Fractals: A Resonance between Art and Nature

Abstract

Fractal geometry has experienced spectacular success in quantifying the complex structure exhibited by many natural patterns, including coastlines, trees and clouds (1). In contrast to the simplicity of Euclidean shapes, the complexity of fractal patterns results from the recurrence of patterns at finer and finer scales. Here we show that humans display a consistent aesthetic preference across fractal images, regardless of whether these images are generated by nature s processes, by mathematics, or by humans.The study of human aesthetic judgement in general, and of fractal patterns in particular, constitutes a relatively new research field within perceptual psychology. Previous studies of fractal objects have focussed predominantly on how fractal character influences perceived visual qualities such as roughness and complexity. Only recently has research started to quantify people's visual preferences for (or against) fractal content. To determine if there is any universal aesthetic quality of fractals, we have carried out a comprehensive study incorporating three categories of fractal pattern:1. natural fractals -scenery such as trees, mountains, waves, etc; 2. mathematical fractals -computer simulations of coastlines; 3. human fractals -cropped sections of paintings by the artist Jackson Pollock that have recently been shown to be fractal (2). The fractal character of an image can be quantified by a parameter called the fractal dimension, D. This parameter quantifies the fractal scaling relationship between the patterns observed at different magnifications (1). For Euclidean shapes, dimension is a familiar concept described by integer values. For a smooth line (containing no fractal structure) D has a value of 1. For a completely filled area (again containing no fractal structure) its value is 2. For fractal patterns, D lies in the range between 1 and 2 with a value depending complexity and richness of the repeating structure.Here, we used 15 computer-generated images of simulated coastlines (1), 5 each with D values of 1.33, 1.50 and 1.66; 40 cropped images from Jackson Pollock s paintings, 10 each with D values of 1.12, 1.50, 1.66 and 1.89; and 11 images of natural scenes with D values ranging from 1.1 to 1.9. All the images were black and white, and in the case of the natural and Pollock stimuli, presented in such a way as to disguise their identity.Initially, visual preferences for images within the same category and with the same fractal dimension were compared using a forced-choice visual preference task. Participants indicated their preference from one of two images appearing on a monitor. All images were paired with each other and preference was quantified in terms of the proportion of times each image was chosen. At this stage, the most and least preferred images were excluded from further comparisons, which led to the selection of the two most representative images for each fractal dimension within each category.Using these representative images, visual preferences were then compared across fractal di...

Predicting beauty: Fractal dimension and visual complexity in art

“…A priori analysis of several well‐known paintings and their fractal dimensions are listed below. None have a value falling within the Taylor et al (2001) hypothesized preference range.…”

“…Fractal dimension combined with complexity (Gif) is able to account for more of the variance in judgments of perceived beauty in visual art than measures of complexity alone. However, further work is required to explore both the hypothesized ranges of preference in art (Taylor et al ., 2001) and the interplay between complexity, colour, and preference.…”

“…Complexity consistently explains more of the variance (53% across all images sets) but fractal dimension still has something to tell us. Taylor et al (2001) offer a solution to criticisms pertaining to the predictive value of the Berlyne (1970Berlyne ( , 1971 hypothesis. They suggest that pictures in the fractal dimension of 1.3-1.5 will obtain much higher preference ratings at lower fractal dimensions (1.1-1.2) and also at higher dimensions (1.6-1.9).…”

“…Taylor's work may also be useful in addressing some of the shortcomings of the Berlyne (1971) hypothesis (predicting the cusp). Taylor has reported the presence of three categories with respect to aesthetic preference for fractal dimension (Taylor, Newell, Sphehar, & Clifford, 2001). These can be categorized into low preference (1.1-1.2), high preference (1.3-1.5), and low preference (1.6-1.9).…”

Perception of emotion in abstract artworks

“…Additional visual properties of abstract images which might be predicted to influence emotion judgments include symmetry (for review, see McManus, 2005) and similarity to natural images in terms of certain statistical properties (Taylor et al, 2001), Thus, both simple features (such as the presence of a particular color) and global attributes of those features might be expected to influence the perception of emotion in an abstract painting.…”