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beauty in art
The fractal dimension The fractal-dimension (D) is the measure to which a fractal ‘fills a space’, a phenomenon observable at increasing magnitudes. A coastline is a one-dimensional fractal because it is a line (i.e. its topology is one dimensional). The repeating patterns in this line cause it to spread across twodimensional space, and hence the fractal dimension lies between 1 and 2. A mountain is a two-dimensional fractal because it is a surface (i.e. its topology is two dimensional). The repeating patterns in this surface spread across three-dimensional space and hence the fractal dimension is expected to lie between 2 and 3.
organisms. Biophilia possibly explains why of artists and physicists alike. With everwe find aspects of the natural environment widening appeal, they have been referred so pleasing (Kaplan & Kaplan, 1989). to both as ‘fingerprints of nature’ and ‘the Fractals have been demonstrated to new aesthetics’ (Taylor et al., 1999, 2003). characterise the detail and irregularity It is thought that fractals tap into of the natural world (Mandelbrot, 1977). specialist cognitive modules that have They can be found in mountain ranges, developed to moderate information about deserts, coastlines, living things, and that clouds, rivers, trees, such modules are linked plants and animals, with emotional “As viewers begin to as well as in sound regulation (Wilson, understand an artist’s (waves, waterfalls and 1984). More recent message it becomes more rain) and music research also suggests (birdsong and nursery some brain areas are meaningful and less effort is rhymes). responsive to fractal required for interpretation” Mathematical patterns. Hagerhall et al. fractals are infinite in (2008) reported that viewing detail and in length, whereas natural fractal patterns elicited high alpha activity fractals can be fragmented (fractus), in areas of the brain concerned with discontinuous and range-restricted. Fractal attention and visual spatial processing (the analysis has been extraordinarily successful frontal lobes and the parietal area). These in quantifying the complex structure studies support research that suggests that exhibited by many natural patterns, training using fractal shapes could help the enabling precise measurement of development of perceptual concepts of the phenomena as diverse as galaxies and natural, stimulate biophilic responses and sea-shells. The adaptability and beauty trigger aesthetic interest and restorative of fractals have captured the imagination responses (Joye, 2006). The strongest
and architecture. Leonardo, 39, 245–251. Taylor, R.P., Micolich, A.P. & Jonas, D. (1999). Fractal analysis of Pollock’s drip paintings. Nature, 399, 422. Taylor, R.P., Micolich, A.P. & Jonas, D. (2003). The construction of Pollock’s fractal drip paintings. Leonardo, 35, 203. Williams, T., Forsythe, A.M. & Sheehy, N. (2010). Painting by numbers: Fractal
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analysis acts as an early predictor of neurological deterioration in artists. Paper delivered at the British Psychological Society Annual Conference, Stratford-upon Avon. Wilson, E.O. (1984). Biophilia: The human bond with other species. Cambridge, MA: Harvard University Press. Zeki, S (1999). Inner vision: An exploration of art and the brain. Oxford: Oxford University Press.
evidence for the application of fractal patterns in therapeutic environments is that fractal patterns reduce physiological stress (Taylor, 1999). Fractal geometry has established its usefulness in understanding the structure and authenticity of major works of art. Taylor et al. (1999) examined film footage of Jackson Pollock at work and concluded that Pollock was clearly generating paintings with a high fractal dimension, or ‘D’ (see box) and that Pollock was actually able to fine-tune the D value of his paintings. Detailed analysis of sections of Jackson Pollock’s work demonstrated that the fractal dimension of his work increased steadily over a 10-year period. Following this analysis it was possible to deauthenticate recently discovered paintings attributed to Pollock, because the dimension values were not consistent with previous works. Taylor’s work may also be useful in addressing some of the shortcomings of the Berlyne (1970) hypothesis (predicting the cusp). Taylor has reported the presence of three categories with respect to aesthetic preference for fractal dimension (Taylor et al., 2001). These can be categorised into low preference (1.1–1.2), high preference (1.3–1.5) and low preference (1.6–1.9). Humans are consistent in their preference for fractal images in the 1.3–1.5 fractal dimension regardless of whether these fractals were generated by mathematics, humans (e.g. the art of Jackson Pollock) or natural processes (coastlines, trees or clouds). Combining fractal measures with measures of visual complexity explains even more of variance in aesthetic
vol 24 no 7
july 2011