Like earlier discoveries in geometry, fractals could change the way artists and musicians portray the world
From classroom to nightclub, by way of T-shirt and video graphics, fractal designs have made their mark on popular culture. Endlessly interesting and strikingly colorful, the images are now so popular that fractal posters and coffee mugs are a dime a dozen. The bold new geometry of the 1980s is in danger of becoming 1990s kitsch.
Some artists, though, expect fractals to be more than a commercial gimmick. “It’s a new way of looking at space and form,” says Rhonda Roland Shearer, a New York sculptress. “For the artists, nothing is more fundamental.” As precedent, she points to the Renaissance, when the discovery of geometrical perspective transformed the way that painters represented depth on a flat surface. The development of non-Euclidean geometry in the mid-19th century inspired wide-spread stylistic experimentation a few decades later among modernist painters who sought new ways of portraying space.
As yet, fractals have not come close to delivering what Ms Shearer suggests they eventually might. In a way, that is because computers generate fractal shapes so easily. A fractal is a geometrical figure in which the same motif is repeated at several different scales; typically, such a design can be expressed as a simple mathematical formula, recalculated again and again. So, to produce even the most haunting of the images now a available, a computer needs only the smallest of human nudges. Benoit Mandelbrot, the mathematician whose eponymous “set” is the most famous fractal of all, wrote recently: “The artist’s taste can only affect the selection of formulas to be rendered, the cropping and the rendering. Thus fractal art seems to fall outside the usual categories of `invention’, `discovery’ and `creativity’.”
However, fractals do not exist only on the computer screen; they can be seen in the real world, too. Dr Mandelbrot’s equations offer ways to explain the complex shapes of nature in simple mathematics, which is why scientists like them. The same thing appeals to artists. Nathaniel Friedman, a mathematician and sculptor, was startled to read Dr Mandelbrot’s description of the fractals created when granite boulders are broken in half – because he was working on a series of broken-granite sculptures at the time. Now he organizes an annual meeting at the State University of New York at Albany at which artists and scientists talk about new ways in which artifice can explore nature.
Ms Shearer’s work is an example. Mondrain once remarked that the branch of a tree was nature’s inferior attempt to create a line. To Ms Shearer, free of Euclidean prejudice, the branch is a perfectly decent attempt at a fractal. Her own work celebrates the new geometry by introducing fractals in traditionally rectilinear sculptures. Adding gender to her concerns, she has has filled in linear outlines of women doing chores with free-flowing fractals. She see this an act of liberation.
In Paris, fractals have attracted a loose-knit, multinational group of artists who are disenchanted with minimalist and “neo-geo” art, two prominent movements in the 1970s and 1980s which glorified bare Euclidean forms. Again, the artists try to express fractal properties metaphorically, rather than contenting themselves with a literal presentation of computer-generated images. They held a group show last May in Lyons.
Susan Condé, a writer who was the show’s curator, has just published a manifesto on fractal aesthetics. She describes three common principles in the group’s work. One is that the detail within the work is mirrored in its entire shape – hinting at the self-similarity present in fractals. The second hints at another fractal property, the idea of infinity confined in a finite space. The final theme is the portrayal of natural objects with an explicit recognition of their fractal structure.
What the ear hears
Fractal attraction is not limited to the visual arts: several composers are trying to turn the geometry into music. Normally, fractal graphics are generated in spaces where each pointed is defined by co-ordinates: so far along, so far back, so far up. For musical expression, the same x, y and z co-ordinates can be used instead to define pitch, duration and volume. A wiggly fractal line becomes a wiggly fractal noise, each bit related to what came before. Imagine music based on a fractal shape like that of a mountainside. Starting from any point on the mountainside, the next is likely to be slightly higher or lower, not radically different. Similarly, fractal music is more likely to be characterized by slow changes instead of notes jumping all over the place.
Hugh McDowell, once the bassist of the Electric Light Orchestra, a rock group, created software based on these principles to help compose a 12-minute fractal ballet called “Teawaroa”, Maori for great river. He says the music that was produced blended well with the dancers’ movements, which evoked the structure of a system of tributaries feeding into a river. Such tributaries are fine examples of natural fractals.
Chris Sansom has developed a program that produces somewhat “purer” fractal music than Mr McDowell’s; it does not require the notes fit a particular key. For this reason, Mr Sansom’s program appeals to purists like Godric Wilkie, a london composer whose university thesis was a multimedia “opera” entitled “Chaos Is a Five Letter Word”. The work included actors reciting the words of several fractal pioneers; vocalists singing their theorems; visual images of their discoveries and a trio of musicians playing fractally generated themes. Reactions were mixed.
Just as painters other than Mondrian were happy to capture the fractal essence of a branch without knowledge of fractals, so Mr Wilkie has found that much music is already fractal. Any music in which the same theme is repeated by different instruments in different registers or speeds has some claim to a fractal nature. One part of Bach’s Art of the Fugue has the same theme played consecutively at three different speeds – a high degree of self-similarity. For good measure, one iteration of the theme is flipped upside down, a common distortion in fractal patterns.
Mr Wilkie wants to find out whether fractal music is “beautiful” in the same way that visual fractals clearly are. So far, the music emerging from Mr Wilkie’s computer is intriguing, though perhaps a bit disconcerting. “My initial reaction was, no, it’s not beautiful,” he says. “I think it’s that we’re not used to listening to music like it. It’s not as immediately accessible as Bach or Madonna, but it is tantalizingly musical. I find it grows on me, but it does take a few listenings.”
It may be possible to predict how well different types of fractal music will grow on people. Richard Voss, a former IBM researcher who is working on a book with Ms Shearer, once recorded how much variation in pitch existed in more than a dozen samples of music from a variety of cultures. In all of them, he found a characteristic pattern – some pitches recurred frequently, with more unusual pitches adding a bit of spice. From what Mr Wilkie can tell, his fractally generated music approximates this pattern of predictability and surprise reasonably well, indicating that fractal music may, in principle, appeal to people as well as other styles of music do.
For that matter, studies indicate that there is a certain level of visual and acoustic complexity that tends to be most pleasing to the human senses. The discovery of a lowest common denominator that defines aesthetic pleasure should not be too surprising: humans probably evolved to become comfortable with the complexity of their physical surroundings. Fractals may provide an easy way to mimic the level of complexity that people feel comfortable with, whether it be in music or in art. Muzak-makers and wallpaper designers may thus be on to a good thing. But the purpose of art is not to mimic nature — it is to comment on it, and on the humanity that exists within it. What fractals can contribute to that great endeavor remains to be seen.
The Economist, 6 November 1993