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July/August 1999
Jackson Pollock's drip paintings have been subjected to fractal analysis. In the June 3 Nature, Richard P. Taylor, Adam P. Micolich, David Jonas of the University of New South Wales report that the fractal dimension of Pollock's drip paintings "increased steadily through the years, from close to 1 in 1943 to 1.72 in 1952.. You can judge for yourself: several of his drip paintings can be seen on the Web, including Phosphorescence (1947), Lavender Mist (1950), and Blue Poles (1952). For more information see the National Gallery of Art web presentation Jackson Pollock 19121956, and Robert Devaney's Boston University class notes on Fractal Dimension. Fractals in the family room. Cover story in the May 27 Nature was "Escaping chaotic scattering" by David Sweet, Edward Ott and James A. Yorke of the University of Maryland. Their work gives "a simple laboratory demonstration of an interesting type of fractal geometry." In the type of geometry referred to, a planar region is broken up into three or more 2dimensional lakes with the "Wada property:" each point on the boundary of one lake is also on the boundary of all the others. Sweet, Ott and Yorke's demonstration produces, by scattering of light, an image with this exotic geometry. They stacked four reflective spheres in a pyramid, put red, blue and black paper over three of the holes between the spheres, and looked in through the fourth. Here is part of what they saw:
Looking into a pyramid of four reflective spheres. Image comes from David Sweet's "Photographs of Wada Basins," used with permission. Do try this at home! As Sweet says, "You can try this at home with four shiny, spherical Christmas tree ornaments and a flashlight. Just place three of the ornaments together on a table in a triangle (so they are touching) and place the last ornament on top. Look inside the the ornament arrangement and you'll see the fractal. Shining a light inside will help bring out the image." For a long time the "lakes of Wada" were a geometrical oddity that could only be obtained as the limit of a process involving an infinite number of steps, in a class with Antoine's necklace and Alexander's horned sphere. Then they turned up in the analysis of dynamical systems. In any complex dynamical system with three or more attractors the basins of the attractors must manifest this property. For example, in Scott Sutherland's study of Newton's method applied to a fourthorder polynomial, he assigned colors to the four roots, and colored each point in the plane according to the root the method would yield with that point as initial guess. Here is part of the picture he produced for the polynomial: (z+ i)(z + 0.6 + 1.6i)(z + 0.6 .45 i)(z0.866  .5i). The roots themselves are in the small circles in the centers of the largest colored regions. Darkness of shading represents the number of iterates the method takes to get into the small circle.
Each point on the boundary of the blue region has red, yellow and green points arbitrarily close to it; similarly for all four regions. Image by Scott Sutherland, used with permission.
Your cell phone may soon have a fractal antenna. According to a piece by George Musser in the July 1999 Scientific American, antennas in the the shape of a Sierpinski triangle or a Sierpinski square capitalize on their selfsimilarity to work equally well at all frequencies. For information about the Sierpinski triangle and its relation to the "chaos game" see Robert Devaney's Dynamical Systems page at Boston University. Fractal antennas have been studied by Nathan Cohen and Robert Hohlfeld, also of BU. The Bryn Mawr website has an applet implementing the chaos game. The Sierpinsky triangle also occurs as the limit of larger and larger Pascal triangles mod 2. This is illustrated for 2 and other moduli by applets on Jeremy Baer's page Explore patterns in Pascal's Triangle! at the University of Washington. Randomly ravenous rabbits? Suppose that Fibonacci's rabbits, instead of placidly pubescing and procreating to generate the famous sequence 1, 1, 2, 3, 5, 8, of numbers of pairs in successive generations, mutated so that at random, with probability 1/2, a generation would be eaten by the previous generation. This is essentially the ideal population studied by Divakar Viswanath in his Cornell Ph.D. thesis, picked up by Ivars Peterson in the June 12 Science News via Keith Devlin's MAA column New Mathematical Constant Discovered  Descendent of Two Thirteenth Century Rabbits. Viswanath proved that with probability one the number of rabbits in the nth generation will still increase exponentially with n, asymptotically to the nth power of a constant he pinned down to 8 decimal places as 1.13198824.. Visnawath's papers are available on the Web: go to the Mathematical Sciences Research Institute and follow the trail peoplepostdocsViswanath. "Connections between music and science are subtle and uncertain,"  Malcolm Browne (New York Times, May 11). The occasion for his essay was a lectureconcert staged by the Emerson Quartet and Brian Greene, Professor of Mathematics and of Physics at Columbia University. The appropriately shaped venue was the Guggenheim Museum in New York, where "the five blackclad performers sought to demonstrate an underlying kinship between some seminal ideas of physics and music." Bach was paired with Newton, Webern and Berg with superstring theory. Browne quotes Leibnitz: "Music is the hidden mathematical endeavor of a soul unconscious it is calculating." and Leonard Bernstein: "The only way one can really say anything about music is to write music." Here is one more quotation. Fourier, in the introduction to his Analytical theory of heat: "...if the order which is established in these phenomena could be grasped by our senses, it would produce in us an impression comparable to the sensation of musical sound." "Multifractality in human heartbeat dynamics" is the topic of a letter in the June 3 1999 Nature. The work, by an international team of physicists and M.D.s (first author Plamen Ivanov of Boston University) shows that a healthy heartbeat manifests several superimposed fractal patterns with different scaling behaviors, and that loss of multifractality is associated with congestive heart failure. For a survey of fractal patterns in nature, visit the Center for Polymer Studies at Boston University. Tony Phillips 
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