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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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Star Trek II But what of the “real Hollywood?” “It” immediately realized the potential of Voss’s landscape illustrations in my 1977 book, and soon introduced variants of these fractals into its films. This led to a lovely reenactment of the old and yet always new story of Beauty and the Best, since it is taken for granted that films are less about Brains than about Beauty, and since the few brainy mathematicians who had known about individual fractals-to-be, had taken for granted (until my books) that these were but Monsters.

Beastly. The pillars of “our geographically extended Hollywood” were Alain Fournier, Don Fussell and Loren Carpenter. Early in 1980, John W. Van Ness, a co-author of mine in 1968 who had moved to the University of Texas at Dallas, asked me to comment on the draft of his student Fournier’s Ph.D. dissertation. Fournier and Fussell had written earlier to as, asking for the IBM programs to generate fractal mountains, but we did not want to deal with lawayers for the sake of programs that were not documents, and the programs were already too intimately linked to one set of computers to be readily transported anywhere else. Therefore, Fournier and Fussell went their own way, and soon hit upon an alternative method that promised computations drastically faster than those of Voss.

Precisely the same alternative was hit upon at the same time by Loren Carpenter, then at Boeing Aricraft, soon to move to Lucasfilm, and now at Pixar. In his own words in The College Mathematics Journal of March 1984, “I went out and bought [Figure Fractals] as soon as I read [Martin] Gardner’s original column on the subject in Scientific American. I have gone through it with a magnifying glass two or three times. I found that it was inspirational more than anything else. What I got out of it myself was the notion that ‘Hey, these things are all over, and if I can find a reasonable mathematical model for making pictures, I can make pictures of all the things fractals are found in.’ that is why I was quite excited about it...” “The method I use is recursive subdivision, and it has a lot of advantages for the applications that we are dealing with here; that is, extreme perspective, dynamic motion, local control – if I want to put a house over here, I can do it. The subdivision porocess involves a recursive breaking-up of large triangles into smaller triangles. We can adjust the fineness of the precision that we use. For example, in ‘Star Trek,’ the images were not computed to as fine a resolution as possible because it is an animated sequence and things are going by quickly. You can see little triangles if you look carefully, but most people never saw them.” “Mandelbrot and others who have studied these sorts of processes mathematically have long been aware that there are recursive approximations to them, but the idea of actually using recursive approximations to make pictures, a computer graphics-type application, as far as we know first occurred with me and Fournier and Fussel, in 1979...” “One of the major problems with fractals in synthetic imagery is the control problem. They tend to get out of hand. They will go random all over the place on you.

If you want to keep a good tight fist on them and make it look like what you want it to look like, it requires quite a bit of tinkering and experience to get it right. There are not many people around who know how to do it.” While still at Boeing, Carpenter became famous in computer graphics circles for making a short fractal film, Vol Libre, and he was called to Lucasfilm to take a leading role in the preparation of the full feature Figure Star Trek II: The Wrath of Khan. Several computer-generated sequences of this film involve fractal landscapes, and have also become classics in the core computer graphics community. The best known is the Genesis planet transformation sequence. A different company, Digital Productions, later included massive fractal landscapes in The Lost Starfighter, which I saw – without hearing it – in an airplane. I had seen Star Trek II in a suburban movie-house (since I had gone there on duty, my stub was reimbursed). An associate had seen it on a previous day, and had reported that it was too bad that the fractal parts had been cut (adding as consolation that it was known that they always cut out the best parts in the suburbs). Of course, my wife and I immediately saw where the fractal portion started, and we marveled: If someone less durably immersed than the two of us in these matters could be fooled so easily, what about people at large?

Later, when he was interviewed for the summer 1985 issue of La lettre de l’image, Carpenter described the severe cost constraints imposed by his work: “We cannot afford to spend twice as much money to improve the quality of the pictures by 2%.” One would hate to be asked to attach a numerical percentage to quality improvement, but computer costs do keep decreasing precipitously, and there is hope that future feature films using fractals will be affordable while pleasing even the crankiest mathematician.

This Beauty and the Beast episode was enjoyable but drew us into a few scrapes, long emptied of bitterness, but still instructive. we were disappointed that the endless credits of the films never included the word fractal, nor our names. Once excuse was that everyone who mattered knew, so there was no need to say anything. Besides, lawyers feared that, if mentioned, we would have been put in a position to sue for a part of the cake. The world at large does not believe that scientists are resigned tot eh fact that their best work – the principle of mathematics and the laws of nature – cannot be patented, copyrighted, or otherwise protected by law. All that the scientists can expect is to be paid in the coin of public – not private – praise.

Later on, we greeted with amusement Alvy Ray Smith’s tern “graftal.” The differences from “fractal” were hardly sufficient to justify this proprietary variation on my coinage.

Fournier & Fussel and Carpenter are not represented in The Science of Fractal Images. It is a pity that we did not come to know them better. They have hardly ever written to us, even at times when we could have helped, which we would have loved to do, and, in any case we would have linked to follow their work as it evolved.

Midpoint Displacement in Greek Geometry: the Archimedes construction for the Parabola Our scrapes with “our Hollywood” have led to a variety of mutually contradictory impressions. Some people came to believe that the fractal landscapes in Fournier, Fussell & Carpenter 1982 are, somehow, not “true fractals.” Of course they are fractals, just as true as the Koch curve itself. Other people believe that I begrudge credit for “recursive subdivision,” in order to claim “midpoint displacement” – which is the same thing under a different term – for myself. Actually, as the French used to be taught in high school geometry, the basic credit for the procedure itself (but of course not for fractals) belongs to someone well beyond personal ambition, to Archimedes (287 – 212 BC).

The antiquity of the reference is a source of amusement and wonder, but rest assured that his work is amply documetned. A great achievement of Archimedes was when he evaluated the area between a parabola and a chord AB, an achievement that many writers view as the first documented step towards calculus. The technique Archimedes used was to take a chord’s endpoints and interpolate recursively to values of x so that they form an increasingly tight dyadic grid. Using this, Archimede’s was able to derive the rule of upward displacements though the parabola would not have an equation until Descartes devised analytic geometry.

Fractal Clouds

The algorithm Voss used to generate fractal mountains extends to clouds, as described in his contribution to this book. The resulting graphics are stunning, but actually do not provide an adequate fit to the real clouds in the sky. This is the conclusion we had to draw from the work of Shuan Lovejoy.

Lovejoy, then a meteorology student in the Physics department at McGill University in Montreal, wrote to me, enclosing a huge draft of his thesis. The first half, concerned with radar observation,. was not controversial and sufficed to fulfill all the requirements. But the second half, devoted to the task of injecting fractals into meteorology, was being subjected to very rough weather by some referees, and he was looking for help. My feeling was that this work showed very great promise, but needed time to “ripen.” (I was reminded of my own Ph.D. thesis, which was hurried to completion in 1952; I had been in a rush to take a post-doctoral position, a reason that soon ceased to appear compelling.) Hence, my recommendation to Lovejoy was that he should first obtain his sheepskin on the basis of his non-controversial work, and then join me as a post-doctoral student. I argued that he must not leave in his publications too many points that the unavoidable unfriendly critics could latch on to.

I was very impressed by Shaun’s area-perimeter diagram, drawn according to fractal precepts in my 1977 book, which suggested that the perimeters of the vertical projections of clouds (as seen from zenith, for example from a satellite) are of fractal dimension about 4/3. Lovejoy 1982, a paper that limited itself to this diagram and a detailed caption, immediately became famous. A second of many parts of Lobejoy’s thesis required far more work and, finally, I pitched in. Our joint paper came out years later as M & Lovejoy 1985. The illustrations of clouds have ytet to be surpassed. ???

Figure ???? By then, Lovejoy had drifted away from me. He had grown impatient with my refusal to reopen old fights that had been won to an acceptable degree, and by my deliberate preference for seeking “soft acceptance,” with controversy only when it is unavoidable, as opposed to “hard acceptance,” with unforgiving victims.

Clouds seem to pose a severe challenge to landscape painters, but one has achieved fame for his prowess. His name was Salomon van Ruysdaë l (1602-1670), and he brings to mind a question and a story. The question is whether fractal geometry can help us to compare the clouds of Ruysdael with those of Mother Nature. Elizabeth Carter was an undergraduate in meteorology at the University of California at Los Angeles (UCLA), in the group of Professor George L. Siscoe. Her hobby is photographing clouds, and she had found a nice way of getting academic credit for it. The fractal dimension was estimated for many varied clouds’ contours 9as seen from a newarly horizontal direction – not the same thing as Lovejoy’s views form the zenith). The conclusion was that Nature’s clouds’ D’s are far more tightly bunched. In hindsigh, the result was to be expected: the painter chose to paint clouds that are dramatic, yet not impossible, hence his clouds’ D’s are near Nature’s maximum.

Fractal Trees

Before moving to nonlinear fractals, it seemed logical to me, as manager of a tiny fractals group, to perform a few preliminary tests without perturbing the on-going programs.

This is how a Princeton senior, Peter Oppenheimer, came to work with us for a few weeks. He later wrote his senior thesis on fractals and, eventually, he moved to the New York Institute of Technology on Long Island, and became an expert on fractal botany.

Today he has competition from Przemyslaw Prusinkiewicz.

Drawing nonrandom fractal trees is comparatively easy, and there are several in Figure The Fractal Geometry of Nature. Drawing random fractal trees that are not of unrealistic “sparseness” presents a major difficulty; branches must not overlap. Suppose that a random tree is to be constructed recursively. Once cannot add a branch, or even the tiniest twig, without considering the Euclidean neighborhood where the additions will be attached. However, points that are close by, according to Euclidean distance, may be far away according to the graph distance taken along the branches. Therefore, a random recursive construction of a tree, going from trunk to branches and on to twigs, is by necessity a global process. One may be drawn to seek a construction by selfcontemplation, or by obeying the constraints imposed by one’s computer’s better way.

By contrast, space appears forgiving; more precisely, it offers an almost irresistible temptation to cheat. Show a shape described as a tree’s projection on a plane, and challenge yourself to imagine a spatial tree having such a projection. Even when the original spatial branches happen to intersect or become entangled, our mind will readily disentangle them, and see them as a tree.

Now back to planar trees, and to ways of drawing them without worrying about self-intersection. A completely natural method was devised by Tom Witten and Leonard Sander. It came about in what we think is the best possible way, not during a search for special effects, but during a search for scientific understanding of certain web or tree-like natural fractal aggregates. The Witten-Sander method is called diffusion limited aggregation. Most unfortunately, it fails to yield realistic botanical trees, but it gives us hope for the future.

Iteration, Yesterday’s Dry Mathematics, Today’s Weird and Wonderful New Fractal Shapes, and the Geometry Supercomputer Project Now, from fractals that imitate mountains, clouds, and trees, let us move on to fractals that do not. For the artist and the layman, they are simply weird and wonderful new shapes. My brief involvement with Poincaré limit sets has already been touched upon.

My initiation to Julia sets began at age 20, when the few who knew them called the Jsets. This, and the beginning of my actual involvement with the study of iteration of z → z2 + c, have both been described in an invited contribution to Figure The Beauty of Fractals, and need not be repeated here.

But I do want to mention a brief interaction with David Mumford which eventually contributed to a very interesting and broad recent development.

David’s name was known to me, and to everyone else in mathematics, because of his work in algebraic geometry. We met when I came to Harvard in 1979, and, in November 1979, he came to a seminar I gave. After the talk, which was on iteration, he rushed towards me: “On the basis of what you have said, you should also look into Kleinian groups; you might even find an explicit construction for their limit set.” “Actually,” I responded, “I already have a nice algorithm for an important special case.

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