ARTS & IDEAS

        Explaining the Patterns of Life in Fractals

        By EDWARD ROTHSTEIN

           In a tightly braided woman's hair style from Yaounde,
           Cameroon, the coils seem to spread out from the
        neck, split into subsidiary branches, and erupt into
        repetitive patterns. An ivory sculpture of the Mangebetu
        people is also based on geometric shapes in which
        large patterns sprout smaller versions of themselves.
        So is the architecture of the Logone-Birni palace in
        Cameroon.

        Ron Eglash, a computer scientist
        and ethnomathematician at Ohio
        State University, finds designs
        like these all over Africa,
        designs that in their more
        elaborate, infinite form, are
        called fractals. Each subsection
        of a fractal design might be an
        image of the whole; a fractal
        curve generates itself out of
        itself, changing in size but not in
        shape.

        In his new book, "African
        Fractals: Modern Computing and
        Indigenous Design" (Rutgers
        University Press), Eglash
        weaves his own design, showing
        that a fascination with shapes that sprout ever smaller
        versions of themselves is common to many African
        cultures. They appear everywhere: in the layouts of
        villages, in the designs on clothing, in the form of
        headpieces.

        It is a suggestive analysis, though there are reasons to
        qualify Eglash's sweeping portrait. Many of these
        designs are not really fractals but only resemble them.
        He also argues that the designs are a form of African
        mathematics, but this seems an exaggeration:
        mathematics involves not just the creation of a pattern
        but also an examination of the principles behind a
        pattern, which seems to have minimal significance in
        these examples. Eglash is so intent on finding the design
        that he discerns fractal patterns in American black
        voting districts and, somehow, in relations between the
        sexes.

        But this kind of obsession with a repeated pattern -- for
        Eglash and perhaps for the cultures he examines -- is
        not unusual. Such recurrent designs are not merely
        esthetic creations. They are metaphysical
        proclamations about how the world works. No shape is
        arbitrary or isolated; each part resembles another; each
        part resembles the whole. Even the structures of social
        and religious life are affected.

        The Greeks, too, had such touchstones, in their
        fascination with regular Platonic solids like the cube
        and tetrahedron and in their explorations of ratio. A
        design is found within the natural world, and that
        design in turn shapes the world that is found. A design
        is a theory, a theory a design.

        This is also the case in contemporary science, which,
        despite the increasing importance of computer
        simulation and calculation, seems to rely more and
        more on artistic metaphors that invoke design and
        pattern.

        In "The Artful Universe," (Oxford, 1995), the
        astronomer John D. Barrow argues that "the arts and the
        sciences flow from a single source; they are informed
        by the same reality; and their insights are linked in
        ways that make them look less and less like
        alternatives."

        The geneticist Enrico Coen, who has just written "The
        Art of Genes" (Oxford University Press, 1999), uses
        painting as a metaphor to describe how organisms
        generate themselves. Beautiful natural patterns --
        spirals, butterfly wings, rippling waves -- and their
        mathematical origins are explored in Philip Ball's "The
        Self-Made Tapestry: Pattern Formation in Nature"
        (Oxford, 1998). This writer has chimed in with
        "Emblems of Mind" (Avon, 1996), examining how
        music and mathematics create patterns that develop out
        of similar styles of metaphorical thinking.

        This attention to the close relationship between the arts
        and sciences is not new, of course. Two hundred years
        ago, the philosopher Immanuel Kant suggested that in
        attempting to understand the natural world human
        beings treat it as if it were specifically constructed for
        their contemplation. They approach it as if it had a
        particular purpose, as if each of its elements were
        created for a reason, as if nothing about it were
        accidental. Kant argued that man treats nature "after the
        analogy of art."

        He might have added that in constructing interpretations
        of nature, man also applies standards that have evolved
        for judging art and beauty. The mathematician J.W.N.
        Sullivan (who wrote an elegant book about Beethoven's
        string quartets) suggested: "Since the primary object of
        the scientific theory is to express the harmonies which
        are found in nature, we see at once that these theories
        must have an esthetic value. The measure of the success
        of a scientific theory is a measure of its esthetic value,
        since it is a measure of the extent to which it has
        introduced harmony in what was before chaos."

        Scientists from Kepler to Einstein have judged their
        theories not just by their success in accounting for data
        but also by their beauty; not just by their order but also
        by the kind of order they produced. Design becomes a
        standard for discovery.

        Kepler, for example, was determined to find that the
        orbits of the planets around the sun could be inscribed
        in the five Platonic solids defined by the Greeks; when
        the data did not support this esthetic ideal, he found
        something just as good: the shape of every orbit is a
        perfect ellipse. If a theory's design is beautiful enough,
        it can inspire allegiance even when the data seem to
        disprove it. The physicist Paul Dirac famously
        proclaimed: "It is more important to have beauty in
        one's equations than to have them fit the experiment."

        The contemporary fascination with the artistic aspects
        of science taps into these timeless concerns. But the
        fascination is all the greater because the creation of
        natural pattern and design has itself become a subject of
        exploration in genetics and computer simulations.
        Arguments in cosmology, physics and mathematics have
        also become so abstract that even their creators must
        resort to physical and esthetic metaphors to explain
        them; the metaphors are, in fact, inextricably knit into
        the theory. Design becomes a standard for judging the
        theory as well.

        This is clear in "The Elegant Universe" (Norton, 1999)
        by the Columbia University physicist Brian Greene, a
        remarkable book that has been hailed as a classic of
        elegant explication. Greene argues that contemporary
        "superstring theory," an incomplete and highly esoteric
        theory linking subatomic particles and supra-galactic
        transformations, is likely to become the long-awaited
        "T.O.E." -- a Theory of Everything that ties together
        every stray strand of physics and cosmology.

        To explain the theory, Greene must link it to various
        images; indeed, there is no way for scientists working
        in the field of string theory to interpret their own
        mathematical work without using such images. Greene
        suggests, for example, that with superstring theory,
        "musical metaphors take on a startling reality, for the
        theory suggests that the microscopic universe is
        suffused with tiny strings whose vibrational patterns
        orchestrate the evolution of the cosmos."

        There are geometric images that suggest what happens
        in the theory's more exotic and invisible realms:
        surfaces bend and rip, are glued and twisted. Greene
        even invokes "the fine mist above the roaring ocean" to
        describe certain subatomic particles.

        Oddly, the design of the universe springing from the
        theory is not as beautiful or as simple as the images of
        space and time that unfold in the first portion of
        Greene's book, where Einstein's theories are
        exquisitely laid out for contemplation without a bit of
        mathematical complication. In fact, there is something
        almost grotesque about the world conjured by
        superstring theory, a world of nine dimensions in which
        six are curled up on themselves, forming, in some
        unimaginable fashion, knotted spheres of multiple
        dimensions (called "Calabi-Yau" surfaces) that
        invisibly erupt out of every point in ordinary space.

        If such a world could be pictured, it would be
        considered hideously ugly, even perverse, and
        unrelated to anything we commonly understand. While
        Kepler was hoping to find a universe based on Platonic
        solids, while African tribes might seek elementary
        fractals in all aspects of human life, it is hard to believe
        that anyone would seek out a universe of crabbed
        dimensions and Calabi-Yau surfaces unless forced to.

        But that may be part of its beauty. In Greene's
        explanation, the theory has the force of necessity. It
        might explain not only what is but also why things
        cannot be different. Beauty is in the theory rather than in
        the ornate surfaces it studies. Greene even shows how
        often esthetic issues have affected the very direction of
        theoretical inquiry into superstrings.

        Eventually this theory may even affect our
        understanding of beauty itself. In Greene's words, its
        impact could be "monumental": we would find our
        world to be constructed out of "loops of strings and
        oscillating globules, uniting all of creation into
        vibrational patterns."

        We would have to revise our understanding of space
        and time. Perhaps, given the human fascination with
        natural design, we may eventually have to imagine the
        possibility of superstring braids.