The lights are dimmed in the auditorium at the Atlanta College of Art, and Richard Voss is playing a simple melody for the audience. But unlike traditionally composed music, this melody was generated by a computer, and is based on the fluctuations of 30 years of IBM stock prices. Voss, who is in town for the APS Centennial meeting, offers it as an aural representation of Brownian motion as part of his scheduled lecture on fractal patterns in modern culture.
A professor of physics and mathematics at Florida Atlantic University's Center for Complex Systems and Brain Science, Voss specializes in scientific computer graphics and the application of fractals not only to the natural sciences, but to art, music, literature, the stock market, cancer cells, and even the organization of base pairs of DNA. "Mandelbrot's mathematical language of fractals has revolutionized the application of geometric constructs in the sciences, and drastically changed the image of mathematics," he says. "Fractal geometry and chaos theory provide new tools and unifying concepts that bridge the traditional boundaries between science, art and finance."
Voss earned a PhD in condensed matter physics from the University of California, Berkeley, in 1975. He joined the IBM research division that same year, and found himself working closely with the "father of fractals" himself, Benoit Mandelbrot, who first coined the term in 1975 to describe patterns or objects that exhibit similar patterns and structures at different size scales.
Certain fractal geometries exist only in the abstract world of mathematics, but Mandelbrot, Voss and others have found that many fractal patterns provide useful models for the irregular shapes found in nature, which don't follow traditional Euclidean geometries of perfectly regular circles and squares. "The structures we find in the natural world, going from the smallest microscopic scales to the largest intergalactic scales, are mimicked in the structures we see at human scales on Earth," says Voss. "If you want to understand, simulate and ultimately manipulate nature, you need to understand its language, and the language of nature turns out to be mathematics."
It was during the early period of his career - beginning when he was still a graduate student - that Voss discovered that a wide variety of music from different historical periods and cultures followed the same patterns as many natural phenomena, illustrated by the melody line generated by IBM stock prices. In music, this enables one to make a reasonably accurate guess as to what the next note might be in a given piece of music; it can also be used to generate music. Voss' early work has given rise to a proliferation of "composers" of fractal music, using specially designed software to generate the same types of feedback processes used to create fractral images.
The same concept can be applied to the visual arts. In the 1980s, Voss analyzed ancient Chinese landscape paintings from various historical periods at the request of James Watt, curator for Asian art at New York's Metropolitan Museum of Art. He determined the dimension of brush strokes in each painting, and found that those considered superior by art historians (the earlier landscapes) had fractal dimensions comparable to those of typical coastlines: between 1.25 and 1.33. In fact, the difference between the fractal dimensions of the earlier and later paintings turned out to correspond to sociological changes of the time. The early landscapes were done by painters working away from urban areas in the countryside, and hence their work involved objects repeated at different sizes, one of the defining characteristics of fractals. In contrast, the later paintings were done by court artists living in urban areas, and involved brush strokes that were simpler and smoother; the paintings were more of an abstraction of natural objects and struck Watt and his fellow art historians as being much less interesting.
The fact that human beings are drawn to objects that exhibit a specific fractal dimension is not surprising to Voss, or others working in his field. Voss conducted a study with perceptual psychologist Bernice Rogowitz, in which subjects were shown randomly generated fractal cloud patterns and asked to indicate in which cases they perceived objects in the pictures. He found that subjects saw more objects in those patterns with fractal dimensions corresponding to those found in nature. To Voss, this is suggestive of a particular sensitivity for fractal patterns in the human perceptual system, which he does not find surprising. "The human perceptual system evolved in nature, which exhibits many fractal patterns," he says. "We live in an irregular world in time and space that is very much at the boundary between predictable and unpredictable behavior."
However, Voss cautions against attaching too much importance to such measurements of fractal dimension, which can be a difficult property to pin down accurately because of the infinity of different shapes that can be analyzed within any given image. "So specific measurements - including those I have made - need to be taken with a grain of salt," he says.
In 1995 Voss left IBM and joined the Center for Complex Systems and Brain Science at Florida Atlantic University. His recent work involves applying fractal analysis to financial variables, such as seeming randomness in stock price patterns, as well as to digital mammography to improve the diagnostic ability of radiologists, particularly in the early detection of breast cancer. This is an area where early diagnosis has particular importance in terms of increasing life expectancy, but to date the diagnosis of breast cancer from X-ray mammograms has proven to be an extremely difficult perceptual process.
"Science is a series of approximations. What we call the great equations of science, are not something we can prove to be true. Rather, they are those we have not yet proven to be false," says Voss of his lifelong work. "Everything we do is useful, but not quite correct; it is an approximation to the real world." He often closes his lectures with a quote by Pablo Picasso: "Art is the lie that helps us see the truth." In his eyes, this is just as true of science as it is of art.
Fractal images from Sprott's Fractal Gallery website: http://sprott.physics.wisc.edu/fractals.html.
©1995 - 2017, AMERICAN PHYSICAL SOCIETY
APS encourages the redistribution of the materials included in this newspaper provided that attribution to the source is noted and the materials are not truncated or changed.
Editor: Barrett H. Ripin
Associate Editor: Jennifer Ouellette