Fractals finger suspect Pollocks Alison Abbott reports in the February 9 2006 *Nature* on a new mathematical development in the saga of the 32 small "possible Pollocks" recently discovered on Long Island. Large poured works by Jackson Pollock bring prices in the tens of millions of dollars; if these paintings are authentic they are very valuable. But their authenticity, accepted by some experts, has been challenged by others. Enter the physicist Richard Taylor. Taylor had published in 1999 (*Nature* **399** 422) his group's discovery that Pollock's poured works showed (as Abbott explains it) "two distinct sets of fractal patterns. One was on a scale larger than 5 cm; the other showed up on scales between 1 mm and 5 cm." and furthermore "that the fractal dimension of Pollock's works ... increased through the years as the artist refined his technique." In a later experiment, he analyzed "14 Pollock paintings, 37 imitations created by students at the University of Oregon, and 46 paintings of unknown origin." Abbott quotes Taylor: "The only shared thing in Pollock's very different poured paintings is a fractal composition that was systematic through the years." The non-Pollocks, when they had fractal structure, had different fractal characteristics. So it was natural for the Krasner-Pollock foundation to send six of the putative 32 for Taylor to examine. His diagnosis: "I found significant deviation from Pollock's characteristics." The foundation's final judgment has not yet been promulgated. The *Nature* piece has several echoes in the *New York Times*. It gets picked up as a news item by Randy Kennedy ("Computer Analysis Suggests Paintings Are Not Pollocks") on February 9. Their art critic Michael Kimmelman weighs in with "A Drip by Any Other Name" on February 12: "... the curious truth is that while a few drips and splashes can imitate Pollock's touch ... it is nearly impossible to replicate ... the full-scale complex rhythms and overlapping patterns, the all-over, depthless, balletic and irregular space he created." And then on the February 19 Op-Ed page, Professor Don Foster (English, Vassar, "Mind Over Splatter") brings us an academic perspective. He starts out serious: "At the heart of the controversy lie critical questions about artistic meaning and value that have vexed literary scholars no less than art historians." But he leaves us with: "Meanwhile, Jackson Pollock may be chuckling in his grave: if the object of Abstract Expressionist work is to embody the rebellious, the anarchic, the highly idiosyncratic --if we embrace Pollock's work for its anti-figurative aesthetic-- may faux-Pollock not be quintessential Pollock? May not a Pollock forgery that passes for authentic be the best Pollock of all?" Plant growth and the Golden Ratio, re-evaluated The *Botanical Journal of the Linnaean Society* ran in its January 2006 issue an article by Todd Cooke (Maryland) with the title: "Do Fibonacci numbers reveal the involvement of geometrical imperatives or biological interactions in phyllotaxis?" From the abstract: "This paper reviews the fundamental properties of number sequences, and discusses the under-appreciated limitations of the Fibonacci sequence for describing phyllotactic patterns." Apparently golden-ratio giddiness has spread to botany, and this paper aims to be a corrective. Prof. Cooke's main point is that although "it is inescapable that the spiral phyllotaxes of vegetative shoots are overwhelmingly characterized by low Fibonacci numbers," the common belief that "such spiral arrangements are attributable to the leaf primordia being positioned in optimal packing" must be questioned and ultimately rejected. The argument, as I understand it, runs as follows. Suppose consecutive primordia are are arranged at an exactly constant angular difference. If that difference is *exactly* the golden angle, here given as 137.5°, then one does indeed achieve optimal packing. But "even slight variation from the Fibonacci angle disrupt[s] optimal packing." E.g. constant angle 137.45° or constant angle 137.92° don't work. "It is difficult, if not impossible, to imagine any biological system being capable of organizing itself with such discriminating accuracy as a direct response to a hypothetical geometrical imperative for optimal packing. It seems more likely that the spiral phyllotaxes observed ... are the outcome of some biological process, the consequence of which is that such structures tend to approach optimal packing." There are two points here. The mathematical one is shaky. The golden ratio is (supremely) irrational, and the evidence for its occurrence in the likeliest interval between consecutive primordia (*viz.,* the appearance of numbers of spirals corresponding to its rational approximators 2/3, 3/5, 5/8, etc.) is excellent. On the other hand the question whether optimal packing is an "imperative" or a "consequence" does not seem to me to be one that science can answer. The end of this article addresses the identification of the biological process governing phyllotaxis. Cooke refers to the 1992 *Physics Review Letters* paper (**68**, 2089-2010) by Stéphane Douady and Yves Couder, where they "managed to create spiral phyllotaxis on a lab bench" working with mutually repelling ferrofluid drops floating on silicon oil in a varying magnetic field. Presumably something analogous is happening at the growing tip of a plant. "The ... mechanism ... apppears to involve the interaction of mathematical rules, generating process, and overall geometry. In particular, it seems quite plausible that the mathematical rules for phyllotaxis arise from local inhibitory interactions among existing primordia. These interactions are apparently mediated by the expression of specific genes whose products regulate growth hormones ... ." This work was picked up in the "Research Highlights" of the February 9 *Nature*. Mathematical Incompleteness in the *Scientific American* The March 2006 *Scientific American* features a report by Gregory Chaitin, entitled "The Limits of Reason," describing his own work on the incompleteness of mathematics. "Unlike Gödel's approach, mine is based on measuring information and showing that some mathematical facts cannot be compressed into a theory because they are too complicated" and that therefore "... a theory of everything for all of mathematics cannot exist." Chaitin outlines his theory, including the irreducible number Omega: the first N digits of Omega cannot be computed using a program significantly shorter than N bits long. He sketches the argument that computing the first N binary digits of Omega would solve the halting problem for all programs of length up to N; so the uncomputability of Omega follows from Turing's proof of the unsolvability of the halting problem. It follows from its definition that "an infinite number of bits of Omega constitute mathematical facts ... that cannot be derived from any principles simpler than the string of bits itself. Mathematics therefore has infinite complexity, whereas any individual theory of everything would only have finite complexity and could not capture all the richness of the full world of mathematical truth." Chaitin then spends some time pondering the scientific and philosophical consequences of his work. "Irreducible principles --axioms-- have always been part of mathematics. Omega just shows that a lot more of them are out there than we suspected." "If Hilbert had been right, ... there would be a static, closed theory of everything for all of mathematics, and this would be like a dictatorship. ... I much prefer an open system. I do not like rigid, authoritarian ways of thinking." "Extensive computer calculations can be extremely persuasive, but do they render proof unnecessary? Yes and no. In fact, they provide a different kind of evidence. In important situations, I would argue that both kinds of evidence are required, as proofs may be flawed, and conversely computer searches may have the bad luck to stop just before encountering a counterexample that disproves the conjectured result." Tony Phillips Stony Brook University tony at math.sunysb.edu |