My purpose in this paper is to offer an indirect critique of some of the assumptions of visual culture-and, by extension, cultural studies- regarding how visual processes work. Drawing on studies of the cognitive neurophysiology of vision, I argue that visual processes, including responses to visual arts as well as the apprehension of "reality," are not innate or instinctive (objects or scenes are not just "out there" to be objectively perceived) but are conditioned by artifacts in our environment. All artistic styles, both representational and "non-representational," abstract sets of geometrical invariants, which in turn attune our sensitivity to them by generating neural pathways devoted to the perception of those sets of invariants. To demonstrate this process, I examine the history of the reception of visual arts that rely on fractal geometry, a recently postulated branch of geometry that models real and imaginary processes such as chaos, turbulence, generative growth, infinity, complexity, and indeterminacy and is characterized by shapes presenting self-similarity, scaling, reiteration, and recursivity. I ultimately argue that fractal art, particularly as used in African artistic design since ancient Egypt, may signal an entirely different mode of visual realism than Euclidian geometry, which is the basis of most Western artistic representations, including photography. While conventional geometry is concerned with representing objects, fractal geometry models processes, including the visual process itself.
Fractal Reception
Since the founding of fractal geometry by Benoit Mandelbrot twenty-five years ago, there has been a growing iterative feedback loop of fractal art analysis, ranging from Hugh Kenner's 1988 study of Ezra Pound's Cantos to recent analyses of the fractal dimensions of Pablo Picasso's Les Demoiselles d'Avignon and Jackson Pollock's drip paintings. Many of these analyses suggest directly or indirectly that the appeal of fractal art arises from an innate response to fractals, which have been called "the basic building blocks of nature's scenery" and even "the fingerprints of God" (Taylor, "Fractal Expressionism"). Physicist Richard Taylor has extensively analyzed the fractal dimension of Pollock paintings in such publications as Nature, Leonardo, and Scientific American and he, as well as other researchers, have attempted to determine the fractal dimension or "D value" considered "most pleasing" to viewers. Fractal dimension is conceived of as lying between the first and second dimensions and is measured by numbers ranging between 1 and 2. A flat line, such as the horizon, is 1; clouds and early Pollock paintings are 1.3; tree branches and late Pollock painting are 1.9. Taylor et. al. reason that the preferred range they found, of 1.3-1.5, also corresponds to "prevalent patterns found in the environment," and suggest that people's preferences are "set . . . through continuous visual exposure to patterns characterized by this D value" (Taylor et. al.). Frederick Turner has traced ALL aesthetic appeal to the presence of fractals in nature: "the iterative feedback principle which is at the heart of [fractal processes such as] evolution is the logos and the eros of nature; and it is what we feel and intuit when we recognize beauty" (128).
In fact, however, there is strong evidence that fractal appreciation is not innate or instinctive. The initial reception of artworks now considered fractal was characterized by, shock, horror, hostility, and derision. In 1956 Time magazine labeled Pollock "Jack the Dripper." Early viewers of Les Demoiselles d'Avignon were appalled by the "hideousness of the faces" of the "monstrous" women in the "terrible picture," which Picasso did not exhibit until nine years after completion, when reviewers called it a "nightmare" (qtd. in Miller 125). In The Fractal Geometry of Nature, Mandelbrot stresses the predominance of negative aesthetic assessments of fractal shapes as "'monstrous,' 'pathological,' or even 'psychopathic'" (36)-an attitude reflected in a long history of western mistrust of concepts central to fractals such as irrational numbers and infinity. As Ron Eglash observes, in "Plato's philosophic cosmology, spiritual perfection was seen as the higher level of transcendent stasis, and illusion and ignorance were the result of life in our lower realm of changing dynamics ('flux,' which in ancient Greek also means 'diarrhea')" (204). Paradoxes such as alogos [the opposite of Turner's logos], or irrational numbers like pi that go on forever, and Zeno's paradox, which pits a tortoise in a losing race against the hero Achilles, "were unnerving, even shocking to philosophers who depended on rationality as the gateway to religious perfection" (204). In restricting reference to infinity as a limit to be approached rather than a proper object of mathematical study, Aristotle succeeded in shelving such concepts until the 13th century, when Leonardo Fibonacci described the Fibonacci series, an additive iterative loop based on adding the last number to the new total (i.e., 1, 1, 2, 3, 5, 8, 13, 21 . . . ). Nevertheless, infinity still didn't become a proper object of study until 1877, when Georg Cantor created the Cantor set (also called Cantor dust), a recursive subdivision of a finite line to create an infinite set of lines having zero length: by removing the middle third of the original line, the middle thirds of the resulting two lines, and so on, one may, in effect, produce a model of infinity bound (fig. 1). In 1904, Helge von Koch added lines to a "seed" pattern, rather than erasing them, creating what he called "pathological curves" (also called von Koch snowflakes) which could then be measured by plotting (fig 2.). The paradox of infinite length fitting into a bounded space was resolved by Felix Hausdorff, who conceived of the pathological curve as having a "fractional dimension" lying between the first and second dimensions. It wasn't until 1977, however, that such "pathological" shapes were positively assessed, when Mendelbrot discovered that, far from being pathological, they characterized many shapes found in nature, such as clouds, coastlines, branches, and turbulence. In reading a study of long-term river fluctuations by British civil servant H. E. Hurst, Mandelbrot realized the only way to characterize these fluctuations was with the same kind of scaling measures used on pathological curves. Mandelbrot named the new geometry "fractal geometry," created computer-generated fractals from algorithms with irrational numbers, and found practical applications for fractal geometry in fields such as the stock market, meteorology, medical imaging-and art. Several computer-generated and non-computer-generated fractal art movements have emerged since 1975, with various manifestos published such as that of Susan Condé, curator of a fractal art exhibit in Lyon, which names three basic principles of fractal art: self-similarity, in which the details of the work are mirrored in the entire shape; infinity bound within finite space; and portrayal of fractally-structured natural objects.
Thus, the history of the reception of fractals suggests that responses to art forms may not be innate or instinctive, but a matter of discerning particular features or constraints, or what James Gibson called affordances, in our environment that are important to survival within that environment-including our cultural environment. Artworks that reinforce recognition of particular constraints generate a feedback loop that increases the ability to discern the necessary constraints by which features may be perceived and understood in a social context, which in turn may increase the ubiquity of such objects in the environment. We perform the former operation by generating recursive neural pathways-called a "bootstrapping process"-as our perception and memory interact with our artifactualized environment. Or in the neurophysiological terms of Erich Harth, "The visual images displayed at topographically organized areas such as the lateral geniculate nucleus (LGN) in the thalamus, or visual projection area VI in the cortex, reflect not just the messages conveyed by the retina but the attentional and selective influence of higher cortical areas. . . . [W}e are dipping into a reservoir of enormous complexity containing traces of everything we have ever seen, heard, or thought" (104). And, one might add, we are always adding more to that reservoir.
Here I'd like to insist that ALL representational styles-from photography to abstract expressionism-necessarily reinforce attentional constraints of particular informational substructures available for perception in the environment. In her study, Varieties of Realism: Geometries of Representational Art, perceptual psychologist Margaret Hagen. draws on Gibson's work to analyze the geometries of various representational painting styles in terms of the geometric invariants or "the kinds of information an artist can transmit about objects and scenes within the conventions of [each] style" (178). As Hagen observes, "It is the task of an artist or of the artist's culture to select from among the options [available to the optic array] a set of informational substructures for depiction. The reasons directing choice may vary considerably from culture to culture" (114). While Hagen leaves speculation on those reasons up to art historians, she strongly argues against developmental concepts of art in relation to "realism," stressing that "any composition faithful to the varieties of visual experience will look realistic to viewers after a degree of exposure because what they see represented in art they have seen before" (270).
Thus, "fractal appreciation" may be more a function of the increased prevalence of fractal representations in our environment than an innate recognition of nature's beauty. But the matter is even more complex, for fractal art is not a recent historical development, but has been, as we shall see, a dominant artistic principle in Africa since ancient Egypt, usually flying under the attentional radar of western perception, but, when noticed, often negatively assessed. Since the era of commodity slavery, African cultures have been characterized by Europeans as monstrous, pathological, and psychopathic. The fractal layout of traditional African urban areas was perceived by European visitors as unstructured, haphazard-above all "irrational"-justifying colonial intervention to "rationalize" entire cities-and cultures. The recent recognition of fractal geometry now allows us to see the fractal principles at work in such layouts, which are generated from reiterations of basic seed patterns, like Kock snowflakes. This recognition of the constraints of fractal geometry allows more than just a new appreciation of African design, however. African fractals may in fact signal an entirely different mode of visual realism. While conventional geometry is concerned with representing objects, fractal geometry models processes, including, crucially, visual processes.
African Fractals
Armed with what might be termed a fractal perspective furnished by the new need to discern fractal constraints or affordances, Ron Eglash, in African Fractals: Modern Computing and Indigenous Design, brings to light the pervasiveness of fractal characteristics in African artistic design, which, he argues, is unique to Africa and not just a general characteristic of "primitives living close to nature" (53). He looks at a wide range of fractal principles in such African cultural production as rectangular, circular, and branching fractals in the layout of towns and villages; recursivity and scaling in hairstyles, architecture, and sculpture; complexity in textiles that show order moving to chaos and back to order; the use of binary numbers-the language of computers-in divination; and algorithmic patterns in the lusona sand drawings of Chokwe children, who are taught increasingly complex Eulerian path patterns during initiation rites: "by using more complex lusona, the iterations of social knowledge passed on in the initiation become visualized by the geometric iterations" (68). He ultimately seeks to demonstrate that African mathematical and knowledge systems are not merely copying from nature but are applied mathematics reflecting a awareness of concepts such as infinity, chaos, complexity, and indeterminacy, which seem new, even postmodern, to us but are "traditional" to African societies and embedded within African practice. Thus the ancient Egyptian creation myth depicts all life as continually generated from the infinite unfolding of petals from a lotus flower growing out of a sea of chaos, as in the alabaster chalice, called the wishing cup, found in Tutankhamen's tomb. The inscription reads "May you spend millions of years . . . with your face to the north wind and your two eyes beholding happiness" (fig 3). And in the modern "vodun religion of Benin," Eglash writes, "the snake god Dan represents the cyclical order of nature . . . in two ways: as an abstract force, he is represented as a feedback loop. As a concrete manifestation, his body is always oscillating in a periodic wave" (142). In vodun, as in other African religions, we find iconic representations of "the stabilizing force of negative feedback," such as Dan (Akan), Legba (Vodun), and Nommo (Dogon), countered by the "disruptive force of positive feedback," represented by trickster figures such as Ananse (Akan), Legba-Oshu, and Ogo (Vodun). Oscillating between order and disorder are the "fractal" figures such as Nyame (Akan), iconically represented by the "turbulent waters of Tanu," Shango (Vodun/Yoruba), represented by forking (branching and bifurcating) lightening, and Amma (Dogon), "described as an expanding spiral, like a whirlwind" (166). "This combination of opposing feedback loops also appears to be at the heart of the conditions that sustain self-organizing structures . . . and it is through this interaction that sustained complexity can arise" (166).
Continuing to counter the notion that Africans are merely copying fractals from nature, Eglash distinguishes between the "mimetic reflection" of European artistic tradition, in which art "holds a mirror to nature," and the "self-conscious abstraction" or "mathematical modeling" of African traditions. It is important to remember that BOTH traditions involve abstraction, but one may perceive a different order of abstraction between a tradition that relies on geometric projection of visual information about objects onto a two-dimensional plane and traditions that rely on fractal geometry, which models not objects so much as processes: feedback, interaction, generation, chaos, fluidity, change. Indeed, vision itself is considered an interactive, bootstrapping process rather than the apprehension of a scene or object that is passively "out there." In characterizing their attitudes toward art and vision, Africans stress art as a multimedia spectacle of interactive performance rather than a solitary appreciation of art forms and "seeing" as a bootstrapping activity rather than the comprehension of a static scene or object. Susan M. Vogel reports, "in the Baule language, words for looking at artworks include words commonly used in the West for theater-'watching,' and 'attending' . . . The usual experience is less one of seeing the work than of apprehending it in a multisensory way that involves what one knows as much as what one sees" (111)-in short, embedded in the very way Africans speak and think about vision is the insight that seeing as a process of "dipping into a reservoir of enormous complexity containing traces of everything we have every seen, heard, or thought." Fractal design, which models this bootstrapping visual process, invites a high degree of conscious participation on the part of the viewer to generate and play with meaning and are thus a constant reminder of the interactive, contingent, and collectively-constructed nature of the world and of the generative power of art-called ase in Yoruba-which "tends to be multifocal, characterized by a shifting perspective" (Drewel and Drewel 6-7).
Eglash's study may provide, in turn, a shifting perspective on European fractal history. As it turns out, temples in ancient Egypt, such as the one at Karnak, were designed according to the number series that would come to bear the name of Fibonacci, who was raised and educated in North Africa and also traveled in Egypt, "collecting mathematical knowledge" (Eglash 205). As it turns out, Georg Cantor's cousin, Moritz, was a leading expert in the geometry of Egyptian art and it he may have introduced Georg to temple design motifs on column capitols that represent the Egyptian origin story of the lotus flower generating the universe out of chaos (fig. 4). As it turns out, the river whose long-term fluctuations Hurst studied, which caught Mendelbrot's eye, was the Nile, whose movements had been recorded by Egyptian scribes for centuries before British civil servants arrived on the scene to impose order.
Finally, let us recursively return to Picasso's Demoiselles, whose African origins have been debated, deconstructed, and denied, notably by Picasso himself. Perhaps not the last word, but most cogent to my argument is Arthur I. Miller's recent book Einstein/Picasso: Space, Time, and the Beauty that Causes Havoc. In his analysis of the "scientific, mathematical and technological roots" of the painting (whose secret "code name" while in progress was "The Egyptians)" Miller convincingly demonstrates that Picasso did not merely copy African and other "primitive" artifacts, which he had seen before in various places (85). Rather, Picasso's famous visit to the African art exhibit at the Trocadero came at a crucial juncture in his work on a painting in which he was struggling to formulate the answer to a conceptual problem involving a "new mode of representation" based on the new, non-Euclidian geometry introduced by Henri Poincaré (94). As Miller writes, "Picasso's epiphany at Trocadero provided the insight into why he had to proceed as he did with the [painting] and why the abstract language of geometry was necessary. . . . If Picasso had not appreciated the importance of the African statuettes he had seen [before], now he was in need of such an idea. The multiple planes and edges of the masks were the perfect way to geometrize the multiplicity of views. Having gleaned the crucial insight from Poincaré's instructions on how to view the fourth dimension, Picasso transformed and elaborated his vision in dramatic fashion" (106).
While one may surmise from the foregoing that fractal geometry as well as western art history owes a debt to Africa, one may also surmise that the need to see Africa as the irrational other to the western self during the period of commodity slavery and colonialism may have also delayed our full recognition of fractal geometry. Either scenario suggests that art, and perception itself, may not be a matter of art imitating nature but a process of art making the world-a highly contingent and indeterminate world-and, in that regard, perhaps the best of all possible worlds.
Final note: Although discussed in terms of African cultural practices, this perception of vision as generative and culture-inflected is not culture-bound: "seeing fractally" is not the property of an "indigenous" people but a response to the intense and multi-lateral cultural interaction that has characterized African environments since ancient Egypt. Just as post-Renaissance Europeans "learned to see" three-dimensional objects on a two-dimensional plane by generating the recursive neural pathways necessary to perceive the projected illusion (that became for them reality), each human is capable of generating the neural pathways that would enable us to "learn to see" the fourth dimension-by imagining processes unfolding through time-in abstract fractal designs that project reality as socially constructed though our own mutually interdependent actions and perceptions. If environmental conditions warrant it. And if capitalism doesn't appropriate that, as well.
Works Cited
Drewel, Henry J. and Margaret Thompson Drewel. Gelede: Art and Female Power among the Yoruba. Bloomington: Indiana UP, 1983.
Drewel, Margaret Thompson. Yoruba Ritual: Performers, Play, Agency. Bloomington: Indiana UP, 1992.
Eglash, Ron. African Fractals: Modern Computing and Indigenous Design. New Brunswick: Rutgers UP, 1999.
Gibson, James J. The Ecological Approach to Visual Perception. Boston: Houghton Mifflin, 1979.
Hagen, Margaret. Varieties of Realism: Geometries of Representational Art. Cambridge: Cambridge UP, 1986.
Harth, Erich. "The Emergence of Art and Language in the Human Brain," Journal of Consciousness Studies 6: 6-7 (1999): 97-115,
Kenner, Hugh. "Self-similarity, fractals, cantos." ELH 55(3,) 1988: 721-730.
Mandelbrot, Benoit. The Fractal Geometry of Nature. New York: W. H. Freeman and Company, 1977.
Miller, Arthur. I., Einstein, Picasso: Space, Time, and the Beauty that Causes Havoc. Basic Books, 2002.
Taylor, Richard. "Fractal Expressionism..". Physics World. Vol 12, no 10, 25 (1999).
Taylor, R. P., B. Spehar, C.W.G Clifford and B. R Newell. "The Visual Complexity of Pollock's Dripped Fractals." http://www.materialscience.uoregon.edu/taylor/art/info.html. (accessed 11/29/02).
Turner, Frederick. 1999. "An Ecopoetics of Beauty and Meaning." Biopoetics: Evolutionary explorations in the arts, ed. B. Cooke and F. Turner, 119-37. Lexington, KY: ICUS, 1999.
Vogel, Susan M. Baule: African Art, Western Eyes. New Haven: Yale University Press, 1997.
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