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«People and Events Behind the Fractal Images When asked to write this article I, without space limitation, have unleashed a flood of recollections ...»

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Please come to my office, and I shall show you.” David came over and saw the algorithm that was eventually published as M 1983i, as told earlier. The exact words of our conversation are, of course, lost, but I should remember their substance as follows: “This is so simple, that Poincaré should have seen it, or someone else singe Poincaré. Why did this discovery have to wait for you?” – “Because no one before me has used a powerful new tool, the computer!” – “But one cannot prove anything with a computer!” – “Sure, but playing with the computer is a source of conjectures, often most unexpected ones. The conjecture it has suggested about Kleinian limit sets has been easy to prove; other are too hard for me.” – “In that case, would you help me learn to play with the computer?” – “With pleasure, but we would have to get help from my latest IBM assistant, Mark laff.” Soon afterwards, it became clear that Mumford had to seek associates closer by, in Cambridge. He was tutored by my course assistant Peter Moldave, and started working with David Wright, then a Harvard graduate student in mathematics, who ceased, at that point, to hide his exceptional programming skills. Eventually, Mumford became thoroughly immersed in computer, first as heuristic tools, then for their own sake.

He became instrumental in helping the awareness of the computer-as-tool idea that spread among mathematicians. The resulting needs grew so rapidly that, after barely eight years, the Figure National Science Foundation had established a Geometry Supercomputer Project! The charter members are F. Almgren (Princeton), J. Cannon (Brigham Young), D. Dobkin (Princeton), A. Douady (ENS, Paris), D. Epstein (Warwick), J. Hubbard (Cornell), B. Mandelbrot (IBM & Yale), A. Marden (Minnesota), J. Milnor (IAS, Princeton), D. Mumford (Harvard), R. Tarjan (Princeton & Bell Labs), and W. Thurston (Princeton). At the risk of sounding corny, let me confess that the opening of this project was a high point in my life. In 1991 it expanded, and its name changed to Geometry Center.

The next topic to be discussed concerning iteration is my fruitful interaction with V. Alan Norton, a Princeton mathematics Ph.D. in 1976, who was in my group as a postdoc in 1980-82, and stayed on the research staff at IBM Yorktown. He was one of the two principal “illustrators” of Figure The Fractal Geometry of Nature, as seen in that book’s very detailed picture credits. He has achieved great renown, starting with Siggraph 1982, for splendid quaternionic Julia set pictures.

Norton also worked on the end-papers, without legend, for The Fractal Geometry of Nature Figure. How these end-papers came about is a tale worth recounting. They involve an important problem from the theory of iteration of analytic functions, an artifact due to inherent limitations of the computer, and two decorative touches.

The original graph was unbounded, and Norton introduced a decorative touch:

inversion with respect to a circle. I loved the result; unfortunately, while bounded, it did not fit neatly on a double page spread. Hence I imposed a second and more arbitrary decorative touch: the horizontal stretching of the graph to fill the available space.

The serious mathematical problem that had led me to this graph was the use of Newton’s method to solve the equation exp(z0) = c. The solutions are known from calculus, but Gaston Julia had shown in 1917 that Newton’s method is a fruitful ground for the study of the iteration of functions of a complex variable z. Chapter 19 of The Fractal Geometry of Nature Figure examines the iteration of z2 + c and other polynomials. This end-paper relates to the iteration of the transcendental function z – 1 + Ce-z.

In Arthur Cayley’s pioneering global studies of iteration in 1879, the interest in iteration had arisen from the application of the Newton-Raphson method. (Peitgen et al.

tell the story, and illustrate it, in The Mathematical Intelligencer in 1984.) Cayley began by solving z2 = C, which proved easy, then went on to try z3 = C, which stumped him by exhibiting three “gray areas: that he found no way of resolving. Julia, in 1917, had found many acts about these areas, and John H. Hubbard had shown us his revealing earliest graph of the corresponding Julia set. It was natural for us, in late 1980, to play with zp = C, and then view exp(z) = C as a suitable limit of zp = C for p→inf. We made many interesting observations about this limit case, but the study was far form complete and publishable when we moved on to very different work.

Finally, and unfortunately, the non-fractal bold boundaries between the background and the solidly colored areas in the end-papers of The Fractal Geometry of Nature Figure are an artifact. The study of transcendental functions’ iterates leads very quickly to enormous integers, hence soon reaches intrinsic limits beyond which the computer takes its own arbitrary actions.

Devaney, Barnsley, and the Bremen Book, The Beauty of Fractals

Out perception of the iteration of transcendental functions as a difficult and very rich topic was confirmed by several eminent mathematicians, such as Robert L. Devaney. No wonder, therefore, that one should see striking resemblances between our end-papers and his beautiful and widely seen illustrations and films. Bob’s papers on the iteration of transcendental functions had already brought admiring attention to him, but we did not become fast friends until we started bumping into each other constantly on the fractal Son et Lumière traveling shows.

The life orbit of Michael barnsley has also crossed mine, and then they stayed in the same fractal neighborhood. The amusing background, in this instance, is in the public record, and I will not repeat it. I first read about it in James Gleick’s book, Chaos: The Birth of a New Science. There I found out how it came to be that Michael burst into my house one day, full of enthusiasm, and lovely tales. Later, we held a few meetings at the Atlanta airport (of all places!), and since then it has been a pleasure to keep up with his work and that of his many associates.

Now back to the pictures of Julia and Mandelbrot sets in Figure The Fractal Geometry of Nature. During the summer of 1984, we were tooling up to redo them in color, with Eriko Hironaka as programmer, when mail brought in, hot off the press, the June issue of the German magazine Geo. We realized immediately that much of what we were proposing had already been achieved, in fact achieved beyond our aspirations, by Heainz-Otto Peitgen, Peter H. Richter, Dietmar Saupe and their associates. Their fractal pictures in The Mathematical Intelligencer earlier in 1984 had been most welcome, but those in color had not yet supplied ample reason for enthusiasm. The color pictures in the 1984 Geo showed a skilled and artistic eye, coupled with a sure hand, one that had gained experience but had not become lazy or hasty. They were unmistakably the outcome of the search for perfection I had admired earlier in the work of Voss, and always attempt in my own.

I wrote to the Geo authors at the University of Bremen to congratulate them, to tell them of the change of plans their success had provoked, and to express the hope of meeting them soon. They told me about a fractal exhibit they were planning, and for which they were preparing a catalogue that eventually led to their book Figure The Beauty of Fractals, and they asked me to write the personal contribution mentioned earlier in this paper. That this book was fated not to come from me or my associates, it is still a delight that it came from them. I gained these new friends when they invited me to Bremen in may 1985, to open the first showing of their exhibit, and I participated in this exhibit in several other cities as well. Our joint appearances have since then have become too numerous to count. There are no anecdotes to tell, only very pleasant events to remember.


I am reminded that, only a while ago (the sting has disappeared, but the memory remains), no one wanted to scan my early pictures for longer than a few minutes, and this would-be leader of a new trend had not a single follower. Then, very slowly in memory, yet almost overnight in present perception, fractals became of so wide interest that Siggraph started devoting lectures, then full day courses to them. The first makeshift appearance of fractals at Siggraph came in 1985 under my direction, the second in 1986 under Peter Oppenheimer, and the third in 1987 and led to The Beauty of Fractals and

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