: "I don't know how he can be so sure of what we gave thought to and what we didn't." Brooks says he respects Mandelbrot's achievements as a popularizer and does not object to the set's being named after him. "It makes more sense than 'the thing with the big cardioid,'" he says, recalling how he and Matelski had referred to the set. "I just wish Mandelbrot were a little more gentlemanly."
Matelski, who works at the Hartford Graduate Center in Connecticut, notes that neither he nor Brooks asked Krantz to credit them with having discovered the Mandelbrot set. (Krantz confirms that another mathematician drew his attention to their paper.) But now that the issue has become public, Matelski says he and Brooks should be acknowledged as co-discoverers with Mandelbrot.
"You don't have to fully exploit the mineral resources of a continent to discover it," Matelski was quoted as saying in the Hartford Courant, a newspaper that reported on the dispute in December . "All you have to do is kneel down and kiss the beach."
A subtly different claim of precedence has been made by Hubbard, who is now considered one of the world's experts on the Mandelbrot set. In 1976, he explains, he began using a computer to map out sets of complex numbers generated by an iterative process known as Newton's method. Hubbard says he did not realize it then, but he had found a different way of generating the Mandelbrot set.